What criteria define an optimum minimum wage? Is there even such a thing?

The reason why you haven't found a satisfactory answer is that there isn't one. There are several and depending on who you ask you will get a different answer.

Some people will say that the optimum minimum wage is 0 and that the state should note interfere in the matter. Laissez-faire.

Some people wants the minimum wage to cover basic living expenses. What that entails varies widely but is usually connected to the general economic level of the country. In some countries it covers only the cost of basic food for 1-2 people. In others it covers rent, utilities, food, a tv, a mobile phone.

Some people want the minimum wage to be as high as possible. Some unions.


The problem is basically to derive the cost function of a firm. What you need to do is minimize costs given the output. That is
$$
\min\limits_{K,L} w \cdot L + r \cdot K
$$
subject to
$$
K^{\frac{1}{4}} \cdot L^{\frac{3}{4}} = Q = 120.
$$
You can solve this using Lagrange or the implicit function theorem (similar application as the MRS).


Can anyone explain the connection between complete market and perfect risk sharing?

Idiosyncratic shocks are those for which each individual face a different shock – while there is no uncertainty in the aggregate. Aggregate shocks are those which affect all individuals within an economy. When you have complete markets –– those on which there are securities for each possible combinations of the idiosyncratic shocks –– individuals will trade to insure optimal levels of consumption across all possible states. So, to get the value of risky options one only need to look on aggregate shocks –– as ultimately these are the ones which will determine prices.


There's this from Polling report showing raw survey data (question, sample data, and responses) back to 2012.

Additionally this paper by Lorenzoni and Pidgeon and this paper by Leiserowitz (both 2005) discussing said data in a more academic context. You would need to do some substantial extrapolation to get time-series out of these though.


Weakly monotone preferences with singleton indifference curves: do any of them admit a utility representation?

What you're asking for is equivalent to finding an injective function $f:\mathbb R^n\to\mathbb R$ that is monotone in the sense that if $x$ is coordinate-wise at most as big as $y$, then $f(x)\le f(y)$.
As a first step, notice that this is equivalent to finding such a function from $(0,1)^n\to\mathbb (0,1)$.
And this is easy by interleaving the digits of the coordinates of $x$, i.e., if $x=(x^1,x^2,\ldots,x^n)=(0.x^1_1x^1_2\ldots,0.x^2_1x^2_2\ldots,\ldots,0.x^n_1x^n_2\ldots)$, then let $f(x)=0.x^1_1x^2_1\ldots x^n_1x^1_2x^2_2\ldots x^n_2x^1_3x^2_3\ldots x^n_3\ldots$.
This is clearly injective and we prove monotonicity as follows.
If $f(x)>f(y)$, then they differ first at some $x_i^h>y_i^h$ for which $x_j^h=y_j^h$ for all $j<i$.
But then $x^h>y^h$, so $x$ cannot be coordinate-wise at most as big as $y$.

ps. Note that $f$ is almost surjective and almost continuous; the only issue is with numbers that have a finite decimal expansion. To see that an injective $f$ cannot be continuous, consider $f^{-1}(\mathbb R_-)$, $f^{-1}(0)$ and $f^{-1}(\mathbb R_+)$ where wlog. $0$ is an inner point of the image of $f$.
By continuity, these are two open sets and a point, so they cannot form a partition of $\mathbb R^n$ for $n\ge 2$. (Here we didn't even use monotonicity.)

It is, however, possible to make $f$ into a monotone bijection.
This can be achieved as follows.
First we show the statement for $f:[0,1)^n\to [0,1)$.
Write every number in its finite decimal form, if it has one, i.e., it shouldn't end with $999\ldots$.
The number $f(x)$ will start with the first few decimal digits of $x^1$.
If $x^1_1\ne 9$, then $f(x)$ will start with $x^1_1$, and then we take the first few digits of $x^2$.
If $x^1_1=9$ but $x^1_2\ne 9$, then $f(x)$ will start with $x^1_1x^1_2$, and then we go to $x^2$.
And so on, we always go until the first non-$9$ digit of $x^1$ before we start taking digits from $x^2$.
Then we repeat this for $x^2$, then for $x^3$, etc. (in a circular order).
This finishes the construction for $f:[0,1)^n\to [0,1)$.
To make it into a function $\mathbb R^n\to\mathbb R$, just apply a monotone bijection $g:\mathbb Z^n\to\mathbb Z$ to the integer part of the numbers.
(Such a $g$ is easy to construct by induction and by taking larger and larger cubes around the origin, $0^n$.)
Such a $g$ gives a partition of $\mathbb R^n$ into cubes that are isomorphic to $[0,1)^n$.
We can combine $f$ and $g$ to obtain a final function that will map $x$ to $g(\lfloor x\rfloor)+f(x-\lfloor x\rfloor)$.


Consumption function with negative intercept?

I assume that you ran a linear regression to find the $C(Y)$ function and by negative intercept you mean $C(0)<0$. This does not contradict anything because what you have is an estimate of the $C(Y)$ function based on empirical data that was probably centered around some non-zero value of $Y$. So it is quite possible that the function's estimate is incorrect far from the observed range of $Y$, especially if you were using a linear regression which is a very basic functional form.

If the actual $C(Y)$ consumption function, not the estimate, were such that $C(0) < 0$ then indeed we would have a contradiction. But as long as you are clear that what you have is an estimate for $C(Y)$ this problem does not arise.


How to handle the missing values' issue of newly listed stocks?

I have found some posts to help answer your question.   In a post on ResearchGate.net,
https://www.researchgate.net/post/How-to-deal-with-missing-value-in-a-time-series-stock-market-data
One of the responses that I think has the best answer for your is to carry forward the previous data . However,  as the person stated,  there is a problem with that method.    “--- if you are developing a trading system, for example, and assume that prices are "tradable", then your analysis will be flawed since you can't trade at a "carried forward price." My solution to this problem is this: create a 'locf' price and as a separate variable, a "tradable price", which when data is missing, is (approximately) the next price you could have realized. “
However,  Mazin A. M. Al Janabi  posted a link on that page which has a better answer for you.  https://www.researchgate.net/profile/Mazin-Al-Janabi/post/How-to-deal-with-missing-value-in-a-time-series-stock-market-data/attachment/5b73a9633843b0024405fcb1/AS%3A659744100450305%401534306659149/download/Dealing+with+the+Problem+of+Missing+Data-+A+short+Note.pdf
He states that you either not include the missing data or you can calculate the price from a geometric mean of the previous and other price data. However using average prices will underestimate the volatility of the stock price data.
Here is another response to a similar question on Stackexchange: https://quant.stackexchange.com/questions/41171/how-to-deal-with-missing-value-in-a-time-series-stock-market-data
I have also posted a similar answer on missing data but not missing stock price data How do I fill gaps in my data?
Either case I think it is VERY important that you document that you have missing data, report which observations are missing, and what method you used to remedy it.


The Shapiro-Stiglitz model. Why under full employment do unemployed workers find jobs instantly?

I think that the confusion is caused by the different meaning of "full employment" in both contexts.

In the Shapiro-Stiglitz model, full employment in the sense that $\bar{L}\rightarrow LN$ implies that the job finding rate $a\rightarrow \infty$, which means no frictional unemployment.

In the other definition, full employment is merely defined by any remaining unemployment being only frictional (or structural/voluntary, not important here). Situations of no full employment could exist, for example, due to minimum wages, according to this definition. This view considers the unemployment arising from workers being laid off and waiting some time to find a new job (frictional unemployment) as a technological necessity that cannot be avoided so as to further increase efficiency.

Last but not least, please note that your quote from Romer concerns the labor supply curve only, while the employment rate will be determined in equilibrium.


Disclaimer: I still have to educate myself on utility theory (same as I disclosed here).

  Would production possible without the government: t= 0?

It cannot be determined with the information given. The problem's definition of $T$ is unclear or incomplete, as the only component of $T$ mentioned in the problem is the physical capital tax rate $t$.

Assuming that the only source of government revenue is the physical capital tax, then no, production would not be possible if $t=0$ because --as you point out--, the latter implies that $T$, $H$, and therefore $Y$ are zero.

But that prompts the question: Does the definition of production $Y$ even make sense if  = ? Obviously not: Production should still be possible because private physical capital $K$ is not affected by the condition $t=0$.

  Do households have incentives to rent the physical capital to the firm
  if t= 1?

Yes, they may have an indirect incentive even if $t=1$: The real wage $w$. If nobody rented physical capital, there would be no demand for labor. Consequently, no household would earn income from labor even if a household's utility function indicates a preference for work over leisure.

I gather that the problem is a textbook exercise, and/or that not all the assumptions are reflected in the inquiry. Thus, a portion of my answer possibly goes beyond the scope of that chapter or section.


Distribution of US after-tax money income by household size

You can find this information here. They provide an overview of their results, together with a link to the data and its description. The data is also available through LIS (second item).

Interestingly, this does not only include the US, but also many other OECD countries, in a homogeneous methodology (great for comparisons). Additionally, this is a longitudinal study of each country. In total, they have data for 27 developed countries between 1978 and 2013.

The data is very comprehensive, and it is at the decile level. It includes:

Entire population, decile cut-offs, equivalised household income
Working age population, decile cut-offs, equivalised household income
Entire population, decile means, equivalised household income
Working age population, decile means, equivalised household income
Entire population, decile cut-offs, per capita household income
Working age population, decile cut-offs, per capita household income
Deflator and PPP


GNP v. GDP as a true measure of national vibrancy

What used to be called GNP is now called Gross National Income (GNI) as being more descriptive of what it is measuring.

So GDP measures what is produced in a year while GNI measures incomes in a year, with the difference being some cross-border flows of money unrelated to exports and imports of goods and services.

Personally I prefer GDP as it reflects what is happening in a country, but it depends on what you regard as vitality.


That is illegal because an I(1) series wanders ( doesn't have a constant mean )  and an I(0) series doesn't so they can't be set equal. In order to obtain a valid time series regression, the order on the LHS has to be the same as the order on the RHS.

Note that if both sides are I(1), then a time series regression is okay but only if what is on the RHS is cointegrated with what is on the left. If they aren't cointegrated and both are I(1), then that's what is called a spurious regression. ( i.e: a non-sensical regression ).


How do transactions in different currencies work?

Let me explain you in the most simple way how it work behind the scenes...

Credit card issuers and payment networks charge cardholders an international transaction fee if the purchase has been made abroad or an overseas bank has to interfere into the transaction process. (it is roughly for about 1 to 3% of the transaction amount, some issuing bank can even avoid charging it). Each credit card issuer uses its own currency conversion rate (per each transaction). I.e. in e-commerce when the transaction occurs (merchant gives you the option to process the transaction in foreign or local currency) and the merchant gets to choose the exchange rate. Paying in local currency means your credit card network will handle the conversion (less costly).
The exchange rate for the customer’s home currency is calculated by adding the wholesale interbank rate and the merchant’s foreign currency fee.

E.g. in the US  international  e-commerce (when accepting payments from other countries with other currencies) merchants have to have an international merchant account/ foreign currency payment gateway but they supposse higher risk of chargebacks (you can file it as a customer) and that is why the rate per transaction tends to be higher /up to 6% for merchant).

What you as a customer do not see is a process behind the scenes --Till the transaction settle, is cleared and authorized it can take even 24 hours or even for about 2 days => one thing is you see on your bank statement the money has been deducted from your account but for e-commerce where the manual confirmation of any order or reservation is needed it has not been settled on merchant’s account  yet and that is why you have to wait until it is authorized and settled (vendor's account set up plays  a vital role as well) and issuing bank , acquiring bank and bank network communicate first and accept or decline the transaction (you can come across a pending transsaction mainly if an an AVS mismatch occur, it is not your failt but it takes time to bank to communicate and validate it, as you have to submit an signed authorization form to prove your billing address).

That explains as well why if you decide to return a product or service is harmful to any business  as the handling and bank transaction fees are always associated with the purchase as the customer will not bear it (has the right to return goods) and you are protected as customer anyway, from merchant’s point of view they have it stipulated in contracts with banks and partners.

You as a final consumer just see a money charged form your card (from the statement from your bank), you can check with retailer to see where is based and which payments options offer and mainly with your bank how much charge for int payment.


What exactly is L in a Cobb-Douglas production function?

It can be both. The number of working hours are of course easier to interpret. You can also make the assumption that every employee works some fixed number of hours, say 8 per day (or 220 per month, or 38 a week, etc). In this case if you denote by $n$ the number of workers and $L$ the number of hours worked you have of course
$$
n \cdot 8 = L
$$
and then
$$
A \cdot K^{1-\alpha} \cdot L^{\alpha} = A \cdot K^{1-\alpha}  \cdot ( n \cdot 8 ) ^{\alpha} = 8^{\alpha} \cdot A \cdot K^{1-\alpha}  \cdot n ^{\alpha} = \left(8^{\alpha} \cdot A\right) \cdot K^{1-\alpha}  \cdot n ^{\alpha}.
$$
So you can choose $L$ to represent either quantity as long as you take it into consideration when choosing/calibrating $A$.  (Provided that some conditions like $n \cdot 8 = L$ hold or are a reasonable approximation of reality.)


Why does the profit function in standard neoclassical theory have exactly one maximum?

Mathematically, most neoclassical  models assume that the profits are concave. This guaranties the uniqueness of the maximum. In economic terms, the Neoclassical theory usually assumes that the law of diminishing returns holds. Thus, the more you hire workers, the more you produce, but at a diminishing rate.

I am not so sure what your professor has in mind when he brings up the "static nature of the theory". There are many dynamic neoclassical models and most of them have a unique maximal profit that also arises from having an appropriate form of diminishing returns, i.e. concavity of the profit function.


Utility function $u(x_1, x_2) = x_1^2x_2$.

  Q. Derive the demand for $x_1$ and $x_2$ as a function of $p_1$, $p_2$ and $m$.

Here are the demand functions for $x_1$ and $x_2$:
$$x_1(p_1, p_2, m) = \frac{2m}{3p_1}$$
$$x_2(p_1, p_2, m) = \frac{m}{3p_2}$$

  Q. Illustrate the equilibrium in a diagram when $p_1=1$, $p_2=1$ and $m=12$.

  Q. Suppose $p_1$ rises to 3. Calculate the substitution effect and income effect.

If $p_1$ rises to 3, the new equilibrium choice is $\left(\frac{8}{3}, 4\right)$. To find the Substitution effect and Income effect using Slutsky approach, we will find the equilibrium at new set of prices when the consumer has just enough money to buy the old equilibrium bundle i.e. we will find the demand at prices $(3,1)$ when income is $m' = 3(8) + 1(4) = 28$. Substituting this data in the demand functions, we get the equilibrium choice as: $\left(\frac{56}{9}, \frac{28}{3}\right)$. Here is how the situation looks in graph:

Substitution effect = $\displaystyle\frac{56}{9} - 8 = -\frac{16}{9}$

Income effect = $\displaystyle\frac{8}{3} - \frac{56}{9} = - \frac{32}{9}$

  Q. Explain your exact results using the appropriate Slutsky equation.

Slutsky equation:
Change in Demand =  Change in Demand due to substitution effect + Change in Demand due to income effect

$$\Delta x_1 = \Delta^s x_1 + \Delta^i x_1 = -\frac{16}{9} - \frac{32}{9}
 = -\frac{16}{3}$$


A dollar is money which can be used in the United States to purchase goods and services and to repay debts. It can sometimes be used for these purposes in some other parts of the world.

If a bank says it has an unlimited amount of dollars, then it will presumably be willing to lend them to customers, who will then want to spend them.

The bank will then either have to give the customer dollar bills (Federal Reserve Notes which presumably it does not have in unlimited amounts) or be willing to transfer dollars to another bank on the instruction of its customer.

The other bank may then expect this to be settled.  Unless there are also flows into the first bank to offset the amount owed, it can only settle with dollar bills (but faces the same issue as before) or by instructing a third bank where it has an account to transfer dollars to the second bank.  This can continue up the chain until an instruction to transfer funds reaches the U.S. Federal Reserve Banks, where the top commercial banks hold their reserve deposits.

Other banks will not be impressed with the first bank simply saying it has unlimited dollars if it does not have accounts with other banks with dollars to support the transfers.  Writing numbers in your own books does not change the numbers in other banks' books.  The constraints that this causes is one of the constraints on lending and so on the creation of money.


How to approach rigorous probability theory from an economics background?

If you want a good book with emphasis on rigorous then An Introduction to Probability: Theory and Application, by William Feller is good source.
The book starts completely from the first principles and covers also a lot of applications in statistics. Arguably the book is more suited to graduate as it has a steep curve - the content difficulty increases rapidly, but I think that it can be suitable even at Msc level - especially if you are doing more research oriented Msc.
If you are would be interested in something more focused on econometrics then a primer in econometric theory by John Stachurski is good source although it does not cover probability so broadly as the above source.


Why is the ten year interest rate lower for Spain/UK bonds than for US?

Inflation forecast is higher in the US than the rest of countries. US here: https://www.statista.com/statistics/244983/projected-inflation-rate-in-the-united-states/

Spain here: https://www.statista.com/statistics/271077/inflation-rate-in-spain/

Japan here: https://www.statista.com/statistics/270095/inflation-rate-in-japan/

UK here: https://www.statista.com/statistics/270384/inflation-rate-in-the-united-kingdom/

Because inflation is higher, central banks will raise interest rate higher. Long term rates are higher in consequence.


How important are capital requirements for controlling broad money supply?

On different banks, and at different times, different regulatory constraints may bind on their ability to create money:

Reserve requirements relate bank lending (bank assets) to the quantity of reserves held by that bank at the central bank.
Capital requirements relate bank lending to the amount of equity capital of the bank.
Liquidity requirements relate the asset market liquidity of a bank's assets to the tenor of a bank's funding.
Stress tests attempt to model the liquidity and/or solvency of a bank under an adverse scenario that affects the value of assets on its balance sheet and/or its funding.

In today’s environment, capital requirements and liquidity requirements are the constraints most likely to be binding on US banks, though stress tests were also binding on some banks in the recent past. Reserve requirements don’t currently bind in the US, as banks are awash in reserves, and as noted in the question, many other countries do not have these requirements.

The fact that capital requirements are a brake on money creation is quite straightforward: if a bank that is constrained by capital requirements wishes to lend more, it must first raise costly equity capital. As a result, capital requirements are by definition a structural constraint on money creation.

Having said that, capital requirements are not a tool that one uses primarily to control the money supply. Instead, they're used as a prudential measure to ensure the solvency of banks. This is because, in the framework of instruments and targets, capital requirements are a very poor instrument with which to affect either of the two main targets of monetary policy, inflation and unemployment.

They're a poor instrument for these purposes for at least three reasons: (1) the relationship between capital requirements and money creation is in no way linear— as noted before, capital requirements are binding on different institutions at different times, so it's difficult to predict the effects of tightening or loosening them; (2) changing capital requirements has a lot of consequences (such as the potential impact on bank solvency, and cost to banks of raising additional capital or shedding assets if a tightening of requirements puts banks over the new, lower limit); and (3) capital requirements affect only banks, while over half of financial intermediation occurs outside banks.

Conversely, interest rates are pretty good at affecting broad money supply, and in a really efficient way, as marginal lending opportunities are compared to the price of money. At the same time, interest rates are pretty useless as a prudential instrument, as demonstrated in the early 2000s.


How to obtain standard surveys / measures to use at end of experiment to "control for altruism" etc?

In experimental economics, the classic ways to measure/control for participants' social preferences are variants of the following games:

The ultimatum game measures a participant's preference for fairness or justice.
The dictator game measures a participant's other-regarding preference.
The trust game measures the level of trust and trustworthiness among a group of participants.

Risk preferences (and time preferences/impatience) can be elicited using either of the following:

Multiple price lists (see Holt & Laury (2001))
BRET (Bomb Risk Elicitation Task)
Convex Time Budget (CTB) method (for time preference, see Andreoni & Sprenger (2012))

The games other than CTB are simple enough to be ran in the "survey" form, e.g. with pencil/paper. There are also free computer programs (e.g. zTree) that implement those games.

NB. If your main audience is economists, you'd also want to run these games in an incentivzed way, namely participants get paid and their earnings are directly related to their choices in those games. Responses to survey questions, especially the hypothetical type, do not get much credit from economists.


Correlated Equilibrium for Rock Paper Scissors

No, the unique Nash equilibrium is the unique correlated equilibrium by a general property of two-player zero-sum games pointed out in:

  Forges, Françoise. "Correlated equilibrium in two-person zero-sum
  games." Econometrica (1986-1998) 58.2 (1990): 515.,

For every action of player 2 that is played with positive probability, the conditional distribution over player 1's actions is an optimal strategy for player 1 and vice versa. Since optimal strategies are unique in RPS, the conclusion follows.


How does fed cutting federal fund rate affect student loan

The Fed funds rate is the rate at which commercial banks can borrow reserves on the overnight market (see this explanation at Investopedia). As such it affects all other interest rates banks charge since when they can borrow more cheaply they can also lend money cheaply to  consumers.
This affects among others also student loans (when they are provided by commercial banks - some places have government provided and subsidized $0\%$ interest student loans). However, of course if you already had a loan in a past it will only be affected if you took it at variable interest rate or if you are taking out a new loan, the fixed interest rate loans won't be changed retroactively.


Risk Premium in the Expected Utility Theory

Is there any (economic) rational for the first-order expansion of the RHS? And for its different neighborhood evaluation?

As for your first question:
This is a purely mathematical tactic in order to obtain an (approximate) equation for $R$. The expansion of first order on the RHS is motivated by this fact, i.e. to bring $R$ alone "in the surface".  The reason why the LHS is subjected to a second-order expansion is in order for something to be left (the variance term). Higher-order expansions of the LHS can certainly be applied.

As for your second question, you are making a mistake. The center of expansion for the RHS expansion is the same as that for the LHS expansion, namely $w-E(x)$ (or equivalently, around $R=0$). It is meaningless (and it fails) to expand a function around its exact argument. Specifically we have

$$u\left(w+E(x)-R\right) \approx u\left(w+E(x)\right) + u'\cdot [(w+E(x)-R)-(w+E(x))] = u\left(w+E(x)\right) - u'\cdot R$$

Finally, why not consider a second-order expansion on the RHS? We would then get

$$u\left(w+E(x)-R\right) \approx u\left(w+E(x)\right) + u'\cdot [(w+E(x)-R)-(w+E(x))] \\+\frac 12 u''\cdot [(w+E(x)-R)-(w+E(x))]^2 $$

$$= u\left(w+E(x)\right) - u'\cdot R + \frac 12 u''\cdot R^2$$

Then we would obtain a quadratic polynomial in $R$,

$$\frac 12 u''\cdot R^2 - u'\cdot R - \frac 12 u''\sigma^2_x = 0$$

$$\ R^2 - \frac {2u'}{u''}\cdot R - \sigma^2_x = 0$$

This has roots

$$R_1,R_2 = \frac {(2u'/u'') \pm \sqrt{(2u'/u'')^2+4\sigma^2_x}}{2}$$

$$\implies R = \frac{u'}{u''} + \sqrt{\left( \frac{u'}{u''}\right)^2+\sigma^2_x}$$

You can totally validly use this expression for $R$, but I guess you understand why the simpler one is used instead.


What is the different between a Sales Tax and a Value Added Tax?

Economically speaking the taxes are equivalent. There should not be any difference in collected tax between them in principle. However, in practice these differ in terms of what administrative burden they impose on firms and more importantly how easy it is to avoid paying taxes under each system. This can ultimately have some effect on the revenue that is collected.
Arguably under VAT government gets to collect more information that can be used during audit to stamp out tax fraud. On the other hand since under VAT companies get rebates on VAT they payed their supplier there are opportunities for companies to commit some additional tax frauds such as trading fictitious software between each other to reduce their tax cost or even receive VAT rebate from government.
This being said it is argued that on the net VAT is more efficient, especially when government wants to set consumption taxes relatively high (as when taxes are small the incentive to commit fraud is small as well). You can find some discussion of further pros and cons of each system in Zodrow (1999).
Addendum: as Brian points out in comments in practice most nations with sales tax have smaller coverage than VAT but this is a result of political decisions rather than any inherent property of either VAT or sales tax. There is no a priori reason to think VAT would solve the political issues itself. Moreover, even though VAT is efficient at higher rates it’s mainly the tax rate rather than coverage that creates the efficiency argument.


If renewable energy is now cheaper than non-renewable energy, why are we still worried about the climate?

To some extent, your suspicions are correct (notice some of the nuance you could add to your four main concluding points):

Renewable adoption trends are slow.

Not just slow, but there is a wide dispersion of adoption based on point of energy consumption. (e.g., Think how many gasoline vehicles there are in the world and how long it will take to replace or modify them all.)
Transaction costs are a big deal. (e.g., Maybe a company just built a new "clean" coal plant. This was not a 1-Click buy transaction on Amazon.)
Cashflow problems. (e.g., Individuals, organizations, and nations do not all have equal access to capital at low interest rates.)
There is the landlord-tenant split-incentive barrier. (e.g., Think of all the incandescent bulbs and inefficient oil heaters that sill exist in rental units. Landlords generally just want to spend as little as possible and renters often focus on the monthly rent, not what kind of heater is in the basement.)
There are regional and political pressures. (e.g., It is hard to convince all of West Virginia to abandon coal when coal mining is part of the culture.)

Intermittent renewable energy is a storage challenge.

This is a technical problem, so it is less daunting. Though the hurdles do exist to some degree, storage does not have as many of the same social or structural hurdles to implementation.

There are bound to be assumptions and approximations in every energy cost analysis.

But even if you assume the report's pricing analysis is 100% accurate, you should still be concerned about climate change.

The world will move toward carbon neutral by economic forces.

This will happen at drastically different rates for developed and developing nations without significant cooperation.

Embedded in the more nuanced arguments that you could make in points 1 and 4 is the more pithy answer by @quarague. If you've just built a greenhouse gas emitting energy generator, the chances are that it will be cheaper to continue operating it than to build a new renewable energy generator (whether or not you include the decommissioning costs, and ignoring the negative environmental externalities).

However, you have yet to mention the most important cause for concern:
The dynamics of climate change are complicated by long time delays because greenhouse gas in the atmosphere is a STOCK (the greenhouse gas emissions are a FLOW). So, even if the entire world flipped a switch today to stop all greenhouse-gas emissions, climate change would still occur because the atmosphere already has elevated concentrations of greenhouse gases, and so there are many current and future climate change effects that are baked in (sorry for the pun).

Think of filling your bathtub with water. The tub is still full of water when you turn off the faucet, just as our atmosphere will still be full of greenhouse gases even if we suddenly stop emitting them.

If you'd like to simulate this and get a better intuition for the problem, you can try the online En-ROADS interactive simulation by Climate Interactive and the MIT Sloan School of Management. And, for your reference, even if you rapidly cut carbon entirely, there are still serious consequences (e.g., temperature increase and sea level rise).


Is it possible to transfer information through strategic games?

What you are implying is thoroughly studied by what is called algorithmic game theory. There you can find the concept of a learning algorithm. In game theoretical scenarios where the mathematical equilibrium is difficult to evaluate, those algorithms are used to get an approximation.

What they do (in a simplified way)
1) Player A starts with an array of available strategies, each to be played with equal possibility.
2) In every iteration he randomly chooses one strategy and plays against his opponent.
3) The result comes as a feedback and according to an update function/algorithm, the array of strategies is re-evaluated/recalculated.
4) This is repeated many times, hopefully to converge to either one strategy which will be a nash equilibrium, or to a mixed-strategy profile.

It is a very interesting topic, connected a lot with machine learning and AI.
Here are some links to get an idea:

1) Game Theory & Learning for wireless networks

2) On Learning Algorithms for Nash Equilibria


Accelerator and Multiplier effects at full employment

Grigory, I am glad you are skeptical of this 'theory'.

I am not sure I understand you line of argument correctly, but given the assumptions it seems to be correct. Accelerator cannot be sustained in long-run due to fixed factors of production (think land, labor force, etc). Accelerator theory is the sense you are presenting it is not present in modern macroeconomic literature or graduate curriculum.

Let's then turn our attention to the assumption that 'multiplier' at full capacity. The old Keynesian theory indeed assumed that the multiplier is higher when there is high unemployment. Indeed, this was a reason for advocating for public spending during recessions.

However, modern evidence do not seem to support this view. For example Ramey & Zubairy (2018, Journal of Political Economy), show that the multiplier does not vary significantly with the business cycle if one controls for expectations. Keep in mind, however, that those things are notoriously hard to measure. An ideal experiment, that would allow us, economists to measure the multiplier, would consist of an unexpected debt-financed government spending in the absence of central bank. You can see that all of these assumptions are typically violated (taxes and central bank's action matter in particular).

So the bottom line is, always take simple, undergraduate-level models with a pinch of salt. Some are logically inconsistent and most (if not all) are consistent with data.


Derive a cost production function give prod f, only K

Here you have to express $K$ in terms of $Q$, since the cost depends on the number of units of capital employed and not on the number of products produced. You have
\begin{equation}
Q = K^a \Leftrightarrow K = Q^{\frac{1}{a}}
\end{equation}
so your cost function will be
\begin{equation}
C(Q) = c_0 + rK(Q) = c_0 + rQ^{\frac{1}{a}}
\end{equation}


Why can we see the Current account as variation in NIIP?

In reality the world is much more complicated than those two equations, but the simple theory is that

the current account net flow is the trade balance in that period plus net primary income in that period, and net primary income is the return on investments abroad less the return on foreign investments at home.  The $i^wB_{t}$ term suggests net primary income can be modelled as an implicit rate of return or interest rate on the net investment stock and this implicit interest rate is the same worldwide for all kinds of investment income in both directions (reality is different, in that different types of invest in different places produce different returns)
the change in the International Investment Position net stock over a period should be equal to the net financial flow in that period (reality is again different as there are also revaluations of the value of the components of existing investment positions, for exchange rate changes and other reasons, and these are not reflected in flows)


Can someone explain graphically why MRS is invariant under monotonic transformation?

You're right that it's a bit counterintuitive that the shape of the indifference curves shouldn't change when you transform the utility function. The reason is that you are transforming along an axis that is perpendicular to the plane where the indifference curve lives.

Let's imagine we have two goods, x and y, and let's say that the original utility function is given by $U(x,y)=x^\frac{1}{3}y^\frac{2}{3}$. We can create a 3D plot of the utility function where $z = U(x,y)$. Any transformation $f(z)$ will work on the z dimension. In the following example, the transformation $f(\cdot)$ is multiplying the utility function by a positive scalar

If you look down on the $x-y$ plane from above, you will notice that you can plot an indifference curve through a point on that plane and it will look identical before and after the transformation. Note that it will not be the same indifference curve because the utility level will change under the transformation. Going back to the multiplication example, let's look at the indifference curve (red line) that goes through the bundle $(x,y) = (10,10)$ (red dot). Regardless of the multiplier, the indifference curve maintains its shape but the associated utility level increases from 10 to 20.

I have also included other level curves, for utility levels 4, 8, 12, ... 40. Note that as the factor with which we multiply the original utility function gets larger, these level curves get squeezed into the lower left corner.

You can see the same thing happening with other monotonic transformations, for example translation (adding a constant) and power transformation.


When and to what extent should the money supply of a pegged currency be included in the "parent" currency?

During the Irish Pound-Sterling pegging era, we again had separate monetary authorities, but currency union. (...) In this case GBP was mostly accepted for trade in Ireland, while IEP was rarely accepted in the UK. Would GBP be in the IEP money supply, but the IEP not in the GBP?

Let the states and the institutions declare what they want. To study economic phenomena we must look at what economies actually do. Think of something totally unrelated: each state has a specific tax structure, a range of various taxes, progressive, proportional, whatever. The official structure matters, because it anchors the effective taxation, but effective taxation itself matters more.
In your case, the Irish Economy incorporated in its workings the official monetary union, while the UK Economy didn't.

So one has to go per case, which creates a testable hypothesis:

a) postulate an hypothesis based on "casual" (i.e. true but unquantified) observations like the 2nd sentence in the above quote
b) derive the theoretical implications of the hypothesis
c) send your theory and your theoretical results out to meet the data.

For your example, assume that UK increased steadily its money supply to accommodate a growing economy, while the Irish economy was stable (or stagnant, depending on your philosophical point of view), or grew at a lower rate. If we add the GBP quantity in the money supply variable of the Irish Economy as equivalent, $1:1$ to the IEP, shouldn't we observe excessive inflation? We should. Did we?  Let's say "partly". Then we can estimate the effective ratio between the two currencies as regards their effect on the Irish Economy.


Natural Capital is an analytic accounting approach: it is a system of putting a monetary value on environmental assets such as ancient woodland. It is also the name of those assets, within that approach. So it's one way of internalising the externalities of environmental change, within macroeconomic indicators.

For more, see in particular the work of Paul Ekins, including "A Framework for the practical application of the concepts of critical natural capital and strong sustainability" (pdf)

The philosophy of decroissance - of deliberately engineering a decline in traditional economic growth indicators - is one attempt to find an optimal and sustainable economic path, taking into consideration both the traditional economic indicators such as year-on-year changes in GDP, and changes to the population's wellbeing, together with changes to the national asset base, which includes changes in Natural Capital.


The general answer is no, a closed market with only money circulating within the market, and that money never leaving it or entering it, cannot increase its aggregate price level - provided that the quantity of goods being sold on that market also remains constant. Price wise however, it can display variations over time, so share values can change - even if the money circulating that creates them doesn't.

The main reason market prices grow and shrink in the real world is that not only are new companies always being listed, but money is entering them and sometimes leaving them at different rates. In a typical western economy for example, the money supply is continuously growing due to bank lending, a percentage of that growth it being invested in the stock market through pension funds, and so there is a linkage between monetary growth and the market.

Consequently any time some kind of conduit between lending and stock market purchases gets created, things tend to get exciting. cf. 1929 US Crash.


Value function iterations with hyperbolic discounting

Refer to Harris-Laibson 2001.
They show the validity of the contraction mapping theorem in a neighborhood of the exponential consumption path.


What is technology in the Ricardo model?

this is attributed to better computer-production technology in the US. But what is that technology? Is it better physical capital, in the form of computer-building machines? Is it better human capital?

In a sense yes. For example, typical Cobb-Douglas function with capital labor and human capital would look like this:
$$Y = A K^{\alpha} L^{\beta} H^{1-\alpha-\beta} $$
Where A would be the technology that actually augments the production capabilities of the different factors ($K$ capital, $L$ labor and $H$ human capital).
You could have a technology that also augments just single factor, for example $ Y = K^{\alpha} (A_LL)^{\beta} H^{1-\alpha-\beta} $ or each factor could have its own separate technology. However, you should not confuse the technology with the stocks of capital, labor and human capital. This being said, you can consider $A$ as a factor that makes capital labor and human capital 'better'.
The technology here is considered in the loosest sense possible. For example, information technology such as skype or zoom allows people to give lecture online to large audiences making their stock of human capital more productive. Some idea about how to improve production process would also be considered. For example, if there are some teaching techniques that allow teacher to teach more efficiently that could be again considered a technology.
Most undergraduate textbook models will not explicitly include production functions in presentation of the Ricardian or Heckscher-Ohlin models, rather they will just assume that certain number of inputs just give you some outputs, but the production functions are included in more complex presentation of these models. See for example some graduate text such as Feenstra's Advanced International Trade.
Furthermore, the production function does not need to be necessary Cobb-Douglas, and it is not necessarily the most appropriate production function for the international trade models but it makes exposition of technology relatively easy so I used it as an example.

Is it related to TFP?

Yes it actually is the total factor productivity. The $A$ is also called the total factor productivity (see pp72 Bruda Wyplosz Macroeconomics). Also, the way how total factor productivity is measured is that people try to estimate the production function such as the one above by putting there all inputs in terms of capital, labor and so on  and using that to actually estimate what is the effect of technology.


Today's value of the Louisiana Purchase?

The video has a transcript with the references. The \$0.25B figure is obtained from here (after adjusting for inflation). Unfortunately, the author does not provide a source for the \$1.2B figure.

However, there are estimates of the value of land elsewhere. For example, here. Their estimates on a map:

These estimates consider the value of land only, removing the value of structures on that land (which, as you say, makes a big difference; see paper). They use hedonic regression methods to separate land and building values (check from page 6 of document).

I leave to you the task to compute the precise value of the Louisiana Purchase. (you could ask them to send you the data on the map).


Fisher Effect - inflation and stock return

It’s simpler if you start with a bond. If the yield increases (expected return), the price goes down. Future returns are higher, but if you were holding it before the price change, you get a capital loss. Your return to maturity will still be the previous (lower) yield; the capital loss acts as a drag on the future expected returns.

If we assume that the only thing that changes in the stock valuation, an increased discount rate causes a fall in price, with a higher expected return later.

However, the situation for equities is more complicated. The fair value is the discounted value of future cash flows. If expected inflation is higher, why are not the cash flows also higher? Therefore, you need to model the relationship between cash flows and inflation.


How do I adjust for heterogeneous labor markets when calculating the wage premium?

It all depends on how you define the 'wage premium between different areas'.

The average NY resident makes $51,000 more per year than the average IL resident.

There are 110 medical workers and 90 natural gas workers in NY and IL combined, so you could try weighting the wage premium of medical workers by 110/200 and the wage premium of natural gas workers by 90/200 to give a combined wage premium of $45,500. This will tell you how much more or less a person chosen at random from the whole population will earn if they work in the same industry that they are currently working but in a different area.


Is it possible for disposable income to be lower than consumption expenditure?

the first graph is 1970-2003, the second is 2003-2015


From an economics perspective, what are the ramifications of a currency with fixed money supply?

FooBar is quite right that unless you expect GDP growth to stop, fixed nominal supply currencies will lead to deflation.

A moderate degree of currency inflation serves a number of useful functions in the economy. The most obvious are:

It induces people to spend their money before it loses its value. In
a deflationary environment there is an incentive to put money under
your mattress and spend it in a year when it has greater purchasing
power. If everbody does this then the lack of demand will lead to a
decrease in overall economic activity (i.e. a recession).
It provides a weapon against downward nominal rigidities. For example, workers are generally reluctant to accept a nominal pay cut, even if market conditions are such that the current wage is above the equilibrium level. Inflation means that their employer can simply increase wages at less than the inflation rate so that the real wage is decreasing.
It erodes the real value of nominally denominated debt. Now, this is obviously only a pseudo-advantage because (whilst it benefits debtors) it harms creditors. However, this kind of erosion of debt may be desirable if national economic stability is threatened by high debt levels. Also, since debtors are usually poorer on average than creditors, it can reduce inequality, which may be a normative objective for the government.

Without inflation you miss out on these benefits. The first benefit might not seem like a big deal if you think you can simply set the rate of inflation at zero percent. But it is very hard to hold inflation constant at some target level so attempting to hit zero inflation will almost certainly result in occasional lapses into deflation, with the attendant negative economic consequences.


How much impact does memory have on risk pricing?

I'm not sure what you mean by risk pricing, but because you mention a repeated game, you might find the following papers useful. They present some theory about repeated games in which players have limited memory.

One - Memory in Repeated Games
Repeated games with one-memory
Reputation and Bounded Memory in Repeated Games with Incomplete
Information


Cournot's game profit maximisation first order condition question

To elaborate on @HerrK’s comment:

The product rule of differentiation is
$$\frac{d}{dx}fg=f’g+fg’$$
The text in your question states that you are trying to maximize the quantity
$$[a-Q-c]q_i$$
So, as you predicted, we must differentiate this and find when the derivative equals $0$. If we differentiate this with respect to $q_i$ and use the product rule, setting $f=a-Q-c$ and $g=q_i$, we obtain
$$-\frac{dQ}{dq_i}q_i+[a-Q-c]$$
since the derivative of $a-Q-c$ with respect to $q_i$ is $\frac{dQ}{dq_i}$ (since $a,c$ are constant) and the derivative of $q_i$ w.r.t $q_i$ is $1$. Setting this equal to zero gives
$$-q_i\frac{dQ}{dq_i}+[a-Q-c]=0$$
or, equivalently,
$$a-Q-c=q_i \frac{dQ}{dq_i}$$
which is the last line of the text.


Calculating Time to Balanced Growth Path

a) Your calculations are correct, but in order for consumption to be positive, so for
$$
c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg)k_t > 0,
$$
you will need to additional conditions. The first is the obvious $k_t > 0$. If there is nothing to gain interest on, there will be no growth and no consumption. The second is the more nuanced
$$
\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1 > 0.
$$
This is actually a necessary condition for the existence of an optimal solution.  The inequality may be reformulated as
$$
\beta \cdot R^{1 - \gamma} < 1.
$$
If this does not hold, then given any consumption path the consumer would gain by pushing off all consumption one period further. Since there is no infinitely distant timeperiod, no optimum exists.

b) On reaching the balanced consumption path:
Perhaps there is a trick with the indeces. $c_0$ seems ill defined unless there is a $k_{-1}$. If consumption only starts at $t = 1$, the above conditions are satisfied and the consumer is rational then
$$
\forall t: \ c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg)k_t
$$
defines the optimal consumption path as I assume $k_0$ is given and
$$
\forall t: \ k_t = k_0 \cdot (\beta R)^{\frac{t}{\gamma}}.
$$
Hence
$$
\forall t: \ c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg) \cdot k_0 \cdot (\beta R)^{\frac{t}{\gamma}}.
$$
It is straightforward to check that this path is feasible (if the conditions set out in a) are met) and balanced.

A note on reaching the balanced consumption path: This concept usually exists when there is some conflict between balancedness and optimality or when the consumer is boundedly rational. (E.g. Solow model.) Here this does not seem to be the case. However when such conflicts arise there is usually a never-ending (Is this the correct spelling of this word? Please correct if wrong.) convergence towards the balanced path. The distance decreases in every period but never becomes 0.


Growth Accounting with variable factor shares

Interesting question. In effect, while factor shares were thought to have remained fairly stable over a long time (the first of the Kaldor's facts), more recently they have varied, particularly in the direction of a fall in the labour share.

This short paper from (2012) shows that under such scenario, a growth accounting exercise which assumes constant factor shares leads to inconsistent results. The abstract reads:

  Recent evidence shows that factor shares are not constant. Consequently, growth accounting exercises rely on a false assumption and a measurement problem arises. We propose an empirical methodology to solve the measurement issue and estimate TFP growth.

A more complete and recent analysis from the same author (plus another co-author) is here (and a theoretical model with these insights is here).

As you can see, this is a very topical issue, and there seems to be very little empirical analysis so far. I only found this one.   As the authors in the second article linked conclude,

  any efforts to align empirical growth research with the reality of the factor share data cannot be carried out using standard techniques. New techniques need to be developed.

There might still be some time until a satisfactory empirical method is developed.


Do all prices just reflect absence of information?

It's a deeper discussion, but you could argue that profits reflect information disparities between agents.  However, in the most idealized circumstance, prices themselves arise from scarcity resulting from physical constraints of the system, not informational constraints.


I gave you an extremely vague lead in my other comment. Thinking about it a bit harder, I beieve you are right with having a look into the matching literature which I know a bit less.
Here is a setting that might be of interest: Kelso & Crawford, Econometrica 1982.
They study $m$ workers (=buyers in your setting) and $n$ firms (=items), where a worker can only work at one firm, but a firm can hire many workers. Firms pay salaries to workers (=accept payments for a unit). If that is a setting that appeals to you, go straight to "Matching with contracts" by Hatfield & Milgrom, AER 2005.


The concept of rational expectations can indeed be confusing.

One of the nicest pieces that I read online on rational expectations is the following:

https://economictheoryblog.com/2012/05/18/rational-expectations

It very nicely describes the concept of rational expectations in very simple words.

The most important thing one needs to understand is that rational expectations do not imply that all people act optimal under all circumstances, it rather implies that people behave optimally given the information they have.


It is important to distinguish between optimal pricing from Uber's perspective and optimal pricing from an efficiency standpoint. If Uber was the only company matching drivers and customers, it would adopt a monopoly pricing strategy on selling rides to consumers and buying rides from drivers. These rates may be very far from the efficient rates. It is therefore important to state what you mean by a "better" pricing model.

From the perspective of Uber/Airbnb, profit maximizing pricing strategies may differ because of many reasons. AirBnB is in a business where the quality of a service can vary greatly and is often unobserved to them. Uber in contrast matches customers with services that have little to no variation in quality (speed, safety), which are fairly well observable due to their GPS data. But even depending on expected company growth may lead the two companies to have optimal pricing strategies closer or further from efficient prices. Finally, the competition in each city may be very different for Uber, since cab rates are fixed by a central planner. The competition of AirBnB usually has decentralized pricing except where a hotel chain has a local monopoly.

Even from an efficiency standpoint, there may be large advantages from a centralized pricing strategy in the market for rides. Most urban areas have fixed cab ride rates to increase transparency. Anybody can get into a cab without worrying about being charged an unexpectedly high amount. This can greatly enhance market outcomes.


How are fund managers usually compensated against benchmark?

The format varies. Many firms just charge a straight fee as a percentage of assets, and/or as a percentage of in/outflows e.g., mutual funds.

For fund managers that are not hedge funds, returns are compared to index returns if there are any incentive bonuses. Individual fund manager’s bonuses are often based on this relative performance, even if the client is not charged. Better returns presumably brings in more clients for the fund, and so the fund firm is more profitable, even if it does not charge performance fees.

The hedge fund 2/20 model was 2% of net asset value, plus 20% of returns in excess of some target (discussed next). There would be other mechanisms, such as a “high water mark” rule, which would means that the variable compensation would not be paid until any losses are recovered. The 2% fixed fee always needed to be charged to keep the expensive hedge fund operation going. Since hedge funds typically used leverage, the 2% would have been smaller as a percentage of assets.

The exact terms would be negotiated, the stories that I have seen is that typical terms are now less generous than the 2/20 numbers, but that varies from fund to fund. Very popular hedge funds might charge more.

Hedge funds were supposed to diversify from equity market risk, so the return target was often either fixed (e.g., 8%), or a money market rate plus a spread (e.g. Treasury bill returns plus 5%). That would change based on the structure of the fund.

As an example, if the target rate of return is 8%, and the fund returned 13% (5% more), the fund would get 20% of the 5%, which is 1%.


What exactly does "carriers of utility" mean?

This expression is used for the arguments of the utility function, that is, for the objects the agent "cares about". In the canonical model, the only carrier of utility is wealth, but in behavioral economics a lot of things can be considered (beliefs, wealth compared to a reference point, ...).

For instance, saying that the carrier of utility for John is the difference between John's wealth and Lisa's wealth means that the utility function of John is of the form $u(x-y)$ where $x$ is John's wealth and $y$ is Lisa's wealth. It is another way of saying that the only thing that matters to John is his relative wealth compared to Lisa, and not his absolute wealth.


What does it mean when capital is immobile as opposed to having full mobility?

Capital Mobility is usually defined, simply,  as the ability to move Capital out of one asset into another.

If a car manufacturing firm (Carmo) is deeply invested in Honda Engines and has built their production line and engineering team around that engine. Their capital is heavily invested in assets, labor and tech related to Honda.

Now let's say tomorrow, Honda has a billion recalls on their entire line due to some severe engineering flaw. Carmo would like to switch to Toyota engines as their somewhat interchangeable with Honda Engines but this would require heavy asset replacement. Carmo's ability to respond to market preferences is hindered by the cost of replacing their asset bucket.

In summary, Capital Mobility refers to firm's capability in transferring capital from asset to asset to produce goods demanded by consumers.


Marc Lavoie, page 265 of Post-Keynesian Economics: New Foundations: "In fact, the SFC  approach is perhaps a misnomer, as several other theories relate stocks and flows in a consistent way. The peculiarity of the post-Keynesian SFC aporoach is that its models truly integrate the real and monetary sides."

In other words, do not focus on the name "stock-flow consistent models," the (alleged) contributions go beyond stock-flow consistency.

The usefulness of SFC models depends upon your theoretical perspective.

If you are already a post-Keynesian economist, SFC models appear to offer a rigourous, standardised framework for setting up economic models. One of the problems facing post-Keynesian approaches is that there are considerable differences of opinion in how to approach economics, and the mathematical models used appear quite different.

The hope was that some of the distinctions might disappear if cast into the same mathematical framework. In practice, there remain considerable differences in the behavioural equations between various post-Keynesian schools of thought, so SFC models have not been able to completely unite the various schools of thought. (The text by Lavoie discusses this at greater length.)

If you are not post-Keynesian, the usefulness depends upon your views about mainstream macroeconomic models, which I will summarise as dynamic stochastic general equilibrium (DSGE) models here as a shorthand. Please note that there is a massive number of variants of DSGE modelling; my comments here reflect the views of an external observer who is not worried about the distinctions between those models.

(Since SFC models have an explicit economic model structure, they are not comparable to unstructured models that are generated by a statistical fitting procedurs.)

From a post-Keynesian perspective, DSGE models embed a great many assumptions, pretty much all of which have been criticised. For example, utility maximisation, the production function, rational expectations, general equilibrium, representative households, price determination, and even the distinction between randomness and uncertainty have all come under attack by post-Keynesians. (Since "DSGE models" vary, not all of these assumptions appear in all models. However, it is probably safe to say that almost all "DSGE models" incorporate at least some assumptions that at least some post-Keynesians object to.) SFC models do not embed those assumptions, and so have an advantage if you view those assumptions as incorrect.

However, even if you move past the arguments over the assumptions, the tractability of SFC models is an advantage versus pure DSGE models. Standard DSGE models have to make simplifying assumptions in order to remain tractable, such as not having a financial sector. Although certain DSGE models can relax such assumptions, what typically happens is that only some of them can be relaxed at once.

In practice, many central banks have had to use hybrid models, that only incorporate some behavioural equations taken from DSGE theory. Although it might not meet a purist's definition of a "SFC model", you could easily create such a hybrid model from a SFC model base. From the perspective of applied mathematics, I would be hard-pressed to give a formal mathematical reason why such a model is not a member of an extended family of "SFC models." (It would not be classified as a DSGE model, as there is no assumption of equilibrium. The behavioural equation is just a bolted on assumption, independent of equilibrium arguments.)

However, if you insist that macro models must reflect the optimising choices of households, SFC models are not going to help you very much.


Can integrals be interpreted simultaneously as aggregates and averages? (Mas-Colell et al. 1995, Proposition 4.C.4)

I talked to my teacher to try and get a handle on the question.

You are correct that usually

$$\overline {f(x)} = \frac{1}{b-a}\int^b_a f(x) dx$$

But for the demand function there are two arguments, $w$ and $p$.

$$x(p) = \int_0^\bar W \tilde x(p,w) dw$$

The left hand side is no longer a function of $w$. The integral is to simplify wealth out of the picture, but we aren't averaging over wealth.
All consumer preferences are identical. For the sake of argument, suppose there were $n$ consumers. We aren't multiplying by $n$ on the inside of the integral, so we don't need to divide out anything (like $n$) through the outside of the integral. The "aggregation" is occurring over all consumers.


Indifference curves passing thru few points

The first sentence means that Tommy's mother's utility function over a bundle $(x,y)$ equals
\begin{equation*}
u(2,7)-u(x,y)=|x-2|+|y-7|
\end{equation*}
since she measures the disutility of this bundle with respect to the most preferred one by the "sum of the absolute values of the differences".

Therefore the utility that she associates with the point $(4,5)$ equals
\begin{equation*}
u(4,5)=u(2,7)-(2+2)=u(2,7)-4
\end{equation*}

You are asked to find the indifference curve that passes through the point $(4,5)$, i.e. the set of bundles $(x,y)$ that satisfy
\begin{equation*}
u(x,y)=u(4,5)
\end{equation*}
which you can write
\begin{equation*}
u(2,7)-u(x,y)=u(2,7)-u(4,5)=4
\end{equation*}

The question is therefore: which of the options a), b), c), d) (if any) contain only bundles $(x,y)$ such that $u(2,7)-u(x,y)=4$? You simply have to compute $u(2,7)-u(x,y)$ for all the proposed bundles, given the formula above, and check whether the value equals 4 or not.


If automation implicitly increases wealth, why is it generally considered undesirable?

Short-term or long-term, Property rights and distributional issues exist, "evil capitalists" or not, and they will always result in strong scrutiny of any change.  Remember that what is "short term" for society is mid- or even long-term for individuals.

Moreover, there exists the transitional pain: whether we can optimize it downwards, and who is going to feel it.

All these cause the heated arguments and the opposing positions.

But your post starts with a strange assertion: that "almost all times" it is assumed that automation has a negative economic impact. If this was so, it seems a little strange that automation marches triumphantly for the last two hundred years, and it continues to do so. If this was so, people now must live in a much worse material situation than two hundred years ago. Is this the case?


Why will Australia increase its interest rate only after the US increases its interest rate?

Australia's cash rate is already quite low. There's no need at the moment for the RBA to drop the interest rate any more, and as you can see from past decisions it's been unchanged for a while. Some economists think the RBA will move up soon, but that's uncertain - I wouldn't say it too confidently anyway - the bank is meeting its inflation targets and appears to be maintaining stability. Some are predicting later in 2015 anyway.

With low rates already, the article is stating that the RBA is watching the Fed to see what it does. The Aussie dollar is going down, you're correct. And while this isn't good for  importing businesses, businesses using overseas intermediatary goods or my eBay Christmas shopping, it will increase competitiveness for our exporting / manufacturing businesses and, in Glenn Steven's words, allows for 'more balanced growth' across the economy. Also mentioned is that the Australian dollar is higher than its fundamental value and the RBA wants it to go down even more.

If the Fed cuts their rates more then the RBA may need to act because there will be higher demand for Australian investment as a result of higher returns, resulting in currency appreciation, which the RBA doesn't want any more of.


Are the retirement plans insured in the USA?

The Pension Benefit Guaranty Corporation is the government entity responsible for insuring private sector pensions against the event that a corporation cannot pay their obligations.


Calculating the amount paid for a loan in today's dollars

The monthly payment formula tries to ensure that you pay a constant amount each month. Hence, in real terms, you are actually paying more up front as $117.46 today is worth more than in 20 years assuming positive inflation. The more you pay up front, the less you have to pay in total. Even if you managed to correctly calculate payments such that they were constant in real terms, you would end up paying more in nominal terms than if payments were constant in nominal terms.


Productivity and wages in a task-based model

I guess you are referring to a situation in which abundand supply of high-skilled workers decrease productivity such that individual $w_h < w_l$
Considering Equation (5) in Acemoglu & Autor, $H$ and $L$ capture supply of both groups of workers. If $\frac{H}{L}$ becomes sufficiently high, I don't see why the relation between both wages should not become lower than 1.


Why did the US credit rating downgrade in 2011 decrease US government bond yields?

My guess is this is a coincidence. That is, the effect of the credit rating agencies was either approximately zero, or if it was substantial and negative, that this was swamped out by other factors.
If bond ratings have a causal effect on yields, that effect is usually for smaller issues and smaller issuers where the market depends on the rating agencies for critical information. For the big issuers the banks are well informed on the issuer and need the rating agency less.
For example, consider the paper The relationship between credit default swap spreads, bond yields, and credit rating announcements (JBF 2004). When they look at CDS spreads, which are mostly for bigger issuers, they find:

By pooling all observations we find a significant (at the 1% level)
increase in the CDS spread well in advance of a downgrade event. In
the case of reviews for downgrade and negative outlooks there is a
significant (at the 1% level) increase in the CDS spread during the 30
days preceding the event. CDS spreads increase by approximately 38 bps
in the 90 days before a downgrade, by 24 bps before a review for
downgrade, and by 29 basis points before a negative outlook. When
observations are pooled there are no significant changes in CDS spread
during the 10 business days after any type of negative event.

I interpret this as saying that bond markets seem to react to the information on these larger issuers that the credit rating agencies are also using to change their ratings and at the same time they are considering those changes. However, the actual rating change, the information in that rating, and the reasoning behind it doesn't matter much. What holds for large corporate issuers is going to be even more true for a large issuer like the US government.
However, there is alternative evidence from the sovereign market in Europe. Credit rating agency downgrades and the Eurozone sovereign debt crises (JFS 2016):

This paper studies the reaction of the Euro's value against major
currencies to sovereign rating announcements from Moody's, S&P and
Fitch CRAs during the Eurozone debt crisis in 2010–2012 based on event
study methodology combined with GARCH models. We also analyze how the
yields of French, Italian, German and Spanish government long-term
bonds were affected by CRA announcements. Our results reveal that CRA
downgrades, watchlist and outlook announcements had no impact on the
value of the Euro currency but increased exchange rate volatility.
At the same time, downgrades as well as negative outlook
announcements increased the yields of French, Italian, and Spanish
bonds and even affected the German bond's yields. This shows that
the monetary union has led to a breakdown of the consequences of the
rating shocks between currency value and sovereign bond yields. The
reason is that part of the rating shock is absorbed by an internal
repricing of sovereign bonds.


I don't know if your statements are true. By what margin is the German stock market small? And the only bank that did matter on the international level (Deutsche Bank) is far from being large and powerful.

That being said, your notion might have to do with the fact how the German economy has been organized since WWII. The Rhine capitalism used to be, among others, characterized by strong institutional links between banks and industry. Whereas firms would finance on the stock market in the U.S., they received credits from banks in Germany who, in turn, often held stocks of these firms. This construct was called "Deutschland AG", but has been deteriorating since the early 2000s at the latest.


Historical GDP of world economies back to 1945: What are the most complete and reliable data sets?

As far as I know Angus Madison's GDP figures are one of the most reliable sources of historical data, even if data construction is as always diputable. The Review of Income and Wealth published a rigorous debate between Angus Maddisson and Carsten A. Holz regarding Maddison's estimates on China's growth.

You may also find useful the  Historical Bilateral Trade and Gravity Data set (TRADHIST) provided by the CEPII, which partly uses Maddison's data. TRADHIST gathers more than 1.9 million bilateral trade observations for the 188 years from 1827 to 2014. TRADHIST also provides about 42,000 observations on aggregate trade, and about 14,000 observations on GDPs and exchange rates respectively. This data set is publicly available.


All you need for this particular question is the following. Let $\mathbf{X}$ be a $T \times K$ matrix, $\mathbf{w}$ a K-dimensional vector and $\mathbf{y}$ a T-dimensional vector, then

$$
\begin{eqnarray*}
\frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{y}}{\partial \mathbf{w}}&=&
\mathbf{X}^{\prime}\mathbf{y}\\
\frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}}{\partial\mathbf{w}}&=&2\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}
\end{eqnarray*}
$$
Back to your original post, we will first rewrite it in order to apply the rules provided. The first element in the expression we want to differentiate is the same as $\phi'(\mu-R_f\mathbf{1})$, hence
$$
\frac{\partial \phi'(\mu-R_f\mathbf{1})-1/2\alpha\phi'\Sigma\phi}{\partial \phi}=
(\mu-R_f\mathbf{1})-\alpha\Sigma\phi
$$
and equating to 0 you get the desired result.


Why does supply (and demand) shift left and right as opposed to up and down?

Both ways of thinking are correct. They are two equivalent and complementary ways of thinking about the same thing:

Shift left (right): At each price, the quantity that producers are willing and able to supply has gone down (up).
Shift up (down): To get producers to supply each quantity, the required price has gone up (down).

Example where the supply curve shifts left or (equivalently) up:


CES production function with non constant returns to scale

Another possibility is
\begin{equation}
Y=\left[ aK^{\mu_K\frac{\sigma -1}{\sigma }}+\left( 1-a\right) L^{\mu_L \frac{\sigma -1%
}{\sigma }}\right] ^{\frac{ \sigma }{\sigma -1}},
\end{equation}
with $\mu_K+\mu_L \neq 1$.
Edit: The function is not homothetic in $(K,L)$, and as mentioned by Giskard, the elasticity of substitution is not longer constant, unless $\mu_K=\mu_L$, in which case we obtain the specification proposed by cel.


GDP to employed people ration - solow swan model

GDP per worker is the ratio of GDP (Y) to the number of people employed (N), it is Y divided by N. Some books denote it as little Y (y). Divide both sides of the production function by N (remember to use the rules of dividing exponents for N on the left hand side i.e. subtract the exponents, for N it will be 2/3-1 = 1/3).

Simplify the equation, so on the left hand side you should have (K/N) raised to the power of 1/3. You can replace this fraction with little k (k), raised to the power of 1/3. The little k represents capital per worker then.

1) $$ Y = A K^{1/3} N^{2/3} $$

2) Divide by N

3) $$ Y/N = A K^{1/3} N^{2/3-1} $$

4) $$ Y/N = A K^{1/3} N^{-1/3} $$

5) $$ Y/N = A (K/N)^{1/3} $$

6) $$ y = A (k)^{1/3} $$


Dynamic Optimization : Resource Stock as a Sink

"Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?"

No. $\mu$ can be thought of as the shadow price of natural resources. Being a "price", it has to be non-negative.
Note that if we set $\mu=0$ the problem reverts back to the standard model, which has an intuitive explanation: if "we don't care" about natural resources, their "price" is zero.

To see this from another route, for the standard CRRA utility function

$$u(c) = \frac {c^{1-\theta}-1}{1-\theta}$$

we would get

$$\frac {\dot c}{c} = \frac{1}{\theta}[A-\rho] - \frac{1}{\theta}\frac{\mu}{\lambda}\gamma A$$

So if $\mu <0$ we would obtain a higher consumption growth rate, which doesn't sound very plausible if by  $\mu <0$ we would want to express that "we care" about natural resources and so we would lower capital accumulation that hurts them, and consequently lower the consumption growth rate.


Why the prices for very popular items do not increase?

Some online-shop does practice price discrimination.

However, the seller must play the price game carefully to maximize their profits:

Lower price means lower profit per units but may increase revenue and increase overall profits
Higher price is more profitable but hurt sales
Shelter for the rainy day
Competition

In addition, influence factors such as inflation, hot money also come in play.  For example, during economy bubbles, people don't mind paying extra $100. However, once the market shrink, you will face a huge retract of revenue. Worst still, the manufacturers need to served bank debt because they borrow money to expand the business.

Even items like EDM ticket may jack up the price(due to annual scarcity) , it may risk of losing patrons in bad time; too profitable, it will attract competitor enter the market.


Is the satisficing choice function rationalizable if the ordering isn't observable?

Yes, of course. That is rationalizable just means that there exists some preference ordering that gives rise to the choices. Observability has nothing to do with it.


Theory about supply of money, prices, and output?

The first thing that comes to mind is classical economics' Say's Law

  In a market equilibrium, aggregate production will match aggregate quantity demanded.

In this case, this law on markets could be applied to your money market. Of course, Say's law has its weaknesses and doesn't always uniformly hold, given that markets aren't always in equilibrium. It is probably still a good guiding principle for the money market though. For example, debt and fractional reserve banking seem to indicate, as you note, that money supply doesn't match money demand, but people are consumption smoothing creatures. Their behavior may be matching their demand over multiple periods with supply (their time-path of income) over multiple periods.

A simpler explanation: money supply is also more than just the money base of course. Modern measures of the money supply are meant to capture the reserve rate.


What are examples of natural monopolies that don't have overwhelming economies of scale?

One answer I found are natural monopolies that form on the basis of significant network effects. While economies of scale decrease the cost per unit as you can produce more units, network effects increase the value per unit as you sell more units. These two things together seem like they cover everything, but I haven't read any sources that mention whether there are other situations than economies of scale and network effects. Btw, I'm including "economies of scope" in my mention of economies of scale, since they're so similar.


Recommended ratio of sells of software of basic and advance edition

I like this question because it provides an opportunity to showcase how one can use the simplest of mathematical analysis, full of simplifying assumptions, and still get something useful/applicable.

So let's call "advanced" and "basic" the two versions of the software. Each customer buys one unit of the one or the other. Let's assume/approximate the demand function as linear and let's assume away cross-price and competition effects.

Assuming linearity is acceptable as a local approximation.

Assuming no competition effects is to assume either that the software is seriously differentiated from the competition, or that we examine the case with competition's prices fixed.

The worst assumption is the one assuming away intra-cross-price effects: we do expect that if we raise the price of the Advanced version, potential customers may turn to look to the Basic version (and, possibly, but less strongly, vice versa).

Anyway, with these assumptions we have the demand functions

$$Q_A = a_A - b_AP_A,\;\;\; Q_B = a_B - b_BP_B$$

and the Revenue function

$$TR = P_A\cdot Q_A + P_B\cdot Q_B = a_AP_A - b_AP_A^2 + a_BP_B- b_BP_B^2$$

Interpretation of the coefficients comes later. Maximizing this function with respect to the two prices is straightforward, and one gets

$$P_A^* = \frac {a_A}{2b_A},\;\;\; P_B^* = \frac {a_B}{2b_B}$$

Then optimal revenue share for the Advanced version can be shown to be

$$S_A^* = \frac {a_A^2}{a_A^2 + (b_A/b_B)a_B^2}$$

Now, $a_A$ is the maximum number of customers that we would have if we gave the Advanced version for free, and likewise for $a_B$. These quantities are routinely estimated/assessed by businesses. The beta coefficients are trickier, because they reflect the marginal loss of customers as we increase the price by one currency unit. But again, a business can and does make educated guesses on this matter also (perhaps considering bigger increases than one unit), based on past experience, and knowledge of the market and the potential customers.

So we see that the abstract and simplistic $S_A^*$ can be applied to an actual situation, and of course we can form some intervals by saying "$a_A$ is from... to..." etc. and so get a sense of where the optimal customer mix may lie.

The first step to make this a bit more realistic is to include an asymmetric intra-cross-price effect: namely, if we increase the price of the Advanced version, potential customers may turn to the Basic version, but if we increase the price of the Basic version we lose potential customers to the competition. In this case the demand function of the Advanced version remains as before, but the demand function of the Basic version becomes

$$Q_B = a_B - b_BP_B + c_BP_A$$

It is a bit more complicated but the final expressions are still workable. One of course would here have to assess the magnitude of $c_B$ also.


Solution Method for Infinite-Horizon Maximization Problem

Your first question (regarding constraints on the parameters) can be answered through first and second derivative analysis. In order to satisfy strictly increasing, we need $u'>0$ and to satisfy strictly concave, we need $u''<0$. What does this actually mean?

$$u'(\frac{}{})=\phi p\theta c_t^{p-1}e^{\theta c_t^p}>0$$
Since we know that $c_t^{p-1}e^{\theta c_t^p}>0$ automatically, this reduces to:
$$\phi p\theta>0$$
Next:
$$u''(\frac{}{})=\big[\phi p\theta (p-1) c_t^{p-2}+\phi p\theta c_t^{p-1}\big]e^{\theta c_t^p}<0$$
Because we know that $e^{\theta c_t^p}>0$ and $\phi p\theta c_t^{p-1}>0$, this must mean:
$$\phi p\theta (p-1)c_t^{p-2}<0$$

By virtue of $(1)$ and the fact that $c_t^{p-2}>0$, we know that:
$$p-1<0$$
or
$$p<1$$
Now,  what we can do is think about it critically. If $p=0$ then utility does not change when consumption changes. This is clearly untrue. If $p<0$ then utility decreases when consumption increases which also seems incorrect. Therefore, our condition for $p$ is:
$$0<p<1$$
Now, what does this mean for $\phi \;\text{and}\; \theta$? It means:
$$either\quad\phi , \theta>0\quad or\quad \phi ,\theta<0$$
Now, this looks like a typical exponential utility function, so we know that in these cases, both the coefficient for the exponent and the coefficient in the power will be negative (I will not bother proving this as I am sure it is somewhere in literature). Therefore we have the following three conditions for the parameters:
$$0<p<1$$
$$\theta <0$$
$$\phi <0$$

Now, the next step is to show that we do in fact have constant consumption. Taking our Euler Equation approach, we substitute in the constraint and take two terms in the sum,
$$...+\beta^t\phi e^{\theta (Rw_{t-1}-w_t)^p}+\beta^{t+1}\phi e^{\theta(Rw_t-w_{t+1})^p}+...$$
Differentiating w.r.t $w_t$ and doing some simplifying, we get:
$$(Rw_{t-1}-w_t)^{p-1}e^{\theta(Rw_{t-1}-w_t)^p}=\beta R(Rw_t-w_{t+1})^{p-1}e^{\theta(Rw_t-w_{t+1})^p}$$
Substituting consumption back in, we get:
$$c_t^{p-1}e^{\theta c_t^p}=\beta Rc_{t+1}^{p-1}e^{\theta c_{t+1}^p}$$
Now, since it is specified that $\beta R=1$ in the problem, we can see that the only way this equality holds is if $c_t=c_{t+1}=c^\star$

What does this mean for wealth? Will wealth also be constant? Well, let's conduct a thought experiment and suppose not. Suppose wealth increases each period ($w_{t+1}>w_t\;\forall t$). This implies that as $t\rightarrow \infty$, we will see wealth grow to be infinite as well, which is wasteful and would not satisfy the transversality condition. Now suppose wealth is decreasing ($w_{t+1}<w_t\;\forall t$). We can clearly see that eventually we will not have enough wealth to sustain constant consumption. Therefore we can see that $w_t=w_{t+1}=w^\star$. Now, what are the relationship between these two? Well, going back to our condition for wealth, we can see
$$w^\star=Rw^\star-c^\star$$
Rearranging, we get:
$$c^\star=(R-1)w^\star$$

Because we have steady state wealth, we know that $w^\star=w_0$

Now, knowing that we have steady state consumption, we know our utility function will be the same value for every period (discounted by $\beta$). So our value function simply becomes:

$$V(w)=\frac{\phi e^{((R-1)w_0)^p}}{1-\beta}$$

Therefore, we do not need to use the method of undetermined coefficients to find the value function.


In Singapore, a government agency called the Housing and Development Board (HDB) builds housing (mostly apartment buildings). HDB is not meant to be profitable. It loses money every year and is subsidized from the government budget.
HDB ensures that its buildings have access all the standard infrastructure nearby - mass transit, retail stores, schools, medical and recreational facilities, etc before tenants move in.
HDB also ensures that the tenant population is diverse and no ethnic group is disproportionally overrepresented in any location (i.e. they will reject a prospective tenant in a location belonging to an ethnic group that they already have too much of).
About 80% of the Singapore population lives in HDB apartments.
The tenants have an option to sign a 99-year lease. About 90% of the tenants choose this option. They're not quite "owners" (the government is the owner and the landlord), but close enough with the 99-year lease - they can re-sell the apartment, hopefully at a profit. The remaining ~10% tenants rent short-term, usually because they can't afford the 99-year lease.
The government has a Central Provident Fund (CPF). CPF withhold 37% of everyone's income into 3 accounts: retirement (like U.S. Social Security), medical care, and housing. for most people, the money accumulated in their CPF account is enough to cover downpayment and most of mortgage payments. Singapore citizens (but not permanent residents) receive certain government-funded subsidies when buying the 99-year leases.
This Wikipedia page has more details and history.


Examples of the use of Vector Autoregressive Models

There is this study from Banque de France using a Bayesian VAR to analyse and forecast macroeconomic indicators, authors are Gergely Ganics and Florens Odendahl.


In general, it is not:

  Despite the focus on ‘jobism’, jobs are a cost of production and
  consumption, not a benefit, and should be minimized

Mark J. Perry @ AEI

  Public discourse tends to be carried out in terms of jobs, as if a great objective was to create jobs. Now that’s not our objective at
  all. There’s no problem about creating jobs. We can create any number
  of jobs in having people dig holes and fill them up again. Do we want
  jobs like that? No. Jobs are a price and we have to work to live.
  Whereas if you listen to the terminology you would think that we live
  to work. Now some of us do. There are workaholics just like there are
  alcoholics and some of us do live to work. But in the main, what we
  want is not jobs, but productive jobs. We want jobs that will be able
  to produce the goods and services that we consume at a minimum
  expenditure of effort. In a way, the appropriate national objective is
  to have the fewest possible jobs. That is to say, the least amount of
  work for the greatest amount of products.

Milton Friedman in a 1980 lecture

  Under most circumstances, wages paid to people employed on a project
  should not be counted as benefits, although this practice is
  disturbingly widespread. As mentioned above, the opportunity cost of
  labor used on a project is the output that that labor would have
  generated elsewhere, either in paid work or in household production.
  Including wages paid to labor on a project as a benefit implies that
  that labor would have produced nothing otherwise and that the agency
  making the wage payments does not have standing.

Zerbe and Bellas (page 117-118):

I say in general, because there are some evidence that "jobs are good for you", and if increasing jobs diminishes the number of long term unemployed, that might be considered a benefit, even though it diminishes the quantity of leisure and home production they consume.


What Math is used in Development Economics?

This does depend on whether you're looking to do applied econometric work, in which case statistics is definitely the way to go or something theoretical, in which case a good understanding in the maths you listed will go a long way. Of course it's always better if you understand both - since you're doing a master's in math, you should already be quite fluent with the math required, so I'd go with econometrics.

There's a growing amount of literature available, but if you have time to spare, I'd give the topics in Nafziger's "Economic Development" a glance: http://tabesh.edu.af/Books/economic/Economic%20Development.pdf


Is it possible to have constant marginal cost and decreasing average cost simultaneously?

Yes, if there are non-zero fixed costs, and constant marginal cost, then average cost decreases strictly monotonically with quantity, asymptotic to the marginal cost.


UK Government Coronavirus Job Retention Scheme Funding

It is not correct to say that the Coronavirus Job Retention Scheme is essentially providing loans.

A little background.  The UK government currently has two major schemes to support businesses and through them their employees: the Coronavirus Business Interruption Loan Scheme for small / medium businesses together with a similar scheme for large businesses, and the Coronavirus Job Retention Scheme.

As their names suggest, the former is a loan scheme, the latter isn't.  As you imply, the cost to the government of the Coronavirus Job Retention Scheme will have to be funded somehow, probably initially by extra government borrowing, but perhaps eventually by increased taxation. Depending on the form of taxation, some of the burden of that extra taxation might fall on the businesses and/or employees who benefited from the Scheme.

However, there is no reason to expect that there would be any close match, at the level of individual businesses, between their receipts from the Scheme and any extra taxation they might have to bear.  Corporation Tax, for example, is subject to complex rules which bear little relation to eligibility for receipts under the Scheme.  It would not be correct therefore to regard the Scheme as offering loans with an expectation of repayment.


World Trade Organization (WTO) rules prohibit export subsidies and China joined the WTO almost 15 years ago (in December 2001). As a condition of membership, China had to eliminate export subsidies in the late 90s. So, strictly speaking, China can't subsidize its exports.

However, Dani Rodrik considers that China subsidizes them indirectly through the exchange rate by maintaining an undervalued currency (here and here). Rodrik adds two interesting thoughts. First, the undervalued currency boosts China's growth and this, in turn, is good for the world's recovery and the alleviation of poverty. Second, China could maintain its growth without trade imbalances if it could introduce industrial subsidies to offset a rising yuan. So, it be would better to subsidise tradables directly than to subsidise them indirectly through the exchange rate.


In order to get coefficient for the materials in the Levinsohn-Petrin model you need to set the gross option true.
 mod1LP = estprod::levinsohn_petrin(data = DataA, va ~ l | k | m, id = "ID", time = "Year", bootstrap = TRUE, gross = TRUE)
     summary(mod1LP)

    Call
estprod::levinsohn_petrin(data = DataA, formula = va ~ l | k |
    m, gross = TRUE, id = "ID", time = "Year", bootstrap = TRUE)

Coefficients
   Estimate Std. Error z value  Pr(>|z|)
l 0.1033754  0.0067384  15.341 < 2.2e-16 ***
k 0.5859219  0.0048667 120.394 < 2.2e-16 ***
m 0.4808831  0.0143580  33.492 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Bootstraped standard errors.

The other two functions unfortunately don't collect the coefficients for proxy variables.
An alternative is to find some different proxy and use $m$ as a control (I do not necessarily endorse such option from scientific perspective but if you are hell bent on having coefficient for $m$ its an option). In your dataset you don't have any additional variables beside $l$, $k$ and $m$ (aside from COGS that is always N/A) so I will just create a new variable to show it would work:
 DataA$m2<-(DataA$m)^2

mod1OP = estprod::olley_pakes(data = DataA, va ~  l |  k | m2 |  m, id = "ID", time = "Year", bootstrap = TRUE, gross = TRUE)
 summary(mod1OP)

    Call
estprod::olley_pakes(data = DataA, formula = va ~ l | k | m2 |
    m, id = "ID", time = "Year", bootstrap = TRUE,
    gross = TRUE)

Coefficients
    Estimate Std. Error z value  Pr(>|z|)
l  0.1057613  0.0040007 26.4360 < 2.2e-16 ***
k  0.4643984  0.0308697 15.0438 < 2.2e-16 ***
m -0.0458770  0.0048684 -9.4234 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Bootstraped standard errors.

   mod1W = estprod::wooldridge(data = DataA, va ~ l | k | m2 | m, id = "ID", time = "Year", bootstrap = FALSE, gross = TRUE)
     summary(mod1W)

Call
estprod::wooldridge(data = DataA, formula = va ~ l | k | m2 |
    m, gross = TRUE, id = "ID", time = "Year", bootstrap = FALSE)

Coefficients
   Estimate Std. Error t value  Pr(>|t|)
l 0.1210159  0.0075386  16.053 < 2.2e-16 ***
k 0.4670722  0.0135809  34.392 < 2.2e-16 ***
m 0.2595440  0.0158134  16.413 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Again such course of action does not seem to make sense to me but it gives you the desired output.


The mathematical derivation is straightforward: set up a Lagrangian for the utility maximization the consumer solves subject to a monetary constraint, then divide its partial derivatives with respects to the quantities one with another, and there you have it: MRS equals the price ratio.
The intuition behind it is that, given fixed prices (that is, a price taking situation), the ratio of the marginal utilities must also remain constant: suppose by contradiction this isn't the case, and that some good provides relatively high marginal utility. Then the consumer would profit from deviating to consuming a basket in which the quantity of said good is higher, since the increase in utility would be higher than the costs incurred.


Euler equation through tangency conditions

In answer to your first question (can you get the Euler equation without a Lagrangean?), the answer is 'yes'. At least, there are less formal ways of deriving it. In general, assuming that a consumer chooses to consume strictly positive quantities of two perfectly divisible goods $x$ and $y$, then it must be that:

$$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$

(If this were not true, then the consumer could increase their utility by shifting spending a little more on one good and a little less on the other.)

The so-called 'Euler equation' is nothing more than an application of this result, viewing the 'goods' and $C_t$ and $C_{t+1}$. If you delay your consumption by one period, you put your cash in the bank for one period, earning real interest at rate $r_{t+1}$. Hence, if we normalise $P_t = 1$, then effectively $P_{t+1}=1/(1+r_{t+1})$. Consumption tomorrow is cheaper than consumption today (assuming $r_{t+1} > 0$) since delaying your consumption allows you to earn some interest.

We assume that utility is constant, time-separable and exponentially discounted (as in your set-up). Thus, we need to remember to put a $\beta$ before $MU_{t+1}$. We now write $u'(c_t)$ for $MU_x$, $\beta u'(c_{t+1})$ for $MU_y$ and $(1+r_{t+1})$ for the price ratio.

Finally, observe that if firms maximise profits taking the cost of capital $r_{t+1}$ as given with total depreciation (as you assume), they hire capital until $f'(k_{t+1})=1+r_{t+1}$. Plugging in yields the Euler equation.

In answer to your second question (how can you derive it with a Lagrangean?), I would recommend that you try the following steps:

First of all, you should really have a summation before your Lagrangean (summing over an infinite number of periods). This is a dynamic problem.
Differentiate the objective function with respect to $c_t$.
Roll forward by one period and combine the equations. You should get:

$u'(c_t)/\beta u'(c_{t+1}) = \lambda _t/\lambda _{t+1}$

Then differentiate the objective function with respect to $k_{t+1}$. Don't forget the $f(k_t)$ term! This may look irrelevant, but the Lagrangean should be an infinite sum so it shows up when you 'roll forward' by one period. This should give you:

$\lambda _t = \lambda _{t+1}f'(k_{t+1})$

Plug that in and you have it! The equation on which so much pointless macroeconomics built.