To summarise crudely: the tariffs are punitive, to give all producers within the EEA as much of an advantage as the trade negotiators could deliver. And because the EEA represents a wealthy group of around half a billion consumers (very many of whom are not individually wealthy; but collectively are wealthy), the EU has huge negotiating power on trade agreements: probably about even with China and the USA, and superior to everyone else. So countries outside the EEA tend to get a poor deal in general.

There is a wide range of tariffs set for goods and services for countries accredited to the WTO in their own right (the UK is not) - as you can see from your linked site. But each of the EEA's significant trading partners seeks a bilateral deal with it, because the pain is too high otherwise.

That site - the one you quite reasonably describe as "unbelievably complicated" - is as transparent and as clear a description of tariffs as I've seen anywhere. And yet it still has vastly over-simplified matters: there's a host of regulation that goes with the tariffs, to ensure that what's imported into the European Economic Area (EEA) is safe and fit for purpose.

Within individual trade publications, there was some discussion on levels of tariffs imposed on countries outside the EEA for that particular trade: so, for example, the Financial Times had quite a lot of coverage on the potentially devastating effect on the financial sector that provides so much of the UK's income; and various manufacturing industries talked about the existential threat that Brexit poses to them; but as you can see from your linked site, general-purpose publications didn't have a chance of getting into the detail across all goods and services.


What happened to the Zimbabwe cash dollar after the introduction of Dollarization?

Normally in order to full dollarization to occur there will be some conversion rate at which debt denominated in old currency is converted into new one. For example, according to the World Bank definition of full dollarization:

All government and private debt under full dollarization is denominated in dollars, and both public and private accounts must be converted to dollars. To make the conversion, countries must set the rate at which old debts, contracts, and financial assets will be converted to dollars.

However, I suppose technically the rate 1 Zimbabwean dollar =  0 U.S. dollar would be possible. This seems to be the case according to Mpofu (2015) [emphasis is mine]:

Zimbabwe is viewed as a dollarized
economy given the dominance of the currency among
the other currencies used in the country and the fact
that the Zimbabwean Government conducts all its
business using the US dollar. In the same month that
the multi-currency regime was introduced in 2009,
the Zimbabwean government suspended the use of
the Zimbabwe dollar as legal tender and decreed that
all wages, prices of all goods and services and
transactions be done in U.S. dollars. They further
decreed that all Zimbabwe dollars be removed from
circulation with no compensation provided to
currency holders, a decision that was viewed as
indicating that the currency had no store of value.

This suggest that in the case of Zimbabwe simply the value of Zimbabwean dollar vis-a-vis US dollar was set to zero. However, I briefly looked over several papers discussing this episode (Noko (2011)]3,  Kairiza (2009), Kramarenko, Engstrom, Verdier,Fernandez, Oppers,  Hughes, ... & Coats,(2010), because this question caught my interest, but none of them seems to explicitly state what happened to private debt. This issue aside the sources mention that US dollars were already widely circulating in the Zimbabwean economy even without government sanction, so my guess is that most likely already most debts were denominated in US dollars already.
PS: An interesting tidbit is that actually Zimbabwe did not just dollarize they adopted multicurrency regime where besides dollar, the rand, euro, pound, metical, and kwacha also became official tenders but dollar was without doubt the most prominent one.


It is better to think of it as a "saving" rather than as a"surplus".
Also, it is better understood if we imagine heterogeneous consumers for whom a threshold price exists, a "maximum willingness to pay". Then at  a given price level, some consumers are willing to buy the product and are expressing their demand for the product, while others are out of the market, because the price is still above their maximum willingness to pay.

If the price falls more towards the equilibrium,  they enter the market.
But the ones already in the market would be willing to buy the good at a higher price. With the added consumers/higher output, they buy the good in a lower price than their maximum willingness to pay, and in that sense, they "save".

Imagine that the market demand function is build gradually not by the same persons increasing their quantity demanded, but by the addition of new consumers (this is easier imagined if we think of durable goods of which most consumers will usually buy just one unit or not at all).


When a country, or perhaps a country's central bank, prints money it is increasing the money supply. When the amount of money available in an economy increases, the value of the currency will decrease. So think about the capital flight from the Euro to the Kroner that is causing the value of that currency to spike. Printing money can basically cause a 'reversal' of sorts. That is, it places downward pressure on the value of the currency.

So Denmark's banking authority might have some large stash of Kroner to meet the current demand. It can sell Kroner to counter-balance the new fluctuation in demand to keep its currency's value from spiking. Denmark might make this decision to avoid making its exports more expensive etc. However, it certainly does not have an unlimited supply of money. If it was willing to print an unending amount of money, Denmark's banking authority would actually  be manufacturing hyperinflation.

I've written this in a non-techinical fashion. Please let me know if you want more details.


Checking incentive compatibility of a mechanism

An example:
Student type $t_1$ has 10 points to bid based on her GPA. Her favourite courses are A,B,C,D,E.
Student type $t_2$ has 0 points to bid based on his GPA. His favourite courses are F,G,H,I,J.
Student type $t_3$ has 1 point to bid based on her GPA. She ranks the courses alphabetically, A being best, Z being worst. (Or if there are only fifteen courses then O, as that is the 15th letter.)

Assume sixty $t_1$ type, sixty $t_2$ type and one $t_3$ type students. If all types were to subscribe to their favourite kind of courses, then A,B,C,D,E would have sixtyone applicants and hence they would be oversubsribed. F,G,H,I,J would have sixty applicants and would be full. $t_3$ may lose the bidding war in a reasonable equilibrium of round 2: If all type $t_1$ students bid 2 points on each course they will all get their favourite picks while $t_3$ type, who can only bid 1 point will lose out on all picks. Now she can only pick K,L,M,N,O, as F,G,H,I,J are already full. She would have been better off if she had picked F,G,H,I,J in round one, as there she would have one the bidding for at least one of the courses. Thus the 'truth-telling' strategy is not an equilibrium and hence the mechanism is not incentive compatible.


Is there a model for how much a philosopher produce in his lifespan?

Here is a somewhat creative (and very imprecise) answer:

Let's assume that Philosophy creates no direct value for society (which seems implicit in your question). However, Mathematics and various other scientific disciplines to some extent originated from Philosophy. Most recently, Linguistics originated as a field of Philosophy. Based on this we can estimate the marginal product of a philosopher. You will need two numbers:

A: Take the marginal value of each invention generated by these fields. For example, speech recognition on your cell phone would not have been possible without Linguistics and thus Philosophy. You can estimate the marginal value of speech recognition by your willingness to pay for a phone with or without speech recognition. Do this for all products and services of an economy. (You can also take market prices and make some strong assumptions about the economy, such as competitive markets, etc.). Using time discounting, you can estimate the value of having a product one year earlier.

B: You must now assign the marginal contribution of a philosopher to the invention of linguistics to the speech recognition software. The marginal contribution seems to be infinite, since without linguistics there would be no speech recognition. However, you may simply assume that each additional Philosopher speeds up the process of the invention of linguistics by an equal amount. Thus, if you would have hired one more philosopher, the time needed to invent Linguistics would have been N/(N+1) the time N philosophers needed to actually invent linguistics.

Combining the information from A and B for all products gives you the marginal product of each philosopher. Note that this is only a rough answer and one would still need to carefully formulate the assumptions. However, it gives you a rough idea of how you could go about finding the marginal product of an otherwise unproductive philosopher.


Can we have a Non-Reflexive Preference Relation?

Rationality requires the following:

Completeness

For all $x, y \in X$, either $x \succsim y$ or $y \succsim x$ or both.

Transitivity

For all $x, y, z \in X$, if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.

Also note that if $x=y$ then completeness implies that $x\succsim x$. So reflexive preferences follow from completeness. So I would say Completeness and Transitivity are required for rationality. Reflexive preferences just follow from complete preferences.


How to determine the Steady State of this model?

In general, a steady state in discrete time is that the first difference is zero at some $t=t_0$, and that it remains so: $$y_t-y_{t-1}=x_t-x_{t-1}=0$$ for all $t\geq t_0$.

Hence, in a steady state, $y_t$ and $x_t$ are constant at some values $y$ and $x$. Thus, $y=\beta(y+\gamma z)$, which implies $$y=\frac{\beta\gamma z}{1-\beta},$$ and $x=\rho x+y$, which implies $$x=\frac{y}{1-\rho}=\frac{\beta\gamma z}{(1-\rho)(1-\beta)}.$$


What do economist state as evidence that "structural reform" works?

A rather positive view of structural reforms

In their recent Economic Policy Reforms 2016, Going for Growth Interim Report, the OECD reviews the main growth challenges faced by OECD and selected non-OECD countries and takes stock of the progress made since 2015 in the adoption and implementation of structural policy reforms to address these challenges.

The chapter 2 in particular reviews the main issues related to the short-term impact of structural reforms in different macroeconomic contexts and takes stock of existing theoretical and empirical studies. It lays out the case of reforms introduced in "normal" and "bad" times. It identifies the main channels through which structural reforms influence rather favorably short-term activity through consumption, investment and net exports. The chapter discusses

Model-based assessments taken from studies that make use of Dynamic Stochastic General Equilibrium (DSGE) models for the analysis of specific reforms
Empirical analysis based on aggregate or sectoral data. Look in particular at Bouis, R., O. Causa, L. Demmou, and R. Duval (2012), "How quickly Does Structural Reform Pay Off? An Empirical Analysis of the Short-Term Effects of Unemployment Benefits Reform", IZA Journal of Labor Policy 2012, 1:12.
Empirical analysis based on micro studies. In this category, a recent paper that is not reviewed in the OECD report but in a VOX CEPR column on Eurozone rebalancing: Are we on the right track for growth? Insights from the CompNet micro-based data.  Their results are "in line with the postulate that the crisis and ensuing structural policies may be generating ‘cleansing effects’." The author are cautious, however. "It is obvious that the evidence presented is still preliminary and only suggestive of the potential use of the dataset."

Look also at Bouis, R., O. Causa, L. Demmou, and A. Zdzienicka (2012), "The Short-Term Effects of Structural Reforms: An Empirical Analysis", OECD Economics Department Working Papers, No. 949. They conclude

  This analysis indicates that the benefits from reforms typically take time to fully materialise. When significant effects are found in the short run, reforms seldom involve significant aggregate economic losses; on the contrary they often deliver some benefits. The absence of major depressing effects does not lend support to the view that reforms should be in general accompanied by substantial macroeconomic policy easing in order to deliver some short-term gains. Nevertheless, there is also tentative evidence that some labour market reforms (e.g. of unemployment benefit systems and job protection) pay off more quickly in good times than in bad times, and can even entail short-term losses in severely depressed economies.

A rather negative view of structural reforms

However, when Krugman sees influential people calling for structural reform as the universal answer to all economic problems, he gets angry. He considers that Structural Reform is the Last Refuge of Scoundrels. For Krugman, traditionally, structural reform was offered as an answer to the problem of stagflation, which makes a fair bit of sense according to him.

Dany Rodrik also opposes structural reforms notably in his paper Goodbye Washington Consensus, Hello Washington Confusion?. He argues that ``evidence has come a more skeptical reading of the cross-national relationship between policy reform and economic growth'' (Page 5).

Moreover, Acemoglu and Robinson (2010, chap 5) document that

  The  attempt  to  induce  African  countries  to   implement institutional  reforms  such as reducing  distortions was not a success   (van  der  Walle,  1993,  2000),  mostly  for  the  reason that international  financial   institutions  (IFIs) did  not  take into  account  the  political  rationale  for  the  inefficient policies they  were  trying  to  reform.  The  most  dramatic  example  of this  is   discussed by Herbst (1990) and Reno (1998).  They argued that attempts by IFIs to  induce   downsizing   of   the   public   sector,   for example  by closing  down unprofitable parastatals,  had  played  an  important  role  in  creating  civil war  in   Sierra  Leone  and  Liberia.


About intertemporal elasticity of substitution

To close this one: As I wrote in my initial comment, one should consider the derivative with respect to $s$, not $c(s)$. This first means that anything containing $t$ has a zero derivative. So we have

for the numerator

$$\frac {d\ln(c(s)/c(t))}{ds} = \frac {c'(s)}{c(s)}$$

and for the denominator

$$\frac{d[\ln(U'(c(s))/U'(c(t)))]}{ds} = \frac {U''(c(s))\cdot c'(s)}{U'(c(s))}$$

Forming the quotient of the two we get after simplifying and adding the minus sign the desired

$$\lim_{s \to t}\sigma = -\frac{U'(c(t))}{U''(c(t))c(t)}$$


Is a strong foundation in advanced microeconomic theory essential to study environmental economics at the graduate level?

It depends on what you like to do in environmental economics. There are lots of people who are working on moral hazard issues on environmental economics by principal-agent models.

More specifically, an interesting question arises from non-point source pollution. Let's say farmers are using pesticides which cause pollution. Probably, you don't know specifically which of farmers are responsible. How could you create "incentives" in order to avoid this situation(pollution) ?
You can look at the Segerson (1988), published in Journal of Environmental Economics and Management in order to understand this issue more precisely.

Another field could be about social choice theory (and decision theory) in environmental economics, especially on equity between individuals and generations. These issues can be included in microeconomics but more in applied math as they use topology etc. (Geir Asheim, Stéphane Zuber's papers could be interesing for this subject.)

Just to finish, I think there are more things to do in environmental economics at a macroeconomic level. (Note that my opinion could be biased as I am working at macro level.)


In US gov treasury auction, can the highest bidder force high prices for all others

No. The auction uses the highest yield, which means the lowest price (for the same coupon). That is, for bonds, price up -> yield down (and vice versa). (The only tricky bit is that the coupon is not known until the auction concludes.)

For a discussion of U.S. Treasury bond structure: link to TreasuryDirect.

Not all bonds are issued via an auction process (one alternative is “tapping” issues). Sweden is an example of using a multi-price auction - link to Swedish auction description.


How does the U.S agriculture subsidy not constitute export subsidies under the WTO?

Certain types of agricultural subsidies are permitted under the WTO's Green Box. These include direct income support for farmers, and environmental and regional development programmes. However the rules state that, to qualify, such subsidies must be no more than minimally trade-distorting.  Whether such subsidies (in the US and other countries) actually meet that requirement is a matter of controversy, as discussed in this ICTSD Information Note which also shows (in Figure 1) the very high Green Box proportion of US subsidies.


Efficient market hypothesis and different performances between countries

Efficient market hypothesis does not in itself predict that stock market returns will equalize among different stock markets so it should not really be part of the question (efficient market hypothesis only refers to informational efficiency of markets). Rather this is something that would be predicted by trade/international macro models that predict factor return equalization (i.e. Rybczynski, Stolper-Samuelson, and factor-price
equalization theorems).
These theories would predict equalization of expected returns to capital due to capital flows between countries. If a country's stock market performs better than some other country capital should flow from the worse performing country to the better performing country until supply and demand for capital equalizes expected returns.
However, in real life expected returns wont always be equal even if they will exhibit tendency to equalize. As you mentioned in your question this can be due to various institutional reasons (Schularick, & Steger, 2008), but it can also be due to imperfect capital mobility, due to market failures, some investments might be geographically specific and so on. The literature on this is extremely broad I suggest looking for keyword factor price equalization on google scholar.
Furthermore, note that link you provided from that site just shows nominal ex post observed returns from stock markets between different countries. That in itself does not actually provide any evidence that ex ante expected returns are not equal between those countries.


I am not an expert in micro theory. This answer is based on my limited knowledge.
Usually you don't. If the supply curve is drawn by stacking all the units of supply according to their marginal cost, as what happened in your figure, then the supply curve is monotonic by definition.
However, while the same goes to demand curve, I have actually seen non-monotonic demand curve in Becker 1991. In that paper, Becker shows that you can have non-monotonic demand curve if there is social influence so that a consumer's demand depends on demands by other consumers.
In analogy, I conjecture that if there is some externality of production so that a firm's marginal cost depends on other counterparts to some degree, there might be non-monotonic supply curve. But I don't know such a paper. I expect some other experts could give a real example from the literature.


The first order conditions equate marginal revenue per factor to the price of that factor:
\begin{align}
p\cdot\alpha\frac{y}{x_1} &= w_1\\
p\cdot\beta\frac{y}{x_2} &= w_2,
\end{align}
Where I used the property of power function $(x^n)'_n = n \frac{x^n}{x}$.
Divide the second FOC by the first to get the relation between the relative prices and the relative factor demands:
$$\frac{\beta}{\alpha}\frac{x_1}{x_2} = \frac{w_2}{w_1}. \tag{A}$$
From this relation we can draw two conclusions:

Rewrite (A) in log form:

$$- \ln \frac{x_2}{x_1} + \ln \frac{\beta}{\alpha} =\ln \frac{w_2}{w_1},$$
And using the log definition of elasticity $\epsilon_y^x = \frac{\mathrm{d}\ln y}{\mathrm{d}\ln x}$ we come to the conclusion, that relative factor demand is decreasing in relative factor prices with unit elasticity:
$$\frac{\mathrm{d}\ln x_2/ x_1}{\mathrm{d}\ln w_2/w_1} = -1.$$

Multiply both sides of (A) by $\frac{x_2}{x_1}$ :

$$\frac{\beta}{\alpha} = \frac{w_2x_2}{w_1x_1}. \tag{B}$$
Rearrangement (B) says that the expenditures on different factors are proportional to their respective input elasticities, i.e. if our total spending on factor 1 is $\\\$\alpha$ then we must spend $\\\$\beta$ on factor 2.
The total cost $C$ is allocated in the same proportion, i.e. for a general Cobb-Douglas production function, spending on factor 1 is
$$w_1 x_1 = \frac{\alpha}{\alpha+\beta}C,$$
or simply $w_1 x_1 =\alpha C$ if $\alpha+\beta=1$, i.e. if the production function is homogeneous of degree 1.


What are some historical examples of laissez-faire capitalism?

There is an ongoing project to try and estimate how close a society is to the ideal, see the indices of economic freedom by the Heritage Foundation and the Fraser Institute. The ones who made They score countries from 0 (all-pervasive state control) to 10. Hong Kong (8.91) and Singapore (8.71) are currently the two countries closest to the ideal, according to the Fraser Institute.

These projects only look at societies from 1970 and onward, however.

A few western countries during parts of the 19th Century have, in a way, been relatively close to the ideal, such as the US up till the Progressive Era and Sweden (around 1860 to 1910). However, both societies restricted the rights of married women and the southern states upheld slavery and later on Jim Crow laws against blacks. Still, the economy was mostly coordinated through private effort, though there were legal barriers for women and blacks to get into the "commanding heights" of these economies.

For a description of the decline of laissez-faire in the US, see Murray Rothbard's The Progressive Era and Robert Higgs' Crisis and Leviathan, which should complement a general economic history of the era.

Mancur Olson, an economist and critic of laissez-faire, argued in The Rise and Decline of Nations that (probably the last) 50 years of British India was among the closest to a laissez-faire ideal. During that time India had hardly started developing. He says that the caste system was a drag on the economy, but he didn't see castes as being in conflict with the freedom to enter any business or trade.

Another way to study laissez-faire systems is to look at specific markets, when not heavily regulated. The Voluntary City shows examples of the private production of urban infrastructure, roads, commercial courts, police, social insurance (healthcare, sick benefits, life insurance), and primary education in 19th century US and England. For examples of the private production of coins, see Good Money; for bank notes (and a free banking sector) see The Theory of Free Banking.

Also, for the record, I'm an ancap.


What are "the fundamental categories of economics"?

Cost covers all the inputs you mentioned and markets don't need to exist. Rent seeking is a perfectly valid alternatives to markets. So having to choose (cost/spend) between different alternatives (profit) is all there is. You can even include time as a cost and reduce it to two factors.


Can econometrics test for correlation or causality between prices and corruption?

Some related studies:

Al-Marhubi, F. A. (2000). Corruption and inflation. Economics Letters, 66(2), 199-202.
The analysis is based on cross-country data consisting of 41 countries from Asia and Latin America for which data is available on four alternative indices of corruption (two from Transparency International, one is the Business International index, and the fourth from another economics study). Note that numerically, these are essentially non-corruption indicators -they range from $0=$ maximum corruption to $10=$  no corruption. The panel covers the period 1980-1995. Four alternative OLS regressions were run (each with one corruption indicator present), with heteroskedasticity-robust standard errors. In all cases a statistically significant positive correlation between corruption and inflation was found.

How large a correlation (to judge its economic significance also)?  The dependent variable was the logarithm of inflation. The value of the coefficient on the corruption variable was on average $-0.22$. Given how the indicators are defined, the minus sign says that  less corruption = less inflation. The actual value roughly says that $1/5$ of inflation was associated with corruption. So if the inflation was, say $10$%, 2 percentage points can be attributed to corruption. Not small.

Braun, M. & Di Tella R., (2004). Inflation, inflation variability, and corruption. Economics & Politics, 16(1), 77-100.
They examine the reverse relationship: inflation facilitates corruption. They present a theoretical model, and also empirical evidence. For the theoretical model they write:

  "(...)high variability of inflation can make over-invoicing by
  procurement officers and under-invoicing by salespersons easier
  because it makes auditing more expensive to the principal".

So here there is a causative theoretical argument as to why unstable prices increase corruption.

Their data relates to 75 countries and for the period 1980-1994:

  Algeria, Argentina, Austria, Bahamas, Bahrain, Bangladesh, Belgium,
  Bolivia, Botswana, Burkina Faso, Cameroon, Canada, Chile, Colombia,
  Costa Rica, Cote d’Ivoire, Cyprus, Denmark, Dominican Republic,
  Ecuador, Egypt, El Salvador, Ethiopia, Finland, France, Gambia,
  Germany, Ghana, Greece, Guatemala, Haiti, Honduras, Hungary, India,
  Indonesia, Israel, Italy, Jamaica, Japan, Jordan, Kenya, Korea
  (South), Luxembourg, Madagascar, Malaysia, Malta, Mexico, Morocco,
  Myanmar, Netherlands, Niger, Nigeria,Norway, Pakistan, Paraguay, Peru,
  Philippines, Portugal, Senegal, Singapore, South Africa, Spain, Sri
  Lanka, Suriname, Sweden, Switzerland, Thailand, Togo, Trinidad and
  Tobago, Turkey, United Kingdom, USA, Uruguay, Venezuela, Zimbabwe.

Here the dependent variable is the corruption index. They use the International Country
Risk Guide (ICRG), that ranges from $0$ to $6$. Here too, a higher score means less corruption. They regress it (among various controls), on inflation variance (since what they want to test is whether an unstable price system introduces noise into transactions that facilitates corruption). They find statistically significant regression coefficients with values around $0.5$. They even find that inflation variability correlates more strongly with corruption, than inflation itself. Lots of references.

Dreher, A., & Herzfeld, T. (2005). The economic costs of corruption: A survey and new evidence. Public Economics, 506001.
A more general study on the costs of corruption, looking also at inflation. For their empirical study (71 countries, 1975-2001), they also use the ICRG index. They regress the level of inflation on this index, and they find a positive sign on the coefficient. Since the index is as described above, a non-corruption index, higher values mean less corruption. So what they find is that less corruption is correlated with higher inflation. Irrespective of whether this can stand reason, their findings are also incredibly large: if you move up the ladder by one point in the ICRG scale (less corruption), the corresponding inflation will be 10 percentage points higher(i.e. if it was, say, 5%, it will become 15%). This is too large to be believable even if the direction of association is to be accepted. Despite this, this is a survey, so lots of references.

Blackburn, K., & Powell, J. (2011). Corruption, inflation and growth. Economics Letters, 113(3), 225-227.
This is a theoretical model. as the authors write:

  "We present a model in which the embezzlement of tax revenues by public
  officials leads the government to rely more on seigniorage to finance
  its expenditures. This raises inflation which depresses investment and
  growth via a cash-in-advance constraint."

So here we have a causative theoretical argument about why increased corruption causes increased inflation.

Bittencourt, M. (2012). Inflation and economic growth in Latin America: some panel time-series evidence. Economic Modelling, 29(2), 333-340.
Four Latin American countries (Argentina, Bolivia, Brazil, Peru) for the period 1970-2007. As the authors note, these four countries account for $\approx 70$% of total GDP and population in South America (for 2009). The dependent variable on their regressions is the growth rate, while both inflation and a composite "political situation" variable are included as regressors. The "political situation" variable again increases with increased transparency and controls on power, and so it can be expected to correlate negatively with corruption (i.e. the higher its value, the less corruption). For our purpose we need to look not at the regression results but at the correlation matrix of the regressors: there we see that the political situation variable is negatively correlated with inflation, with correlation coefficient $-0.142$ (so again, more corruption is associated with more inflation).

Overall, it appears that scholars have empirically detected a positive correlation between the level of corruption and the level of inflation, and also offered theoretical insights on how this can come about, although one theoretical model argues in favor of a causal effect from inflation to corruption, while the other from corruption to inflation. Since both arguments seem reasonable, one could think that it may be the case of a vicious feedback spiral.


Predicting GDP contribution from market share

They are not necessarily proportional.

GDP contribution is the value added by a company. You need to deduct the cost of inputs that are either imported or bought from another entity that is included in the GDP.

Even if two companies seem similar the value added can differ.

Let's compare two imaginary sneaker companies.

CheaperSneaker buys/manufactures sneakers from Asia for \$10 and sell them to department stores for \$20. \$3 are spent on labor and \$5 are used to buy various services from domestic companies. Profit is \$2.

Sales: 20
GDP contribution: 5  (25% of sales)

FancySneaker also buys/manufactures sneakers from Asia. The cost is \$12 per pair. \$5 are spent on labor and \$20 are spent on domestic services (mostly ad campaigns and PR firms). The sneakers are sold to stores for \$50. Profit is \$13.

Sales: 50
GDP contribution: 18  (36% of sales)

The GDP contribution is not the same.

As a general rule of thumb, highly profitable companies and companies with high domestic labor costs contribute a higher proportion of revenue to GDP.


How is it traditionally defined the minimum wage legislation in economics?

It depends. Some countries use hourly wage (Australia, Morocco, UK, US), other use daily (Mexico), other monthly (Argentina, Brazil, Chile, Colombia, France, Peru, Spain), others annually (Bolivia). (this is not an exhaustive list)

Countries without a minimum wage established by law are: Denmark, Iceland, Italy, Norway, Sweden, Switzerland. They do however have minimum wages negotiated at the trade union level, even at industry level.

Given this information, the "traditionally" adjective you are using is not very helpful.


What is the difference for player 2 between accepting nothing or rejecting? None. Therefore, we may say that from the point of view of player 2, he is indifferent between accepting nothing or rejecting.

What is done, as in Rubinstein's (1982) Bargaining Game, is to assume that in indifference, player 2 accepts receiving nothing, which also has more implications when the game is extended to an extra round where players change positions in case of player 2 rejecting the proposal on the first round, or when is included an outside option.

EDIT:

Following the comment from denesp, I will complete my answer.

The previous reasoning is applied mainly to SPE. If we want to see if there is any other equilibrium, we can start by representing the game in normal form, such that, we can identify two equilibria.

Player A offers 2, and Player B accepts
Player A offers 0, and Player B accepts.

However, the last one is not SPE because precisely in Rubinstein's model (and in other bargaining models), it is assumed that if player B is faced with two options that have the same payoff (when involves accepting, and the other rejecting), then he accepts.


What are some currencies that are not pegged and fixed?

According to the table from this website, these are countries that peg their currencies to the Euro:

Bosnia and Herzegovina
Bulgaria
Comoros
Denmark
Sao Tome and Principe

Other than these, Bhutan fixes their exchange rate to the Indian rupee, Lesotho and Namibia fix their exchange rates to the South African rand.


Difference between multinomial logit and conditional logit

It would be much better to have at least a head of your dataset to be sure, but from the description conditional logit is the way to go (though, in practice, random coefficients logit should be used to avoid IIA - independence of irrelevant alternatives).

General form: Think about both models as special cases of some general discrete-choice model. I've got accustomed to convention that index $i$ is for individuals and $j$ is for alternatives.
$$ \pi_i(Y = j|\mathbf{z})= \frac{\exp{(\boldsymbol{\beta}^\top}\mathbf{z}_j)}{\exp{(\boldsymbol{\beta}^\top}\mathbf{z}_1) \: + \:... +\: \exp{(\boldsymbol{\beta}^\top}\mathbf{z}_J)}$$
Notice that $\mathbf{z}_j$ and $\boldsymbol{\beta}$ are vectors of length $J \times K$, where $K-1$ is the number of features [for details, see Agresti(2002:300)]. It is important to understand that we have one term for each alternative in the denominator of this softmax function.
We can derive mlogit and clogit by setting to zero different elements in these vectors. Thus we get different models.

For simple mlogit with $J = 3$ choices (alternatives) and one individual characteristic (chooser's attribute) we can find probability of choosing each alternative. In example, the probability of choosing second alternative by some individual is:
$$ P_{i2}(x_i)= \frac{\exp(\alpha_2 + \beta_2 x_i)}
{1 + \exp(\alpha_2 + \beta_2 x_i) + \exp(\alpha_3 + \beta_3 x_i)}$$
It is important to note that there is the function of one variable $x_i$ (it doesn't change between alternatives for the same individual). Another point is that coefficients are different for all alternatives. Finally, we may use intercepts $\alpha_j$ in mlogit, because they are different and can be identified (if we set $\alpha_1$ and $\beta_1$ to $0$ - that is why 1 in the denominator).

For clogit the situation is quite different, because we have different values of the same feature for each alternative. So the model with 3 alternatives and one feature takes the following form (i.e. probability that individual $i$ chooses second alternative):
$$ P_{i2}(\mathbf{x}_j)= \frac{\exp(\beta x_2)}
{\exp(\beta x_1) + \exp(\beta x_2) + \exp(\beta x_3)}$$
Here we deal with the function of three variables ($\mathbf{x}_j$). As far as these values are different it is enough to fit only one $\beta$, which is the same for all alternatives. As such, we do not need a base alternative (no unity in the denominator). Finally, we cannot identify any intercept in such a model, because any $\alpha$ will simply vanish in the softmax ratio ("thanks" to exponent property).

What things are also important for both models? The common way to fit both models is maximum likelihood, though in conditional logit this likelihood is also conditional (it simply means that we impose the condition that one and only one alternative should be chosen by each individual).

How to understand in practice, which model to apply? Well, you will not get results if you put data for one model in another model command (statistical software will complain that there is no variability in data).
If you have your data in long form, it might look something like this:
Y   c_logit_feature    m_logit_attrib
--------------------------------------
0           3                 10
1           5                 10
0           7                 10
--------------------------------------
1           3                 12
0           5                 12
0           7                 12
--------------------------------------
...
--------------------------------------
0           3                 18
1           5                 18
0           7                 18
--------------------------------------

Usual package will be able to fit clogit from such setting (using last column as grouping indicator). But mlogit usually wants different form and this fact (in my experience) confuses most of users. If we label our three alternatives as "a", "b" and "c", mlogit setting derived from the above table is going to be:
label  m_logit_ready
--------------------
"b"         10
"a"         12
...
"b"         18

Modern statistical packages (like R or Stata) are able to work with different formats but the general idea stays the same.
Hope that helps.


How did Soros make money by shorting the pound?

As a preamble, it is important to keep in mind the context in which this occurred:

Two years earlier (in 1990), the UK government had made the controversial and largely political decision of becoming a member of the Exchange Rate Mechanism (ERM);
eventually, the macroeconomic reality presented a dilemma to UK
policymakers: (1) to devaluate the pound, or (2) to raise interest rates;
the UK government discarded devaluating the pound because the extent of an effective devaluation would discredit the UK government's ERM decision and be politically detrimental;
but UK policymakers also refused to raise interest rates, since that would strangle UK's already troubled economy and trigger another crisis (e.g., borrowers ending up with negative equity).

Soros was able to close his short position profitably because:

when the UK decided to enter ERM,
the UK unilaterally chose a sterling/DMark exchange rate which was recklessly high from the start (and impliedly, without anticipation or consideration of complications that could subsequent ensue in the economy);
the UK government stubbornly embraced the ERM as the magic solution
for its economy, and it ignored the macroeconomic growing disparities between German economy and that of the UK & members of the ERM;
the Deutsche Bundesbank decided to raise its interest rates so as to curb the inflation in Germany as well as the costly process of Germany's reunification;
the UK ministers' political stand was so arrogant that they naively presumed they could compel the Deutsche Bundesbank to tailor its policies so as to help the UK ministers avoid the discredit and embarrassment for having made the wrong decision as to joining the ERM;
after Italy devaluated its currency in 1992, there was mounting pressure or expectation that the UK would have to follow suit due to the systemic imbalances that were ensuing;
Soros (or his company) was not the only entity selling/shorting pounds in the critical days of September of 1992;
in the morning of September 16, the UK government raised rates from
10% to 12%, thereby fueling the selling of pounds because the markets
interpreted the abrupt raise of rates as a sign of weakness;
hours later, the UK government desperately raised the rates to 15%;
short time later that day, the UK government suddenly desisted from its attempt to keep the pound within the ERM range, at which point the pound went in free
fall (also the increase to 15% was withdrawn).

Buying pounds back so as to close Soros's short position simply could not reverse the free-fall of the pound that resulted from UK government's prolonged stubbornness/arrogance and its subsequent, sudden erraticism.


If the players are symmetric and the core is nonempty, then $x_i = v(N)/n$ for all $i$ is a Core element

The center allocation $z_{i}:=(v(N)/|N|)$ for all $i \in N$ is in the core of the symmetric game $v$, whenever $z(S) = \sum_{i \in S} z_{i} \ge v(S) $ holds for all $S \subseteq N$. Now, if $v(N)/|N| \ge v(S)/|S|$ for all $S \subseteq N$, then it holds $z(S) = \sum_{i \in S} z_{i} = |S|\cdot v(N)/|N| \ge v(S)$. Thus, $\mathbf{z} \in C(v)$.

We observe that for a symmetric game $v$, the core is non-empty if $v(N)/|N| \ge v(S)/S$ holds for all $S \subseteq N$. Otherwise, the core is empty.


Age effect when controling for individual fixed effects and time fixed effects

If I understand your question correctly, then in the current set up you describe, the answer is "maybe"--meaning, age might be effecting your estimate as an omitted variable, but you are currently not accounting for it if is doing so in anycase.
To clarify, individual fixed effects are simply controlling for omitted variables that vary among individuals, but not across time. These omitted variables could include such things as age, gender, religion, or whatever, but until they are explicitly accounted for in your model, they are just omitted. In your specification, what the fixed effects model is really doing is just splitting up your estimation of the regression line into multiple lines with their own intercept term, $\alpha_i$. An illustration to make it more clear:

In this image, you have OLS, which is trying to fit a regression line to all the data points. The coefficient you are trying to estimate, $\beta$ is the slope of this line.

Here are those exact same data points, but with a fixed effects model. What the fixed effects model is essentially doing is grouping the different entitities (or in your case, individuals) in the data together, and estimating a separate regression line for each of them. The $\alpha_i$ term is the intercept for each of those entitites/invididuals. The invididual fixed effects assumes that the omitted variables could take on different values for one individual compared to the other, but their effect on that individual themselves will be negligeable or only changing at a constant rate over time. In your model, you can see that the $\beta$ you are trying to estimate does not have the $i$ subscript, which means that you are trying to estimate a single, similar slope, for all these different regression lines, which in theory, should be moving in parallel (as can be seen in the image).
If you believe that one of the omitted variables in your model is actually having an effect on your estimates, such as age, these lines would no longer be parallel, and you would need to account for the omitted variable explicitly as you would do in OLS. So for example, assuming in your model $X$ is your regressor of interest, if you wanted to control for the "age effect", your specification could be something like:
$$
Y_{it} = \beta_1 X_{it} + \beta_2 \text{Age}_{it} + \alpha_i + \gamma_t + \epsilon_{it}
$$
Until you add this control for age to your model, your $\beta$ estimate is not accounting for systematic differences in the different age groups in your data, which, assuming there are differences by age groups, would mean your data is biased.
(*Note: images taken from this helpful document)


Difference between "Debt Held by the Public" and "Total Public Debt Outstanding"?

The difference are the intragovernmental holdings, which consist of  treasuries that are owned by government agencies and are thus not counted as government debt - it's a debt that the government owes to itself.

From Wikipedia (https://en.wikipedia.org/wiki/Intragovernmental_holdings):

  In the United States, intragovernmental holdings are primarily
  composed of the Medicare Trust Fund, the Social Security Trust Fund,
  and Federal Financing Bank securities. A small amount of marketable
  securities are held by government accounts.


Why does $\frac{MP_L}{MP_K} = \frac{w}{r}$?

This is a result of firm maximization.  Consider a problem that a profit maximizing firm faces --it must rent capital, $k$, at rate $r$ and hires labor, $l$, at wage $w$ to produce goods ($F(k, l)$) for which we can sell at a normalized price of 1.

$$ \max_{k, l} F(k, l) - w l - r k $$

Then if we just think about the firm's maximization, we can take derivatives w.r.t $k$ and $l$ to get:

$$ F_k(k, l) - r = 0 $$
$$ F_k(k, l) = r $$

and

$$ F_l(k, l) - w = 0 $$
$$ F_l(k, l) = w $$

Notice that by definition $MPL_K = F_k(k, l)$ and $MP_L = F_l(k, l)$.  If we take the ratio of the two equations then we get what you initially introduced --that is:

$$ \frac{F_l(k, l)}{F_k(k, l)} = \frac{MP_L}{MP_K} = \frac{w}{r} $$

Edit: As mentioned in another answer, this is imposing some assumptions on the behavior of $F(k, l)$.  These assumptions are $\frac{\partial F}{\partial k} > 0$, $\frac{\partial F}{\partial l} > 0$ and $\frac{\partial^2 F}{\partial k^2} < 0$ $\frac{\partial^2 F}{\partial l^2} < 0$


Heckscher-Ohlin with different technologies

As the companies doing the production will minimize costs in equilibrium
$$
\begin{align*}
|MRTS_c(K_c,L_c)| & = r/w \\
\\
|MRTS_w(K_w,L_w)| & = r/w
\end{align*}
$$
will hold. (Or you can also use the general form of your equations (1),(2),(5) and (6) to get this.) Given your particular production functions the marginal rates of technical substitution are
$$
\begin{align*}
|MRTS_c(K_c,L_c)| & = \frac{L_c}{K_c} \\
\\
|MRTS_w(K_w,L_w)| & = \frac{L_w}{K_w}.
\end{align*}
$$
Taken together, this means
$$
\frac{L_c}{K_c} = r/w = \frac{L_w}{K_w}.
$$
It follows that
$$
L_c + L_w = (K_c + K_w) \cdot r/w.
$$
The total amount of Labor and Capital are given in your exercise, and these will determine the factor price ratios as well as the $L_c/K_c = L_w/K_w$ ratio. Once you plug those in, your equation system will become linear.

Notice that this is true both in autarky and in free trade equilibrium, we have not used demand functions so far at all. Setting the appropriate demand equal to the appropriate production will give you the remaining equations you need to determine the equilibrium price ratio.


Strict monotonicity of the (internal) utility function is not assumed but is implied by GARP thanks to Afriat's theorem

GARP is equivalent to:

  There exists a continuous, strictly increasing, and concave utility function
  which rationalizes the data

The equivalence does not hold for SARP. SARP implies GARP, but not vice-versa.

Recall the actual definitions:

Directly revealed weak preference; $x^a \ge^R x^b \; \text{if} \;p^ax^b \le w^a$
Indirectly revealed strict preference:
$$ x^a >^I x^b\; \text{if } \; \exists x^{i_1}, x^{i_2}, \ldots, x^{i_m}
\; \text{such that} \; x^a \ge^R x^{i_1}, x^{i_1} \ge^R x^{i_2}, \ldots , x^{i_m} \ge^R x^b \\ \text{with at least one of these inequalities strict}$$
GARP: if $x^a \ge^R x^b$, then not $x^b >^I x^a$
an (internal) utility function $u:\mathbb{R}^n_+\to \mathbb{R}$ rationalizes the data $\{x^i, p^i, w^i\}_{i=1}^n$ if

$$ \forall x \in \mathbb{R}^n_+, \; p^ix \le w^i\; \text{implies}\; u(x^i) \ge u(x).$$

So you can argue just "by inspection" that GARP is a "sham" set up just so we get a strict (internal) utility that rationalizes the data. (Why else ask for one of those inequalities to be strict? It's actually a key to the proof.) Apparently, the GARP axiom was formulated by Afriat himself, although he called it "cyclical consistency", so... draw your own conclusions how he may have come up with GARP.

The selling point of Afriat's result, as I understand it, is that you can check the "rational expectation" that the agent is utility maximizing (making optimal choices) by merely looking at a sample of its choices (i.e. testing this sample for GARP); furthermore, if the test is positive, you can also derive an utility function that is "as good as" the one that presumably generated the choices. There's an issue that the method is not robust to measurement error though.

    The revealed preference principle of Samuelson … elaborated by Houthakker … easily gives a condition for the rejection of the hypothesis of existence. But the principle has been absent by which the hypothesis can be accepted or rejected on the basis of observed choice of the consumer, supposed to be finite in number; and, in the case of acceptance, a general method is needed for the actual construction of a utility function which will realize the hypothesis for the data. (Afriat, 1967, 68)

    Afriat’s approach … was truly constructive, offering an explicit algorithm to calculate a utility function consistent with the finite amount of data, whereas the other arguments were just existence proofs. This makes Afriat’s approach much more suitable as a basis of empirical analysis. Afriat’s approach was so novel that most researchers at the time did not recognize its value. (Varian, 2006, 101)

  [...] Houthakker had proven that if demand functions satisfied SARP there would always exist a rationalizing utility function, but provided no practical way to find it, while GARP techniques provided a way to construct such a utility function.


Can't a country grow its GDP without foreign investment?

No, you can definitely grow GDP without foreign investment or even trade.

There are 3 basic ways to grow GDP. One is increasing human capital, which includes increasing population as well as allowing women to work. Basically, anything that increases the number of people in the labor force.

Second, we have physical capital, which is where foreign investment comes in. This includes machines and computers, etc.

But both of these eventually have diminishing marginal returns. The final way is innovation, which means more efficient ways to do things, as well as new markets and ways to use goods, etc.

But foreign investment also helps with innovation, especially in developing countries, as the investment usually involves bringing innovation from foreign countries to the developing one.

So while it is not necessary, it is definitely helpful.


Ttest and heteroscedasticity problem in no intercept model

You are generally better off asking statistics questions at cross validated, simply because there are more statisticians there. Also, you will get a better answer if you supply some more detail of what you want to test exactly, and what program you use if you want hints on where to read up on these things

I assume that you want to test whether two coefficients that you found in a linear regression are equal or different from another, where your regression model has no intercept. That is an example of a linear restriction. These kind of restrictions are generally tested with F-tests. You can read up on them a.o. here (which implements them in R).

How to implement them depends on your stats program.


The calculation is correct but the explanation is not simply 'equal the equations and solve for Y'.
Following Blanchard et al Macroeconomics: A European Perspective, 2nd ed. ch 5. by definition

LM curve represents all the combinations of output and interest rate for which the money market is in equilibrium, i.e. the demand for money equals the supply of money.

As given by your question the demand for money is given by $M/P$. Now in your question you do not explicitly label $a+bY−cr$  as a supply for money - but it looks like a supply for money. The reason why you equate the equations is because you want to equate supply and demand for money. If this would be given by some more complex relationship, for example you are given some extra equation for how $M$ or $P$ behaves, you would have to use that too and equating the two equations above would not be enough.
In addition, it is not necessarily to solve it for $Y$, you could as well solve it for $r$. In fact by convention in economics prices go on $y$-axis, so $r$ goes on $y$-axis and output $Y$ would go on $x$-axis, so if you would want to plot LM curve you would actually solve for $r$ not $Y$. This being said if you want to know what is effect of parameters and variables on output (which often is the case) it is not a mistake to solve it for $Y$ instead of $r$.


When the federal reserve buys bonds or other assets from banks, do the banks make a profit?

I am not sure about this, but the Fed mainly trades with Primary Dealers. These are often subsidiaries of banks, but could be independent financial entities.

If the dealer is selling from its own inventory, it will make a gain or a loss depending on the price it originally bought the bond at. Since that may be some time ago, there is no certainty about profits.
The dealer may effectively act as an intermediary. It will buy from clients who are bond investors, and then sell to the Fed. If the trading desk of the primary dealer does a good job, they have a good chance of locking in a small profit. (Making recurring small profits by intermediating trades is the business model of investment dealers.)


Let’s assume that the price index is denoted $p_t$, with $p_1$ the first month, and $p_4$ the fourth. To calculate the 3-month annualised rate between the 1st and 4th month, we note that :
$p_4 = (1 + r)^{\frac{1}{4}} p_1,$ where $r$ is the annualised inflation rate.

Or: $(1+r) = (\frac{p_4}{p_1})^4$.

We cannot calculate the price index level from the 12-month change data; you need either index levels or 1-month percentage changes.


Finding equilibrium of perfect competition in the short run with a cost function

In these problems, you are generally dealing with identical firms - all of whom will supply the market according to their marginal cost curve.  If we were solving for a long run equilibrium the first thing we would do is get the supply curve - which can be done by finding the optimal mix of capital and labor via:

MPL/W = MPK/r

This will get you L = 4K.

In this situation you can skip to:
$$
Q = \sqrt{3}\sqrt{L}\\
L = Q^{2}/3
$$

Plug those into $C(Q) = Kr + Lw$

$$
C(Q) = 108 + 3Q^{2}\,
$$

The fixed cost is \$108 (but that is sunk in the short run, so the firm will ignore it), the variable cost is $3Q^{2}$, making the marginal cost $6Q$.  For the market, this will be $\frac{Q}{1.5}$ (the horizontal sum of all firms' production)

Marginal revenue in this market is just the price since we assume it is perfect competition, so setting market supply = demand gives you
$$
360 - 2Q = \frac{Q}{1.5}\\
Q = 135\,
$$

$Q/n = q$  tells us that each of the 9 firms must produce 15 units for the market to supply 135 total units.  According to the demand curve, the market price is \$90 when 135 units are sold.

Total surplus can be found by calculating the area of two triangles:

$$
CS = (1/2)135*(360-90) = \$18,225\\
PS = (1/2)135*(90) = \$6,075\\
TS = \$24,300\,
$$


Edit1:

Based on the additional information you provided I still believe that my recommendations are valid and that you should analyze your missing data, and attempt to fill it before using a regression to create imputed values.

If you believe the following assumptions about your imputation regression and data analysis to be true, then you are probably in good shape:
1. The specification is correct and based on economic theory in the literature.
2. You correctly identified your missing data as MCAR or MAR.
3. You correctly ran your regression in matlab. (Note: I use SAS and Stata so I'm not able to provide specific advice on how to do your regression.)

Original Answer:

If you are using a regression to fill data gaps in another regression you may want to take a step back. By using estimated values alongside actual values you may run into problems when you try to interpret results. Imputation via regression does not store any way to analyze the precision or accuracy of your imputed values. Your final model may be over identified in that your final regression will not account for the uncertainty in the imputed values.

Data gaps - which it sounds like you are dealing with - are a thorn in the side of econometric analysis. When dealing with data that has significant gaps in it, it is very important to ask why these gaps exist. If it is because of bad data collection on the part of something or someone you can influence, you should call that person or company to find out why there is a gap in their data. Chances are that they either know and have a solution, or had no idea and need to find a solution. Of course sometimes they may not be supporting the data set anymore, or in a rare subset of cases simply don't care that their data is bad.

For those times when you absolutely cannot fill data gaps you have several options. One, which is sometimes advisable if you have significant resources at your disposal (ex. a Fortune 500 company) you can collect the data yourself and make a dataset that works for your specific needs (or pay someone to do it for you).

If you can't fill the gaps, and you can't collect the data yourself, you are stuck with a set of lesser options. The first step is to indent oft the nature of your missing data.

MCAR - Missing Completely At Random
If X is your variable of interest, and contains missing data, and the reason that the data is missing is independent of the non-missing data and independent of other data in the dataset, then your missing data is MCAR.
MCAR data is pretty harmless, and is common in big datasets. If it comprises less than 5% of your data, you can usually just delete the observations. If it is greater than 5% you will probably want to use imputation (more on this below) to fill the gaps. So if your data were MCAR, your strategy of using regression imputation is actually applicable. However, you should be aware that regression imputation isn't a sure fire strategy. As I mentioned above it can lead to over identification.

MAR - Missing At Random
The difference here is that the reason the data are missing is independent with the other data in the variable of interest but may depend on other data in other variables.
For MAR data you should use multiple imputation.

NON-IGNORABLE DATA
This type of missing data isn't missing at all. For example, say you want to find out if your son is dating a girl at school. So you give him a questionnaire. It asks: "Are you dating a girl at school?" He leaves the question blank. That would tell me: Yes he may be dating a girl at school.

For non ignorable data, the reason they are missing is dependent on the othe data in the variable. In our scenario, my son's non response was actually a form of implied response.

Non ignorable data should not be imputed nor should they be ignored. Every effort should be made to fill them. If you still cannot fill them you should consider switching data sets or, if that is not feasible, at least make detailed note of these missing data in your research.

IMPUTATION TECHNIQUES
For more details on imputation techniques, I recommend consulting your statistical package's documentation.

BASIC IMPUTATION METHODS(for MCAR only):
Hot decking - This is essentially just swapping in a randomly selected similar value from the non-missing data. I don't recommend this technique. It can lead to false conclusions and is statistically unsound.

Mean substitution - Replace missing values with the mean of the non missing data. This is better than hot decking but not by all that much. It is however an option if you just need an early idea of what your model results might be (ex. for informal paper proposals to colleagues).

Regression imputation - This is basically what you did. It has issues with overidentification but is not all that bad in the scheme of things assuming you are using a good model for these estimates.

Multiple Imputation (best option for MAR) - The most rigorous form of imputation requires that you draw your imputations from many datasets, using a particular procedure such as the one from Donald Rubin's An Overview of Multiple Imputation.

Many statistics packages have imputation methods programmed in already as they are quite technical to set up.

Overall, you have to balance your amount of missing data, type of missing data, time constraints, and need for highly accurate results, and make a decision on what you want to do. As in the rest of economics, this decision is a trade off.


What is the relationship between inflation and imports/exports?

As requested in comments:

It is a matter of which other factors are being held constant. If the relative-inflation-adjusted exchange rate (i.e. the real rate) stays constant, there may be no supply or demand effect on export and import volumes.

However exchange rates do not exactly follow relative inflation in different economies; other factors such as interest rates and investment returns, balance of payments differences, and sentiment can also have an effect.


"Trade", in economics, usually means trade between countries.  So if goods made in, say, the Netherlands are sold in Belgium, that counts as trade.  But if goods made in, say, California are sold in Florida, that doesn't count as trade.  Because the US is a large and populous country, it has a high proportion of internal transactions that count as part of its GDP, a measure of its economy, but do not count as trade.  Many smaller countries, on the other hand, will have a share of world trade exceeding their share of the world economy.


Can one Calculate Fixed and Variable Cost From TC and Quantity Alone?

As said in my comment,

  The fixed cost is usually defined as the cost when quantity is equal to zero, and the variable cost as the total cost minus the fixed cost.

Hence, if $TC(q)$ is the total cost for the given level of quantity $q$, then $FC=TC(0)$ is the fixed cost, which is a constant independent of $q$; and  $VC(q)=TC(q)-FC$ is the variable cost.

For example, if $TC(q)=10+q$, then $FC=TC(0)=10$ and $VC(q)=TC(q)-FC=q$.


Saudi Arabia motivations for OPEC production cut

You could think this in terms of an oligopoly. As an imperfect market, firms get more profits than possible in a competitive market because decisions around levels of production influence prices. All members of the OPEC have an incentive to reduce production as a whole (to raise prices), and an incentive to increase production individually (to earn more profits at those higher prices).

You can also consider political issues like production outside the cartel: they don't hold a monopoly and the higher the OPEC raises prices, firms with costlier ways of extracting oil (like fracking in the US) can turn a profit and compete with them, which naturally reduces profits (and value).

While the IPO may be a possible explanation, you can think of it in terms of oligopolies and still get a reasonable answer.


Does a monotonic transformation of a homothetic utility function imply the preference relation on the set of consumption bundles is still homothetic?

Yes.
We know that a monotonic transformation of a utility function still represents the same preferences and as the old utility function represented homothetic preferences the new one does, too.

As an easy example you could look at Cobb-Douglas utility functions of the form $u(x,y) = a\left(x y\right)^\alpha$. For $\alpha = \frac12$ the utility function is homogeneous of degree 1, but for every $\alpha,a>0,$ the preference relation is homogeneous of degree 1.

We never used homothetic as a property of the utility function to answer your question (as it works for every utility function).
A homothetic utility function is, to the best of my knowledge, not very clearly defined, I have see two different versions:

Homothetic as different name for homogeneous of degree 1
A homothetic utility function is a utility function that represents
a homothetic preference relation.

Which means, with definition 2 Cobb-Douglas utility functions with equal weights are always homothetic, with definition 2 only some of them are:
$$u(tx,ty) = (tx)^\frac12 (ty)^\frac12 = t xy = t u(x,y)$$
and some are not
$$u(tx,ty) = (tx) (ty) = t^2 xy \neq t u(x,y)$$


The MU question you quote is poorly worded. Here the proper answer would be:
C Indeterminate with the given information
First we can visualize it as shown on the diagrams shown below. Consumer surplus is the difference between willingness to pay (given by demand)`and price they actually pay so for all consumers it is the area below demand curve and above market price i.e. the blue area on the picture. Producer surplus is the difference between market price and price at which they are willing to supply so for all producers it is the area below market price and above supply curve i.e. the red area on picture below (you can see Mankiw Principles of Economics Ch 7 for more discussion on consumer and producer surplus).  Without any price celling the market will be at equilibrium where supply equals demand so there will be no shortage. After price celling producers are only willing to supply $Q_S$ while people demand $Q_D$ resulting in shortage of the good of size $Q_D-Q_S$. Shortages can result in waiting lines as usually some non-market mechanism then takes over.
We know that total surplus of society shrinks because $CS+PS$ is definitely smaller as the total colored area shrinks. However, if we only look at $CS$ it is possible that it increases because consumers both loose part of their original surplus and gain part of the surplus that was originally belonging to producers. Whether $CS$ increases or decreases depends on whether the gain  from appropriating part of producer surplus offsets the loss to $CS$ from consumers that no longer can get the good anymore.
Possibly, the question has a mistake and it was supposed to be total surplus where answer b) would be correct, or there are some unstated assumptions, such as that people experience disutility from queuing (reasonable but not explicitly mentioned in the question) which would lower CS further as it would impose additional costs on consumers.


Is that derivation for $k(t+1)$ correct? Technically, you never reach the steady state, but only asimptotically as $t\rightarrow\infty$, but at infinity the $A(t)$ will also be infinite because it grows exponentially with time. I suspect that $k(t+1)$ should depend on $(1+g)$ instead of $A(t)$.
Usually these models are expressed in units of effective labor, that is, dividing the capital stock $K(t)$ and the labor $L(t)$ (and hence, the production and consumption) by $N(t)$ and $A(t)$.


How to get the percentage-change form of the quantity equation?

I think the author is just taking logs of the original equation.  The natural log approximates percent changes for small changes.  More info here.  Using that and properties of the log, $\ln(ab)=\ln(a)+\ln(b)$, you get that formula.


Lexicographic preference relation cannot be represented by a utility function

We can say more generally that lexical preferences are not representable using a continuous utility function. Lexical preferences are not continuous. Note the definition of a continuous preference relation.

The preference relation $\succeq$ is continuous if for any sequences of consumption bundles $(x_{i})_{i \in \mathbb{N}}$ and $(y_{i})_{i \in \mathbb{N}}$ with $x_{i} \to x$, $y_{i} \to y$, and $x_{i} \succeq y_{i}$ for each $i \in \mathbb{N}$, then $x \succeq y$. That is, continuity preserves the relation at the limit point.

Consider $(x_{i})_{i \in \mathbb{N}}$ defined by $x_{i} = (\frac{1}{2^{i}}, 0)$ and $(y_{i})_{i \in \mathbb{N}}$ defined by $y_{i} = (0, 1)$. Clearly, $x_{i} \succeq y_{i}$ for each $i \in \mathbb{N}$. However, $x_{i} \to (0, 0)$ while $y_{i} \to (0, 1)$. So this preference relation is not preserved at the limit point.

Even more generally, no utility function represents the lexical preference relation. I prove for $\mathbb{R}_{+}^{2}$, but this argument extends to $\mathbb{R}_{+}^{n}$ by projecting into $\mathbb{R}_{+}^{2}$.

Proof: Suppose to the contrary that some utility function $u : \mathbb{R}_{+}^{2} \to \mathbb{R}$ represents $\succ_{lex}$. We thus have $u(x, 1) > u(x, 0)$, as $(x, 1) \succ (x, 0)$. We construct the interval $I(x) = [u(x, 0), u(x, 1)]$. Now for any two distinct $x, y \in \mathbb{R}_{+}$, $I(x) \cap I(y) = \emptyset$ as we have either $x > y$ or $y > x$ (so WLOG, we have $(x, 0) \succ (y, 1)$).

Define $\mathbb{I} = \{ I(x) : x \in \mathbb{R}_{+} \}$, and let $\phi : \mathbb{R}_{+} \to \mathbb{I}$ be given by $\phi(x) = I(x)$. Observe that $\phi$ is an injection, as each $I(x), I(y)$ are disjoint for distinct $x, y$.

Note that $\mathbb{Q}$ is dense in $\mathbb{R}$. So there exists a rational number in each interval. Define $\tau : \mathbb{I} \to \mathbb{Q}$ such that $\tau(I(x))$ returns a rational number contained in $I(x)$. So $\tau$ is an injection. We have $\tau$ composed with $\phi$ an injection, which implies $|\mathbb{R}_{+}| \leq |\mathbb{Q}_{+}|$, a contradiction. QED.


Interpretation of Demand and Cost Curves

Your calculation is not wrong. First a more easy and more intuitive way of calculating marginal revenue is just to take derivative of total revenue.
Total revenue is price times quantity $TR=P (Q) Q$ where price is still function of quantity as at higher price people demand lower quantity of goods (I am expressing everything in terms of Q since that is the choice variable in your model). Substituting for quantity we get:
$$TR=  (100/Q)Q \implies MR = \frac{dTR}{dq} = 0$$
Hence in this case they are indeed zero. Furthermore, in this case the profit maximization problem of monopolist does not even have a solution. Profit is given as total revenue minus total costs so:
$$\Pi = TR-TC = (100/Q)Q -4Q$$
If we would try to derive the first order conditions for maximization we would just get that 0=4 which means there is no solution. To make that clear I even programmed a visualization of your problem in R, this is how it looks like where green curve is demand, red line is marginal revenue and blue line is marginal cost.

So there are three options I see here:

This is a trick question and you teacher expects correct answer to be that there is no solution (no solution in standard non-negative real numbers that is as it could be argued that quantity produced should be as small as possible but not zero (infinitesimal) from right side - however thats not a standard real number).
You misread the problem or made mistake in part of the problem which asked you to first create cost function or demand from some text.
Your teacher made a mistake when making up the problem.

Hence my advice is first make sure you read the set up thoroughly, if there is no mistake there then contact your teacher and ask if its possible that the teacher made a mistake since MR is zero in your case. If there is no mistake then its a correct solution.

response to question in comments:
Price elasticity of demand is defined as:
$$\epsilon = \frac{d Q}{dp}\frac{p}{Q}$$
So in your case it will be:
$$\epsilon = -\frac{100}{p^2} \frac{p}{\frac{100}{p}}= -\frac{100}{p^2} \frac{p^2}{100} = -1$$

Addendum: In response to comments I also added simulation for profits of the monopolist in this situation assuming that the profit function is:
a) $\Pi = \frac{100}{Q} Q -4Q$  - in red color
b) $\Pi = 100 - 4Q$ - in blue color


Let $N$ denote the set of consumers. Given a set of possible allocations $A$, an allocation $a$ is Pareto-optimal if no allocation $a'$ exists for which
$$
\forall i \in N: a \preceq_i a'
$$
and
$$
\exists j \in N: a \prec_j a'.
$$
So Pareto-optimality exists without any notion of property rights, you don't need to know the initial endowment is, who has what in the beginning. Given an initial endowment $\omega$ an allocation $a$ is individually rational for consumer $i$ if
$$
 \omega \preceq_i a.
$$
An allocation is in the contract curve if and only if it is individually rational for all consumers and is also Pareto-optimal. If the allocation $a$ is not individually rational for a given consumer $i$, he would not sign the contract which would take him there from $\omega$.

Thus the contract curve is indeed a subset of all Pareto-optimal allocations. Unless the initial endowments are given, it is not possible to determine the contract curve.


Why recent crises all started with a housing bubble?

In a fractional reserve monetary system loans create money and loan repayments destroy money. Most bank lending in modern times is for the purpose of purchasing real estate. So the amount of money that exists in the economy is closely tied to the enthusiasm for purchasing real estate. You could not really make this kind of statement for any other class of asset - this is why the housing market in particular has such an impact on the economy. During the upswing of a housing bubble the money supply will be growing fast, but when the bubble bursts the money supply will fall (as existing mortgages are repaid) leading to recession. The whole point of QE is to counteract this fall in the money supply.

Confusingly economists use the word "credit" to describe (most of) the money we use in the economy, so many people do not realise that the "credit crunch" was in fact a "money crunch".

Hyman Minsky is the most famous economist that would subscribe to these kind of views. You might also want to look up the term credit cycle.

By the way, none of these views are mainstream. Mind you the mainstream economists did not see the last crash coming, so you should not really expect them to explain what happened.


In a standard cheap-talk setting, a sender (S) has better information on a state of the world and wants to communicate this information to a receiver (R) who then takes an action. However, S and R prefer different actions conditional on the state.
Importantly, S is free to send any message independent of what she knows. That is, she cannot commit to a signal given the true state. This is a complicated strategic setting, and communication may fully break down - in a babbling equilibrium any message is ignored, and R decides based on his prior.
Even equilibria with information transmission can be bad for S.

KG give S more commitment power in the sense that she can design a signal structure and the signal realization is truthfully communicated to R. This commitment assumption helps to see this as a sort of mechanism design problem. Instead of setting rules on transfer payments, the designer manipulates the informational environment. In this problem of information design, the commitment to the signal structure determines the meaning of a signal such that "babbling equilibria" are circumvented.

Can R be "persuaded" to do something S wants when R is a rational Bayesian and well aware that S wants to manipulate him? Perhaps surprisingly, the scope for manipulation is quite large. KG show how to optimally do this. The prosecutor-judge example illustrates the paper very nicely.
Essentially, KG interpret the signal structure as a distribution over posteriors and a signal realization then determines R's posterior. Can S just use any such distribution? No, S is restricted to a Bayes plausible distribution, the expected posterior must be equal to the prior! I guess that is why the "Bayesian" is important. Of course, R being a Bayesian is not a unique feature and quite standard, but in the same sense this word shows up all the time when analyzing standard settings, think of BNE, PBE, BCE...

However, I don't like the term persuasion so much. When I think of persuasion, misrepresenting and concealing information is part of it. This is explicitly excluded in KG! Depending on the application, I find the assumption on S's commitment power quite a stretch for a persuasion. If you just consider the theory without a persuasion application in mind, a smooth way around a discussion about commitment power is to consider a setting with symmetric information: S doesn't know the state herself but designs the information structure.


What country has the highest requirements for loans?

A good starting point would be the "getting credit" subindex of the Doing Business rankings compiled by the World Bank.  There is also an explanation of the methodology.  This is supposed to be a test of the business environment, i.e. rules laid down by the government and central bank rather than about the behaviour of market participants.


Difference between capital and current account

You can think of the capital account as "exports of liabilities" - a capital account surplus means that a country is exporting liabilities (borrowing) from abroad to pay for imports of goods and services and/or for income transfers abroad (a current account deficit). An increase in foreign assets owned by the US is an import of liabilities: the US is buying liabilities from rest of the world by sending them goods and services (or reinvesting the returns from the assets they already own). The increase in foreign asset holdings corresponds to a current account surplus (as it must), but they are not the same thing.

The capital account and the current account of the balance of payments measure the same thing: the absolute change in the external debt stock (defined generally to include all liabilities, equity is in here as well) of a country. They just go about the measurement in different ways: the capital account measures the change in the external debt stock directly by looking at financial asset holdings, whereas the current account notes that an increase in the external liability stock of a country has to come from either a net import of goods and services or income transfers (interest payments, dividends, etc), so it measures these instead. In practice, measurement errors mean that the two come out to be different, but as a matter of accounting they have to be equal.

The reason increases in holdings of foreign assets is in the capital account is simply because that's what the capital account measures, and it's not in the current account because that's not what the current account measures. The question is a bit like asking "Why is rental income not in the expenditure method for calculating GDP?" Rental income is in there, but due to the way the expenditure method measures GDP, it is measured implicitly as part of expenditures.


What types of 'market mechanisms' are said to exist besides auctions?

Other methods or mechanisms by which goods are traded in a market economy:

1. Seller posts price.

Probably the most familiar method to most people. Examples: McDonald's posts a price of \$1 for a hamburger, an airline posts a price of \$400 for a ticket, etc. Anyone who's willing and able to pay the posted price gets the good.

2. Buyer posts price.

Previously somewhat uncommon, but thanks to the internet most people are now familiar with this method. Example: Buyer of a ticket to tonight's game posts the price he's willing to pay on craigslist. Anyone who's willing and able to sell at the posted price gets to sell.

3. Seller and buyer both post prices.

Mr X announces he's willing to sell one Berkshire Hathaway share for \$300,000. Mr Y announces he's willing to buy one for \$300,000. The New York Stock Exchange (the intermediary) then matches them together and executes the trade.

4. Bargaining.

No prices posted. Typically, the buyer approaches the seller, the seller sizes up the buyer, offers a price, and the two then higgle and haggle till a mutually agreeable price is reached. Such bargaining could be over a US\$0.20 bottle of water on the streets of Hanoi or a billion-dollar takeover deal on Wall Street.

I'm not sure if your book's characterization of auctions is quite right. For example, a trade on the NYSE is quite anonymous but I don't think we could consider it an auction. (Of course, one can define an auction so as to subsume all of the above mechanisms, but in that case we have wasted the word auction.)


How technology will help economists to do better predictions?

I'm an economist by training who also is a programmer and works with a lot of data scientists, so I've some insight into this area.  One of my projects at the Urban Institute is trying to bridge the gap between these fields, and part of our work, supported by the Sloan Foundation, is publicly available here.

To some degree the answer is yes, but there are huge obstacles as well.  Some are surmountable with work, like differences in terminology, programming platforms (knowledge of SAS or Stata versus R or Python) and conventions (do others in your field understand and/or accept the results of machine learning models?).  Others are much more difficult to surmount, such as the size of data available.

Essentially, the fields that make extensive use of AI and machine learning have huge amounts of data.  They can afford to, say, set aside a million observations for training the model and then test the model on the other nine million.  Economists, and social scientists in general, however, are often lucky if they have 100 observations.  Many macroeconomic variables are annual, or quarterly.  Some are monthly, which means if you can get 30 years of observations then you've got 360 time periods!  Data-rich by social science standards, but not even a rounding error in many data science models.

That said, computing power and sophisticated data collection are making it more and more feasible to approach social science questions with big data.  Social media is a major area that social scientists are just beginning to reach into.  However, some areas of economics will likely remain closed off to data science methods due to the nature of the data.  Your fiscal stimulus example might very well be one - the relevant data just isn't produced that fast.

So the answer to your question is yes, with reservations and a hope for the future.


Where is there more money: Bank Accounts or Brokerage Accounts?

At the time of writing this answer, I would note that the question is somewhat confusing. The issue is that the question uses the common definition of the word "money" which corresponds to the term "wealth" in economics, while "money" has a narrow technical meaning (referring to the "money supply aggregates").

This section is the part that is confusing.

  For example, look at the largest US company by market cap - Apple. It
  is currently worth over 800 billion US dollars. I've heard there is
  about 1 trillion total US paper dollars. So Apple would cost you about
  80% of all US paper money. I've also heard people say it is just
  'funny money' and the 800 billion is not 'real money'.

(Please note that I have not validated these numbers; the exact values are not important.)

Paper dollars (technically, "currency in circulation") is one form of financial instrument, and is a component of the various money aggregates. It is easy to work with, because it has a fixed value. The shares of a company (equity) are another financial asset, but they are not a component of the money supply. (That is, they are not "money" from the perspective of economics.) The value of equities change continuously during the trading day. If there is good news, the price of a stock could double, doubling the wealth of all of the holders. This means that a shareholder's brokerage account value could double, without any "money" changing hands.

Is this "real" money? It is certainly wealth. But it would be effectively impossible for all Apple shareholders to sell at the same time at the current share price, and so the market value cannot be converted into "money" at the current market capitalisation. This is not a problem for U.S. paper money - it can always be redeemed at par (for a bank account balance). Therefore, there is a logic to not treating the market value of a company as the same thing as the equivalent amount of money in the bank.

As for the comparison of sizes, any numbers I give will be outdated when later people read this answer. I will instead send you to a good source of data - the U.S. Federal Reserve's "Flow of Funds Report" (the Z.1 Report). Link to Z.1 Report

This report offers a quarterly summary of financial statistics for the United States. It contains data on flows - the value of transactions in instruments for the quarter, as well as balance sheet data ("stock variables" in economics jargon).

I would recommend just looking at the balance sheet tables for now. In particular, you can look at the table for "Households and Nonprofits" (table L.101). It has a breakdown of household (and nonprofit) balance sheets. (Link to html table for households.)

As of the end of 2016, households held \$15874.2 billion (\$15.87 trillion) in corporate equities; that does not include indirect holdings. Meanwhile, bank deposits and currency was only \$1237.2 billion, (or \$1.24 trillion). In other words, equity holdings swamp narrow "money" holdings (currency and bank deposits) for households. (Total bank deposits are much larger. I just wanted to compare direct holdings; the reader will have to look at the tables in their entirety to see total asset and portfolio sizes.)

The other balance sheet tables will also be useful to find the relative sizes of financial asset holdings by various sectors. This may help answer your concerns.


Is inequality increasing? Does it correspond to increasing social harm?

Regarding your first question:

Usually public economists assume that utility is a concave function of income. Therefore, the utility gain from one additional dollar of income is lower for a rich person than for a poor person. This is consistent with risk averse behavior of individuals which we often observe when individuals face risky choices.

If utility is concave in income and the total income is fixed, then the optimal utilitarian distribution would be that all individuals receive the same amount of income. Any deviation from this optimal income generates some loss in social welfare, which you may call "social harm". However, in many cases the optimal utilitarian distribution of income will still be unequal if a higher level of total income can be reached by allowing for inequality. Reasons for this may be asymmetric information or labor market incentives. Thus, higher inequality alone does not necessarily mean lower social welfare. But higher inequality with lower or equal income means lower social welfare in a utilitarian framework.

Disclaimer: notice that if your definition of social welfare is not utilitarian, none of this applies.

Extended reading: in case you are interested in this subject, I recommend you to read an introductory textbook on public finance or public economics.

For a more detailed survey of how social welfare and inequality are related, see:

Lambert, Peter J. "Evaluating impact effects of tax reforms." Journal of Economic Surveys 7.3 (1993): 205-242.


A government running like a literal business

This is an extremely implausible scenario.  The conditions for it to work would be extremely demanding:

The government would need to run businesses in which it has the
potential for a sustainable competitive advantage, giving it the
opportunity of earning profits on an ongoing basis. That might be
based on a natural resource endowment (cf Saudi Arabia and oil) or
on a technology which others are, for whatever reason, unable to
copy. For most countries, such opportunities are limited.
The government would need to run the businesses efficiently, so as
    to realise those profits. This would be challenging due to:
    a) political necessities conflicting with efficiency; b)
    rent-seeking behaviour by managers and workers in the businesses.
The profits realised would need to be sufficient to fund the
        government's expenditure.  For a typical developed country with high
        government expenditure on education, health, welfare, defence, etc.,
        this would require improbably large profits.  To give an idea of scale, UK government spending in 2015/16 was £772 billion. By contrast, the world all-time largest corporate earnings (Vodafone in 2014) were less than 10% of that at £59 billion.


Oligopoly kinked demand graph doesn't make a sense to me

As the graph notes, the red segment of the demand curve is relatively inelastic, meaning that compared to the blue segment, the red segment of demand is relatively insensitive to price changes. This does not mean that price elasticity anywhere on the red segment is less than 1 (which implies inelasticity in the absolute sense).


You can also find Kenneth Judd's book on numerical analysis here. It is one of the very known and notorious books in that field.


Simple General Equilibrium Model of Public Goods: Intuition?

You can take MRS as the willingness to pay (Take the other good in MRS as money). If the good is a private good that costs 3, your willingness to pay should be 3 because you are buying it for yourself. However, a public good is used by all payers, so we only need the sum of the willingnesses to pay to be equal to 3. (You are simply splitting the bill)


The concept of expansion of an information structure

You have to be careful what you can say is that if $\underline{S}$ be given by $(T^1,\pi^1)$ and $S$ is any information structure, say given by $(T^2,\pi^2)$, then there exists an expansion $S^*$ of $\underline S$ given by $(T^1\times T^2, \pi^1\cdot \pi^2)$ where $\pi^1\cdot \pi^2$ is the product distribution of $\pi^1$ and  $\pi^2$. Note that I have constructed the expansion, thus shown its existence.

The information structures, $S$ and $S^*$ are different since their support is in different spaces, $T^2$ vs $T^1\times T^2$. However, they are informationally equivalent. The equivalence here is in terms of the distribution of posterior beliefs induced by $S$ and $S^*$. Alternatively, if $\succsim$ is the Blackwell partial order, then $S\succsim S^*$ and $S^*\succsim S$. Your intuition, however, is correct, any information structure will be "essentially" an expansion of $\underline S$.


What affects the interest elasticity of investment?

Demand:

Expected return on investments. If the expected return is much higher than interest rates, then elasticity will be low. When interest rates are close to the expected return elasticity will be high.

Supply:

Money supply. When money is scarce the elasticity will be low. When money is abundant the elasticity will be high.


Mechanism design with multi-dimensional types (here: the willingness-to-pay for each object) is a notoriously difficult problem. Even when you abstract from bundling as you do by assuming that the seller only wants to buy one of the goods.
Your problem is studied by John Thanassoulis in "Haggling over substitutes", JET 2004. Unfortunately, there is no "nice" solution. While it is always optimal to post a fixed price in the single-good case, fixed prices are not optimal with two distinct good and randomization is part of the optimal seller strategy:

Using a model of substitutable goods I determine generic conditions on tastes which guarantee that fixed prices are not optimal: the fully optimal tariff includes lotteries. That is, a profit maximising seller would employ a haggling strategy. We show that the fully optimal selling strategy in a class of cases requires a seller to not allow themselves to focus on one good but to remain haggling over more than one good. This throws new light on the selling strategies used in diverse industries. These insights are used to provide a counter-example to the no lotteries result of McAfee and McMillan (J. Econ. Theory 46 (1988) 335).

Gabriel Carroll looks at a simpler problem in "Robustness and Separation in Multidimensional Screening", Econometrica 2017, where the seller does not know the joint distribution of the willingness-to-pay and considers "worst-case" profits.

A principal wishes to screen an agent along several dimensions of private information simultaneously. The agent has quasilinear preferences that are additively separable across the various components. We consider a robust version of the principal's problem, in which she knows the marginal distribution of each component of the agent's type, but does not know the joint distribution. Any mechanism is evaluated by its worst‐case expected profit, over all joint distributions consistent with the known marginals. We show that the optimum for the principal is simply to screen along each component separately. This result does not require any assumptions (such as single crossing) on the structure of preferences within each component. The proof technique involves a generalization of the concept of virtual values to arbitrary screening problems. Sample applications include monopoly pricing and a stylized dynamic taxation model.

If you punch the title into youtube you will find two of his talks about the paper. His introduction may give you an idea on how other researchers have failed to find a cute solution.


What is the importance of Epstein-Zin preferences?

This is only a quick answer, unfortunately. The key intuitive insight for Epstein-Zin is that they separate two distinct properties of preferences: risk aversion ("I'd prefer less uncertainty to more uncertainty*") and intertemporal substitution ("I may want to shift consumption forward or backwards in time**").

In the very popular Constant Relative Risk Aversion class of preferences (CRRA), risk aversion and intertemporal elasticity of substitution are tied together as inverses of one another. Recursive preferences, and specifically Epstein-Zin, use certainty equivalence in a clever way to split out the parameter which controls intertemporal substitution from the parameter which controls risk aversion in a static gamble.

The static risk aversion parameter is embedded in the function which imposes the certainty equivalence, and the intertemporal elasticity of substitution parameter is imposed over today's certain consumption and the certainty equivalence value for utility of consumption tomorrow.

That's my attempt to describe it intuitively/verbally. It's much more precise in mathematical form -- one good exposition that I like a lot is Francios Gourio's Asset Pricing field's course notes (EC745 is the course number). You can currently find these notes, titled "Lecture Notes on Macroeconomics and Finance
Ec 745," on his website here; see section 8 on page 36.

Work through the math a few times and hopefully it will suddenly "click." The analytical idea is really quite clever. Gourio goes on to discuss how to actually estimate these models, which is very helpful.

(*The proper definitions involves preferences over "lotteries," but I think discussion of that would cloud what we care about here.)

(** by a certain percent, as a function of the interest rate.)


How do excessive foreign reserves cause inflation?

Because you are taking other currencies off the market and/or selling your own currency, and increasing the supply of your own currency relative to others.


Hint 1: So I think you are making a mistake by substituting their production technology into their utility: when you went from $\max 6(1-l_A)+c_A$ to $\max 6(1-l_A)+4l_A$, by doing this, you are assuming that $A$ only consumes what she produces.

Hint 2: Let me give you a Pareto optimal allocation: $B$ spends his $1$ hour working to produce 24 units of consumption and gives them to $A$ who spends her 1 hour in leisure. This allocation is feasible (there are enough resources to do it, given the technology), and it is Pareto optimal because $A$ is the happiest, so there is no way to improve $A$, and in order to improve $B$, you would have to decrease $A$'s utility. Of course, this is allocation is not envy-free, but it is a Pareto optimal allocation.

Hint 3: I would first find all the possible allocations in order to then see which ones are Pareto, and then which ones are envy-free.


What if only the government could create money?

When you dig down, this questions is very nearly a duplicate of this:

Implications of abolishing Fractional Reserve Banking on mortgages and interest rates

but based on the comments, it seems that the connection isn't obvious.

This question asking about what would happen if only a government institution could create money, but equally you say the proposal is not to have a fixed money supply.  They are, in practice, the same thing.

The central bank makes the money base, M0, and then the banks create the money supply, with some multiplier to this.  In a full reserve system, that multiplier is 1.0, and so the money supply is always equal to the money base.

In your question, we consider having a M0, but also have a state run bank that chooses the money multiplier to some number, that may or may not be 1.0.

From that point on things become almost identical - it doesn't really matter from the point of view of the economy, whether you make M0 ten times larger or smaller, if you offset it with changing multiplier.  It should be the total money supply that matters - the money base never leaves the central bank, and so may be thought of as a "construct" onto which the monetary system is hung.  The important point is that the government gets to directly control the money supply.

Before the era of quantitative easing, when a central bank wanted to increase the money supply, they lowered rates and that increased the multiplier.  With quantitative easing, they simply increase M0 and force the money supply up in a 1:1 ratio.  Either works (for a given value of "work").

There is, however, one very important difference:

With full reserve banking borrowing money is very expensive - it is economically the same as capital
With a state run lending bank, borrowing could be done at any rate the government chooses, including at an economic loss.

So, in your system, you could have a state run bank, that effectively subsidises business and/or people's mortgages using money raised from taxes or elsewhere.  This would be like an extreme version of the UK forcing RBS and Lloyds to increase lending.