Muth exposition of the rational expectations hypothesis

Muth assumes a model of

  "...short-period price variations in an isolated market with a fixed
  production lag of a commodity which cannot be stored".

It is useful to remember that the model's equations are expressed as deviations from equilibrium values. So in a bit more clear notation than the original (a star denotes long-run equilibrium value)

$$\begin{align} & D_t-D^* = -\beta (p_t-p^*) & {\rm (Demand)}\\
& S_t-S^* = \gamma (p^e_t-p^*) + u_t & {\rm (Supply)}\\
& D_t = S_t,\;\; D^* = S^* & {\rm (Market \;Equilibirum)} \end{align}$$

Production is determined one period before, based on expected future price, but final supply is also subject to random shocks, $u_t$, with $E_{t-1}u_t =0$. $p^e_t$ is expected price but we have not yet make any assumption on how it is formed, or to what is equal.

Eliminating quantities through market equilibrium we obtain

$$p_t-p^* = -\frac {\gamma}{\beta} (p^e_t-p^*) - u_t \tag {3.2}$$

Taking expectations conditional on time $t-1$ we obtain

$$E_{t-1}p_t-p^* = -\frac {\gamma}{\beta} (p^e_t-p^*) \tag {3.3}$$

Rearranging and subtracting $p^e_t$ from both sides we see that equation $(3.3)$ leads to

$$p^e_t-E_{t-1}p_t = (1+\gamma/\beta) (p^e_t-p^*) \tag {3.3a}$$

If  $\gamma / \beta =-1$ we obtain, without making any assumption on how expectations are formed but as a solution to the model, that $p^e_t=E_{t-1}p_t$. But this is uninteresting, being a very specific configuration of demand and supply responses. Assume then that $\gamma / \beta \neq -1$.

Then this way to write the relation (not in Muth's paper), shows clearly that if $$p^e_t \neq E_{t-1}p_t \implies p^e_t \neq p^*$$
and that
$$p^e_t = E_{t-1}p_t \implies p^e_t = p^*$$

Throughout the paper Muth treats $E_{t-1}p_t$ as the theory's prediction, a best prediction (and it is, in the sense of being the minimizer of mean squared error of prediction). Given this Muth argues as follows: if "market expectations" $p^e_t$ (i.e. some concept of "average", "prevailing" expectations) were not equal to the "best" prediction, then recurring pure-profit opportunities would exist, for someone that used $E_{t-1}p_t$ as his own expectation, while all others used some other expectations formation rule. But, is it reasonable to argue that the market as a whole is outperformed by some "wise man"? Is it reasonable to argue that the firms and the businessmen and any other people whose livelihood depends on the workings of this specific market, would not really try hard to be as efficient and as accurate as possible regarding their predictions? It doesn't sound too convincing, especially since we are talking about the collective wisdom of all market participants here.

So making the assumption $p^e_t = E_{t-1}p_t$ (i.e. imposing the RE Hypothesis) appears reasonable, and this leads to

$$p^e_t =p^*$$

(remember the right-hand side is long-run equilibrium price, not next-period's - we are not looking at period-by-period perfect foresight here).

Now use this result on the initial equations describing the market, and eventually obtain the determination of the short-run equilibrium price as

$$p_t = p^* -(1/\beta)u_t$$
This happens because we have imposed REH. In other words the imposition of REH brings about the result that current equilibrium price remains "attracted" and "chained" to long-run equilibrium, fluctuating randomly but not explosively.

Also we have

$$p_t = p^e_t -(1/\beta)u_t$$

which also means than in unconditional expected-value terms

$$E(p_t) = E(p^e_t)$$

"On average" (intertemporally), the price expectation will equal the actual price.

In one move Muth obtained two extremely powerful results:
a) Markets do not explode
b) Market participants on average and "as a whole" predict correctly.

And really, if markets did tend to explode rather than not explode, they would not be around for thousands of years, as they are. And if market participants were predicting consistently poorly, we would have seen much more personal financial ruins than we do.

What REH does not do well, is in helping model and analyze short run and transitional dynamics. It remains a long-run concept, a "long-run view" if you will, and this is why Adaptive Learning emerged, and this is why we are currently researching (in a frenzy), other expectations formation hypotheses.


Is it reasonable to consider government bonds as money

Short answer: Yes, there's another important reason that government bonds are often considered money, which is the role they play as a store of value.

There are three dimensions of money that form the long-standing textbook definition of money (see Mankiw, “Principles of Economics, 8th Edition.” p. 605):

Use as a medium of exchange (i.e., its value for settlement of claims).
Use as a store of value (the degree to which it is a "safe asset”).
Use as a unit of account (the degree to which it is used to denominate claims).

In the modern framework different assets are considered to fulfill these dimensions to differing degrees, which is why different measures of money include or exclude different assets.

Thus, the state of the art is not to think of assets as being either "money" or "not money," but to think of different assets as being more or less "money-like," or having differing degrees of "moneyness."

Dimension three, for example, is generally fulfilled only by physical cash and claims on banks. Claims aren't settled for "20 units" of Treasury bonds, they're settled for a certain par value of those bonds, denominated in dollars. So clearly Treasury bonds are not at all money-like along this dimension.

Dimension one, the one you raise in your question, is fulfilled by Treasury bonds to a greater degree than one might expect, though still only to a very limited amount. While Treasuries are not used for settlement in the real economy, they are used for settlement somewhat frequently in the financial system, which is responsible for a disproportionate amount of financial transactions (though even in finance they are used far less often than settlement for cash). For example, Treasuries are commonly used in securities lending transactions to obtain other securities, and they're often used to meet default and liquidity fund obligations to central counterparties.

Where Treasury bonds shine, though, is on dimension two, their use as a "store of value,” or as a “safe asset.” Per Gorton:

  Safe assets play a critical role in an(y) economy. A “safe asset” is an asset that is (almost always) valued at face value without expensive and prolonged analysis. That is, by design there is no benefit to producing (private) information about its value. And this is common knowledge. Consequently, agents need not fear adverse selection when buying or selling safe assets. Safe assets can easily be used to exchange for goods or services or to exchange for another asset. These short-term safe assets are money or money-like.

The “short-term” part of this refers to the fact that their usefulness as a store of value rises as their residual maturity (and thus their duration, or sensitivity to changes in interest rates) falls. Once their residual maturity falls below three months, they're treated for accounting purposes as being equivalent to cash, because their value isn't likely to change significantly in response to changes in the interest rate environment (this is true under both US GAAP and IFRS):

  Cash and cash equivalents comprise cash on hand and demand deposits, together with short-term, highly liquid investments that are readily convertible to a known amount of cash, and that are subject to an insignificant risk of changes in value. Guidance notes indicate that an investment normally meets the definition of a cash equivalent when it has a maturity of three months or less from the date of acquisition.

Importantly, Treasury bonds solve a very real problem for many firms that have large cash balances, which is that for large-value claims, they are actually more money-like than bank deposits. This is because there's a limit on deposit insurance (currently in the U.S., \$250k per bank, per account type, of which there are two, so \$500k per bank). As a result, for firms that hold large amounts of cash, bank deposits become a relatively poor form of money, because they're not eligible for the government guarantee. As a result, if the bank fails, they're exposed to both credit and liquidity risk (they may not get all their money back, and even if they do, it may not be available for quite a long time).

This is actually why firms frequently either hold balances in institutional money funds that do repos against government securities, or lend them in repo themselves— it allows them to upgrade what would otherwise unsecured claims on banks to claims that are secured by Treasury bonds, highlighting the "safe asset" role that these bonds play.

A very good summary of this concept, the "hierarchy of money," can be found in Pozsar's "Shadow Banking: The Money View." As he discusses in the paper, for many financial market participants, safety (dimension two) is more important than settlement value (dimension one) (emphasis mine):

  [T]here is a strong case for the introduction of a new set
  of monetary aggregates that track the supply of money and money-like claims held not for real
  economy but for financial economy transactions. The Federal Reserve’s M2 aggregate measures the
  money demand of households and has been used to analyze growth dynamics and threats to price
  stability. Because the bulk of money claims included in M2 are insured, it was built following a
  hierarchy based on transactional liquidity. But for institutional cash pools, money begins where M2
  ends, and as the crisis has shown, intra-system holdings of uninsured money market instruments can
  pose threats to financial stability. Institutional cash pools hold money claims mostly for portfolio
  management reasons and because they are too large to be eligible for deposit insurance, their focus is
  on money claims’ safety – that is, proximity to the government – first, and transactional liquidity
  second.

This is, of course, why the Fed had to create the Reverse Repurchase Program. Without it, cash lenders in repo were occasionally so eager to exchange their "cash" in the form of claims on banks for claims on Treasuries that they were willing to accept negative real (and, in certain circumstances, negative nominal) yields on general collateral repos against Treasuries— they were literally willing to pay to hold Treasuries rather than bank deposits. As a result, the Fed set up the RRP to allow their counterparties to do unlimited amounts of repo at a fixed rate, setting a floor under short rates in money markets.

Finally, there is a good working paper out of the Federal Reserve, authored in part by former Fed governor Jeremy Stein, that goes into further detail on the use of Treasuries as money-like claims.


What is meant by credit-based investment?

As the name suggests, credit based investment is investment that takes place based on funds that are themselves borrowed.

An example of credit-based investment is leverage.


True or False: During the Japan Bubble, one sky scraper in Tokyo was worth more than the entire state of california?

Brian is right. The claim was that, if you took the cost per square meter of land area in Tokyo based on market rates, the Imperial Gardens in Tokyo were worth more than California.

Source: http://articles.chicagotribune.com/1989-11-19/business/8903110641_1_land-prices-national-land-agency-rockefeller-center

Obviously, this is a classic example of a fallacy that arises from extrapolation— if that additional land area had been made available for sale, it would have significantly increased supply and caused prices to drop accordingly.


Is DARA utility implying CRRA most of the time?

Let me turn my comment into a quick answer:
Using the notation of the article you quoted $A(c)$ is the absolute risk aversion and $c A(c)$ the relative risk aversion. If $A(c)$ is decreasing, the preferences fulfill DARA. If $c A(c)$ is constant, the preferences fulfill CRRA. If CRRA holds, then $A(c)$ must be decreasing in $c$.

If we take any $A(c)$ such that $c A(c)$ is decreasing and $A(c)$ is positive, then $A(c)$ will also be decreasing. In this case, the preferences are both DARA and DRRA. You asked whether DARA implies CRRA in almost all cases. From this analysis, it seems to be rather the case that CRRA is the exceptional case.


How does a bank convert one currency to another?

Yes, a reserve is needed to ensure the integrity of the money issued (as always).

On the international foreign exchange market, many millions are traded at a time. Otherwise, with smaller amounts of money, the costs of transferring that money (e.g. via SWIFT) would be too high for the amount transferred. That is why investors and small businesses do not trade in the forex market themselves—they go through the financial intermediary.

Banks communicate with each other using computers and telephones. If you collectively put all these many communications together, you get the concept of an international interbank foreign exchange market.
Prices are offered for foreign exchange deals. If the price is fairly high, the bank may not get a sale quickly—they will have to wait, or lower their price. If the price is lower, they will be more likely to find a buyer.
This question is related to the reserve requirement. If banks need to increase their reserve, they will buy securities denominated in that currency (government bonds, stocks, and other financial instruments), as these securities can easily be transferred from other locations. Vice versa: If the financial institution has too many illiquid assets (giving a lower leverage), they could choose to then sell these assets to other financial institutions who need them. So, buying and selling currency is not inextricably tied up with buying and selling reserves, as there is some play between the two. You asked about physical cash—cash is related to the demand for currency (so, yes, some cash is transferred, but not much as not everybody wants to draw cash). And you asked about precious metals—well, those are transferred even less, and only really serve as a 'reserve' to give integrity to fiat money generally.

The rest of my answer is copied from https://kevinkotze.github.io/if-2-forex/ , which comes from Kevin Kotzé's web page https://www.economodel.com/international-finance

1.1 Communications and funds transfers
The enormous volume of trade in the foreign exchange market requires an extensive communication network between traders and a sophisticated settlement system to transfer payments in different currencies between the buyers and sellers of different currencies. Traders are able to obtain information that is provided by major commercial distributors such as Reuters and Bloomberg. The traders are then able to contact each other, to obtain actual prices and negotiate deals. In addition, they could approach a foreign exchange broker to broker a deal, or they can trade on an electronic brokerage system, where quotes on a screen are transactable. When a trade is agreed upon, banks communicate and transfer funds electronically, using systems such as the Society of Worldwide Interbank Financial Telecommunications (SWIFT), which confirm trades and facilitate payment.
As Cross-Currency transactions may involve the simultaneous exchange of currencies, there is a risk that only one leg of the transaction may be completed, due to the possibility that parties use different systems in different countries that operate out of different time zones. This is known as cross-currency settlement risk, or Herstatt risk. Recently, foreign exchange dealers, encouraged by the BIS, have developed a number of practices to limit settlement risk. These measures include: firstly, banks now have strict limits on the amount of transactions they are willing to settle with a single counterparty on a given day. Secondly, banks have started to engage in a variety of netting arrangements, in which they agree to wire the net traded amounts only at the end of a trading day. Thirdly, settlement risk is eliminated if the exchange of the two monies occur simultaneously in a process known as payment versus payment (PvP).
More recently, we have witnessed the foundation of the Continuous Linked Settlement (CLS) Bank, which is owned by the world’s largest financial groups. CLS is the largest multi-currency cash settlement system, eliminating settlement risk for over half of the worlds foreign exchange payment instructions and its members include central banks, large commercial banks and other large corporations. The CLS daily settlement cycle operates with settlement and funding occurring during a five-hour window when all real time gross settlement systems are able to make and receive payments. This enables simultaneous settlement of the payments on both sides of a foreign exchange transaction. Each member holds a single multicurrency account with CLS, which has a zero balance at the start and the end of trading day. The settlement of the payment instructions and the associated payments are final and irrevocable.


Does the government have a role to play in certifying people's abilities for the labor market?

Information is certainly not a public good. Job applicants know their skills with certainty, while employers do not know the skills of applicants. The application process is designed to increase the employers information set so that they can select the best candidate. However, applicants have an incentive to lie, so this process may not always be effective.

Certification, on the other hand, is a credible quality signal that cannot be (easily) faked. This massively reduces the employer's search and opportunity costs. This is not just theory: many job applications simply require a university degree, and virtually all require at least some type of certification.

Whether the government has to play a role in the certification process is another question. An ideological rather than an economical one, if you ask me. Regardless of the answer, it should be noted that many advanced certification standards are upheld by non-governmental bodies. Even if public universities are considered part of government, they are usually accredited by non-governmental organisations. And so are most driving schools, cooking schools, professional degrees, etc.

The question seems to have changed a lot since this answer.


NPV and PDV are two different names for the same concept. If you were to receive M every year and the discount factor is $\delta$ then NPV = PDV = $\sum_{i=0}^{\infty}\delta^i M= \frac{M}{1-\delta}$.


I believe there are different goals to undergraduate and (research-oriented) graduate programs.
Most undergraduate students will not pursue a career in research. For them, different proof techniques may not be of first-order importance. In my eyes, they should learn basic stuff like incentives matter; thinking in terms of marginal changes; supply and demand; market power can be bad; what can be done with OLS; although the market is great it can fail with externalities or public goods; why does inflation matter; how to model asymmetric information; correlation and causality; solving a dynamic game backwards and credible threats; and so on. All of that can be taught without using heavy mathematical machinery. There are enough important concepts to teach and you can only do so much. However, I agree that it should be made clear that taking a few math classes can be very helpful if a career in research is planned.
Moreover, I think that I would not have enjoyed pure math without economic applications as an undergraduate. Usually, I become interested in some economic problem which then incentivizes me to learn new mathematical techniques. My impression is that most undergraduate students have even stronger feelings about that. I fully understand that it is hard to catch up on basic mathematical ideas as a grad student, but it would not have been much easier as an undergrad without a conviction to do research.
Different institutions have different approaches. Therefore, most PhD programs have a mandatory first-year that is quite heavy on proofs and mathematical techniques. Often there is also an additional math camp. The idea is to get everyone on a similar level.


The Bureau of Labor Statistics Local Area Unemployment series of data provides monthly figures at the state level from 1976-2014 downloadable as Excel, CSV, or images.

I looked into the following Series:

LASST100000000000003
LASST100000000000004
LASST100000000000005
LASST100000000000006

LAUST100000000000003
LAUST100000000000004
LAUST100000000000005
LAUST100000000000006

Where LAU meant not seasonally adjusted, and LAS was seasonal, with the 3,4,5,6 standing for labor force, employment, unemployment, and unemployment rate, respectively, and the 10000000000000 (I believe) indicating the region I chose (in this example, the State of Delaware).

You can use the One Screen Search Tool to find whatever you need.


In the UK, does the rate of annual price increases in food consistently beat inflation? How can we tell?

The short answer is: No, the changes of food prices in the UK sometimes beat inflation, but not always.

To answer the question, you can simply look at the long term development of food prices in real terms. Note that "real" in the economic context generally refers to something being adjusted for inflation.

The graphic below, shows this development. It was taken from the Department
for Environment Food & Rural Affairs. You can access the underlying data here:

Now, why is it so often informally and apparently wrongly claimed that the price of food always goes up? I cannot provide you with a definitive answer, but I would assume that it is connected to the negativiety bias and the fact that most people generally tend to forget about inflation. Why do they forget? Well, maybe the italian economic-historian Carlo Cipolla was actually right.

Lastly: Does the fact that food prices have indeed risen in real terms in the UK, during certain periods, mean that human progress has indeed been going backwards, for a while?
No, not necessarily. We need to be aware that not only the price of the food we consume changes, but also its quality. For example: Regulations on food retailers, that grow evermore tightly, could lead to a continous increase in food prices but also reduce the number of foodborne illnesses. Human prosperity could therefore be growing, even while food prices increase.
This also means, that in some cases, it might indeed be rational to bet on an increase of food prices in real terms. Just dont stockpile food, the warehouse costs will almost certainly outweigh the gains and most food doesnt tend to get better as it ages.


Why aren't open market operations a zero sum transaction?

When we talk about the money supply, we are often referring to credit that is circulating in the economy at a moment in time. In other words, the more assets the Federal Reserve takes on, the more the money supply will increase at that moment. Indeed if the Federal Reserve decided to reduce its balance sheet by letting treasuries mature and the US treasury absorbed an equivalent amount in taxes to pay off its debt, the money supply will shrink back to where it started before the government issued those treasuries. But that will be far into the future if it even ever happens, and so for the present time open market operations do nominally increase the money supply.


According to the third row in this table from the American Enterprise Institute (they credit the Census Bureau but I could not find the same indicators in their data)

your hypothesis is incorrect: the Mean Number of Earners per Household is increasing in income for each Quintile.

I am surprised by the average number of earners in the highest quintile, but I think it is not very surprising that the poorest people are poor because they have trouble finding stable employment.


Is net neutrality not important in a competitive market of internet providers?

Yes, if there were more competition then net neutrality would not be such a big issue. Any throttling would cause an ISP to lose customers to competitors.

Competition is a main assumption behind why markets work. However, in this case an ISP has market power which breaks the assumption. The market power allows them to easily put smaller ISPs out of business.

Here is a link to a rather long article on how ISPs get this market power. Essentially, the reason is that starting an ISP is entering an economy of scale, which requires a large amount of capital to start: http://arstechnica.com/business/2014/04/one-big-reason-we-lack-internet-competition-starting-an-isp-is-really-hard/


Books on Technological Progress and Growth

I'd recommend ADVANCED
MACROECONOMICS
Fourth Edition by David Romer.

Its a textbook for advanced macroeconomics, however it covers the topics in a very precise way. Id recommend reading chapters 1  in order to review the basic Solow model and then chapter 3 to see how the model changes when accounting for human capital and endogenous technological growth.

Hope this helps.


Yes, this has actually been quite an active area for research within the consumer search literature. As a starting point, I would recommend looking at the following:

BAYE, M. R., AND J. MORGAN (2001): “Information Gatekeepers on the Internet and the Competitiveness of Homogeneous Product Markets”, American Economic Review, 91(2), 454–474. These authors have a model that is interesting because it explains the persistence of price dispersion, even in environments like price comparison sites where one would ordinarily think that all consumers will buy from the lowest-priced merchant. The intuition is closely related to that of Varian's classic model of sales, if you know that. The idea is that some firms price low to sell to the consumers on the price comparison site, whilst others price high to sell to less-informed consumers who do not use the price comparison site and therefore do not compare as many prices.
BAYE, M. R., J. MORGAN AND P. SCHOLTEN (2004): “Price Dispersion in the Small and in the Large: Evidence From an Internet Price Comparison Site”, Journal of Industrial Economics, 52(4), 463–496. (older working paper link). This is interesting because they test various models (including the one mentioned above) against data from a price comparison site.
De Los Santos, B., A. Hortaçsu, and M. R. Wildenbeest (2012): "Testing Models of Consumer Search Using Data on Web Browsing and Purchasing Behavior." American Economic Review, 102(6), 2955-80. Not directly related to the question, but an interesting take on a similar issue. There are two classic models of consumer search: sequential and fixed-sample. In sequential search, consumers visit a merchant, evaluate its offer and decide whether to search again; repeating his process until they find an offer they are willing to accept. In fixed-sample search consumers first decide the number $n$ of merchants they will visit; they then visit precisely $n$ merchants and buy the best offer among them. Given dispersed prices, sequential search is optimal and the fixed-sample strategy is not. These authors test the two models against data from online retailers and find evidence to suggest that consumers use a strategy closer to the fixed-sample end of the spectrum.

A couple of extra references to finish up:

BAYE, M. R., J. MORGAN AND P. SCHOLTEN (2006): “Information, Search, and Price Dispersion”, in T. HENDERSHOTT (ed.) Handbook of Economics and Information Systems, Elsevier Science, 323–371 A nice survey of theories of search and price dispersion that can give you an idea of what work has been done on these general phenomena (and possibly some ideas of what can be empirically tested?).
ELLISON, G., AND S. F. ELLISON (2009): “Search, Obfuscation, and Price Elasticities on the Internet”, Econometrica, 77(2), 427–452 A study of how firms can obfuscate the search process to maintain market power in a frictionless context like a price comparison site. Ellison and Wolitzky also have a nice theoretical paper on this phenomenon in the RAND Journal of Economics.


NE equilibrium with lobbying of cournot producers

Since the choice of $q_i$ can be conditioned on $(s_i,s_j)$, strategies in this game are of the form $(\hat s_i, \hat q_i(s_i,s_j))$. For the given values $\alpha=3$ and $k=1$, the SPNE can be calculated as the profile where $s^*_i=1/3$ and $q^*_i\equiv 1$. Indeed, production levels $q_i^*=1$ are the unique NE in all subgames, independently of the chosen $s_i$-levels in the first stage. Therefore your suggested "other SPNE" is not an SPNE.

But consider the following strategy of player 1: $\hat s_1=1/3$ and $\hat q_1(s_1,s_2)=\left\{
\begin{array}\ 1 & \ldots & s_2 = 1/3+\epsilon \\ 2 & \ldots & s_2 \ne 1/3+\epsilon \end{array}
\right\}$, where $\epsilon>0$. Let player 2's strategy be $\hat s_2=1/3+\epsilon$ and $\hat q_2(s_1,s_2)\equiv 1$. Then $(\hat s_1,\hat q_1)$ is a best response to $(\hat s_2,\hat q_2)$, and provided $\epsilon$ is small enough that deviating in the first stage doesn't pay for player 2, $(\hat s_2,\hat q_2)$ is also a best response to $(\hat s_1,\hat q_1)$. Therefore the corresponding strategy profile is a NE with higher total lobbying than in the SPNE (but with the same production levels), benefiting player 1.

However, it is not subgame perfect, since if player 2 deviated in the first stage, player 1 would not carry out his threat to double production in the second stage. Technically speaking, the NE induces production levels $\hat q_1=2$ and $\hat q_2=1$ in all subgames following a deviation of player 2 in the first stage, but these production levels are not in equilibrium in these subgames.


Is the mid-market currency exchange rate something real? Or are bid and ask rates the real rates?

Indeed the bid and ask spread is very real. The liquidity taker cannot buy and sell at the same price. The market maker extracts a "liquidity premium" for providing liquidity.

When you buy with a market order, you pay the ask price. When you sell with a market order, you get the offer price.

(OTOH, what you mean by "real rate" is not at all clear.)

So, in your hypothetical example, assuming all else constant---yes, after making the round trip trade, you will end up with less than you started. This already happens at, for example, the currency exchange booths at an airport. The booth is acting as the market maker.

Now you can also buy with a limit order. That is, post a offer price and wait. If your order gets hit by someone's sell market order, then you get to buy cheap. Downside is execution risk---you may have to wait a while, possibly a very long while.

Some further comments:

The mid-quote between the best bid and ask could be a vacuous quantity when the spread is equal to the minimal price increment.
The spread is the difference between the best bid and best ask. If an market order size is larger than number of limit orders at those best prices, it will deplete those orders at the best prices and hit orders at the second best price and so on. So if an order size is large or market is thin at the best prices, the spread is a conservative indication of transaction cost. Professional investors split their large orders into smaller ones in order to minimize price impact.
How much the spread matters depends on the type of trader. For a retail trader who occasionally transacts small orders, the cost is probably negligible. On the other hand, slippage is an important issue for institutional investors. Also, on inter-dealer platforms where orders are placed in units of million, spread is closely watched.


Geographic mobility of workers within Germany, as compared to within the US?

Here is a figure documenting annual (I think) internal migration rates in Germany:

Internal Migration in Germany, 1995-2010: New Insights into East-West Migration and Re-urbanisation  (Sander (2014))

This table show 5 year migration rates in many countries, including the USA:

Cross-national comparisons of internal migration: An update on global patterns and trends (United Nations Department of Economic and Social Affairs (2013))

Depending on methodology, annual and 5-year rates can't exactly be compared because some people will move to another area and move back within five years, so asking people if they live in the same state as 5 years ago won't give the same result as asking someone if they lived in a different state last year for five years. That said, if we can make some rough comparisons if we treat the five year rate as roughly comparable to five times the annual rate. The USA rate is about 9 percent every 5 years or 1.8 percent per year. This would be higher (but not much higher) than the West-East, East-East, and East-West German internal migration rates but quite a bit lower than the internal migration rate within West Germany. It might be that this converts the 5-year rates to annual ones that are too low, but the sizes seem roughly comparable. This is just two sources but they provide little or no evidence for Feldstein's claim that the German rates are far lower than in the US.


What's effect of introducing infinitely high denomination bank notes during hyperinflation?

This question seems analogous to "if I think the word infinity, does my head explode?" The answer is no. Infinity is an abstraction. You do not have to accept a piece of paper promising infinity. I would even hesitate to accept an incredibly large finite bill.

On tricky countries buying the world:
You do not buy foreign things in your own currency, first you purchase foreign currency with your currency and then you buy the foreign goods with foreign currency. Printing excessive amounts of money is not only inflationary, it also deteriorates the exchange rate of your currency to other currencies. The most your currency has to offer is being able to buy everything in your country. The total value of your currency cannot exceed this. Hence others are unlikely to trade more (e.g. the world) for it.


Your first answer would be correct conceptually if this was an interior solution. (The math is incorrect because 2*0.9=1.8 and not 0.18).
However, note that the price falls below the average variable cost. So the firm is better off not producing at all both in the short and long term.
As for your second question: you confuse revenue with profit. Revenue is $p*q$, profit is $p*q-c(q)$
For good measure here are two graphs: one depicts profit for q=0 to 20. Clearly it's negative for the full range
And here's one depicting average variable costs which you can see never falls below 4.


Why would China's housing be in a bubble?

This issue has been at the fore among China watchers for many years. While bubble classification is often subjective (even the Fed concedes):

[The] econometric detection of asset price bubbles cannot be achieved
with a satisfactory degree of certainty.

However, there is mainstream consensus that China's real estate sector likely has cause for concern. The below are not intended to be smoking guns, but may be useful heuristics for intuition：

Property developers are typically among the most leveraged operations
in China (leverage has been associated with previous "bubbles" like
US in 1929 and 2000.)
As you mentioned, property transactions as a result of speculation is
increasing versus organic demand from end-users of property
(also have divorce legal loopholes to acquire more real estate)

China began a deleveraging process several years ago but rebooting the economy after economic shutdown in 1Q20 required stimulus. However, recently, PBoC has been tapering liquidity after a banner year in financial assets and even some alternative assets (like real estate). As mentioned, leveraged companies often feel the brunt of a tightened liquidity environment first. China Fortune Land was perhaps the first casualty, defaulting on billions of dollars in 4Q20. A few days ago (at the time of writing) the financial media also picked up a similar situation unfolding with an even larger developer:
https://www.bloomberg.com/news/articles/2021-03-25/rating-cut-ignites-fresh-concern-over-chinese-real-estate-firms
Housing vs credit vs gold
To get at your question more directly, I think it's still fair to say that credit is at play here. The latest macro leverage figure from December 2020 prints 270% of GDP. (M6404532 is the code if you have a Wind financial terminal). Ultimately, this number plays a large role in guiding policy decisions. In the last year alone, property restrictions and land developer bond financing rules were tightened in an attempt to bring leverage down and cool the sector as a whole. Still, there is a lot of nuance and market quarks to be aware of here; this is where the 'art of policy' comes into play.
On the one hand, regulators want to preemptively de-risk the financial system and develeraging is one of several paths forward.
However on the other hand, real estate is a vital part of the fiscal income of many regional governments -- many of which have very troubled financial situations. Many provincial governments turn to non-standard credit to bolster their fiscal budget: real estate developers and other local SOEs are often the issuers. Global ratings agencies are not currently allowed to rate these local government financing vehicles, and local rating agencies tend to be rather generous, so this could explain partially the necessity and propensity of a 'bubble' to emerge and sustain itself.
The notion of 'cultural obsession' has its role too. After all, capital markets are somewhat new to China; the SSE has only been around for 30 years. We can see that the median investor portfolio tilts towards real estate more so than most:

However, after we peel back the financial engineering, China's housing bubble is not so different from the US experience. Or, it could arguably be characterized as a combination of cultural obsession and leverage.
Let's try and tie this to gold. Any bubble, again, as the Fed has conceded, is subjective. However, if we look at what prices reflect on the various portions of the economy, you could find conflicting signals.

In a reflationary scenario, or if deleveraging turns out to be
successful, the real estate and financial markets could march forward
into the next few years unscathed. This might suggest the bubble was
in gold.
Alternatively, if we view gold as a safe-haven asset and succeeds in
preserving wealth after a vicious correction, then life in a
rear-view mirror makes the bubble appear to be in assets (financials/alts).

Ultimately, if we subscribe to the belief that capital markets produce informative prices, then the prices in these different asset classes could reflect diverging views of the direction of the economy. Still, in a world of  momentum traders and Fed interventions, fundamentals aren't the only thing at play. For this reason we should pay attention to who is investing with stewardship or high-conviction from those who are trying to turn a quick buck. I'm sure there are plenty of speculators in the gold market.


The reason for that is that if you check carefully the model an important assumption is that investment is fixed $I=\bar{I}$ once you relax this assumption and allow $I$ be some variable function of interest the paradox of thrift disappears in long run.
This does not meant that fixing $I$ in the short run is incorrect assumptions as during recession you will get liquidity trap in a short run. Banks are scared to borrow money because they can’t distinguish good lenders from bad. So in short run all the excess savings during recession get absorbed into banks hoarding liquidity and no matter how much you save $I$ is fixed. Even without recession another way of thinking about it is that in short run it’s hard for companies to change their investment profile.
However, in long run once the $I$ is some variable function of interest rate and of course it has to equal public and private saving  $I=S+ T-G$ the paradox of thrift disappears ($I=S$ only if you assume government runs balanced budget $T=G$ which is fine but I wanted to point out that you should not forget about public saving).
Also I basically took the answer to this one from Blanchard et al. Macroeconomics an European Perspective. If you want to understand this deeper I really recommend checking the book, it’s in my opinion the best undergraduate macro book on the market. If you google hard enough you can even find a free pdf version...


Can anyone solve this indifference curve question?

Consider the utility function $u(x, y)= -|x-5|-|y-5|$.
Indifference map for $u$ is as follows :


As mentioned in the other answer, the Lagrange multiplier is the marginal effect on the value (optimized) function, when the constrained is "relaxed" marginally. In your case then it should be interpreted "how much studying time changes as the required average expected grade..." increases?

Well, this is a good example to show that how we set up the Lagrangian function matters in order to interpret what we get meaningfully. Your is a minimization case so we have
$$\min T = t_1+ t_2 \\
s.t\;\;\; \displaystyle m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}=0$$

How do we write the Lagrangian? Do we write

$$\Lambda = t_1+ t_2 + \lambda\left[m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\right]$$

or

$$\Lambda = t_1+ t_2 - \lambda\left[m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\right]$$

Since the constraint is an equality constraint, you may hear that "it doesn't matter". Mathematically, it indeed doesn't, but it does matter when the time comes to interpret the value of the multiplier. And, maybe in Utility maximization problems we don't much care because utility is ordinal. But in a problem like the one the OP is solving, the objective function is measured in a very real and measurable and quantifiable unit, time.

Now how we should understand "what happens when the constraint is relaxed"? In the specific minimization problem, it intuitively means "what happens when the required $m$ is reduced". (Since "relax" has a "desirable" connotation, and so we would desire to be asked for less effort, a lower target).

I guess it is now clear how the obtained value for $\lambda$ can be interpreted. And do ponder what variant of the Lagrangian is the appropriate one for your case.


Logit/Probit with numbers different than 0 or 1?

I don't see how Logit/Probit models are relevant here. They are designed to estimate conditional probabilities, not what you describe.
It appears you postulate a relation
$$S = f(P, C)$$
namely supply being a function of price and cost, and where you treat price as a "control" This could be written as a linear regression model, as a first-order approximation. Why don't you think estimating that model?


Which of Anscombe-Aumann's axioms imply the Sure-Thing principle?

As a first remark: the Anscombe-Aumann axioms, in particular Independence, are defined over acts taking the state space to a linear space (generally simple lotteries over consumption objects). Even when we consider the restriction of the model to purely subjectively uncertain acts, we still need to employ the full model or we will lose information.

That being said: Lets let $S$ be a finite state space, and $X$ a finite set of alternatives. Let $\Delta(X)$ denote all the lotteries over $X$ and $f: S \to \Delta(X)$ is an act. For an event $E \subseteq S$, let $f_{-E}g$ be the act defined by
$$f_{-E}g \begin{cases} f(s) \text{ if } x \in E \\ g(s) \text{ if } x \notin E. \end{cases}$$

Now, we can say that our model satisfies the sure thing principle if $f_{-E}h \succsim g_{-E}h$ and $f_{-E^c}h \succsim g_{-E^c}h$ then $f \succsim g.$ This definition is valid for all acts, not just ones without objective risk, but clearly you can consider only the relevant projection.

Assume the antecedent of the STP. From $f_{-E}h \succsim g_{-E}h$ and independence we have that
$$\frac12 f_{-E}h + \frac12 f_{-E^c}h \succsim \frac12 g_{-E}h + \frac12 f_{-E^c}h.$$
Notice we can rewrite this as
$$\frac12 f + \frac12 h \succsim \frac12 g_{-E}f + \frac12h$$
and, applying independence again, we get
\begin{equation}
\tag{1}
f \succsim g_{-E}f.
\end{equation}

In an analogous fashion, from $f_{-E^c}h \succsim g_{-E^c}h$ and independence we have that
$$\frac12 f_{-E^c}h + \frac12 g_{-E}h \succsim \frac12 g_{-E^c}h + \frac12 g_{-E}h.$$
Again, we can rewrite as
$$\frac12 g_{-E}f + \frac12 h \succsim \frac12 g + \frac12h$$
and, applying independence again, we get
\begin{equation}
\tag{2}
g_{-E}f \succsim g.
\end{equation}

Combining (1) and (2) via transitivity yields the desired relations. Going back to the prefatory remark, notice that to apply independence, we need to mix acts, appealing to objective risk. Thus, even when $f$, $g$, and $h$ have no objective risk, we still need risky acts to serve as an intermediary in the proof. In a sense, this is the grand insight to the whole AA framework---using objective risk to get around the necessity of an infinite state space by using the linearity of expectations to force the STP.

Notice only independence and transitivity were used. This should indicated that even state-dependent EU (where monotonicity / state-independence fails) or Bewley EU (where completeness is relaxed)  will still satisfy the STP.

Edit in response to a comment: Lets call the above notion of the Sure Thing Principle STP1 and say the preference satisfies STP2 if $f_{-E}h \succsim g_{-E}h \iff f_{-E}h' \succsim g_{-E}h'$ for all $f,g,h,h'$. Then if  $\succsim$ is a preorder, it satisfies STP1 if and only if it satisfies STP2.

First assume STP2 holds and that $f_{-E}h \succsim g_{-E}h$ and $f_{-E^c}h \succsim g_{-E^c}h$. Then by STP2 we have
$$f = f_{-E}f \succsim g_{-E}f \qquad \text{ and } \qquad g_{-E}f = f_{-E^c}g \succsim g.$$
Transitivity implies $f \succsim g$; STP1 holds.

Next, assume STP1 holds and $f_{-E}h \succsim g_{-E}h$. Define $\hat f = f_{-E}h'$ and $\hat g$ analogously. By definition
$$\hat f_{-E}h = f_{-E}h \qquad \text{ and } \qquad \hat g_{-E}h = g_{-E}h,$$
so our assumption is identically that
\begin{equation}
\tag{3}
\hat f_{-E}h \succsim \hat g_{-E}h.
\end{equation}
Further $\hat f_{-E^c}h = \hat g_{-E^c}h = h'_{-E}h$ so we have, by the reflexivity of preference, that
\begin{equation}
\tag{4}
\hat f_{-E^c}h \succsim \hat g_{-E^c}h.
\end{equation}
Now we can apply STP1 to (3) and (4) to obtain that $\hat f \succsim \hat g$, which, given their definition, exactly what we need to show for STP2 to hold.


How to explain negative correlation between quantity sold and expenditure on advertisement?

In the output above, only psn is significant (based on t stats). Our Professor told us that we should consider only significant coefficients in the demand function. I am finding it difficult to agree with. In this case, if I consider only psn in my demand function, I am perhaps violating the null hypothesis, which is based on F stats (that all coefficients are 0). Also, if I consider only psn I am essentially negating the combined effect of all three variables, and basically choosing a different model that what I fitted.

It is important to take into account significance because you cannot directly observe data generating process. Regression does not tell you what $\beta_0, \beta_1 ... $ are, it tells you what they appear to be based on fitting line that minimizes the sum of squared residuals through data. Yes in expectations (provided all OLS assumptions hold) $E[\hat{\beta}] = \beta$ but this result only holds in the limit as the number of observations go to infinity (see discussion in Verbeek. A Guide to Modern Econometrics or really any econometrics textbooks for more details).
In this particular case the $F$-stat (with 3 an 11 degrees of freedom) is 4.99 which is sufficient to reject the null of joint insignificance i.e. $H_0: \beta_0 = \beta_1 = ... \beta_k = 0$ so that is not a problem.
However, you cannot reject the null hypothesis that $\beta_3$ (effect of advertising) is zero. Consequently, just taking the results of the regression you present at face value (which is not necessary always appropriate in real research but might be in classroom setting), the correct answer would be "advertising has no significant effect on quantity sold, we cannot reject the hypothesis that the true effect of advertising is 0".

adv has a negative coefficient, which is against my theoretical understanding. As I have read that it has a positive correlation with the quantity demanded. How can I explain this relationship?

This could be because you have some reverse causality. Demand equations are endogenous systems and should be modeled with simultaneous equations, not just simple regression (see this explanation for how to do that in R). Using simple multivariate regression in such cases is likely going to result in biased coefficients.
Possibly when demand is low firms ramp up spending on advertising, while when demand is high and firms quickly sell their inventory they might  not want to spend extra money on advertising. This creates reverse causality issue which biases the $\beta_3$ estimates that no longer give you accurate effect of adv  on $y$ (quantity sold). It is possible that advertising spending increases sales, but higher sales are simply associated with less spending and the second effect from sales $\rightarrow$ adv simply dominates.
This being said you mention that this is an exercise imposed on you by your teacher not real research. It is possible your teacher just wants to test if you can interpret coefficient estimates properly not if you understand concepts such as reverse causality (if you did not covered this in your class then probably your teacher is not interested in discussion of that). You should ask your teacher about their expectations regarding the answer.


How to estimate a constant elasticity of substitution for a country in a solow model including human capital and scarce resources

macro123,

You might want to have a look at MRW (1992) https://academic.oup.com/qje/article-abstract/107/2/407/1838296/A-Contribution-to-the-Empirics-of-Economic-Growth and the literature on growth regressions in the handbook of economic growth and the handbook of macroeconomics. In addition, Acemoglu's textbook on economic growth will provide you with some useful references as well as a clear and concise introduction to the topic. Although, I believe that you'll also be a little bit disappointed as you'll find that something close to what you wish to do has already been done.


Burdett and Mortensen (1998), Equation (22), Integral by parts question

The authors apply in the second step the exact same rules they used in the first step, regarding the calculation of the total differential of an integral, taking also into account that the integral is taken over a function of the integrating variable and not just the integrating variable itself.

The first step is

$$d[u(x|F)]=d\left[\int_{b_0}^{x}\frac{m}{1+k[1-F(b)]} dH(b)\right]$$

$$=\frac{m}{1+k[1-F(x)]}dH(x) \tag{1}$$

and re-arranging we get the first result. Applying the same logic, we have

$$(m-u(b_1|F))\cdot d\big[G(w)\big]=kd\left[\frac{\int_{b_0}^w [F(w)-F(x)]du(x|F)}{1+k(1-F(w))}\right] $$

We are considering the differential with respect to $w$, since this is the argument of the $G$ function. To clearly indicate this we will write $d_w$. To clarify the picture write

$$kd_w\left[\frac{\int_{b_0}^w [F(w)-F(x)]du(x|F)}{1+k(1-F(w))}\right] \equiv k d_w\left(\frac{A(w)}{B(w)}\right) = k\frac{B(w)d_wA(w) - A(w)d_wB(w)}{[B(w)]^2}$$

We see that the denominator is exactly as the final expression of the authors, so it is of no concern. We turn to the numerator.
One by one,

$$ d_wA(w) = d_w\left[\int_{b_0}^w [F(w)-F(x)]du(x|F)\right] $$

$$= [F(w)-F(w)]du(w|F) + \left[\int_{b_0}^w d_w[F(w)-F(x)]du(x|F)\right]$$

$$= 0 + d_wF(w) \int_{b_0}^w du(x|F) + 0$$

We applied here Leibniz rule for differentiation w.r.t. a limit of integration under the integral when the integrand is also a function of the integrating limit. The second zero result comes from the fact that $d_wF(x) = 0$ ($F(x)$ does not include $w$, so as we change $w$ the change in $F(x)$ is zero). So

$$B(w)d_wA(w) = [1+k(1-F(w)]d_wF(w) \int_{b_0}^w du(x|F) $$

$$= d_wF(w)\int_{b_0}^w [1+k(1-F(w))]du(x|F) \tag{2}$$

where we have moved the term inside the integral since it does not depend on the integrating variable which is $x$.

Going to the second term, we have

$$d_wB(w) = -kd_wF(w)$$

So

$$-A(w)d_wB(w) = d_wF(w) \int_{b_0}^w k[F(w)-F(x)]du(x|F) \tag{3}$$

Therefore

$$B(w)d_wA(w) -A(w)d_wB(w) = d_wF(w)\int_{b_0}^w [1+k(1-F(w)]du(x|F) \\+ d_wF(w) \int_{b_0}^w k[F(w)-F(x)]du(x|F)$$

The $d_wF(w)$ is a common factor and the two integrals can be written as one due to linearity, so

$$B(w)d_wA(w) -A(w)d_wB(w)=\\ = d_wF(w)\int_{b_0}^w\Big([1+k(1-F(w)] + k[F(w)-F(x)]\Big)du(x|F)$$

The term $kF(w)$ in the integrand cancels off and we are left with

$$B(w)d_wA(w) -A(w)d_wB(w) = d_wF(w)\int_{b_0}^w[1+k(1-F(x))]du(x|F)$$

But using the re-arranged $(1)$ and substituting we get

$$B(w)d_wA(w) -A(w)d_wB(w) = d_wF(w)\int_{b_0}^wmdH(x)$$

$$= md_wF(w)\cdot \Big(H(x) \big|_{b_0}^w\Big) = md_wF(w)\cdot \Big(H(w) - H(b_0)\Big)$$

But since $H()$ is a CDF in $(b_0, b_1)$, it follows that $H(b_0) =0$.

And we are done. In short, don't be afraid to push through the mathematical calculations.


Advice on method for analyzing the effects of increase in minimum wage

The output from the model you mention would be not accurate at all for several reasons, including amongst other reasosns:

Your post you mention you want to use labor costs as predictable variable - there is no such thing as predictable variable in econometrics. If you by that mean predicted variable (i.e. dependent variable) it would be wrong to have labor costs as dependent variable. Even if you would want to include it as a predictor looking only at labor costs is not that useful in this case since firms can shift increases in labor costs on their customers instead of reducing employment, and indeed research shows that most of the labor cost increases by minimum wages are just shifted onto consumers (e.g. see Harasztosi & Lindner, 2019). Furthermore, labor costs can increase for plethora of other reasons completely unrelated to minimum wage.

Unless you live in a country with public access to firm level data on employment having 1 country and 6 years is not enough. As a rule of thumb you need 30 observations per independent regressor. Given that data for most controls you need will be avaiable at best at monthly frequency, but most likely at quarterly/yearly frequency you will simply not have enough data points for running parametric model (such as regression). You can expand your dataset by including multiple other countries, in such case having 3 years before and 3 years after might be enough.

Controlling only for 2 other variables beside for dependent variable is not enough and you will likely have model suffering from omitted variable bias.

If you want to analyze a particular increase in minimum wage quite common method in the literature is differences in differences (DiD). There are also more complex techniques, but simple regression would not cut it here (even if it is just for fun, if you care at all about quality of results) and  DiD is not that much more complex.
Your regression specification would look like:
$$y_{it}= \delta_i + \lambda_t + \rho D_{it} + X_{it}'\beta+ \epsilon_{it}$$
where $y_{it}$ will be your dependent variable, $\delta_i$ country fixed effects, $\lambda_t$ time fixed effects, $D_{it}$ is a dummy that tells you whether country $i$ did implement increase in minimum wage at time $t$ or not, and $\rho$ will tell you the effect of this policy. Finally,  $X_{it}'$ is a vector of control variables. You want to control for anything else that you think  might affect the employment in the country.
This is slightly more complex because you will need at least one extra control country that did not increased its minimum wage at the same time (although this should not be that hard to find). In addition, you might want to add more countries into the data set if for no other reason then just to have more data.
Now what you actually want as your dependent (i.e. predicted) variable is not labor costs but unemployment if that is what you are interested in. The reason for this is that research regularly shows that minimum wage increases lead to increases in consumer prices which means that most of the increase in labor costs is shifted on the shoulder of consumers  (see Harasztosi & Lindner, 2019). Hence, even if your model would find large effect on labor cost it might have zero to none effect on employment in the industry. So if employment is what you are after you need employment as dependent variable.
The above would still not be likely good enough, on its own, to be publishable (there are more issues that would need to be addressed) but it would at least count as some first preliminary model from which you can draw some tentative conclusions. You might also want to have a look at some econometrics handbook before doing this to learn more about how to estimate these sort of models. For the model above I think Mostly Harmless Econometrics  by Angrist and Pischke, chapter 5 on DiD would be sufficient reading (or have a look at some online tutorials).


How can you know which is the "current dollar value" of a dollar value from another decade?

There are quite a few websites which will do a point-to-point calculation for you, if all you want is the comparison of the CPI.  For example, https://www.measuringworth.com/uscompare/

Saying how much a dollar was "worth" at a past point, however, is not a very precise question unless it's made more specific.  The price of computers, for example, have fallen even as they have become more powerful -- and even as the price level has risen.  It's simply not accurate to say that a five-hundred-sixty-three dollar computer today (of which there are many choices) would have cost only $1 in 1774!

I recommend this essay for a lot more detail on the matter.


How is it not arbitrage that EURIBOR (EUR) is around -0.5% and LIBOR (USD) is around 0.5%?

Because of exchange rate and uncovered interest rate parity (UIP). By the UIP the following relationship should hold between nominal interest rates and exchange rates:
$$ (1+i_{t}^$) = \frac{E_t(S_{t+1})}{S_t}(1+i_{t}^€)$$
In short, the ratio of expected future nominal exchange rate ($E_t(S_{t+1})$) and current nominal exchange rate ($S_t$) should make the interest rates in US ($i^{\\\$}$) and Eurozone ($i_{t}^€$), equivalent eliminating any arbitrage opportunities.  If the equality above would not hold, then there indeed would be an opportunity for an arbitrage, but as soon as people would take advantage of the arbitrage opportunity trading on forex markets would move exchange rate to the proper position. You can learn more details on intuition behind UIP in Copeland Exchange Rates and International Finance.
In real life the above equation will sometimes be violated but empirical evidence shows trading typically eliminates arbitrage opportunities very fast. Usually within hours or days, or few weeks at the most save for some special circumstances (see discussion in Nelson International Macroeconomics and Finance in first chapter).
In addition, to the above mentioned UIP which may not always hold there is also related concept of covered interest rate parity (CIP) where expected future exchange rate is replaced with value of futures for the exchange rate and thus given by:
$$ (1+i_{t}^$) = \frac{F_{t}}{S_t}(1+i_{t}^€)$$
When it comes to intuition and mechanism of the model there is not much difference between UIP and CIP but evidence shows CIP is being violated less frequently (again see the Nelson reference above). This is precisely because value of future contracts is easier to measure and thus any possible arbitrage will closed much sooner or even prevented altogether by movements in price of futures.


Extending the Muth-Mills Model to Different Income Groups?

One way to expand the model is to allow for different income classes as stated in assumption (1) below. Let the model be completely standard with all the usual assumption of the monocentric city model as laid out in Brueckner (1987) the paper you cite.
I will only present the basic idea of the expanded model. Hence I do not give proof of existence of equilibria, proof of uniqueness, nor a comparative static analysis or numerical procedures to solve for equilibrium given values of exogenous variables.
I will allow myself to assume Cobb-Douglas preferences with residents utility given as
$$U(c,h) = A\left(\frac{c}{1-\alpha}\right)^{1-\alpha}\left(\frac{h}{\alpha}\right)^\alpha,$$
where $c$ is composite good and $h$ is square meters of housing.
It is well known that the Marshall demands are $c^\star=(1-\alpha)I/p_c$ and $h^\star=\alpha I/p_h$ with $I$ being income. Plugged into utility function results in the value function
$$V(p_c,p_h,I) = \frac{AI}{p_h^\alpha p_c^{1-\alpha}} = \frac{I}{p_h^\alpha},$$
having assumed that $p_c=1$ and $A=1$. The value function tells us that the max price a consumer can pay for a unit of housing $h$ if she is to attain the utility level $u$ is
$$\frac{I}{p_h^\alpha} = u \Leftrightarrow p(I,u) = \left(\frac{I}{u}\right)^\frac{1}{\alpha},$$
found by inverting the value function to express price as a function of income and utility (which offcourse is the well known bid-function).
The next step is to find the bid-lot size which is the amount of housing consumed by a resident attaining the utility level $u$ having income $I$. To derive this I will simply use that the Marshall demand $h^\star_M(p,I)$ with bid-function $p(I,u)$ substituted for price is the bid-lot size $h_b(u,I)$ [same holds offcourse for the Hicks demand]. Hence the bid lot size is given as
$$h^\star_M(p(I,u),I) = h_b(I,u) = \frac{\alpha I}{p(I,u)} = \alpha I^{1-1/\alpha}u^{1/\alpha},$$
which is decreasing in income because when consumer gets higher income and utility attained is held constant the price offered for housing is increase and hence less housing consumed. Importantly this is not a description of equilibrium behavior, what is being said is that if the consumer has higher income she can offer a higher price for a unit of housing while retaining utility level $u$ but that will require her to substitute away from housing over to consumption of $c$.
I now add the linear geometry and linear transport costs of standard monocentric city expositions. Hence the city expands from center at $d=0$ to potentially infinity $d \in [0,\infty)$ and transport costs are $t(d) = t d$ such that income net transport is $I(d) = w-td$.
I now introduce the extending assumption that
Assumption (1): There are two income classes $j=\{1,2\}$ with incomes exogenously given $w_2>w_1$ and size of population income segments exogenously given as $N_1$ and $N_2$.
and for ease of exposition I will let housing be exogenously
Assumption (2): At location $d$ the housing is given as $H(d)$,
the second assumption is not so restrictive as it may appear (but that is easier to illustrate later).
Given the wages I have income net transport for the two classes $I_1(d) = w_1 - td$ and $I_2(d)=w_2 - td$ which are plugged into bid function $p(u,I)$ and bid lot size $h_m(u,I)$ to get the main result that
$$(3)\ \ \ p_j(u_j,d) = \left(\frac{w_j - td}{u_j}\right)^{1/\alpha} \ \ \ \ h_j^m(u_j,d) =\alpha (w_j - td)^{1-1/\alpha}u_j^{1/\alpha},$$
which is really not new in the sense that it is completely like the model with homogenous residents as in the Brueckner (1987) model just adding Cobb-Douglas preferences.
To illustrate the use of these functions select parameter values for $(w_1,w_2,\alpha,t)$. Assume existence of equilibrium with equilibrum utilities $u^\star_1$ and $u_2^\star$ and select values for these ($u_2^\star > u_1^\star$ rich get higher utlity in equilibrium). Also select a value for the alternative use of land $r_a$ referred to as the land rent (not to be confused by the rent paid for land by producers of housing if that sector was endogenized in the model).
In the figure below the bid function $p_j(u_j,d)$ of the high and low income are illustrated. Residents will live where they bid the highest value for housing. The city expands to where the bid-rents of either income class is higher than the land rent.

Two observations are made:
Observation (1): The low income resident will live in the center of the city (this requires housing to be normal good)
Observation (2): There are perfect income stratification with low income in one area and high income in another area
To close the model the housing market must be in equilibrium. Let $[0,\hat d]$ be the locations where the low income reside. Then for $d \in [0,\hat d]$ housing demand is given as $h_{j}^m(u_{j},d)$ per person and units of housing is $H(d)$ so population density at $d$ must be $H(d)/h_{j}^m(u_{j},d)$ and integrating over $[0,\hat d]$ this must be equal to $N_1$ the number of low income residents I have exogenously stipulated to live in the city. So equilibrium conditions includes that
$$N_1 = \int_0^{\hat d} \frac{H(t)}{h_{1}^m(u_{1},t)} dt$$
and
$$N_2 = \int_{\hat d}^{\bar d} \frac{H(t)}{h_{2}^m(u_{2},t)} dt.$$
As is hopefully apparent from the figure above it is the case that: Given two utilities $(u_1,u_2)$ the bid functions determine $\hat d$ and $\bar d$ - where $\bar d$ is the value of $d$ where $p_2(u_2,d) = r_a$ - and given these values the integrals can be calculated to get $N_1(u_1,u_2)$ and $N_2(u_1,u_2)$. When the equilibrium is unique these equations
$$N_1 = N_1(u_1,u_2) \ \ \ N_2=N_2(u_1,u_2),$$
will be invertible such that equilibrium utilities can be found from the exogenously given $N_1$ and $N_2$.
Proof of existence and uniqueness of equilibrium for a model with $K$ income classes and standard restrictions on the utility functions (not assuming Cobb Douglas) are given in
John Hartwick (1976) Comparative Statics of a Residential Economy with
Several Classes Journal of Economic Theory 13, 396-413.
which also include comparative statics of the model.
Proof of existence and uniqueness of equilibrium for a model is also given in
Fujita (1989) Urban economic theory: Land use and city size
which also offers an algorithm for solving for the equilibrium.
See also
Puga and Duranton (2015) Urban Land Use
for discussion of main results of this and other extensions and empirical relevance of the models (but not detailed exposition).
Note on housing supply: Instead of assuming housing to be exogenously given I could have followed Brueckner (1989) and assumed land to exogenously given $L(d)$ and then housing be produced by constant returns to scale production under perfect competition using capital and land with the capital supply being infinitely price elastic and having price $r_K$ in which case housing supplied is a function of $d$ through $p(d)$. So housing supply is a function of the price $H(p)$ but at any distance $d$ the price residents are willing to give is set by the bid function which is a function of $d$ hence either way we get $H(d)$.
Here is some really fast written R-code implementing the solution algorithm in Fujita (1989):
library(rootSolve)

a <- 0.35
t <- 200
w <- c(6000,12000)
N <- c(500,500)
J <- 2
x <- rep(1,J+1)

u <- rep(170,J)

# Function to make bid-max lot size for agent of type j
# with income and utility (w_j,u_j).
# Bid-max lot size is a function of distance to center d
# given type specifics (w_j,u_j).

make_g <- function(u_j,w_j,t,a)
    {
        g <- function(d)
            {
                nominator <- a*u_j^(1/a)
                denominator <- (w_j-t*d)^((1-a)/a)*.6*pi*d
                bid_lot <- nominator/denominator
                # return should be 1/bid_lot
                # but denominator can be 0 so do not calculate it
                # and use it as denominator instead
                denominator <- ifelse((w_j-t*d)>0,denominator,0)
                return(denominator/nominator)
            }
        return(g)
    }

# Solve the integral of g with respect to r from x_L = x_j to x_U = x_j+1
# such that the integral equals N_j

# Initiate with j=1 so x_l is set to 0 and u_j is some guess
j <- 1
x_L <- x[j]
u_j <- u[j]

    # Get exogenous income and population size for relevant segment
    # of population.
    w_j <- w[j]
    N_j <- N[j]

    g <- make_g(u_j,w_j,t,a)
    v <- function(x_U) {return(N_j-integrate(g,x_L,x_U)$value)}
    x[j+1] <- uniroot(v,c(x_L+0.0001,500))$root

# Initiate j=2
j <- 2

    # Find u_2 given I_1,I_2 at x and u_1
    u[j] <- u[j-1]*(w[j] - t*x[j])/(w[j-1]-t*x[j])

x_L <- x[j]
u_j <- u[j]

    # Get exogenous income and population size for relevant segment
    # of population.
    w_j <- w[j]
    N_j <- N[j]

    g <- make_g(u_j,w_j,t,a)
    v <- function(x_U) {return(N_j-integrate(g,x_L,x_U)$value)}
    x[j+1] <- uniroot(v,c(x_L+0.0001,500))$root

# Calculate price at x[j+1]
j <- 3
p_edge <- ((w[j-1] - t*x[j])/u[j-1])^(1/a)
x
u
p_edge

p_max <- (w[1]/u[1])^(1/a)+10
plot(x[1]:(x[J+1]+1),type="n",ylim=c(0,p_max),ylab="housing price",xlab="distance to city center")
colors <- c("blue","red")
for (j in 1:J)
    {
        x_L <- x[j]
        x_U <- x[j+1]

        d <- seq(0,x[J+1],length.out=100)
        u_j <- u[j]
        p <- ((w[j] - t*d)/u_j)^(1/a)
        points(d,p,type="l",col=colors[j])
    }

abline(v=x[2],col="grey")
abline(h=p_edge,col="black")
legend(12,20000,c("bid function (low income)","bid function (high income)","sorting threshold",
"land rent"),
col=c("blue","red","grey","black"),cex=0.7,bty="n",pch=24)
#savePlot("monocentric.jpg",type="jpg")


What is the difference between a free trade agreement and a preferential trade agreement?

A free trade agreement stipulates free (cero tariff) trade between countries/states. In practice, this also includes broader provisions, such as agreements on movement of capital, goods and people (such as NAFTA). A free trade agreement will also mostly include all or a large portion of goods.

In contrast, a preferential trade agreement is much less broad covering preferential (i.e. low or lower other countries) tariffs for a set of products or services. A preferential trade agreement can also just be unilateral or for a particular amount of years, etc.

A free trade area is an area inside a country or state where free-trade rules (in general, but depending on the country and circumstance) apply. These areas are intended mostly for the import and export of goods, and are useful for a country that imports a good as an input by which another good is produced, thus lowering tariff costs for the producers.

A free trade agreement is, of course, not a physical place (see explanation above).

Further, I recommend simply wikipedia in this case...


Why is the Australian Government financing the stimulus via debt rather than the RBA?

As noted in comments, the reason why the RBA acted the way it did most likely comes down to its legal framework. Before the 2020 crisis, very few countries allowed their central bank to directly fund the fiscal arm of the government, which I will call the Treasury. I had often seen Canada cited as the only developed country that allowed the central bank to buy government bonds at auction (I am unsure that no other example exists). I think such financing was allowed in the 1945-1970 period in more places.

I know very little about the legal framework the RBA operates under. I will instead just discuss the economic effects.

The Modern Monetary Theory (MMT) view is that the addition of banks into the loop does not matter. Having the Treasury issue bonds that are just bought back bu the central bank - which is owned by the Treasury - has exactly the same economic effect as the Treasury directly borrowing from the central bank.

If we look at the mathematical models used by other economic theories, they say the same thing, since the final balance sheets are the same. The usual description of this is that this is “money financed deficits.” However, the argument seems to be that the difference is that behaviour of the fiscal arm will change, and so there is a greater bias towards inflationary policies.


Recall that the total differential $df$ of a function $f : \mathbb{R}^n \to \mathbb{R}$, $x = (x_1, \dots, x_n) \mapsto f(x)$, is defined as

$$df = \sum_{i = 1}^n \frac{ \partial f }{ \partial x_i } dx_i = \sum_{i = 1}^n f_{x_i} dx_i = \nabla f \cdot \left(dx_1, \dots, dx_n\right)^\top$$

Let $X$, $Y$, $Z$ be variables, and let $a$, $b$, $c$, $k$ and $n$ be constants. Then the following rules are valid:

$$\begin{alignat}{3}
dk & = 0 && \qquad \text{(constant rule)} \\
d(cX^n) & = cnX^{n-1} dX && \qquad\text{(power rule)} \\
d(aX + bY) & = a\cdot dX + b \cdot dY && \qquad\text{(sum rule)} \\
d(XY) & = Y \cdot dX + X \cdot dY && \qquad\text{(product rule)} \\
d(XYZ) & = YZ \cdot dX + XZ \cdot dY + XY \cdot dZ && \qquad\text{(product rule for three factors)} \\
d\left( \frac{X}{Y} \right) & = \frac{ Y \cdot dX - X \cdot dY }{ Y^2 } && \qquad\text{(quotient rule)} \\
d(X^{-1}) & = -\frac{ dX }{ X^2 } && \qquad\text{(inverse rule)} \\
d(X \circ Y) & = X'(Y) \cdot dY && \qquad\text{(chain rule)}
\end{alignat}$$

When evaluating around $(X, Y, \dots) = (X_0, Y_0, \dots)$ (e. g. around the steady state of a model), replace $X$ by $X_0$, $Y$ by $Y_0$ etc. in the above rules. For $X = X_0$ or $Y = Y_0$ respectively:

$$\begin{align}
d(cX^n) \biggr\rvert_{X = X_0} & = cnX_0^{n-1} dX \\
d(X^{-1}) \biggr\rvert_{X = X_0} & = -\frac{ dX }{ X_0^2 } \\
d(X \circ Y) \biggr\rvert_{Y = Y_0} & = X'(Y_0) \cdot dY
\end{align}$$

For $(X, Y) = (X_0, Y_0)$:

$$\begin{align}
d(aX + bY) \biggr\rvert_{(X, Y) = (X_0, Y_0)} & = a\cdot dX + b \cdot dY \\
d(XY) \biggr\rvert_{(X, Y) = (X_0, Y_0)} & = Y_0 \cdot dX + X_0 \cdot dY \\
d\left( \frac{X}{Y} \right) \Biggr\rvert_{(X, Y) = (X_0, Y_0)} & = \frac{ Y_0 \cdot dX - X_0 \cdot dY }{ Y_0^2 }
\end{align}$$

For ($X, Y, Z) = (X_0, Y_0, Z_0)$:

$$\begin{align}
d(XYZ) \biggr\rvert_{(X, Y, Z) = (X_0, Y_0, Z_0)} & = Y_0Z_0 \cdot dX + X_0Z_0 \cdot dY + X_0Y_0 \cdot dZ
\end{align}$$

As an example, consider the IS equation

$$Y = C((1 - t) \cdot Y) + I(i, E) + G$$

of an IS/LM model of a closed economy with a flat tax rate $t$. Take the total differential around the current state of the economy, using the rules above and setting $Y^v = (1 - t) Y$:

$$\begin{align}
dY & = d\left[ C((1 - t) Y) + I(i, E) + G \right] \\
& = d\left[ C((1 - t) Y) \right] + d\left[ I(i, E) \right] + dG \\
& = C_{Y^v} d\left[ (1 - t) Y \right] + I_i \, di + I_E \, dE + dG \\
& = C_{Y^v} \left[ Y_0 d(1 - t) + (1 - t_0) \, dY \right] + I_i \, di + I_E \, dE + dG \\
& = C_{Y^v} \left[ Y_0 (-dt) + (1 - t_0) \, dY \right] + I_i \, di + I_E \, dE + dG \\
& = C_{Y^v} \left[ (1 - t_0) \, dY - Y_0 \, dt \right] + I_i \, di + I_E \, dE + dG
\end{align}$$

References:

Chiang, Alpha C. and Kevin Wainwright, Fundamental Methods of Mathematical Economics, 4th ed., McGraw-Hill 2005, chapter 8.


The Great Depression vs The 2008 Financial Crisis

2008 was a recession. 1929 was a depression. Simple as that. Unemployment rates did not hit 20% in 2008. People were literally starving in 1929. Big difference.


In the picture below offer curve of individual 1 is given by lines connecting $E$ to $O_1$ and $O_1$ to $O_2$. And offer curve of individual 2 is given by lines connecting $E$ to $O_2$ and $O_2$ to $A$ and $A$ to $O_1$. Set of competitive equilibria are given by the intersection of two offer curves.
$p_X = 0$ and $p_Y = 1$ supports allocation at $O_2$ as competitive equilibrium. Allocations on the line segment $O_1A$ are supported by prices $p_X = 1$ and $p_Y = 0$.


Show that strict preference ordering is not reflexive

Just think about the definition of $\succ$ (in terms of $\succeq$) and the result follows immediately.

Proof:
Suppose $x\succ x$. Then, by definition, $x\succeq x$ and $x\nsucceq x$, which is impossible. Hence, $x\nsucc x$ which means that $\succ$ is not reflexive.


What is the difference between two stage least squares and instrumental variable regression?

The meaning of the words first

Some people use the word "IV estimator" to refer to any estimator that uses instrumental variables. To them, IV estimators contain 2SLS, LIML, k-class estimators, and others, so 2SLS is a special case of IV. For example, the title of Bekker's (1994, Econometrica) paper is "Alternative approximations to the distribution of instrumental variable estimators".

More traditional people mean by IV the particular instrumental variable estimator $(Z'X)^{-1}Z'y$ for the exactly identified case ($Z$ = instrument matrix, $X$ = regressor matrix, $y$ = regressand vector), and 2SLS is a generalization of IV to the overidentified case. But, as Paul says, 2SLS can be expressed as an IV estimator of this second sense because it is $(\hat{X}'X)^{-1} \hat{X}'y$, where $\hat{X} = Z(Z'Z)^{-1}Z'X$ is the instrument matrix.

I personally think it is very fine to leave the meaning of IV estimators ambiguous because the meaning is usually clear in the context and we need not rigorously distinguish them.

As for your question ...

It seems to me that the sentence "2sls is predicting the endogenous variable" means the first stage regression of the endogenous regressor on the instrumental variables (to get $\hat{X}$). The expression "instrumental variables are similar to proxy variables" looks more casual. Proxy variables (e.g., IQ for ability) can be used to solve the endogeneity problem. Instrumental variables are another way of solving the endogeneity problem. In that sense they are "similar".


CobbDouglas: Constant marginal costs and constant returns to scale

Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant.  If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will get increasing marginal cost.


modelling disutility from over consumption

Varian's Intermediate Microeconomics covers a concept called the bliss point.
If the consumed amount of a certain good is under the quantity specified by the bliss point, then the consumer would prefer to consume more of the good, all other things being equal. If the consumed amount is over the quantity specified by the bliss point, then consumer would prefer to consume less of the good, all other things being equal (i.e. it becomes a 'bad').

In case of one good, a utility function that represents this kind of preference is
$$
U(x) = -(x - x_b)^2,
$$
where $x_b$ is the bliss point quantity. For two goods, similar utility functions exist, e.g.
$$
U(x,y) = -(x - x_b)^2 - (y - y_b)^2
$$
or
$$
\hat{U}(x,y) = |x - x_b| + |y - y_b|.
$$
Both $U$ and $\hat{U}$ violates monotonicity, $U$ represents strictly convex preferences, so averages are still preferred to extremes. $\hat{U}$ represents weakly convex preferences.


The terms of art here for laziness like concepts are "shirking" and "effort", particularly  "endogenous effort" or "variable effort". You should also look into "efficiency wages". "Free riding" sometimes appears in this literature but because it is used heavily in the public goods literature it will have more false positives.

This paper may also be of interest:

  Using a sample of male and female workers from the 1992 Employment in
  Britain survey we estimate a model of employees’ self-reported effort
  levels. Effort is modelled as a function of wages, the perceived
  monitoring and supervision environment in the individual's workplace,
  labour-leisure preferences, local unemployment rates and unionisation.
  Using a two-step estimation strategy to account for the endogeneity of
  the wage, we find that effort levels are increasing in wages and in
  preferences for work over leisure. While the extent of monitoring
  seems not to affect workers’ effort levels the ease with which they
  can be dismissed is a motivating factor. Unionisation reduces
  self-reported effort levels which may reflect the outcome of an
  effort-bargain and, in contrast to the efficiency wage shirking story,
  local unemployment rates do not affect effort. Finally, in spite of
  being paid more than women, men report exerting less effort at work.

The Determinants of Work Effort: Evidence from the Employment in Britain Survey
(Clark and Tomlinson (2001))


Macroeconomics: Aggregate Demand, Equilibrium Dynamics and Expenditure Multipliers

In shifting the$\ AD_1 \rightarrow AD$ curves to achieve long run equilibrium, Real GDP will have to increase by 200 billion. Refer to the diagram below, (in purple).

$\ MPC = 0.75 \rightarrow MPS = 0.25$

In this 3 sector closed-economy model, the multiplier can be calculated using :

$$\ K = \frac{1}{MPS+MPT} $$

Assuming MPT (Marginal Propensity to Tax) = 0, then $\ K = \frac{1}{MPS} = \frac{1}{0.25} = 4 $

$\ \Delta Y=K\Delta (AE) $, considering only government expenditure, then
$\ \Delta Y=K\Delta G $

Substituting the values, $\ 200B = 4G \rightarrow G = 50 $

If currently, the $\ AD $ component is $\ AD = 400B $ at $\ MPC = 0.75 $, then we can calculate the original income, $\ Y = (0.75)^{-1} \times AD = (0.75)^{-1} \times 400Bil = 533.33Bil $

Now using this value of $\ Y $, we can calculate the new $\ AD $ component at $\ MPC = 0.6 $, which is $\ AD_{new} = 0.6 \times 533.33Bil = 320Bil $

Hence, using the same approach as in Part 1, the lime-green line is $\ AD_{new} $, the purple section represents $\Delta AD, $ which, as in Part 2, with a new multiplier of $\ K $ calculated using the new $\ MPS = 0.4 $, and substituting those values into $\ \Delta Y=K\Delta G $ will result in your new government expenditure.


The idea behind Pareto optimality is efficiency. On the discrete set, there is no allocation where, compared to $(1,1,1,1,1)$, we could make someone better off and keep the others at least indifferent. This is why this allocation is Pareto optimal. However, any other efficient allocation (in the sense that we are not throwing away resources) will be Pareto optimal here as well.

For the set in $\mathcal R$, we assume a symmetric solution and solve

$$\max_{q\in[0,2]} 5\cdot q(12-(5q))$$
$$\Rightarrow 5(12-5q) -25q$$

The first term is the direct benefit of increasing $q$, the second one is the externality (that we now internalize). The interior solution is given by $q = 1.2$ where everyone gets the payoff $7.2$. We know that the interior solution solves the problem given that the boundaries $0, 2$ both delivered lesser value than $q=1$.

Finally, for the tax, derive the FOC of the agents given a general tax $c$. Then, look for the optimal quantity you want them to have, that is, replace $q_i$ with $q_i^*$, the welfare-maximizing solution, and then look for the value of $c$ that would solve the FOC (and hence make the farmers optimally chose what you want them to).


Expected payment in a second price auction with affiliation

I see nothing wrong with Krishna's expression. If I had to take a guess, it would be that either Menezes and Monteiro define their $f_{Y|X}(y|x)$ differently or they simply forgot to adjust the density on $Y_1 < x$. Both expressions are supposed to be the same.

If you have some random variable $Y$ with cdf $F$, density $f$ and support $[a,b]$
, the following two notations say the same thing
$$\int v(y) f(y) dy  = \int v(y) d F(y) \quad  \mbox{for any function } v(y).$$
If you condition $Y$ on being $Y<x$, you have to adjust the cdf because the new support only goes up to $x$,
$$ F_{Y|Y<x} (y) = \begin{cases}
0 \quad &\mbox{if } y <a, \\
\frac{F(y)}{F(x)} \quad &\mbox{if } y \in [a,x],\\
1 \quad &\mbox{if } y >x.
\end{cases}$$
That way you have an expression that is one for $y=x$.

Hence,
$$E[v(Y_1,Y_1)|X_1=x,Y_1<x]= \int_0^x v(y,y)d\cfrac{ G_{Y|X}(y|x)}{G_{Y|X}(x|x)}
= \int_0^x v(y,y)  \cfrac{ g_{Y|X}(y|x)}{G_{Y|X}(x|x)}   d y.  $$


First, there are substantial reasons to believe the data is not log-normal so it would be improper to assume log-normality.  So, no you should not do that.
You can research the literature on youth wages, it may have information on distributions.  However, that is probably more than is necessary.
There are two primary ways to estimate the standard deviation in economics.  There are more than two ways.
Both of those ways understand the idea of an estimator as being "good" by finding an optimal procedure by integrating the product of a utility function and a probability distribution to create a risk function.  The method with the "best" risk function for a given utility function "wins."
And you probably just thought it was a formula.  There is more than one formula.
The two methods have a slightly different interpretation.  If the method of maximum likelihood was used, then the mean and the standard deviation are the most likely values for the observed data.  All other methods are methods of mediocre likelihood after all.  If the Frequentist method was used, which is the most likely case, then it is an unbiased estimator, so that if you were to repeatedly make measurements, then the value it gives you will be correct, on average, though not necessarily even close for a specific case.
Since you do not know the distribution, you probably used the Frequentist method.
Now as to how to interpret it, you don't really.  It is an estimator.  Without more information, it just sits as is.  Of course, you do have more information.  You have the sample mean and you have Chebyshev's inequality.  You know that 75% of the values must be within two standard deviations and 88.88% must be within three standard deviations.  You can choose any arbitrary set of valuations that you like.  You can look at 2.5 standard deviations, or you can reverse it and solve for a fixed percent.
That will give you a feel for how wide wages really are, in logarithms.
It is here that the two interpretations become important.  If you decide to reverse the logarithmic transformation, the Frequentist estimator is no longer an unbiased estimator.  That is because the logarithm is concave.  You just cannot plug the values in and then reverse the process.  Nonetheless, at this scale, unless you are actually doing scientific work, they are probably not that far off.
The Maximum Likelihood Estimator is invariant to transformation, so you can plug it into the exponent of the base of natural logarithms and get a correct estimate.
Even doing that, however, is more informational rather than interpretational.
Interpretation must always be within the framework of an economic theory.  It has no meaning otherwise.  For example, imagine you were baking a cake and you accidentally added one egg too many.
If you were baking a six-inch diameter cake for a child's birthday then that could have an outsized effect.  If you were baking a cake that is ten feet tall and twenty-five feet wide in a pyramid shape for the winning Super Bowl team, then it's not going to matter.
It also depends on what area of economics you are looking at the data for.  You can use the same data to look at many different types of real-world problems.  In some problems, a wide or narrow spread of wages could be good, while others bad.


Suggested model for dependent variable of different groups

For individual $i$, let $Y_i$ and $X_i$ be the relevant variables and let $D_i$ be the dummy for agricultural sector: 1 if in agriculture and 0 if not.
Then you can run the following regression:
$$
Y_i = \alpha_0 + \alpha_1 D_i + \beta_0 X_i + \beta_1 X_i\times D_i + \varepsilon_i.
$$
Assume that $\mathbb{E}[\varepsilon_i|D_i, X_i] = 0$.
Then if a person is not employed in agriculture, we have:
$$
\mathbb{E}[Y_i|X_i = x, D_i = 0] = \alpha_0 + \beta_0 x.
$$
On the other hand for a person employed in agriculture, you obtain:
$$
\mathbb{E}[Y_i |X_i = x, D_i = 1] = (\alpha_0 + \alpha_1) + (\beta_0 + \beta_1) x.
$$
So you only need to create one dummy. The value of $\alpha_1$ give the difference in agriculture and non-agriculture employment when $x = 0$. The value of $\beta_1$ gives the difference in the marginal effect of $X$ on $Y$ between agriculture and non-agriculture.


Your example does not contradict the theory.

Uncovered interest parity (UIP) indicates the degree and direction of movement of exchange rates in the long term (though I see some economist supposes it's mid term). The intuition behind is that the currency of higher interest rate county has the expectation of depreciation in the future, to the percentage as the difference in the interest rates, to make arbitrage not profitable during a period of time.

Come back to your example. Given the foreign interest rate $R_{F}$ is 7% and and the domestic interest rate $R_{$}$ is 4% (less than 7%), the domestic currency (dollar) is expected to appreciate about 3% in the future according to the model, which is consistent with your calculation (from the current exchange rate $E_{F/$}^{c}$ 0.67 to the future exchange rate $E_{F/$}^{e}$ of 0.69, a 3% change). If the domestic interest rate $R_{$}$ is 8%, while the foreign interest rate $R_{F}$ is still 7% (now domestic interested rate is higher), it is expected that the domestic currency will depreciate for 1%, which is indicated by a change from 0.67 to 0.66.

This theory does not account for short term movement of exchange rate. The underlying assumption implied by this theory is that exchange rate is determined by capital flow, which is driven solely by interest rate spread, and it does not account for market impact of rate hike as well.

In addition, a interest rate hike, according to the theory, will change the direction of appreciation or depreciation only in the circumstance that it causes the interest rate spread to change direction, which means that a rate hike is not necessarily associated with appreciation or depreciation, but only the degree of appreciation or depreciation as the interest rate spread may increase or decrease under a rate hike.


There’s a lot of techniques used, but it is extremely rare that forecasts are on “supply and demand”, rather prices. To go from supply and demand to prices (which is what investors are interested in), you need supply and demand functions, which nobody can really estimate well.

I will first comment on supply and demand analysis for the bond market (the market I am familiar with), then have a few quick comments on price forecasts.

One does see attempts to match up the supply of government bonds (based on projected deficits) to what end investors want to buy, but such analysis is only of a niche interest. For example, governmental debt agencies try to gauge what maturity structure of issuance will match investor needs.

Meanwhile, on the demand side, people look at the effects of regulatory changes, and how much demand that will create for long-term bonds. The UK gilt curve was greatly affected by pension demand for long-term gilts.

There’s a wide range of techniques used in practice to forecast prices (rates in fixed income). If one looks at the debate about market efficiency, most do not work very well (bulk of investors underperform the market average). Some analysts use methods like story-telling or technical analysis, which are generally looked down on by academic studies. However, the bulk of the analysis is fair value analysis.

For example, if I a wanted to gauge fair value for a 2-year bond, I am just going to look at the outlook for central bank policy. This is because the fair value of a bond is the expected cost of financing over the life of the bond (the short rate), which is set by the central bank. If one is interested in corporate bonds, one estimates the probability of default, to get an idea what spread is fair. Equity investors attempt to forecast future dividends and earnings. All of these analyses can be done in a wide variety of ways.


Short-run price competition with capacity contraints

If Firm 1 assumes Firm 2 produces and sells the maximum it can produce, i.e. $\hat{q}_2$, then Firm 1 can aim to maximise its revenue and profit $$\Pi_1=q_1p = q_1(1-q_1-\hat{q}_2)$$ which is maximised at $\left(\dfrac{1-\hat{q}_2}{2}\right)^2$ with $q_1=\dfrac{1-\hat{q}_2}{2}$, at least when Firm 1's maximum production exceeds this quantity. If not, then Firm 1 should continue to produce as much as possible.
So if Firm 1 is big enough then both firms producing as much as possible may not be an equilibrium

As an example, suppose $\hat{q}_1=0.5$ and $\hat{q}_2=0.1$, with a sum of $0.6=\frac35$ as required. If both firms produce at maximum capacity then the price will be $1-0.5-0.1=0.4$ making the Firm 1's revenue and profit $0.5\times 0.4=0.2$

But if Firm 2 produces at maximum capacity and Firm 1 were to produce only $0.45$, this would make the price $1-0.45-0.1=0.45$ and the Firm 1 's revenue and profit $0.45\times 0.45=0.2025$ which is higher than $0.2$, and the Firm 2 will also see profits rise due to the price increase, showing that both firms being at maximum production would not be an equilibrium in this case

With the overall maximum production being $\frac35=0.6$, the equilibrium will break down in this way if $\hat{q}_1\gt \frac25 =0.4$ and $\hat{q}_2 \lt \frac15 =0.2$


Do open market operations permanently increase the money supply?

If these T-bills mature and are repaid then you are correct money supply will decrease. In fact that might be desirable. Just because central bank wants to increase money supply in one year that does not mean they will not want to decrease it at some further period.
When the T-bill matures central bank also have option to then buy new T-bill to replace the old one if they want to keep money supply constant.
Lastly, central banks could also retire the T-bill meaning government never needs to repay it and the increase in money supply is now permanent (or more precisely there might be some other ways how it can be decreased but not anymore through the same instrument so if we are talking only about OMO's it would be permanent).   This is possible but given that it is permanent and central banks might want to not just increase money supply but also decrease it in the future they might prefer just buy new bonds as old ones expire until they decide that they want to decrease money supply.