Is exogeneity guaranteed for a lottery-generated instrumental variable?

The simple answer is no.
Randomness does not imply exogeneity.
In my opinion, the following example from Deaton: "Randomization in the tropics, and the search for the elusive keys to economic development" is quite illuminating.
Assume that you want to estimate the effect of having a railway-station on poverty. We have data on the poverty rate $P_c$ and the presence of a station $R_c \in \{0,1\}$ for a large number of cities $c$. Assume that the true data generating process is given by:
$$
P_c = \alpha + \beta_c R_c + \varepsilon_c,
$$
Notice here that the slope $\beta_c$ is city specific, so the effect of a railroad is heterogeneous across cities.
The coefficient $\beta_c$ cannot be estimated, but you might wonder whether it is possible to estimate the mean of $\beta_c$ over all cities that decide to build a railroad.  $\overline{\beta} = \mathbb{E}(\beta_c|R_c = 1)$. Then:
$$
P_c = \alpha + \overline{\beta} R_c + (\varepsilon_c + (\beta_c - \overline{\beta})R_c) = \alpha + \overline{\beta} R_c + \upsilon_c,
$$
where $\upsilon_c = \varepsilon_c + (\beta_c - \overline{\beta})R_c$ is the new error. Although $\mathbb{E}(\upsilon_c) = 0$, you might be worried that:
$$
\mathbb{E}(R_c \upsilon_c) \ne 0.
$$
Indeed, we have:
$$
\mathbb{E}(R_c \upsilon_c) = \mathbb{E}(\varepsilon_c R_c = 1) + \mathbb{E}(\beta_c - \overline{\beta}|R_c = 1) \Pr(R_c = 1) = \mathbb{E}(\varepsilon_c R_c).
$$
This will only be zero if the decision to build a station is uncorrelated to anything that might otherwise influence the poverty rate in the city. As this is probably not satisfied, we need an instrument.
A random instrument
Assume that the government randomly allocates cities to development areas, captured by the instrument $Z_c$ (equal to 1 if city $c$ is a development area and zero otherwise).
If designations to development areas motivate the building of railways, we have that there is positive correlation between $Z_c$ and $R_c$. Also as $Z_c$ is random, we can assume that $\mathbb{E}(Z_c \varepsilon_c) = 0$. However:
$$
\mathbb{E}(Z_c \upsilon_c) = \mathbb{E}(\varepsilon_c Z_c) + \mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1)\Pr(R_c = 1, Z_c = 1).
$$
The first term on the right hand side is zero by the randomness of $Z_c$. As such, $Z_c$ will be uncorrelated to $\upsilon_c$ iff:
$$
\mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 1, R_c = 1) - \mathbb{E}(\beta_c|R_c = 1) = 0.
$$
This will be the case if the mean effect of building a station over all cities that are a development area equals the mean effect over all cities that build the railroad (whether they are a development area or not).
The last condition can be rewritten as:
$$
\begin{align*}
&\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|R_c = 1),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_C = 1, R_c = 1) \Pr(Z_c = 1) + \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0|R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)\Pr(Z_c = 0|R_c = 1),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0, R_c = 1).
\end{align*}
$$
This will be the case if either:

$\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)$ which means that the mean effect among all cities who are a development area and decided to build a railroad should be equal to the mean effect among those who build a railroad and are not a development area. So if the behaviour (i.e. the decision to build a station) changes due to being designated as a development area, this condition will not be satisfied.
Or $Pr(Z_c = 0, R_c = 1) = 0$ which means that no city that is not a development area builds a station.

In short: randomness of the instrument makes sure that $\mathbb{E}(\varepsilon_c Z_c) = 0$ but does not guarantee that $\mathbb{E}(\upsilon_c Z_c) = 0$.


Can repeated game sustain a discounted average payoff that is not individually rational?

I'm inclined to say $(2,2)$ is not sustainable, though I don't have a formal proof at the moment.

To achieve an average payoff of $(2,2)$, the players must play $(H,H)$ in some stages. But then a player can deviate to playing $L$ and avoid any subsequent punishments by ensuring that he can always get at least $3$, which is better than $2$.


In perfect competition, why is there economic loss if marginal cost > marginal revenue?

There is no loss at Q' and you are indeed right that the yellow area represents the firm profits. However, if you were to compare the yellow area to the blue area you would find that the latter is greater. So profit for the firm is not optimal at Q' but is optimal at Q*. This is what the quote refers to, it says the firm will be generating losses on its marginal production (so on the quantity Q' - Q*). It does not say that the firm will generate losses on production in general. This will only happen beyond MR=ATC or TC=TR


Determine price and cost of intermediate good

One way to close the model is through the labor market equilibrium. The OP clarified that the final-goods sector does not use labor.

Let $D_y = D_y(q)$ be the total demand for the intermediate good, in quantities. Since the market for the intermediate good is competitive it will clear, so we have

$$D(q) = AL \implies L^d = D(q)/A$$

Let $L^s(w)$ be labor supply coming from the utility maximization problem of the households and depending also on the wage. Assume a competitive labor market. Then the wage will adjust to clear it. Here we have

$$L^s(w) = L^d = D(q)/A$$

which gives us a second relation between $w$ and $q$, alongside $w=qA$, and it permits to determine both simultaneously.


Does a drop in GDP per capita, with the population staying around the same, mean people on average are poorer?

Regardless of the population size, what matters in terms of income is real or nominal terms. The gdp per capita may rise despite a constant population for the economy. But is it real terms that’s what matters. In an environment where the inflation skyrockets, gdp per capita is misleading.


How can lending be feasible on blockchain?

So, I don't understand how loans would be feasible on blockchain.

Lending and borrowing money via blockchain is similar but not equal to traditional methods.

The similarities:

Same parties: lender, borrower, broker
Same goals: those with capital wishing to increase their capital, those without wishing to increase their capital, a (optional) agent broker who receives a commission as part of their brokerage services.

The Differences are in how the process works:

I presume you are knowledgeable of traditional lending and borrowing mechanisms: banks, private loans, bond, equity, etc. So I'll cover the blockchain side of things.

You and I would sign up at a blockchain institution where the service of the firm is to match borrowers with lenders. I 'deposit' my blockchain assets and I either allow the firm to look for matches based on my preferences of interest rate, term, risk, and credit (in the abstract sense, not your FICO score); or pick and choose contracts to my liking.

The caveat for blockchains is that there is no known of recourse in case of default. The keystone basis of blockchain tech are the control and usage of private 'keys' (or password) matched to public keys that allow a user to control the 'data' associated with 'wallets'.

Unlike a traditional funding mechanism, if 'keys' are lost due to theft or malice (default), it is currently impossible to recover lost funds. No matter how powerful the judiciary and law enforcement system is in a given jurisdiction, they can't use a court order to brute force a solution to find the lost keys.

In a traditional lending environment, a creditor can go after assets that are tied to your identity and have monetary value (other forms of cash, real estate, vehicles, furniture, etc.) and liquidate them for their monetary worth.

Riskier? Definitely, but this doesn't mean that blockchain tech hasn't been used to facilitate lending arrangements with differing ranges of success.

  To me, the only time it would be feasible to borrow BTC is when I am shorting BTC... Or am I missing something?

You are, the key point here is if the loan is in BTC or fiat currencies (USD, Yuan, Ruble, Yen, etc.) BTC can be the mechanism of which the current market prices of BTC to X is transferred, but the loan would still be valued in USD.

So if you enter into a contract for 10,000USD at 1% over 10 years, (simple interest). Your USD obligations for the contract would be $100 a year, regardless of the USD to BTC rates.

Year 1: BTC/USD might be 1BTC/1USD. Your debt obligation is 100 BTC.
Year 2: BTC/USD might be 1BTC/10,000USD. Your debt obligation is .001 BTC.

The best way to approach the system is to first determine A: if the underlying monetary system is deflationary or inflationary. Deflation benefits the creditor, inflation credits the borrower. But with blockchain lending system: no one can force the borrower who is in default to return the Blockchain based assets. With traditional systems: you have law enforcement and judicial systems available.

Here's something I wrote for LinkedIn.


Why can we write any lottery as a convex combination of the degenerate lotteries?

Pick any set of non-negative $p_1,\dots,p_n$ such that $p_1+\cdots+p_n=1$. The convex combination of degenerate lotteries $L^1,\dots,L^n$ with the $p_i$'s can be written as
\begin{align}
p_1L^1+\cdots+p_nL^n&=p_1(1,0,\dots,0)+\cdots+p_n(0,\dots,0,1)\\
&=\bigl(p_1(1)+p_2(0)+\cdots+p_n(0),\;\dots,\;p_1(0)+\cdots+p_{n-1}(0)+p_n(1)\bigr)\\
&=(p_1,\dots,p_n).
\end{align}
This is just a lottery $L=(p_1,\dots,p_n)$.


How do countries create their own currencies (and give them meaningful value)?

Money is something which has value because everyone believes it has value.

A good way to create a trustworthy currency is to create a representative currency. A representative currency is a currency which is backed by some rare commodity by always guaranteeing that it can be exchanged back into that commodity. For example, a state might decide to create a new currency, let's call it the TirousCoin, and back it with gold. They decide that one TirousCoin is worth one gram of gold. The state guarantees that when you give them 100 gram of gold, you get 100 TirousCoins from them. And when you give the state 100 TirousCoins, you get 100 gram of gold. That means the TirousCoin's value is bound to the world market price of gold. Giving someone 10 TirousCoins is as if you would give them 10 gram of gold which is as if you give them 442 USD.

But many countries go a step further and have currencies which are pure fiat money. Fiat money is money which is not backed by any commodity. It is simply printed by a central bank, lent to regular banks and from there lent to people. And people trust that central bank to not lend too much of it, because that would cause inflation.

But how do you get people to accept your TirousCoins as their tender for their every day business transactions instead of competing currencies issued by other states?

Declare it legal tender by law. This means that you decree by law that  the TirousCoin is valid payment for all debts. When someone thinks that someone owes them, they can ask the state to force their debtor to compensate them with money. They do that by bringing the case to the state's civil court system. But the court system will consider a debt settled if it is paid in TirousCoins. When a creditor rejects the debtor's TirousCoins and demand to be paid in some other currency, the state will consider the case closed and won't assist the creditor.
Have the state do all their financial business in TirousCoins. Pay government employees in TirousCoins, demand that taxes are paid in TirousCoins (including taxes on transactions in foreign currency), pay subsidiaries and welfare in TirousCoins and demand fines in TirousCoins. When people need to acquire TirousCoins to avoid going to prison for tax evasion, the coins gain real value.
Make sure TirousCoins are a stable currency. Many states in financial perils make the mistake of paying their bills in freshly printed fiat currency instead of playing fair and only spend what they receive in form of taxes. This leads to an inflation which destroy trust in the currency and leads people to do dealings in other currencies.

For further watching, I recommend the Extra History's series on the history of paper money.


Hall (2005) mentions Barro's paper twice, but the first one in p. 51 is in the introduction and in order to alert the reader on what to expect. The essential dialogue between the two papers is in p. 56, where, Hall, after presenting his model and its workings, writes:

  "The earlier work implied inefficient outcomes, especially the loss of
  a job under conditions where both worker and employer could have been
  better off with a wage adjustment. The wage norm I consider interferes
  neither with the formation of efficient matches once the parties are
  in touch with one another nor with the preservation of jobs with
  positive surplus. Inefficient separations cannot occur. As a result,
  the model provides a full answer to the indictment of sticky wage
  models in Barro (1977) for invoking unexplained inefficiencies in
  economic arrangements."

In p. 64 moreover Hall writes

  "Stickiness is plausible, because it occurs only within the range
  where the wage does not block efficient bargains from being struck and
  maintained."

From this second quote, one could indeed indirectly validate the OP's second quote - that in order to provide a rationale for wage stickiness, we should restrain ourselves to cases and models where wage stickiness, in order to arise does not require blatant inefficiency of the contracting parties (because, after all, "blatant inefficiency" can explain everything).

The important remark here is that not all sticky-wage outcomes are necessarily inefficient.

But this essentially concurs with Barro's (1977) approach: Barro criticizes specific sticky-wage models where contracts have provisions only on the wage and not on the level of employment (or they have arbitrary such). Barro shows that not providing for employment determination may create dead-wight losses for both parties. On the other hand, providing an optimal rule for employment determination (optimal in the sense of "maximizing the pie", as he writes, available to employers and workers), would eliminate this dead-weight losses, and then, the issue of wage determination is to be considered, which may very well imply fixed nominal wages.
pp 311-312:

  "Given this form of employment rule, which maximizes the total pie
  available to firms and workers, one can then consider optimal (and
  competitive) payment arrangements that would determine the split of
  the pie. For example, it may be optimal for firms to perform the
  insurance function of guaranteeing to the workers, ex ante, a fixed
  (nominal or real) wage or income. In this case long-term contracting
  may account for ‘rigid‘ wages, and may also explain the apparent
  non-wage rationing of jobs."

So, I don't see any change in the interpretation of Barro's paper. Both papers seem to accept that it may be the case that sticky wages arise from optimal, efficient behavior. Barro criticized sticky-wage models available then for not modeling such an efficient way to obtain the stickiness. Hall's accepts Barro's critique on past sticky-wage models, and creates a model where indeed, fixed wages can be a feature of an efficient equilibrium, something that Barro also considers it possible, as the above quote shows.


Why is the purchase of assets not entirely tax-exempt in the year of purchase?

In theory, income taxes tax "wages", "salaries", and "net income".

The design of a tax system is an important political issue.  A good tax system will raise the amount of revenue that the government "needs", and seem fair to both most of the people who vote, and to the people who pay most of the taxes.  If the tax system does not seem fair to the voters, the legislature is likely to be thrown out of office.  If the tax system does not seem fair to the tax payers, the government will have a difficult time collecting the taxes.

There are countries that tax dividends from businesses, but not the profits of businesses that stay reinvested in the business.  There are countries that tax the profits of businesses, but don't "double-tax" the dividends from businesses.  And there are countries that tax both the profits of businesses, and dividends from businesses.  It sounds like you are in a country that thinks it is fair to tax business profits, but does not think it would be fair to pretend that the assets used up by a business are free.

Anyone designing a tax system needs to think about how someone could "game" the system.  If it is possible for business owners to figure out how to not pay any taxes, some of them will.  If none of the business owners pay any taxes, the voters will notice.

Suppose you could re-invest all of your profits in physical assets for your business, and those assets retained most of their value for a long time. (For an extreme case, imagine that you bought a chair made of gold, and most of the value of the chair was the gold.) If you could deduct the full cost of the purchase immediately, then you could avoid all taxes on your profits. This would make the tax system seem unfair to most voters (who earn wages and salaries, not business profits.)

Thus, most governments choose a compromise.  By letting businesses "depreciate" physical assets over the assets over their estimated useful lives, the governments recognize that the decline in the value of the assets is a cost of doing business.  This does not seem completely unfair to most businessmen.  This compromise means that most business people wind up paying some taxes, which makes the system seem fair to people who are paying taxes on wages and salaries.

In theory, "depreciation" is supposed to approximately match the decline in value of an asset due to your business' use of it (or due to the time you hold it, because your business might need it). In theory, you only put up with the decline in value of the asset because the asset helps your business make money in other ways. Thus, the "depreciation schedule" is supposed to match up the cost (of the declining asset) with the revenue (at each time in the future) that it should help you earn.  In theory, the total amount that is allocated adds up to the total decline in the value of the original asset.  If you invest wisely, the incremental revenue you gain (because of the asset) should exceed the cost of the asset.  In theory, the incremental revenue you gain per period (because of the asset) might be proportional to the depreciation per period.

In practice, there are lots of fudge factors.  Many investments are a waste of money.  Other investments are spectacularly good.  Sometimes you can figure out which investments were good in hindsight.  Some things wear out faster than expected, or need to be replaced for unrelated reasons.  Other things are useful (and perhaps even valuable) indefinitely.


What are good books about financial markets?

The classic undergraduate textbook is Financial Markets and Institutions from Mishkin and Eakins. I can wholeheartedly recommend this textbook check some local secondhand textbook stores where you can get it cheap (new book is unnecessarily pricey).

If you are not looking for a textbook I would recommend Ascent of Money by Ferguson- that’s mostly about history and it’s written in extremely simple language targeted at general audiences


Starting with your original equation:

$max_{c_t, m_t, b_t} E_0\sum_{t=0}^\infty U(c_t, m_t)$

s.t.

(1) $y+\frac{m_{t-1}}{1+\pi_t}+\frac{1+i_{t-1}}{1+\pi_t}b_{t-1}=c_t+m_t+b_t+\tau_t$

Here: $R_{t-1} =1+i_{t-1}$ and $1+\pi_t=\frac{P_t}{P_{t-1}}$

Note that in this problem, you have have two state variables, $m_{t-1}$ and $b_{t-1}$, and your main issue have been that you have bunched these together. your Bellman should be:

$V(m_{t-1},b_{t-1})=max_{c_t, m_t, b_t} U(c_t,m_t)+E_t\beta V(m_t,b_t)$

s.t. (1)

You can here use your contstraint to get rid of one control, or you can solve it using the lagrangian. If you use your constraint to substitute for $c_t$, your updated problem becomes:

$V(m_{t-1},b_{t-1})=max_{m_t, b_t} U(c_t(y, m_{t-1}, b_{t-1},m_t, b_t, \tau _t),m_t)+E_t\beta V(m_t,b_t)$

For clarification:

$c_t(y, m_{t-1}, b_{t-1},m_t, b_t, \tau _t)=y+\frac{m_{t-1}}{1+\pi_t}+\frac{1+i_{t-1}}{1+\pi_t}b_{t-1}-m_t-b_t-\tau _t$

FOCs:

$[b_t$]: $-U_{c_t}+\beta E_tV_{b_t}=0 $

$[m_t]$: $-U_{c_t}+U_{m_t}+\beta E_tV_{m_t}=0$

Envelopes:

$[b_{t-1}]$: $U_{c_t}\frac{1+i_{t-1}}{1+\pi _t}\implies V_{b_t}=U_{c_{t+1}}\frac{1+i_t}{1+\pi _{t+1}}$

$[m_{t-1}]$: $U_{c_t}\frac{1}{1+\pi _t}\implies V_{m_t}=U_{c_{t+1}}\frac{1}{1+\pi _{t+1}}$

You should be able to combine these equations such that you find the demand for money. To me, your equilibrium condition (along with the fact that no capital is present in the set-up) impies that there is no saving in real variables, and hence you have nothing that can pin down the real interest rate, and you therefore lack information to find the fisher equation.


Historical sovereign bond yield and CDS rate data download

There was a similar question in the Quant. Finance section (see here: https://quant.stackexchange.com/questions/22819/where-to-get-price-data-on-credit-default-swaps)

People suggested the following options: https://www.datagrapple.com/ (free for subscription), http://www.markit.com/Product/Pricing-Data-CDS (not free)

I also guess that Bloomberg and Thomson Reuters databases should also contain historical CDS rates. Practicioners use the data source extensively. Though they are very expensive for a personal use, some universities buy a license to get access to it and it is worth asking people from the finance/banking/econ department if they have access to the database.


Why is the total wealth of the world not a conserved quantity?

There's no such thing as a law of conservation of wealth. Wealth is created and destroyed in many ways. It is created when society produces goods and saves them. It is destroyed in disasters, or consumed when we use up our savings.It is created when we create knowledge, software, etc.

In a toy example: when a society starts out without any saved food, it produces food, eats half and saves half, then it is richer at the end. The accumulated food is an asset.

There's all kinds of wealth in a society: patents, ideas, buildings, roads, cars, computers, education, etc.

What does not get created or destroyed are nominal claims that we have against each other: if I give you a paper promising to pay you 10, that makes you 10 richer and it makes me 10 poorer but that's it. This means that its pretty had to figure out what is a society's total wealth, because there's all kinds of claims, explicit or implicit that are hard to value.


What is the difference between contract theory and mechanism design?

Contracts are a subset of all mechanisms where agreements are enforcable.

An example of a mechanism that is not a contract:
A second price auction (or Vickrey auction) is a truth-telling mechanism where the enforcability of contracts is not required. In the truth-telling equilibrium no one has any incentive to change their bid, no matter the outcome. This is not because changing the bid would be a violation of contract and result in penalties.


Wealth - Bond Market and Liquidity Preference Framework

These are theories, so need empirical justification. They may not be true, and even if they appear to be true they may still be wrong about the cause and effect.

If you regard your first statement as saying that people with increased monetary wealth will want to see a income increase from this wealth then they will buy bonds and other less liquid investments pushing interest rates down, and your second statement as saying that people with increased non-monetary wealth will want to see an increase in liquid assets such as money so they can spend some of their wealth then they will sell bonds and other less liquid investments pushing interest rates up, you will find there is no intrinsic contradiction. Instead you get an explanation of the existence of secondary markets in bonds and other investments to reconcile the needs and desires of different people


Why hasn't the huge increase in supply provided by Airbnb decreased prices?

To summarize what's detailed below:

Most research finds some small hotel price decreases following Airbnb entry. But the substitution effect between hotels and Airbnb is hardly 1:1.
Furthermore, some research even found increases in hotel revenues in some areas where Airbnb appeared; this is hypothesized to be due to more discriminatory prices that the hotels can practice because Airbnb customers are generally thought to be more  price sensitive and once they largely leave the hotel market, the hotels are left with less price-sensitive customers, on average.
You are further confusing the substitution/competition of Airbnb with hotels with the effect of Airbnb on long-term rents and/or (residential) property prices. The effect of Airbnb on the latter is indeed generally found to be contrary to that on the hotel market, i.e. because Airbnb offers more (and higher-profit) options to residence owners, an increase in long-term rent prices and property prices is observed in some (but not all) areas where Airbnb operates.

Your research (on hotels) is probably not sensitive enough to small effects:

  There is evidence that Airbnb increases the supply of short-term travel accommodations and slightly lowers prices.  [...]

  Airbnb is essentially a positive supply shock to short-term accommodations. Like all positive supply shocks, it should be expected to lower prices. There is some accumulating evidence that Airbnb does exactly this. Zervas, Proserpio, and Byers (2017) examine the effect of Airbnb expansion across cities in Texas. They find that each 10 percent increase in the size of the Airbnb market results in a 0.4 percent decrease in hotel room revenue. They find that most of this revenue decline is driven by price declines. Evidence of the positive supply shock is particularly evident in the 10 American cities where Airbnb’s presence is largest. Dogru, Mody, and Suess (2019) find a negative correlation between Airbnb expansion and hotels’ average daily rates in the 10 U.S. cities with the largest Airbnb presence.

Also, Airbnb and hotels are not necessarily perfect substitutes:

  Further, it is possible that by substituting more strongly for a less-expensive slice of the traditional hotel market—leisure travel as opposed to business travel, for example—that Airbnb introduction might actually be associated with raising measured short-term travel accommodation prices, through a composition effect.

See elasticity of substitution for the theoretical notion.

Also, an MA thesis/study in Norway (not cited in that review) seems to support this idea of hardly perfect substitutability of hotels and Airbnb:

  I find that a 10 % increase in Airbnb supply decreases hotel
  revenue by 0.3 %. This effect is around 20 times smaller than the effect on hotel revenue of an increase in hotel supply.

As for your last claim "In fact, it is argued that Airbnb increases prices!" that source seems to be talking about property prices:

  Our empirical results show that the overall causal impact of Airbnb on property values can be large. For example, in areas within 5km of Los Angeles’s central business district the price increase has been 14%. Within 2.5km of beaches, the price increase due to Airbnb has been almost 10%. This shows that in areas that are attractive to tourists, increases in property values are substantial, while in areas without much tourist demand (e.g. Pasadena), effects are small.

I don't see any contradiction with the findings about hotels.

Also (long-term) rental prices being increased by Airbnb by offering another (short-term) use for the same (residential) resources is consistent with this increase in property prices. Going back to my first (EPI) link:

  The single biggest potential cost imposed by Airbnb comes in the form of higher housing costs for city residents if enough properties are converted from long-term housing to short-term accommodations. If property owners take dwellings that were available for long-term leases and convert them to short-term Airbnb listings, this increases the supply of short-term rentals (hence driving down their price) but decreases the supply of long-term housing, increasing housing costs for city residents.  [...]

  The mirror image of Airbnb’s positive supply shock to short-term travel accommodations is its negative supply shock to long-term housing options.  [...]

  Airbnb—though relatively new—is already having a measurable effect on long-term housing supply and prices in some of the major cities where it operates. For example, Merante and Horn (2016) examine the impact of Airbnb on rental prices in Boston. The authors construct a rich data set by combining data on weekly rental listings from online sources and data from Airbnb listings scraped from web pages. They find that each 12 Airbnb listings per census tract leads to an increase in asking rents of 0.4 percent. It is important to note that this is a finding of causation, not just correlation. [...]

  Barron, Kung, and Proserpio (2018) undertake a similar exercise with different data. They create a data set that combines Airbnb listings, home prices and rents from the online real estate firm Zillow, and time-varying ZIP code characteristics (like median household income and population) from the American Community Survey (ACS). To account for the fact that rents and Airbnb listings might move together even if there is no causal relationship (for example, if both are driven by the rising popularity of a given city), they construct an instrumental variable to identify the causal effect of rising Airbnb listings on rents. Using this instrument, they find that a 10 percent increase in Airbnb listings in a ZIP code leads to a 0.42 percent increase in ZIP code rental prices and a 0.76 percent increase in house prices. They also find that the increase in rents is larger in ZIP codes with a larger share of nonowner-occupied housing. Finally, like Merante and Horn, they find evidence that Airbnb listings are correlated with a rise in landlords shifting away from long-term and toward short-term rental operations.

If you want results that appear more genuinely paradoxical, there is a 2016 paper with those:

  This paper gives an overview of the activities of Airbnb in 14 European cities. Since
  Airbnb provides an online accommodation platform linking property owners and
  visitors, it could potentially affect both the hotel market and the domestic rental market
  in the localities in which it operates. We discuss the structure and the segmentation of
  the accommodation market, and then present some descriptive statistics on Airbnb
  activities in the 14 cities. Finally, we present some estimates of the impact of Airbnb on
  hotels and on rents, among the first estimates for European markets. We find Airbnb’s
  presence in a market has a negative effect on hotel occupancy rates, but a positive
  effect on total hotel revenue and the average daily rate they charge.  In the two cities we
  consider, the platform’s impact on the rental market is ambiguous, suggesting local
  market conditions are important. [...]

  We find that a rise in the number of Airbnb listings in a city was associated over this
  period with a fall in the average hotel occupancy rate, but an increase in the average
  daily rate received by hotels. The combined effect on total hotel revenues was
  ambiguous to slightly positive. On the other hand, the arrival of Airbnb is positively
  correlated with the rental price index in London, but not in Berlin.

However even the seemingly paradoxical divergence between hotel occupancy and their revenues (observed in Europe) has an (elasticity-based) explanation:

  Business travellers are much more able or
  willing to pay for convenience and reliability. For example, Airbnb hosts can cancel
  reservations at short notice and the platform does not penalize the hosts who renege on
  their promises, or compensate the users who find themselves nowhere to stay at short
  notice. The informality of the sharing economy does not (yet) sustain service norms. On
  the other hand, we always expect hotels to observe their promises. By screening out
  budget travellers, hotels may be better able to identify customers with less price-elastic
  demand and thus possibly even raise prices and obtain higher revenue thanks to the
  greater scope for price discrimination. Thus the entry of flexible low cost supply could in
  effect push up the prices and revenues of some hoteliers, though low-cost hotels may
  suffer.

Relevant theoretical notions for this last para: price elasticity of demand (PED); price discrimination.

Unfortunately, empirically testing this latter hypothesis that customers with lower PED go to hotels, wasn't done in that study. This requires more data on the customers than supply-side statistics convey.

There is however some corroborating independent research that confirms that Airbnb customers in the UK have the same age profile as low-cost/economy hotel customers.

  people
  aged 25–34 years are the largest group of users of
  Airbnb.  [...]

  Economy hotels are the most popular among
  people aged 25–34 years.

(Of course, age is not a perfect proxy for income or elasticity of demand.)


All of these answers are true but don't provide an easy solution which doesn't use excel/code. Gini can be fairly easily computed by hand too.

The Gini coefficient fundamentally shows the shaded region above the lorenz curve in order to get a relative gauge of the distance the lorenz curve is from the line of equality. Fundamentally what this shows is the average distance between incomes. with 1 being the highest possible income and 0 being the lowest.

One way to calculate this is to simply calculate the area between the line of equality and the Lorenz curve.

This is clearly pretty difficult to do. Luckily there is an easier method.

Gini is also equal to: (The average variance for all incomes) / (2 * the mean)

The average variance for all incomes can be calculated by constructing a matrix of all incomes and then calculating the distance between each income.

For your example:

  | 1   2   3   4   5
__|___________________
1 | 0   1   2   3   4
2 | 1   0   1   2   3
3 | 2   1   0   1   2
4 | 3   2   1   0   1
5 | 4   3   2   1   0

Then add up all of the distances within the matrix to get the total variance.

0 + 1 + 2 + 3 + 4 + 1 + 0 + 1 + 2 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 2 + 1 + 0 + 1 + 4 + 3 + 2 + 1 + 0 = 40.

Then divide it by the total number of elements within the matrix to get the average variance.

Average Variance = 40 / 25 = 1.6.

Now divide the average variance by the average income times two.

Mean = (1 + 2 + 3 + 4 + 5)/5 = 15/5 = 3

Mean*2 = 6.

Since the gini is equal to: (The average variance for all incomes) / (2 * the mean):

Gini = 1.6 / 6 =  .2666

Which is what the other users got through more complicated methods.


Accounting for household work in the GDP

Are there techniques, regardless of practically how difficult they are, of possibly measuring domestic work?

Yes, measuring household gross production (GDP measures gross product not amount of work), is just matter of estimating how much households produce, in a given unit of time (e.g. quarter, year... etc), and what would be value of that output on the market (or with income approach how much would household get if the household production was supplied to the market). Both can be estimated via various econometric techniques.
Nowadays econometrics is too broad to provide overview of all possible techniques that could be used. However, a commonality among those techniques is that they estimate the value of household production by  inferring it from value of observable output. For example, one simple method just looks at value of production that is done in market (e.g. nanny services) and then assigns the same value to the amount of that kind of output produced by household in house. Other approaches try to measure the opportunity cost of households for home production and so on. You can have look at Dulaney et al. (1992) for overview of various methods, but there is quite a lot of heterogeneity in approaches that can be applied here especially when it comes to the details.

Have such estimates been made by researchers, maybe at scales smaller than the national economy?

Well this was already done on scale of national economy. In fact, Bureau of Economic Analysis (BEA) already provides estimate of value of US household production that is estimated in a way to make it complementary to GDP (i.e. you could add it on top of it - although that does not mean that you should as it is also less precise than just GDP itself, so it depends on a particular research if that is worth while sacrifice). In fact BEA even produces household production adjusted GDP (see their article on that here). Although, most statistical offices around the world don't provide such estimates yet.
Beyond statistical offices, individual researchers were providing estimates for value of household output already back in 70s and possibly even earlier (e.g see Weinrobe, 1974 as an early example), with various degrees of precision and success. However, I don't know any study that would do that on a regional level (and quick google scholar search did not revealed any either), but as mentioned at the beginning that would be actually much harder to estimate.

What are some theoretical approaches to understanding this? I remember Amartya Sen developed a model about intra-household trade. Is this related? Are there other such well known models?

This seems unrelated to the above questions. You can estimate household output without having a theoretical understanding what determines how much output will be produced by households. This being said, in economics there is a specific subfield of economics of family. However, it is beyond scope of any reasonable SE answer to provide literature review of whole field, although generally speaking the theories of household productions are somewhat analogous to the regular ones (e.g. households will have production functions, their stocks of labor and capital, costs, home production competes with production that could be gotten from market, they will try to optimally allocate their resources to maximize utility etc.). You can have look at overview of various theories of household production in Browning & Martin  Economics of the Family or Menchik Household and Family Economics or the classic text Gary Becker Family Economics (although slightly out of date now).


If every company in the economy earns zero economic profit, can they contribute to real economic growth?

The standard measure of the size of an economy is total income. Profit is just one type of income (accruing to owners of firms). Wages that accrue to the workers in a firm can increase, which can be taken from the profit of the firm. In that way, income from other sources can continue to increase even though firms earn zero profit.

Other sources of income are rent (from land), and interest (from owning productively-used capital), but I'll stick to wages to keep it simple.

So, you can see that there are different markets at play—the labour market, the market for land, and the market for capital (or input goods). The important thing is that one does not, for example, simply take profit away from firm owners and give it to labourers. It must be a result of equilibrium in these markets. So, for example, if equilibrium in the labour market determines that wages for that sector must be high (e.g. due to scarcity of skills or an abundance of demand for those skills), then that high wage is just a cost of production that firms in that sector must swallow, in order to produce in that sector. Even though competition drives profits down to zero, all the other firms in that sector will also be having to pay the market wage rate.

I'm answering your question in a highly theoretical fashion, but then again, zero economic profit across the economy is a highly theoretical situation itself.

Edit:

To clarify, I'm arguing that aggregate income can increase through other forms of remuneration, even though profits stay at zero.

If you had a ceteris paribus clause that said these additional income sources remained the same, then economic activity would remain constant (i.e. growth would be zero). Take note that even if the wage rate (or interest rate or price of renting land) remained the same, an increase in the number of firms entering the market would increase economic activity. So, this ceteris paribus clause is not so much talking about the rate of remuneration remaining constant, but rather the aggregate level of income from other sources remaining constant.


Episodes of Housing Crashes and Never Recovering

Maybe the case of Germany before and after WWI could provide you an example. See here. The short story is that there was a big housing shortage before the WWI, which, after the war, which was no solved by the market, because of rent controls, profits controls, and other regulations.

The origin of housing shortage and rising prices:

  Germany's housing problem originated in the period of rapid industrialization,  population  growth,  and  urban  concentration  between  1870
  and  1914. Population mushroomed  from  41  million to  67 million,  and
  by   1910  twenty-six German  cities  boasted populations of  200,000.

This led to:

  high  real  estate  prices, heavy mortgage indebtedness, and high rents

State intervention begun:

  Systematic government  intervention  in the German housing  market
  began  during  World  War  I.

Some thought it would stop after the war:

  After  the war,  many  Germans  would  have  been satisfied  to  permit the wartime measures to  expire,  allowing the economy to  return  to  "normal." With  a return  to  normalcy,  private  enterprise would return to the housing market, eliminating the need for state intervention.

However, this did not happen:

  Plagued by inflation and  a severe capital shortage, the German housing
  industry  failed  to return to normal. The little capital that was available
  commanded exorbitant interest  rates;  until the credit crisis of June 1931
  first mortgage interest rates hovered between  8 and   10.8 %  per cent. ...  Predictably, there was practically no  demand  for new  housing,  and  thus very  little new construction. ...

The state capacity to build was also damaged:

  the war  effort  had exhausted  Germany's  capital supply, the loss of her colonies, merchant marine,   and  industrial  investments   in  Alsace-Lorraine  diminished  the nation's  capacity   for  new  capital formation,
    and   reparations  payments appeared to have first claim  on  any capital formation  the country might manage.

So, albeit not the exact narrative of a modern housing crises, it might be of help.


R's felm() fixed effects vs. factor() within lm()

You are correct in noting that, adding factor(FIPS) in side the regression formula should be equivalent to specifiying factor(FIPS) + YEARxQT in the felm() fixed effects block. I am less certain that your reg2 model that throws up the NA's is working correctly however, as that would typically throw up an error message before getting that result. Without actually seeing your data, I can only presume your issue is related to a bug, or has something very specific to do with your data.
Fortunately, there are lots of different ways to do this estimation, and it might be worth it to try this using different packages to corroborate the correct estimation and better understand what is going on with your specific use of felm().
For example, just to check, I downloaded some sample panel data and ran a FE estimation using a binary for the main regressor. I run a Least Square Dummy Variable model, a FE using the plm package, and then two versions with felm, both placing the entity fixed effect inside and outside of felm's FE block as you did in your example.  In all four cases the coefficient estimate was exactly the same (which it should be).
library(foreign)
library(plm)
library(lfe)

Panel <- read.dta("http://dss.princeton.edu/training/Panel101.dta")

modlsdv <-lm(y ~ op + factor(country) + factor(year), data=Panel)

modplm <- plm(y ~ op + factor(year), data=Panel, index=c("country", "year"), model="within")

modfelm1 <- felm(y ~ op | country + year | 0 | country, data=Panel)

#in this last model, I move the entity FE outside the FE block
modfelm2 <- felm(y ~ op + factor(country) | year | 0 | country, data=Panel)

I only show the coefficient of interest as all the factored-out variables take up a lot of space, but you can see below that they are all equivalent (apart from the SE's in the felm models being different because of how they were clustered).
────────────────────────────────────────────────────────────────────────────
                     LSDV           plm           felm1          felm2
               ─────────────────────────────────────────────────────────────
  (Intercept)    419655690.33
                          3
                 (1320579554.
                        769)
  op             -1434277502.   -1434277502.   -1434277502.   -1434277502.
                        244            244              244            244
                                                          *            *
                 (771947977.3   (771947977.3   (524652684.0   (594276313.4
                         01)            01)            46)          03)
               ─────────────────────────────────────────────────────────────
  N                  70             70             70           70
  R2                  0.386          0.264          0.386        0.386
  logLik          -1609.649                     -1609.649     -1609.649

  AIC              3255.298                      3255.298     3255.298
──────────────────────────────────────────────────────────────────────────


Theory behind financial flows not going to the most developing nations even though they have the highest growth rates

Its called Lucas paradox and its actually specifically about flows of capital not financial flows per se. Also it is not an idea from developmental economics or growth theory but international macroeconomics (or at least that’s where it originated I am sure the idea could get home in other subfields as well).

The paradox arises because classical international macro theory would predict that capital should flow from capital rich countries to capital poor countries as in those the rate of return on capital should be higher. Hence rational investors should flock to those.

The paradox is actually relatively recent as it was proposed just 30 years ago in this paper:

Lucas, Robert (1990). "Why doesn't Capital Flow from Rich to Poor Countries?". American Economic Review. 80 (2): 92–96.

The interesting thing is that this paradox emerged only during 20th century as there is an evidence for large capital flows during the first wave of globalization (i.e. pre WWI) between capital rich an poor countries.

Some possible explanations offered in literature are that this is either due to market imperfections, sovereign risk, difference in institutions (or better said lack of good institutions in poor countries), or missing technology, infrastructure, or some other important factors of production such as human capital in developing countries.

Some of the recent work on this area includes:

Montiel, P. J. (2006, February). Obstacles to investment in Africa: Explaining the Lucas paradox. In high-level seminar Realizing The Potential for Profitable Investment in Africa.

Schularick, M., & Steger, T. M. (2008). The Lucas Paradox and the quality of institutions: then and now. Available at SSRN 1656282.

Azémar, C., & Desbordes, R. (2013). Has the Lucas Paradox been fully explained?. Economics Letters, 121(2), 183-187.

Akhtaruzzaman, M., Hajzler, C., & Owen, P. D. (2018). Does institutional quality resolve the Lucas Paradox?. Applied Economics, 50(5), 455-474.

You will also find other good sources cited inside the articles listed above, especially the first one has a good literature review.


Short term and long term price elasticity of currencies

There is no way to predict that.

A "price" is just the level of the last transaction. What happens in the next transaction does not depend on the previous transaction. So if you sell dollars at 1.10 dollars per Euro, the next person to sell dollars might sell them at 1.12 or 1.08 or some other amount based on a myriad of factors. There is no way to predict what trades might occur in the future, just from knowing the trade you made in the present.


What do we know about the causes of global income inequality?

I have to say I do not believe in simple all encompassing answers. The reasons behind the differences are probably complex and multitude, varying from country to country. Below are some answers that one runs into and I think these probably account for a share of the inequality.

The book Why Nations Fail deals with the first question extensively. It is a highly acclaimed work (though personally I was not always convinced by their reasoning) by renowned economists Daron Acemoglu and James Robinson. Their basic argument is that inclusive democratic institutions lead to greater growth because it safeguards business and economic interests of the people.
Hence they see the biggest obstacle in undemocratic cultural traditions and lack of institutions.
An argument I read somewhere (perhaps it was the Dictator's Handbook) is that developed countries prefer dictators in power to fledgling democracies in other countries. The reason is that fledgling democracies are not very stable, and that is bad for investment. Also dictators may be bribed more cheaply to disregard their people's best interest. (For example they may not implement environmental regulation for a price. The same may be said for democracies, but the price is usually higher.) In this way business interests in developed countries may directly contribute to dictatorships, which according Acemoglu and Robinson is what hinders economic growth.


What is the link between econometric theory and applied work?

I am not exactly sure what you're asking. So...I hope we both understand identification strategy to mean the same thing. If so:

The identification strategy serves as the bedrock upon which you build your argument.

For example, consider some policy implemented by a school district that focuses on improving academic outcomes for low performing students. Say this school district sets a cutoff of a 1.5 GPA. Any student at or below this GPA is placed on academic probation. This assignment to AP triggers a set of responses designed to increase a student's academic success. However, these responses are time consuming. Now, suppose at the same time that a neighboring school district with a similar GPA distribution does not implement this policy.Suppose students can move freely between the two districts.

There are several ways to analyze the efficacy of such a policy. I will consider two.

Assume researcher A decides to investigate the efficacy of the policy by simply conducting some simple trend analysis. Let's suppose he decides that it is reasonable to determine the efficacy of the policy by evaluating the amount of students with a "low" GPA both before and after the policy implementation. This seems like it could serve as an intuitively sound approach.

Let's assume now that when the school implements this academic probation policy that half the students placed on academic probation leave and begin attending the neighboring school to avoid the time demands associated with the academic probation.

The trend analysis by researcher A likely now overestimate the efficacy of the academic probation program. This is because there will be a large decrease in the number of students with a low GPA. However, this will be largely due to those students leaving the school and not because the policy helped raise their GPAs.

Suppose researcher B uses a difference-in-difference approach using the neighboring school district that does not implement the academic probation policy as a counterfactual (where the similar trends assumption holds). His identifying assumption and research strategy will allow him to identify and control for the change in composition brought about by the policy implementation.


They are both OLS estimators but generally $\beta_1 \neq b_1$.
In statistics broadly speaking estimator is just some rule, method, or criterion for arriving at an estimate of the value of a parameter. According to Stachurski: A Primer in Econometric Theory, pp 217:

An estimator is a statistic used to infer some feature $\gamma(P)$ of an unknown distribution $P$. Thus a statistic becomes an estimator when paired with and compared to feature of the distribution. Nothe that there is nothing in the definition of an estimator that implies it will be a sensible estimator of the target feature, let alone a good one.

Consequently, by definition if you use the following formulas:
$$b_1= (X_1’M_1X_1)^{-1} X_1’M_2y$$
$$\beta_1= (X_1’M_2y)^{-1} (X_1’M_1X_1)$$
for an inference they become estimators. Moreover, both of these look like OLS estimator. So in that sense they are the same type of estimator.
However, running the same estimator will not give you the same result when you change position of dependent and independent variable (in fact OLS estimator will be biased if there is reverse causality so it does not make any sense to do it in the first place). Consequently generally speaking $\beta_1 \neq b_2$.
In fact in your specific case:
$$y= X_1b_1+X_2b_2 + u \implies X_1=y\frac{1}{b_1}-X_2\frac{b_2}{b_1}-\frac{u}{b_1}$$
Consequently, we have:
$$\beta_1 = \frac{1}{b_1}; \beta_2 = -\frac{b_2}{b_1}; v = -\frac{u}{b_1}$$
Trivially since $\beta_1 = \frac{1}{b_1}$,   $\beta_1 \neq b_1$, save for some special cases such as when $b_1=1$ for example.


Incentive compatible here means that you have no incentive to lie. A direct mechanism that is not incentive compatible would entice some agents to report a false type.


The basics are of course Cournot, Stackelberg and Bertrand competition, which you can find in any textbook.

If you are referring to needing references for research, then the paper you absolutely must know is Dixit and Stiglitz (1977) "Monopolistic Competition and Optimum Product Diversity". American Economic Review. With over 10k citations, the importance of this paper for research cannot be overstated. Especially for macroeconomics, which interests you, knowing this paper in and out is indispensible. It has been extensively used in international trade (e.g. Krugman's models), economic geography and monetary policy (most DSGE models incorporate it) to name a few fields.

While this paper is a specific form of oligopoly, namely monopolistic competition, it is very general and implicitly incorporates much of Cournot and Bertrand competition models. It is simultaneously relatively simple and tractable for use.

Most papers use a "light" version of Dixit and Stiglitz, however. If you search for explanations of this model online, you will find a lot of ressources that explain the different (light) versions step by step. I would actually read those before reading the paper.

A good book on the Dixit & Stiglitz model is also Brakman, S. and Heijdra B.J. (eds.) (2004): "The Monopolistic Competition Revolution in Retrospect." Because this model is so important, many interesting parts of Dixit and Stiglitz are also explained in Baldwin, R., Forslid, R., Martin, P., Ottaviano, G. and Robert-Nicoud, F. (2003). "Economic Geography and Public Policy."

You usually won't need to know any other model for applying oligopolies to research problems. An alternative is, however, the Salop (circle) model (Salop, Steven C. (1979), "Monopolistic competition with outside goods", The Bell Journal of Economics, 10). It may have an advantage depending on your setting, but is a bit outdated compared to Dixit and Stiglitz. In my experience Dixit and Stiglitz is preferred by the research community.


Why isn't the "annihilator" matrix a zero matrix?

This is more a related thought than anything else, but I thought it might be useful for people looking at this question in the future. Herr K. should write a short answer and have it marked as accepted.

Let $X$ be an $n \times k$ matrix with linearly independent columns. Let $S \equiv \text{span}(X)$.

Regression can be thought of as the following problem: Given an n-dimensional vector $y \in \mathcal{R}^n$, find the vector in $\hat{y} \in S$ that is closest to $y$ -- i.e.
$$\hat{y} = \arg\min_{z \in \mathcal{R}^n} ||y - z||$$
It can be shown that the solution to this problem is given by
$$\hat{y} = Py = X (X' X)^{-1} X' y$$
where we refer to the matrix $P = X (X' X)^{-1} X'$ as the "projector matrix." Then the matrix $M = (I - P)$ allows you to get the residuals of your regression by $\hat{u} = (I - P)y = y - \hat{y}$. As you said, if $X$ is square then $P$ reduces to the identity and the residuals are 0. This is insightful for several reasons.

Geometrically, if $X$ is square and has $n$ linearly independent columns then $\text{span}(X)$ is all of $\mathcal{R}^n$. Then the projection problem reduces to picking the vector in $\hat{y} \in \mathcal{R}^n$ that is closest to some vector in $y \in \mathcal{R}^n$. Obviously, if your vector is coming from the span of the set, the minimum distance choice would be the vector itself. We see this because as you showed $P = I$ in this case. Then $M = (I - P) = 0$ which means you have 0 residuals and you have a perfect fit of your data.
Additionally, this is why empirical economists are careful about "throwing in the kitchen sink" in terms of how many variables are included. If you have as many linearly independent variables as observations then you can get a perfect match whether your exogenous variables are related to your endogenous variables or not.


Does Inflation Make Money Eventually Worthless?

"Worth less" than before - yes, that's exactly what inflation does.

"Worthless" - not quite. No percentage-based reduction in value can make something worth 0, but there are extreme examples from history of times inflation has spiralled so far out of control ("hyperinflation"), money became worth less than the paper it was printed on, and life savings became almost worthless.

1920s Germany is the most famous example, but there are more recent examples, such as Zimbabwe.

This is a real Zimbabwean banknote from 2009 - you don't need the terrifying inflation rates to see that savings of thousands of Zimbabwean dollars made in the early 2000s had become effectively worthless:

But those are extreme examples.

As for the idea of a McDonald's $100 Menu - it wouldn't be unbelievable looking at history.

According to measuringworth.com, $1 one hundred years ago would buy the equivalent of $24 of common retail goods and the equivalent of $99 of the time of an unskilled labourer. A time-traveller from 1914 would expect a "$1 menu" to be high quality dining, worth days' wages for the average Joe. Not so different to how the idea of a $100 menu seems to us today,

This seems like a big change - but even this is a small change by international standards.

The US dollar has historically been one of the world's most stable currencies, partly related to its use in international trading and its status as a reserve currency. In other countries, even countries with relatively stable economies, such changes do happen in a person's lifetime. Here in the UK, it's a cliche for elderly people to complain that "in our day", meals could be bought for sums of money that today would considered small change.

But even though it'd be perfectly believable, it might not happen. Beware of extrapolations that assume trends won't change (cartoon from XKCD):

Anything could change. The US dollar might not keep its current world status. Governments influence inflation, and may sometimes simply change currency values if they feel it's getting out of hand. For example, in 2005 Romania knocked 4 zeros off its currency. 10,000RON became 1RON overnight.


Continuity of preference relation (iff?)

This is a definition. Definitions are always iff.

Consider this property. From now on we call this and exactly this (name of the concept here).


A market is covered if all consumers will choose to buy from at least one of the firms at the prevailing prices.

For example, consider a standard Hotelling model with two firms who are located at opposite ends of the unit interval. Suppose that each firm charges that same price $p$ and that consumers are uniformly located on $[0, 1]$. Finally, suppose that every consumer has unit demand and the same utility function $u = v - tx - p$ where $v$ is their valuation for the good, $t$ is the cost of transportation and $x$ is the distance that they travel to the firm from which they buy. In this example, every consumer will choose to buy a good if and only if the consumer located in the centre chooses to buy the good (since she has the furthest distance to travel). This is equivalent to asserting that

$$ v - 0.5t - p \geq 0 \iff v \geq 0.5t + p .$$

That is, the market is covered when the valuations of the consumers are sufficiently high relative to the price and cost of transportation.


Computing the continuous time survival rate

I think the step

  "...where $P(a, t) = exp(-d(a,t)\Delta)$ is the discrete time analogue of $d(a,t)$..."

is the problem.

In continuous time I guess we have

$$\dot m(a,t) = -d(a,t)m(a,t) \implies m(a,t) = m_0\exp \{-d(a,t)t\}$$

We then have, discretizing,

$$\frac {m(a,t+\Delta) - m(a,t)}{m(a,t)} = \frac {\exp \{-d(a,t)(t+\Delta)\}-\exp \{-d(a,t)t\}}{\exp \{-d(a,t)t\}}$$

$$=\exp \{-d(a,t)\Delta\}-1 \approx -d(a,t)$$

$$\implies d(a,t) \approx 1-\exp \{-d(a,t)\Delta\}$$

So it should be $P(a, t) = 1-\exp(-d(a,t)\Delta)$. Now $a$ changes exactly in the same way as $t$ does (at $t+\Delta$ those of age $a$ at $t$ will be of age $a+\Delta$). So $m$ changes in both arguments together.

$$m(a+\Delta, t+\Delta) = (1-P(a, t))m(a, t) = \exp(-d(a,t)\Delta m(a, t)$$

$$\approx [1-d(a,t)\Delta]m(a, t)$$

I understand the left hand side as "the people that were of age $a$ at time $t$, and are still alive at time $t+\Delta$, where they have become of age $a+\Delta$. Is this what you are after?


Let's guess that the value function is of the form $a + b \ln(k)$.
Then substituting for $V(k) = a + b \ln(k)$ in the Bellman equation gives:
$$
a + b \ln(k) = \max_{k'}\left(\ln(k^\alpha - k') + \beta(a + b \ln(k')\right)
$$
The first order condition is given by:
$$
\begin{align*}
&\frac{-1}{k^\alpha - k'} + \beta b \frac{1}{k'} = 0,\\
\to & k' = \beta b (k^\alpha - k'),\\
\to & k' = \frac{\beta b}{1+ \beta b} k^\alpha
\end{align*}
$$
If we plug this  into the objective function of the Bellman equation, we obtain the following identity:
$$
\begin{align*}
a + b \ln(k) &= \ln\left(k^\alpha - \frac{\beta b}{1 + \beta b}k^\alpha\right) + \beta\left(a + b \ln\left(\frac{\beta b}{1 + \beta b}k^\alpha\right)\right),\\
&= (\alpha + \beta b \alpha) \ln(k) + \ln\left(1 - \frac{\beta b}{1 + \beta b}\right) + \beta a + \beta b \ln\left(\frac{\beta b}{1 + \beta b}\right)
\end{align*}
$$
As this holds for all $k (> 0)$ we can equate coefficients on both sides:
$$
\begin{align*}
a &= \ln\left(\frac{1}{1 + \beta b}\right) + \beta a + \beta b \ln\left(\frac{\beta b}{1 + \beta b}\right),\\
b & = \alpha + \beta b \alpha
\end{align*}
$$
The second one gives a closed form expression for $b$:
$$
b = \frac{\alpha}{1 - \beta \alpha}.
$$
Then substituting this into the first order condition gives:
$$
\begin{align*}
k_{t+1} &= \frac{\beta \frac{\alpha}{1 - \beta \alpha}}{1 + \beta \frac{\alpha}{1 - \beta \alpha}}k_t^\alpha,\\
&= \beta \alpha k^\alpha_t \tag{1}
\end{align*}
$$
This shows that:
$$
k_{t + 1} > k_t \iff \beta \alpha k_t^\alpha > k_t \iff k_t < (\beta \alpha)^{\frac{1}{1 - \alpha}}
$$
So the capital stock will rise as long as $k_t$ is below $(\beta \alpha)^{\frac{1}{1 - \alpha}}$ and it will decrease if $k_t$ it is above this threshold.
According to the dynamic equation (1) above, it would appear that $k = 0$ is also a steady state. However, for $k = 0$, the first order conditions are not satisfied and in fact the value function does not exist. Anyway, for $k_t$ very close, it's value will be below $(\beta \alpha)^{\frac{1}{1 - \alpha}}$ so the stock of capital should increase to the unique steady state.


Adding a non-binding constraint to the objective function

To illustrate what Tirole has done, let's consider a simpler environment.

Consider a utility maximisation problem over two goods, $x$ and $y$. The consumer has utility function $u(x,y) = f(x) + y$, where $f$ is strictly increasing and strictly concave. The consumer's problem is thus

$$ \begin{align}
\max_{x,y} &\quad f(x) + y \\
\text{s.t.} &\quad p_x x + p_y y \le m
\end{align} $$

Given the conditions on $f$, we know that the budget constraint must bind. That is to say that in any solution to the above maximisation problem, it must be the case that

$$ p_x x +p_y y = m $$

We can rearrange the above equality to

$$ y = \frac{m - p_x x}{p_y} $$

We can thus incorporate this fact into our objective function and use the expression derived above for $y$ to replace it. Thus, our optimisation problem now becomes

$$ \max_x \quad f(x) +  \frac{m - p_x x}{p_y}$$

which, in a way, is a simpler problem, given that I now no longer have to worry about the Kuhn-Tucker conditions. The second-order conditions are also now much easier to verify.

Tirole does something similar to this. He uses the fact that a constraint is binding to simplify the objective function by expressing the objective in terms of the binding constraint.


How would privately minted coins enter circulation?

I am unsure about the American situation, but I learned a bit of history about token issuance in pre-Confederation Canada. These comments are based on the notes in a token collector guide.
Since they were privately issued, they would be called “tokens” and not “coins,” as that was reserved for governmental issuance.
In the Canadian colonies, there was a deficiency of low denomination copper coinage, which were officially minted in Britain (and British governors did not really  care). So the locals took matters into their own hands.
The banks were serious about coinage, and the metal content was close to what was implied by face value. Non-banks also issued tokens, but the metal content was quite often below face value. The banks did not like these tokens, and they typically did not accept them at face value, and melted them down. (This means that the tokens with the least copper are the rarest, and worth the most now.)
The tokens entered circulation via being handed out by the firms that minted them. Even if the tokens were too dodgy to be used elsewhere, people could return them to the same shop.


Why is the Argentine Peso still so expensive?

Buying and selling of currencies in Argentina is currently strongly controlled by the government:
http://www.reuters.com/article/2015/02/09/argentina-imports-idUSL5N0VJ4I520150209

This means that you cannot buy as many Dollars as you want, but the ones you buy you get at a government controlled exchange rate, which is actually cheaper. The Argentine government might have to pay 10 Pesos for a Dollar, but they sell it to the importer for 8 Pesos. They do this so that consumer prices do not spiral out of control. Hence the current exchange rate is not what it would be in an uncontrolled market. Further devaluation (controlled by the government) is expected next year, because this subsidized exchange is too costly for the government:
http://www.bloomberg.com/news/articles/2015-03-18/argentine-peso-forwards-at-record-show-devaluation-bet-for-2016

As for your second question: while the Peso may not have any value outside of Argentina, individuals and firms that buy products in Argentina will still have a use for it. For example a US firm importing and selling Argentine beef will trade Dollars for Pesos.


Are US federal reserve notes money, or merely currency?

Federal notes are both money and currency.
Currency can have two meanings:

In context of foreign exchange rate currency might simply refer to a system of money in common use in a particular country. That is, when discussing foreign exchange all USD, GBP, RMB etc are considered currencies.
Outside foreign exchange, currency means notes and coins in circulation. However, again currency is still money.

Money is more broad term in economics for a thing that serves as medium of exchange, unit of account and store of value (see Mankiw Principles of Economics 8 ed. pp 604). Consequently, money can be viewed as a sort of 'umbrella' term that encompasses currency but also much more.   That is all currency is money but not all money is currency. For example, money on deposit account will be considered money but not currency. Checks are also money but not currency. In addition, broad money will include also assets that can be easily converted to traditional money such as treasury bills.
However, most of money does not have special extra properties vis-a-vis currency. There is little practical difference between money in your wallet and money on your deposit account. Sure deposit accounts with debit cards might be more convenient (take less space, most cards now have contactless pay etc) but economically they are not special.  This being said when discussing broad money, such as T-bills, you could justify saying they have extra properties as well.


Study guide (and...) for "Intermediate Microeconomics with Calculus" by Varian

Nothing will be better than going through the textbook. Lecture's slides can be beneficial, you can google newer versions or access older here: http://www.mtholyoke.edu/~mirobins/econ212.html. I also used notes done by Łukasz Woźny, from SGH: http://web.sgh.waw.pl/lwozny/LectureNotes.pdf - they are not entirely based on Varian's textbook but might be helpful as well. There is also an Instructor’s Manual to each of Varian's books where he points out what is the most important from each chapter - also you should be able to find one online, and can also be quite helpful if you want to consume his textbook faster.


Can we successfully pursue the goals of environmental sustainability and reduction of income inequality at the same time (and, if so, how)?

In Denmark the government is both pursuing policies aimed at environmental sustainability as well as income equality. If one country can do it it must be logically possible. So the answer is: yes we can!!


In the typical textbook treatment of the two-part tariff model, income effect is ignored/assumed to be negligible.
Goldman, Leland, and Sibley (1984) provided detailed comparison of optimal nonlinear pricing strategies (of which two-part tariff is a special case) with and without income effect. Their conclusion is

When [income effects] are absent, the aim of optimal pricing is to raise a given amount of profit with the least possible distortion to individual's consumption plans. Therefore increments of consumption are priced according to the  Ramsey rule. When income effects are present, the optimal pricing problem is equivalent to the optimal taxation problem, and the goal is optimal income redistribution.


Basic New Keynesian model with flexible price

I think what you are looking for is a standard RBC-model. Because the main step, going from RBC to New Keynesian, is to include Calvo pricing among other things. This is covered in many textbooks like Gali's, where he first introduces the RBC model and then moves on to the New Keynesian setup.


«public-sector debt must now also include the central bank’s sharply rising debts». It didn't before?


Game Theory: finding nash equilibria of an extensive form game (game tree)

For the SPNE, you want to proceed by backward induction as explained by Lee Sin. For all other NE you want to construct the normal form representation (the usual table for simultaneous games) and solve for NE as if players were choosing their actions simultaneously. Note that the actions of player 1 are $\{U,D\}$, while for player 2 are $\{pp, pq, pr, ps, qp, qq, qr, qs, rp, rq, rr, rs, sp, sq, sr, ss\}$. This will indeed be a very large table.

Another way to do it is simply to look for fixed points directly. For example, assume that Player 2 plays $pp$, then the best response for player 1 is $D$ but if player one chooses $D$, the best response for player 2 is either of the following actions $\{ps, qs, rs, ss\}$ since $pp$ is not in this set, it is not a fixed point. iterate that process for all the actions of player 2.  Try to convince yourself that there is no other NE, except for the SPNE. And think about how would you change the payoffs so that the SPNE is the same, but now there are other NE's. (Hint: would changing the payoff of the terminal node after U,p from (2,1) to (1,1) work?)


Can the demand curve be a rectangular hyperbola, but $E_D \neq 1$?

Let's try a linear demand function, so $Q=a-bP$. Then $\epsilon_A=\frac{\partial Q}{\partial P}\frac{P_A}{Q_A}=-b\frac{P_A}{Q_A}$, so $b=-\epsilon_A\frac{Q_A}{P_A}=0.8\frac{500}{40}=10$. Then $a=Q_A+bP_A=500+10\cdot 40=900$. Thus, your table is compatible with the demand function $Q_D=900-10P$.


Current status and trends in the choice of software for teaching econometrics

In 2008 the American Economic Association (AEA) created a reproducible research requirement about posting the code and the data for their papers. According to analysis done by the AEA Data Editor Vilhuber, Turrito, and Welch (2020), Stata and Matlab were by far the most dominant software used in these submissions (see figure below, the raw data are here). Given the elite status of this family of journals in the profession, I think this is a reasonable proxy for the software used by elite economists and therefore in elite economics PhD training programs. For what it is worth, Stata and Matlab (and in that order) were also the software I was trained to use in my PhD econometrics classes and my work for my economics PhD, about ten years ago, at an R1 school in the USA. The only exception was my Bayesian econometrics class which I believe used R. As an undergraduate (also at a R1 school in the USA), I used Eviews in my econometrics classes and R in my probability and statistics classes.


Does a high positive output gap relate to high unemployment rates?

As denesp said, you're looking at a graph with no output gap.

A positive output gap usually does lead to employment being higher than the natural rate (=unemployment is lower than the natural rate).

Reading these lecture notes might help: http://eml.berkeley.edu/~jaya/lecture/Econ1_lecture9.pdf


What does these "strategy notations" mean?

The above post is pretty close (+1), but has not mentioned the subgame perfect Nash Equilibrium and the notation $s_2^*=8-s_1^*$ might be misleading, as it implies that player 2's Nash Equilibrium strategy is a function of $s_1$, which would mean the best response for player 1 could not be anything less than 8.

First consider the action set for each player (each has the same action set):
$$a_1 \in \{0,1,2,...,8\}$$
$$a_2 \in \{0,1,2,...,8\}$$

First consider the strategy set for each player:
$$s_1 \in \{0,1,2,...,8\}$$
$$s_2(s_1) \in \{f: \{0,1,2,...,8\} \rightarrow \{0,1,2,...,8\}\}.$$
Notice that player 1's strategy set is the same as her action set, since player 1 cannot condition her action on player 2's action before moving. On the other hand, player 2's strategy set is the set of all functions mapping player 1's action set into player 2's action set.

If we consider the definition of a Nash Equilibrium,and carefully consider the strategy sets for each player, we can see there are at least 10 Pure Strategy Nash Equilibria:
$$(s_1^*,s_2^*) \in \{(0,8),(1,7),(2,6),(3,5),(4,4),(5,3),(6,1),(7,1),(8,0),(8,8-s_1)\}.$$
In the first 9 equilibria listed (i.e. those the first poster was describing), firm 2's equilibrium strategy is simply a number, not a function that depends on player 1's strategy (e.g. $s_2(s_1)=4$). Notice these would be the equilibria if this were a static game (meaning player 2 did not observe player 1's choice before choosing an action).

The last equilibrium listed $(8,8-s_1)$ is the subgame perfect Nash Equilibrium. To see why, we must use backward induction. Starting with player 2's move, we have 9 possible subgames (one for each of the possible actions chosen by player 1). For the arbitrary subgame in which $s_1=C$, player 2's best response is to choose the action corresponding to $8-C$. We can generalize this across the 9 subgames and deduce that the strategy $s_2(s_1)=8-s_1$ forms a Nash Equilibrium in each of these 9 proper subgames.

So we have determined that in the subgame perfect Nash Equilibrium, player 2 must choose strategy $s_2(s_1)=8-s_1$. We must now find player 1's best response, which can easily be seen to be $s_1=8$. Thus the 10th Nash Equilibrium listed above is the subgame perfect Nash Equilibrium.

You might be inclined to argue that $(8,0)$ and $(8,8-s_1)$ are equivalent outcomes, but they are not. The former cannot be a subgame perfect Nash Equilibrium since player 2's strategy $s_2(s_1)=0$ does not form a Nash Equilibrium in every proper subgame (e.g. in the subgame where player 1 had chosen $s_1=7$, player 2 would still have chosen $s_2(7)=0$.


Why would a politician sollicit foreign investors to come and buy real estate?

The owners of existing housing certainly benefit from increased demand and higher prices for their assets. The total benefit to society may be positive or negative, depending on whether current owners win more than current non-owners lose (from higher prices if they want to buy and become owners).

Higher price of housing makes new buildings more profitable, benefiting the construction industry.

The politician may favour certain groups over others, for example the construction industry is usually a large donor in election campaigns. So the politician may want real estate prices to rise even if it is on average bad for the country.

`Invest in' need not mean buying existing housing, but it could mean building new buildings and selling the apartments in them. This could increase availability and decrease the price of housing.


In a perfectly competitive industry, why is apple considered best example of commodity?

In a perfectly competitive industry model, out of the examples given, the apple is the commodity with the least differentiation in the market. It is considered to be a homogeneous product.

A person (in this model) is unlikely to care about finding a specific type of apple from a certain company that uses a certain type of advertising. They just want to buy "an apple." All things being equal, they will probably just purchase the apple with the lowest price.

The other examples of shampoo, hotels, and ice cream are heterogeneous  in that these products have many ways of distinguishing their features from competing products. Design, advertising, packaging, utility, and other features stand out in addition to sensory features. Pricing can vary significantly since suppliers can make their products stand out from competing products.

(In reality, apples have a lot of variation, but in the model of perfect competition we just assume apples are of about identical size, taste, texture, weight, shape, etc. Apples are a classic example of a homogenous commodity, but not a perfect one because apples are not identical in the real world. However, they are probably more similar to a consumer comparing products than the other given options.)


Relationship between IRS and Competitive equilibrium

"Most likely" is not a precise term.

Given the available information it is quite possible that an equilibrium does not exist. In that case we can't really tell what will happen.


No. In fact, they mean quite the opposite.
Infinitely elastic, means the elasticity is infinity.
Perfectly Inelastic means not elastic, i.e. the elasticity is 0.
Ineslastic means the elasticity is very low, so between 0 and 1. This means there is "underreaction" (less than proportional).

So if something is infinitely elastic there will be an extremely large reaction.
If something is perfectly inelastic there will be no reaction.
If something is inelastic,  there will be very little reaction.

Let's take the example of price elasticity for an ordinary good where price increases cause demand to fall.

A good with a very close substitute would have a large price elasticity. If it's infinite: At a given price there will be a given demand. As soon as we increase the price even marginally the demand will fall to 0.

A good with no substitute and especially addictive goods usually have a price elasticity of (close to) 0 typically. There is the same demand at all prices. Assume an individual who always must smoke a pack of cigarettes a day. If we increase the price of cigarettes and if his elasticity is 0, then he will continue to purchase one pack a day. Even if we double the price or more, there will be no reaction (in this case reduction).


Why did the price level of government consumption increase in the United States?

The price level of government consumption (education, parts of health care, police etc.) consists mainly of wages.

Since the productivity (as we measure it) of government production often does not increase too much, the price level increases about as much as the level of wages. This, contrary to the price level of consumer expenditure which is driven down by productivity, increases in manufacturing as well as cheap imports.

The price level of investment goods might have gone down because of 'hedonic'calculation, i.e. statisticians calculate the dollar price of one unit of computing power instead of the price of one computer. In the new national accounts, military equipment is however also (and rightly so, IMO, as many military items have a clear second hand value and a well developed second hand market exists) considered to be an investment good, this gear may have another pattern of development than computers.

In a general sense, I suspect that the investment price level shown is only valid for machinery and the like. Most investment is however in land related items like dwellings, roads, bridges and other buildings as well as the prices of these and of the land itself, have not gone down that much (to the contrary, even, though there are quite some differences between the pre- and post-2008 period).

Here is some useful background information for the EU which shows that price levels of investment goods (broad definition) can differ up to 400% between countries.


Can the benefits provided by the Ex-Im Bank of the US be considered a Public Good?

Since you keep stressing the public good part of the argument..

From wikipedia:

  The Export-Import Bank of the United States (Ex-Im Bank) is the official export credit agency of the United States federal government.[1] Operating as a government corporation, the bank finances and insures foreign purchases of United States goods for customers unable or unwilling to accept credit risk. According to its charter, the Ex-Im Bank does not compete with private sector lenders, but rather provides financing for transactions that would otherwise not take place because commercial lenders are either unable or unwilling to accept the political or commercial risks inherent in the deal.

So the service at hand is lending to higher risk customers.

In economics, a public good is

  a good that is both non-excludable and non-rivalrous in that individuals cannot be effectively excluded from use and where use by one individual does not reduce availability to others.

Lending is very much excludable. To the extent that funds of the ex-im bank are limited, it is also rivalrous. Hence no, the services of the ex-im bank are not a public good.

Second order public good

To quote you:

  that might result in what you might call a Public Good  i.e. improved export conditions, job growth, tax revenue,

This part is irrelevant. Either you provide a public good, or you don't. Providing a private good, that may result in benefits for society doesn't make you a special provider that warrants special benefits.

Any producer of consumption goods creates "improved export conditions, job growth, tax revenue,". That doesn't warrant the government sponsoring them, because they create something "akin to a public good".


Is comparative advantage only beneficial with convex utility functions?

What traditionally matters is a quasiconcave utility function (that is, the individual/country at least weakly prefers mixing such that all better sets are convex). I assume that's what you're referring to when you describe the Cobb-Douglass example.

Though I believe your comment about the functional form of utility is overall correct. If, for example, the utility function were
$$
u(x_{1},x_{2})=\max\{x_{1},x_{2}\}
$$
both countries would produce whatever they were more productive in, and neither country would be incentivized to trade.

(But again, I think the key is quasiconcavity of the utility functions, not convexity.)


What could happen if a country changes their currency rate constantly?

At the outset, I'll point out that it would be neither healthy nor realistic for Yolistan's regulator or central bank to arbitrarily modify exchange rates. Issuing decrees about what the new exchange rate is does not guarantee that foreign entities will be willing to perform transactions at that rate.

  Where could this country acquire dollars at such rate or how they do
  it?

Yolistan could acquire dollars through exports, revenues from providing goods and services to tourists visiting Yolistan, money transfers from Yolistanis living abroad, persuade a country to lend money, and/or selling bonds (which is a form of debt).

  Yolistan(yolos) wants to buy Iron from the US(dollars) do they pay is
  Yolos or they have to buy dollars first?

Yolistan needs to buy dollars first, as it is unlikely that the US would want to have reserves in a new and uncertain currency such as Yolos.

  What happens to their economy and currency if that country changes the
  rate constantly to favor Exports/Imports?

Markets would not consider the Yolo a reliable currency (see the
discussions regarding the volatility of bitcoin). Markets expect a
currency to be stable.
Frequent and abrupt manipulations of the Yolo would discourage other
countries (and foreign companies) from signing trade agreements with(and invest in) Yolistan because of (1) concerns about unfair competition, and (2) excessive uncertainty about purchase power, rendering investments
and obligations too risky to commit with.

This answer is not exhaustive.


Informed traders do know the cross section of the privately known signal between each other

Yes, that assumption just means that every informed agent knows the true distributions where signals come from (i.e. roughly speaking, they know the parameters that define them).
The terminology "cross-sectional distribution" comes from the usual interpretation that instead of having literally 2 agents, in reality, you have a continuum of agents that come in 2 types. If that is the case, due to the law of large numbers, the empirical distribution of signals at any point in time will be equal to the theoretical distribution of signals. Therefore, knowing the cross-sectional distribution is the same as knowing the theoretical distribution. Since these distributions are normal, the latter corresponds exactly with knowing the mean and variance of signals of the other (type of) agent.


The assertion of the book is based on the phenomenon of commercial credit - the fact that business-to-business sales almost always are on credit, and the differences between terms-of-credit that a company gives to its customers, compared to the terms of credit that enjoys from its suppliers. It describes the (short-term) phenomenon, peculiar to some, that "the higher the sales and the profits, the lower the cash available".

A simplified bottom-up unfolding of how this can happen is as follows: Assume that a company has steady monthly sales $S$ for the years 2013 and 2014. Assume that the terms-of-credit that extends to its customers is "60 days/two months". Roughly, this means that the Accounts Receivables at the end of, say, year 2013 equal the amount of two months of sales (assume away any seasonality), denote it $AR_{2014} =2S$. On the other side, the company has only a "1 month" period to pay its suppliers. Assume the monthly Costs of Sales is $cS$.

So during year 2014 the Cash flow from the Customers-Suppliers loop has been

$$CF_{2014}=12\cdot S - 12\cdot cS = 12(1-c)S$$.

Now assume that starting from January 2015, Sales go up 20%.

During the year 2015  the Collections will be $2S + 10\cdot 1.2\cdot S$: November-December 2014 Sales collected in January-February 2015, and 10 months of collecting the increased sales amount.

As for Accounts Payable, during 2015 the company will have to pay $CSpay = cS + 111\cdot 1.2\cdot cS$ (we keep Inventories constant in value for simplicity).

So for 2015 we will have

$$CF_{2015}=2S + 1.2\cdot 10\cdot S - cS - 1.2\cdot11\cdot cS$$

$$=[14 - 14.2c]S$$

Depending on the value of $c$ we may have $CF_{2015} < CF_{2014}$ (in the specific example, we will need to have $c >0.91$ , i.e. a very small mixed profit margin $1-c$).

Namely, the combination of sales growth with the gap in the terms of credit, may result in the company having less cash available to cover its other, operational expenses like salaries etc. But even if this is not the case, the increase in Cash Flow may not be enough to cover the increase in other Assets that accompany an increase in Sales: an increase of the value of Inventories, but also possibly investments. Both will require increased cash. Also, operational expenses, like marketing or labor cost may have to increase. If the terms-of-credit gap is large and the sales growth aggressive, the company will require a Cash injection from shareholders or a bank loan to be able to pay all its obligations, even though its Sales has gone up.

The above are compacted nicely into a Balance-Sheet-point of view (where the equality $\text{Assets}=\text{Liabilities} + \text{Equity}$ holds always by construction, given how the terms involved are defined and measured). In fact, the Balance Sheet approach is the way many Finance Directors use to get a quick taste of what kind of financing will the company need next year. Essentially, the increased funds "automatically" injected by increased suppliers credit (due to increased purchases), plus any projected increased retained profits, may not be enough to cover the increase in the Current Assets due to the increase in Accounts Receivable (and possibly the increase in Inventories and in Fixed Assets that accompany the growth of Sales growth).
It may be the case that such an expansion may be profitable in terms of "Profit & Loss" while at the  same time having a negative effect on the Cash Flow, in the short-term.

Indeed, the difference could be covered by running down the cash reserves of the company, but in many cases, companies do not hold so much cash to do that. Then, one has to close the gap by for example, increasing Liabilities, say, taking a loan.


Comparing & contrasting decision problems and normal games

Decision under uncertainty is sometimes called a "game against chance", and can thus be modeled as a two-player normal form game: the decision-maker vs Nature/Chance. The possible states would form a set of pure strategies for Nature, and Nature commits to a publicly known mixed strategy that randomizes over those pure strategies (assume Nature is indifferent between all outcomes so that the mixed strategy is rational). The decision-maker's best response to this known mixed strategy is equivalent to the choice that maximizes his expected utility in the decision problem. In this sense, decision problems are really just a special case of a game.
The above suggests that your statement

A decision problem deals with expected utility, whereas normal form games do not.

is incorrect, since normal form games do deal with expected utility, especially when proper mixed strategies are involved.


Your answer is correct. Given that you have a net profit function once you substract the costs of the events, the way to proceed is differentiating that profit function, taking the behaviour of the other firm as fixed and setting that to 0.

Profit maximization involves producing upto the point where marginal benefits equal marginal costs and that is exactly what you do by differentiating and setting to 0.


Does only "supply/demand" rule justify different prices for a digital good?

One explanation would be price discrimination.

Some people are willing to pay more for software than others. (Typically, we would expect such people to live in richer countries.) As a result, it will be in the interest of software seller with monopoly power to charge different people different prices.

In case this isn't clear, here is an illustrative example. Suppose that Microsoft is deciding how much to charge users for a digital copy of Microsoft Office. Legally, it is the only entity that is permitted to sell Office, so it has some monopoly power. Since we are considering digital copies of Office, its marginal cost will be essentially zero. Suppose that:

In Country A, all inhabitants would pay at most £10 for one copy of Office.
In Country B, all inhabitants would pay at most £200 for one copy of Office.

(Assume that no inhabitant wants more than one copy.)

What prices should Microsoft charge (to maximise profits)? Obviously, it should charge £10 in Country A (or £9.99 if you prefer) and £200 in Country B. Thus, even though the cost of production are the same in both countries, prices will vary greatly between them.

Note that this explanation only works if Microsoft can prevent resale - which it can, since it has a patent for Microsoft Office. Otherwise, someone might start buying copies of Office in Country A and re-selling them in Country B, a strategy which will render Microsoft bankrupt unless it changes its pricing.

Of course, it also assumes that price discrimination is legal.


Question about constant relative risk aversion

Whenever you need to make someone indifferent between $x$ and $y$, it means that

$$U(x) = U(y)$$

a:

Denote amount he would pay by $z$. Paying $z$ to avoid the lottery gives him "certain utility" $u(100.000 - z)$. With Von Neumann-Morgenstern utility as given,
we can denote the lottery utilities as
$$prob*U(10.000 + x) + (1-Prob)*U(10.000-x)$$ (Understand why we can do that!)

b

Without uncertainty, he will have $u(w)$. Then, you need to compute the utility of the lottery as the function of $p, x,y$.

That is, you need to solve

$$U(100000) = prob*U(w+y) + (1-prob)*U(w-x)$$

Now, I'm not sure whether you were meant to say loses x or wins y, I suppose it is loses x or wins y. Anyhow, the value of the lottery is on the right hand side. Can it ever dominate the left hand side, given the additional information for $p$ that you were told in that subquestion?


What lies behind supply and demand model?

The “law of supply and demand” is a feature of economic models with supply and demand curves. These are usually mainstream equilibrium models. It is somewhat difficult to relate these models to real world behaviour. I am in a theoretical camp that is critical of such models; perhaps someone else can give a better account.

As for your questions about stock prices, you probably would need to look at the mechanics of stock trading. Investors post prices at which they are willing to buy or sell, and then others will transact at those prices. (In some cases, market makers are the ones posting the price.)

Those participants can change their posted prices whenever they want, typically in response to news. The prices can move without a single share trading hands. This can be interpreted as a shift in the supply or demand curve.

If someone has a big order to buy or sell, they can clear out the first set of posted prices, and transact against other orders. This will result in a price change. For example, assume there are orders to buy 100 shares at \$10 and \$5. An order to sell 200 shares would first clear the best bid (\$10) and then hit the next bid (\$5). In this way, big orders can change prices.

A longer version of this example.

There are no existing orders for shares in a company.
Investor A puts in a bid to buy 100 shares for \$5. The best bid (highest bid) is now \$5.
Investor B puts in an order to buy 100 shares for \$10. The best bid is now \$10. The bid price solely as the result of the new order; there was no transaction that moved the price.
Investor C wants to sell, and puts in an order to sell 200 shares. The first 100 shares clears B’s bid at \$10, and the second 100 clears A’s bid at \$5. The prices at which transactions happened was the result of an order clearing out the existing order.