The most useful way of thinking of a Computational Economist is one that will be able to estimate economic models in a quantitatively meaningful way. This makes them very useful for central banks which have the need to study the macroeconomy quantitatively and also have the time and resources to invest in a computational economist...

Other ways to look at this question:

The conferences by the society for computation economics are all populated by academic economists or by central bank economists. enter link description here

The job-board sites, mostly ask for data scientists when you type in computational economics. enter link description here

Commentators at the job econ rumors site claim two things: that computational economics is forever going to be the science of the future, but never the presently hot science.enter link description here


consequences of an inheritance tax on family owned businesses

It’s true that inheritance tax could pose a problem for businesses. The reason for that is that even though you are correct to say that the successor to businesses owner (I would avoid word patriarch - many business owners are women) has option of selling equity stake or take a loan this can be sub-optimal or difficult.

For example, selling equity in privately owned firm is not easy as investor can’t ‘see into the firm.’ So it might be difficult to properly value business - the equity share could be sold at very unfavorable price to the owner and going public forces firm to follow a host of special regulations that might be difficult for some firms to cope with - this is one of the reasons why not all firms are public - in principle if there would be no special rules even many family businesses could benefit from the access to capital markets. This being said in that case it’s also difficult to value the firm for tax purposes, so I am not sure how the system would work - in the above I implicitly assumed tax authorities can know the true value of business.

Taking loan could be again difficult. Usually when you are applying for a business loan you have to give some proposal of how you will effectively invest money so the bank can see you will be able to repay it with the interest. If you come to bank applying for a loan to pay taxes because you don’t have enough liquidity to do it yourself, you will most likely be turned away. So there are borrowing constrains. Even if you can get loan it could make the leverage of the firm sub-optimal and hurt the firms efficiency or ability to absorb shocks.

However, this being said you could always make the tax in a way that illiquid assets would be exempt or that the tax would not be payable in lump sum but in installments, or exclude business altogether etc. - although making tax system complex has its downsides as well. Also, all taxes, except for Pigovian taxes, are disruptive to the economy so it’s more of an question if the tax is relatively more efficient (causes less disruptions to the economy) than other taxes and then as you correctly pointed out it’s a question of philosophy and value judgements if the disruptions are worth while to get the revenue that can be used for some public purposes.


Why isn't the Law of Demand true for Marshallian demand?

Marshallian demand is also called uncompensated demand. That is a price changes and you see what changes in what the household buys. This consists of two effects.

The good itself is more expensive
The household is poorer because they can no longer afford their old consumption bundle.

The effect of (1) is always to make you consume less of the good. The effect of (2) is what is ambiguous. An inferior good is one where as you get richer (poorer) you consume less (more) of it. For an inferior good forces (1) and (2) oppose. You consume more of the good because you are poorer and less because it is more expensive. A good (check out Giffen goods) could be so powerfully inferior that the effect of (2) is bigger than the effect of (1). No one seems to be able to find evidence that they exist, but they certainly are possible. Often potatoes in Ireland are used as an example, though they were not.


In some statistical software (not all), in a correlogram the "zero lag" is also depicted - but the zero lag is just the correlation of a random variable with its own self, so it is by construction equal to unity. This appears to be happening in your first graph where the autocorrelation function is calculated.

The second graph is the partial autocorrelation function which calculates the correlation coefficients after the effect of all "previous" lags (i.e. of lower order) has been removed (by linear projection estimation).

First impression? The process is white noise. What lags appear to exceed the  "statistical significance threshold", they nevertheless indicate very small values, never greater  than $0.05$. This is truly economically negligible correlation even if it exists (irrespective of whether it is "statistically significant"), and most likely is due to "finite-sample variation", rather than indication of any actual relation.

And apart from that, do the lags appearing "significant" make sense?
It always matters what the process you are examining is, and what is its "typical properties" based on past scientific work (but don't take this to mean that "past typical properties" always carry over to all the realizations of the process).

In any case, it would be instructive to "follow blindly" the Correlograms and see what you get, by trying various combinations based on the lags that appear statistically significant.


Where do I find the US GDP growth on FRED?

With the link you posted you get quarterly data, so to get the yearly growth rate you need to change the frequency to annual. There are different aggregation methods but if it's not specified in the question you were asked I don't see how one should be more right than another.


During an economic expansion, how do the ratios of Return on Equity and Return on Asset are affected?

If all else were equal, then using more equity financing would reduce return on equity. (This is assuming a normal situation where the return on assets is greater than the cost of interest. If the interest cost is greater than the return on assets, using debt for financing reduces profits. In this case, there are less profits, and a smaller equity base. The net effect on the return on equity depends upon the exact numbers.)

However, this tells us nothing about observed data during an expansion, it just tells us about the difference between two hypothetical scenarios (the scenarios being defined by the level of equity financing).

Aggregate profits are determined by many factors. The effects of those factors change every cycle, and may be unrelated to businesses' financing decisions.

For example, if the wage cost per unit of output (unit labour cost) is rising faster than selling prices, business profit margins will tend to fall (wages will represent a greater share of total income). Conversely, the profit share will generally rise if unit labour costs increase slower than selling prices. There is no law of economics that says which outcome will occur, and so we cannot be sure what is happening to profit margins.

Therefore, we have no way of answering the question of what happens to ROE during expansion, without pinning down what else is happening at the same time.


What does the distribution of the savings rate in America look like?

How much do people save?

Report on the Economic Well-Being of U.S. Households in 2014, Board of Governors of the Federal Reserve System, p. 22


Does Preference have a Hierarchy? A Silly Question

*A little disclaimer, this is more  me ruminating than providing a rigorously proofed response.

I have never thought of preferences being related that way before, especially through the use of hierarchies.

$\textbf{Main response}$

The way I think about the scenario above is through preference weightings, were the values of the goods that can be chosen are weighted according to the good-specific preferences of the agent. This allows the agent to still be a rational econ and maximise their utility, but make sub-optimal financial choices. In a simple scenario similar to your example,  there would be two goods, $X$ and $Y$. $Y$ provides greater returns and  from what I understand above, the agents utility would take an additive form (to ensure that the goods are substitutes)  i.e.
$$
U_1 = 2Q_Y+Q_X
$$
Thus, a utility maximising agent would choose to spend his budget on $Y$ as it provides greater returns.

$\textbf{However}$, we can incorporate the agents preferences for

  a new build property, which is closer to home.

which we can call $\beta_{nbp}$. Thus, as $X$ has characteristics which match the above, the agents utility would now take the form;

$$
\hat {U}_1 = 2Q_Y+\beta_{nbp}\cdot Q_X
$$

Thus, the agents utility now incorporates the financial superiority of good $Y$, as it outperforms good $X$ by a factor of 2, and it incorporates the agents non-financial preferences through $\beta_{nbp}$. For a utility maximising agent to choose $X$ over $Y$, $\beta_{nbp}$ must be greater than 2, which implies that the agents non-monetary preferences are more important than his financial preferences. Thus, by adding preference weighting,it is one way to show that a rational utility maximising agent will choose a financially worse product over another product, as there are other non-financial elements which the agent desires more. However, if the agent had been a profit maximiser the choice of $X$ over $Y$ would not be rational, and in the problem above we would not include $\beta_{nbp}$ in the analysis.

$\textbf{Real life example}$

A real-life example I suppose which is slightly related, would be choosing between an ultra cheap  but crappy mobile(cell) phone and an expensive but good quality phone. On a strictly monetary basis, the ultra cheap one is preferable. However, as soon as you start incorporating quality, the utility-values of the phones change. The change isn't financially, but based on the utility (eg. combined preferences for monetary and non-monetary benefits) derived by whoever is buying the phone, and hence they may choose the better quality phone at the expense of spending more money.

$\textbf{meta-discussion}$

On a more meta-level, I think your question has some really interesting points. It appears to be that there is a misinterpretation between utility and profit maximisation. Considering this definition of utility;

  utility is the want satisfying power of any commodity or capacity of a commodity to give satisfaction.$^1$

It can be seen that utility is not solely reliant on money, but rather is only partially reliant on financial benefits. However, as money is such a large part of our society and can be (relatively)easily measured and compared,  it is used a lot to quantify the non-monetary preferences (through monetary trade-offs, which is a really interesting field of research by itself!). Thus, this quantification seems to lead to a bit of confusion between profit maximisation (which is solely focused on money) and utility maximisation (which is focused on utils and want satisfaction), which I wouldn't be surprised is from the way economics is taught and very rarely explicitly mentions this difference. Hence, economics is not necessarily not just solely focused on money, but it does use money to help compare the costs and benefits of tangibles and intangibles.  This is probably, a reason why economics is rather practical  and often relied upon during policy development, as it provides easily comparable estimates to values which would otherwise be illusive.

$^1$ https://en.wikibooks.org/wiki/Principles_of_Economics/Utility


Choosing Data for CES Production Function

The Penn Wolrd Tables have the data you need.


Are there two deadweight welfare losses associated with a Pigovian tax?

The Pigovian tax is responsible for neither of the deadweight losses in your diagram.

The Pigovian tax has partially, but not wholly, corrected a deadweight loss that was caused by the negative externality.

There is a deadweight loss associated with Pigovian taxes: that is the administrative cost of collecting the tax. This is not pictured in your diagram.


I believe this is the basic case discussed in the Laffont-Martimort textbook "The Theory of Incentives". The choice of the agent is either 0 or 1. You can find a complete description in "The Moral Hazard" chapter, section 4.2. The results are listed later, and while I am not sure, I do not think there should be qualitative differences.


How can all the bitcoin in the world be worth more than McDonald's Corporation?

Bitcoin doesn't have a value. It doesn't create wealth - it destroys it. It therefore has no yield.

It does, however, have a price. It's designed to have tightly-constrained supply, so that those people who got bitcoins early, can get rich by luring more people into buying bitcoin. And demand is just as large as the number of gullible people who can be fooled into buying it. That combination of extremely inelastic supply, and very malleable demand, creates very volatile pricing.

This extreme volatility makes it very impracticable as a currency, as does its extremely inflexible supply.


What is the difference between aggregate supply and GDP?

Aggregate supply is a relationship of price level and output. It is a function, or a curve, or a table. It is not a single value. If we know a particular price level, then we can determine the level of output that would correspond with that. The GDP for 2006 is determined by plugging in the price level of 2006 to the AS curve for 2006, and seeing what output is produced at that price level.


A choice structure can be whatever it wants to be. A choice structure that is rationalizable by a preference relation must (among other things) satisfy Sen's $\alpha$.

So if you want to show that a rationalizable choice structure satisfies $\alpha$ then you can show that any that satisfies WARP will also satisfy $
\alpha$.

I recommend Kreps or the UPitt notes for more info on the relationship between choice structures and preference.

http://www.pitt.edu/~luca/ECON2100/lecture_02.pdf


Predicting $y$ when response variable is $\ln(y)$

You do not find the same answer because of what I suspect to be a typographical error, which would thus be the main reason of your problem: $x_{03}$ would be set to $80$, not $8$. Another possibility, if you keep $x_{03}=8$, is an error in the second estimated coefficient, say, $\hat{\beta}_2 = -0.1$ instead of $-0.01$.

Anyway, one of these modifications solves everything and yields the same result as the solution to this exercise.

Considering this change, with $t_{\alpha/2}=1.96476138969835$, one gets

Method 1

$\text{CI}(E[y_0|x_0])=[e^{6.43618291164626},e^{6.49755798189177}]=[624.020307335178,663.519326788772]$
(the given solution to this exercise)

or Method 2
(as stated In Wooldridge's Intro to Econometrics, in page 212) if we're sure the error terms are normal (and one is extremely lucky)

$$\text{CI}(E[y_0|x_0])=\left[e^{s^2/2}624.0203,e^{s^2/2}663.5193 \right] = \left[ 624.0960,663.6002 \right] $$

however

the method 2 is very unlikely to be correct, since as you mention in your question [...] the (underestimation) correction should not be used to the CI but only for the point estimation.

Why ? I would say because of the dependency betweeen the two terms, knowing the expectations of $e^{s^2/2}$ on the one hand and $\hat{y_0}$ on the other hand does not mean one knows the one of $e^{\frac{s^2}{2} + \hat{\ln(y_0)}}$.


My argument is that he is just looking to find a justification for a policy choice he made, and so there is no reason to believe that his arguments are completely coherent.

We need to understand why he is making such arguments in the first place. If interest rates are purely the result of expectations (term premium equal to zero), the purchases themselves have no direct effect on yields, only as the result of signalling what the Fed will do (the first channel). However, the Fed could achieve the exact same effect by just saying that the economy stinks, and they will not raise rates for a really long time.

In order to find a justification for the decision to purchase bonds (instead of just talking), he needs to argue that the change in supply affects bond yields. The problem with that argument is that the front end of the yield curve does a pretty good job of tracking expectations, regardless of the state of supply and demand in the bond market. (One can look at how market participants use front end pricing as an estimate of what the central bank will do in the next few rate setting meetings across all the major markets as an empirical validation of that assertion.)

Therefore, we need some reason to believe that the short maturity bonds are somehow different from long maturity bonds, i.e., there is some form of segmentation. This way, one can hope that supply and demand affects long-term bond risk premia more than short-term premia (which would sound plausible to most bond market analysts). What matters is that the term structure of the term premia is changing due to the supply effect. One can come up with different verbal explanations for why term premia move, they will end up being indistinguishable in practice. That said, we need to ignore the possibility that other factor affect term premia, such as returns volatility (which would be suggested by portfolio theory).

The obvious issue with this argument is that Fed purchases are not the only factor affecting supply and demand in the bond market. Why are those other factors being ignored?


How do efficient credit markets allocate capital properly?

Under the usual assumptions of decreasing MPK and complete markets in equilibrium $MPK_{Firm_1} = MPK_{Firm_2} = r$. If firm i's MPK is larger than r and firm i does not face constraints with respect to scale, if will simply borrow more capital as it can make a net return of $MPK_{Firm_i} - r$ on every unit for which  $MPK_{Firm_i} > r$. If $MPK_{Firm_i} < r$ firm i will divest until $MPK_{Firm_i} = r$.

There are many reasons for which this nice result can break down in reality (Deviations from the nice assumptions above). For example agency costs (Bernanke, Gertler, and Gilchrist (1996)), where an entrepreneur can only borrow up to the present value his net worth at end of the loan contract. If an entrepreneur does not have enough wealth he cannot borrow enough capital to achieve $MPK_{Firm_i} = r$. Hence when talking about this topic you should make your assumptions known in advance.

With regard to you question about Africa, from the point of view of the setting described above too few capital is flowing to developing countries. This fact has been known for a while and is referred to as the "Lucas paradox" or the "uphill flow of capital" a recent summary on the topic can be found here:

enter link description herehttps://voxeu.org/article/revisiting-paradox-capital


What possible reasons could make arbitrage impossible between some markets?

The seminal paper in this area with over 3,000 citations is by Shleifer and Vishney:

  Textbook arbitrage in financial markets requires no capital and
  entails no risk. In reality, almost all arbitrage requires capital,
  and is typically risky. Moreover, professional arbitrage is conducted
  by a relatively small number of highly specialized investors using
  other people's capital. Such professional arbitrage has a number of
  interesting implications for security pricing, including the
  possibility that arbitrage becomes ineffective in extreme
  circumstances, when prices diverge far from fundamental values. The
  model also suggests where anomalies in financial markets are likely to
  appear, and why arbitrage fails to eliminate them.

The Limits of Arbitrage (Andrei Shleifer and Robert W. Vishny (1997))

There are many versions online, one of them is likely ungated.


The Price and Demand Index in Homothetic Kimball Utility

The problem is the following:
$$
\begin{align*}
\max_{Q, q_\omega} Q \text{ s.t. } &\int Y\left(q_\omega/Q\right) d\omega = 1,\\
&\int p_\omega q_\omega d\omega = y
\end{align*}
$$
The first order conditions give:
$$
1 - \gamma \int Y'\left(\frac{q_\omega}{Q}\right) \frac{q_{\omega}}{Q^2} d \omega = 0,\\
 \gamma Y'\left(\frac{q_\omega}{Q}\right)\frac{1}{Q} - \lambda p_\omega = 0
$$
Then:
$$
Y' \left(\frac{q_\omega}{Q}\right) = \lambda \frac{p_\omega}{\gamma/Q}
$$
So we have that $\delta = \gamma/Q$.

What I don't understand is that the papers using this preference often set price index $P = \delta/\lambda$ and denote $\delta$ as a demand index. Whereas in CES case, the price index is $1/\lambda$.

I think the answer is already given by Baquee and Farhi themselves.
If we define $P = \gamma/(Q \lambda) = \delta/\lambda$ then we can write:
$$
\frac{q_\omega}{Q} = Y'^{-1}\left(\frac{p_\omega}{P}\right)
$$
Using $Q$ as a quantity index, we have on the left hand side the "normalized" for good $\omega$ and on the right we have a function of the "normalized" price. Under this definition, you can therefore interpret $Y'^{-1}$ as a demand function.
On the other hand, if you would use $P = 1/\lambda$  you obtain from inverting $Y'$ in the first order condition that:
$$
\frac{q_\omega}{Q} = Y'^{-1}\left(\frac{Q}{\gamma} \frac{p_\omega}{P}\right).
$$
The right hand side now also involves the quantity index $Q$, so this does not take the "form" of a demand function.
As a conclusion, I think the main idea of defining $P$ in this way is to obtain a nice expression and interpretation for the normalized demands as a function of some normalized prices.


Constant Returns to Scale and Positive Marginal Products

Given $m(u,v)$ is CRTS we have the property that $m(u,v)=\frac{\delta{m}}{\delta{v}}v+\frac{\delta{m}}{\delta{u}}u$. Divide by $v$ to get in terms of $\theta$: $m(\theta)=\frac{\delta{m}}{\delta{v}}+\frac{\delta{m}}{\delta{u}}\theta$.

Thus, it clearly follows that:

$\frac{df}{d\theta}=\mu\frac{dm(\theta)}{d\theta}=\mu\frac{\delta{m}}{\delta{u}}>0$ given that we have positive marginal products


Do Hong Kong highschool teachers earn more than Hong Kong quants?

As you said it doesn’t make sense to compare entry level finance jobs with non-entry level normal jobs. Salary increase in finance is usually quite large - also many have bonus schemes, which many statistics leave out of the “average salary”. Over time definitely the Quant will make more than the school teacher.


Calling 118 118 (UK) - A market failure?

When there was only one telephone service supplier, BT directory enquiries used to be free. If you did not know the number of who you wanted to call, you could not call. Once you got the number from the phone book or directory enquiries, you could make the call and would be charged for it - instant payback for the phone company.

After the telephone market was opened to competition, BT started charging for directory enquiries eventually reaching 40p per call.

After de-regulation of directory enquiries in 2003, hundreds of suppliers have sprung up. The cheapest is currently 40p per minute plus your phone provider's Access Charge. The most expensive is £15.98 per call, plus £7.99 per minute after 60 seconds, plus your phone provider's Access Charge.

Prices for a number of 118 services have risen substantially over the years.

118 118 prices have risen sharply in recent years...

1 July 2015 - £5.74 per call, plus £2.99 per minute after 60 seconds, plus your phone provider's Access Charge.

1 March 2016 - £6.98 per call, plus £3.49 per minute after 60 seconds, plus your phone provider's Access Charge.

1 May 2017 - £8.98 per call, plus £4.49 per minute after 60 seconds, plus your phone provider's Access Charge.

It is no surprise that the most well-known services such as 118118 and 118500 are the most heavily advertised.

It should come as no surprise that the most heavily advertised services are among the most expensive. All that advertising has to be paid for somehow.

The cheapest services have no budget for advertising. In order to advertise, they would have to increase their prices.

What is needed is a third-party to rate and review all of the suppliers and their services and for this information to be freely available and easily accessible.


Proof that two utility functions $u(x_1,x_2) \text{ and } v(x_1,x_2)$ with the same MRS represent the same preference

I think that you need to also assume that the utilities are non-decreasing in the goods. That is, if $x'_1>x_1$, then $u(x'_1,x_2)>u(x_1,x_2)$.
The utility functions having the "same preferences" means
$\forall (x_1,x_2), (x'_1,x'_2): u(x_1,x_2) > u (x'_1,x'_2) \text{ iff } v(x_1,x_2) > v(x'_1,x'_2) $
So let's assume $u(x_1,x_2) > u (x'_1,x'_2) $. Take a curve going from $(x_1,x_2)$ to a point $(x'_1, x''_2)$ by changing $x_1$ to $x'_1$ and having the second coordinate change according the MRS so that this curve has constant value of $u$ [1]. Then since $u$ is constant along this curve, $u(x'_1,x''_2) = u(x_1,x_2)$. Since $u(x_1,x_2) > u(x'_1,x'_2)$, $u(x'_1,x''_2) > u(x'_1,x'_2)$, and from the non-decreasing utility property, we have $x''_2 >x'_2$.
Since this curve was defined by the MRS of $u$, and the MRS of $u$ is equal to that of $v$, we also have that $v(x'_1,x''_2) = v(x_1,x_2)$, and so  $v(x_1,x_2) > v(x'_1,x'_2)$
[1] This can be done, for example, by taking $C = (x_1+ t, x_2+\int_0^tMRS_{x_1x_2}dt)$. Then taking $t=x'_1-x_1$, we get the point $\left(x'_1, \int_0^{x'_1-x_1}MRS_{x_1x_2}dt \right)$


Showing that production technology exhibits decreasing returns to Scale

They are two different concepts I think.

Diminishing product means that holding other things fixed, one unit of extra input($K$ or $L$ here) yields less and less additional output, which you have known.

Meanwhile DRS is saying that if you multiply both $K$ and $L$ by some scalar $ t > 1$, the corresponding output would be less than $t$ times the original output.


Which income dollar is taxed most heavily, including implicit taxation, in USA?

Unfortunately this question is exceptionally complex and depends on factors such has how many children the person has, which state they live in, and which services/opportunities they are taking advantage of.  I read a paper a while back (having a hard time finding it) in which the authors attempted to compute the effective tax rate for all incomes after the various deductions, benefits, alternative minimum tax, and other considerations.  The rate was very nonlinear with many peaks and valleys.  It was not particularly increasing with income.  This is further complicated by state taxes and the fact that states administer many welfare benefits using different criteria from each other.

Most taxes scale in and benefits scale out as income increases.  I believe the most heavily taxed dollar would be the one that takes you from qualifying for one of the various welfare benefits to not.  Many welfare programs have hard cutoffs.  This will vary from state to state, though, and may depend on other factors like what your savings are and how your social worker is feeling that day.

When I was in graduate school I had a stipend of about \$22K and my friend got slightly less.  He qualified for food stamps, WIC, and medicaid, and HUD paid his rent and utilities.  He received about \$30K a year in government benefits (not counting the value of medicaid) on top of his stipend.  I was just over the income line got none of these (well, my wife got WIC when she was pregnant).  So the small difference in our income cost me many thousands of dollars--he lived much, much better than I did.  It was more than an income difference, though.  HUD paid his rent because he had found a cheaper apartment than I was able to and they had a rent-per-square-foot cutoff.  Later I tried restructuring my life in order to qualify for some of these, but the state had changed some of its rules (you had to have less than $2,000 in your bank accounts to qualify).

This is also complicated by the fact that the rules are not applied consistently.  For example, some of the social workers deciding which benefits he got did not count his stipend as income, while all the ones I talked to counted mine.

Your tax rate and marginal tax rate depend on many elements of who you are, where you live, how you file taxes, what consumption choices you make, and luck.


Calculate supply function based on production or cost function

Q1
This is not a straightforward problem. Two things to note:
First, the production function is linear in the inputs. This means that their marginal products are constant, and so are their marginal revenue products (presumably the firm is treated as a price taker in the output market). This implies that if price is sufficiently low, then production will be zero.
Second: the production inputs are not complementary, production can happen using both or either one of them.
The profit function is
$$pQ - C = p(2x_1+ x_2) - 4x_1 - 3x_2 = (2p-4)x_1 + (p-3)x_2.$$
From this we see that
$$p < 2 \implies Q = 0$$
because with any choice of inputs the firms will incur losses.
At exactly $p=2$, the firm can have zero-profits if it uses only input $x_1$ for any output level, so here
$$p=2 \implies Q = {\rm indeterminate.}$$
"Indeterminate" in the sense that it could be zero, or any positive quantity, or even "infinity" (see below) - the firm is indifferent between these choices.
Suppose now that $p>2$.
If price is $2 < p \leq 3$, the firm has an incentive to use only input $x_1$, because using also input $x_2$ will incur some loss, while it can use only input $x_1$ and have profit for every output unit it produces. And then, the more it produces using just $x_1$ the more profit it has. So here
$$2 < p \leq 3 \implies Q = \infty$$
Certainly, one could wonder whether this is meaningful, since
a) there is nothing infinite in this world and
b) as the firm produces more and more output wouldn't that eventually push the price down? How much of the good do consumers are willing to buy after all?
These are valid remarks, but the solution that the supply function will be "infinite" indicates the profit-maximizing tendency of the firm, not what will actually happen in the real world.
If $p>3$ the conclusion as regards the supply function does not change. In theory, the firm can here use a combination of both inputs, but because the marginal profit of $x_1$, $2p-4$ is always greater than the marginal profit of $x_2$, $p-3$, for these range of prices, the firm will here too want to use only input $x_1$ for its "infinite" supply.


What's a "relative pair trade" for currencies?

It means you can make money trading a pair of currencies if both lose value compared to a standard. Likewise you can make money trading a pair of currencies when both increase in value compared to a standard. You mostly care about the change of a currency's value relative to another.
If you think USD will decrease in value 10% versus gold and you think EUR will decrease 15% versus gold, you sell EUR to buy USD and wait to see if you were right. You don't have to compare currencies to gold. There are other standards for what a currency is worth. It doesn't matter what the standard is because your profit depends upon the value of a currency compared to another.


What is the definition of distributionally neutral tax?

It depends on what the writer intended.  Here, just over halfway down section 19.10, is a part with the orange title  Progressive and regressive redistribution with a paragraph saying

When the direct effect of a tax or transfer policy (compared to what would happen in the absence of the policy) is a reduction in inequality, it is called progressive. We have just seen that expenditures are more progressive than taxes. If a policy’s direct effect is a rise in inequality it is called regressive. Policies that are neither progressive nor regressive are called distributionally neutral.

Clicking on the blue bolded words gives the definitions

progressive (policy) An expenditure or transfer that increases the incomes of poorer households by more than richer households, in percentage terms.
distributionally neutral A policy that is neither progressive or regressive so that it does not alter the distribution of income.

The definition of regressive seems to be missing but presumably is something which increases the incomes of poorer households by less than richer households, in percentage terms.  Similarly, presumably all three can also be applied when the policy reduces incomes, with progressive being where the richer households are hit proportionately hardest, regressive being where the poorer households are hit proportionately hardest, and distributionally neutral where all households are hit proportionately the same.  There will be other types, such as those which hit rich and poor proportionately the same but middle incomes differently.
The key to answering the exercise comes from considering the proportionate effects on incomes of a poll tax.
Other people have used distributionally neutral in a different sense: for example Michael Linden discussing a flat percentage tax with a single deductible used it to mean a result with the same distributional effect as the income and payroll taxes it might replace; he meant that the change (rather than the policy itself) should be broadly neutral on the current distributional effect of these taxes


because bitcoins do not generate earnings, then (it is a ponzi scheme)?

What is the reasoning behind the notion that because bitcoin has no intrinsic
  value therefore it's not suitable as currency? Or, to put another way, if no one
  "held" US dollars, but simply used them to spend or invest, would that
  invalidate it as a currency for buying chickens?

The author addresses the difference between bitcoin and fiat currency (such as greenbacks):

  Because of their power to tax, governments can make money by fiat, simply by
  declaring their willingness to accept that money in repayment of tax debts.

I'll be honest and admit that I hadn't heard that argument before and off the cuff, it makes sense.  Because the government can impose taxes and require they be paid in dollars, the demand for dollars will exist at least as long as the government has the power to tax.

I find this part of the argument to be pretty weak:

  As with any kind of asset used as currency, from gold to tobacco to U.S.
  dollars, Bitcoin is valuable as long as people are willing to accept it. But
  in all of these examples, willingness to hold the asset depends on the fact
  that it has value independent of that willingness. Tobacco can be smoked or
  chewed, gold can be used to fill teeth or make jewellery...

The problem here is that the demand of gold for jewelry and other industrial purposes is minuscule and can't come anywhere close to justifying the price of gold (even in down markets.)  If we believe this argument, the price of gold is going to go down so far that it might as well be zero.

The argument that I find convincing for why gold has been used as money for so long is that it's physical properties make it extremely suitable for that purpose.  It's not consumable (e.g. you don't put it in your pipe and smoke it.)  It is extremely stable: the list of fluids that will dissolve it is extremely short, it doesn't rust or sublimate.  It is easy to smelt.  It's heavy; making small amounts easy to measure.  It's purity is relatively easy to determine.

It created a universal standard against which the value of other things could be measured so that people didn't have to determine how many chickens was a cow worth.  Even if you were bartering, the relative value of the items could be determined based on their price in gold.  As evidence of this it is said that the discovery of the touchstone sparked the explosion of trade in the ancient world.  Nowadays very few people accept gold as payment so the demand for gold seems to be mostly fear-driven.  People believe that if fiat currencies falter or fail, gold will return to it's position as the universal measuring stick of value.  Only time will tell whether they are right.

Where does bitcoin fit into this?  It seems to me part the demand for bitcoin comes from illegal activity.  As long as hackers continue to demand ransoms in bitcoin, it's a bit like fiat currency.  It's not a measure of value.  It's value is measured in terms of fiat currencies like dollars.  Bitcoin meets a need: a digital equivalent of cash.  Technically it is traceable but it more or less fits that need.  The biggest concern for bitcoin should be is that there's nothing inherent about it that makes it unique.  Unlike gold, another competing system could be created at any time.  There's been talk of a bitcoin 'fork'.  If someone were to come up with a better system (say something that is impossible to trace) it might very well drop to zero.


Data on Individual Performance of Franchises

It depends on how detailed you your data want, but in the case of McDonald's or Wendy's, you can find some data in their annual reports. They split their total revenues and costs into company-operated and franchised restaurants.

Hope that helps you


When big companies invest on a new idea/product?

Legal perspective

Big companies have no reluctance is taking other people ideas:

usually legal systems does not provide protection for ideas themselves,
even if an idea is protected by intellectual protection rights (IP, as a side note please consider long procedure for application for patents, etc.), litigation is costly and long-lasting. You will have significantly smaller resources to be able to protect your idea (even if you win, by the time of the court ruling the market will be taken).

Thus, an idea is not purchasable for big companies. IP, prototype or a small company with IP is.

Business perspective

Even if you will have purchasable IP/prototype/company, big companies still decide which is the most beneficial: to buy or to reproduce. Most often the latter is cheaper. This is because if a larger company can, as you say, remove you from the market very easily, it probably will. If you are considering building your own company, I suggest seeing Michael Skok’s framework, especially chapter ‘Disruptive business models’ and ‘Getting Behind the Perfect Investor Pitch’. In short: if there are no technical barriers for possible competition, make a business model impossible to be replicated by competitors (real life example: Netscape market position was taken by Microsoft’s Internet Explorer because Microsoft could sell IE with every copy of Windows; Microsoft’s lost huge part of the market to the Google, because Google's business model comprised of delivering free products to consumers and profit based on advertisements).

Please also consider that creating a new market takes not shorter than 3 years from the release of the product, sometimes even 7 years (source: S.Blank, B.Dorf, The Startup Owner's Manual).

If you are interested in methods of evaluation return of investment

However, if you are interested how big companies evaluate new ideas for investment (usually for development by themselves), most important factors are soft criteria. Those depend on the project, please find some exemples of such criteria below:

technology (in short: is it the way to go?)
competition (the questions you’ve described can be good examples of some of questions, which companies have to answer before going into investment)
missed opportunities (what would be the better investment, which the company could miss if it will engage in the project at hand?)
profitability in general (how much the value of the company will increase as a result of the investment? when the project will yield return?)
advisability and viability (in general: labour-, capital- and resource- intensity of the project)
means - what is the funding structure (who is bringing their financial contribution? will the gathered means be sufficient?)
in cases when capital is to be transferred to an outside entity (e.g. while investing in smaller company) - credibility of the team, which will lead development works (can they deliver? what were their previous projects? what is their commitment to the project?),
‘extras’ (additional cost-, income-, manufacturing- and/or public image- advantages)

There are many interesting mathematical models to calculate return on investment (ROI, ROE, Payback Period, Accounting Rate of Return, Net Present Value, Internal Rate of Return etc.) but these are only used after all the soft questions can be answered positively.

Hope it helps. I intended to make a small comment, but ended with few paragraph answer :)


Difference-inDifference model with Discontinued Treatment

You presumably should be using all 3 years for the data. Given that you had the appropriate conditions for diff-in-diff (parallel trends), and you're looking at a standard case of the treatment only being relevant in treated periods (no lagged effects), then the additional data should not hurt your estimates.  Additional years of untreated data will help you more appropriately measure the de-facto difference between the two groups, and therefore more precisely measure the effect treatment.
Take a look at panel data and difference-in-differences application - you add fixed effects for all $t$ and all $i$, and create a DiD variable for the the interaction of the treatment and control variable.  It doesn't, under ideal conditions, matter what the timing of treatment is.  If you do have a reason to believe there are lagged effects you should, of course, account for that with lags/leads of treatment, which I believe the other answer is concerned about.


How do I apply the distance-weighting matrix in a spatial autoregressive model

The weight matrix that you have illustrated needs to be row-normalized, i.e. the row-sum must be one in order for the rho parameter to have the proper parameter bounds, i.e. between (-1/lambda,1), where lambda is the smallest real eigenvalue of the W matrix.

When the weight matrix is row normalized and there is a change in the dependent variable, the shock in that other country will be an average of the surrounding values.

I'm not sure if this helps but a good resource is the LeSage and Pace text, Introduction to Spatial Econometrics, CRC Press 2009.

You can also visit www.spatial-econometrics.com to download free PDF books on spatial econometrics as well.


Does Nash Equilibrium predict the existence of vaccine reluctance?

It depends on how you build the game. You may have games with several Nash equilibria, with a dominant strategy equilibrium (which is a special case of Nash equilibrium) or no equilibria at all.
Let us consider a game with 3 players, where herd immunity is achieved at 40% and where players incur a small cost for taking the vaccination and make a big gain when herd immunity is achieved. In this case you have 3 Nash equilibria (when 2 players decide to get the vaccine, nobody has an incentive to deviate).
In another game you may have 3 players who don't care about herd immunity, but that want to become immune (say the disease can come back at any time). In this case you have dominant strategy equilibrium: everybody gets the vaccine.
Finally let's construct a game with 2 players, where herd immunity is achieved at 40%. In this scenario players incur a small cost for taking the vaccination, and a bigger cost if they are the only player who takes the vaccination [1]. Again, herd immunity gives a gain. This can be modelled with a payout matrix such as:

PlayerA Vax
PlayerA No Vax

Player B Vax
1,1
2,-1

Player B No Vax
-1,2
0,0

then the game doesn't have a Nash equilibrium (you may have noticed that this is the same payout matrix of the prisoner's dilemma).
To answer your question, a game that includes vaccine reluctance may have several equilibria or none at all. Vaccine reluctance is an a-priori to your model, which can be factored in when building the payout matrix. Then you can analyze the game an try to predict possible outcomes.
[1] For instance if players are members of an anti-vax community/group, they may incur a reputation cost if other members don't get the vaccine.


In this 3-way currency exchange, who loses, and by how much?

If the starting balances were

A = 1USD
B = 1.2EUR
C = 1000JPY
D = 1.5USD

then the ending balances are

A = 1.5USD
B = 1USD
C = 1.2EUR
D = 1000JPY

so the simple answer to where the 0.5USD came from is D.

A more complicated answer would be some combination of B, C and D depending on how the exchange rates rebalanced to iron out this arbitrage opportunity.

For example if 1000JPY dropped back to 1USD, then D could do some exchanges with B and C leaving them where they started and D with 1USD, i.e. D lost the 0.5USD

But if the EUR/JPY changed so that 1.8EUR = 1000JPY, then C ends up with -0.5USD, if you do exchanges to restore the JPY and EUR to where they started.

If USD/EUR changed so that 1.5USD = 1.2EUR, B ends up with -0.5USD.


Vegetarians vs shoppers of humane meat/eggs/dairy - Who has the biggest impact on the factory farming meat industry?

I don't think there is right or wrong. Here is an example of what could be behind some of these arguments.

Let us assume that there are two buyers in the economy: Singer and Foreman. Foreman enjoys meat but he prefers humane meat (other things equal). Let $V$ be the value that Foreman derives from a unit of humane meat and $v$ the value from normal meat.

Singer is a consequentialist and has as a goal to reduce the meat consumption. Singer is willing to pay $1\$$ to reduce the total consumption of meat in unit and $\beta\$<1$$ to reduce the total consumption of humane meat. If Singer was alone, he would simply refrain from eating. Because Singer is a consequentialist he does not care who eats the meat (this simplifies the analysis).

Now, suppose that there are scale economies in the production of humane meat so that the first unit costs $C$ and the second costs $C'$, while the production of normal meat has constant returns and the marginal cost is $c$. Both markets are competitive.

If $V-C<v-c$, then in isolation Foreman would buy normal meat. Singer would have a disutility of $1$. If Singer buys humane meat then Foreman would do so too if
$V-C'>v-c$ (which may be possible since $C'<C$). Singer would then have a disutility of $2\beta+C'$ (the price of the meat plus the total disutility of two units). If $2\beta+C'<1$, then Singer judges the world to be better if he buys humane meat and he proceeds to do so.

Thus, if activists consider 'humane meat' to be much better than normal meat and there are substantial scale economies in its production, they may prefer to buy it as a way of inducing others to do the same. If they have agent-specific morality (e.g. they prefer a world in which others buy meat than a world in which they buy meat but they have the same world consumption), then participating in the humane meat market may be a less desirable option.

Finally, notice that there are other ways in which Singer could improve the world. For instance, he can buy stock in humane meat consumption and swallow the losses that come from selling at a price $p=c+V-v<C$. This gives Singer a disutility of $\beta+(C-c)-(V-v)$ which is lower if $\beta$ is sufficiently high.

(For simplicity I assume that Singer does not consider Foreman's consumption of utility as deserving moral praise so he does not include Foreman's extra utility in his computation. If he did, then buying humane meat would be even more desirable.)


Basic Fundamentals question on regression analysis based on economic theory:

You never consider either $r^2$ or significance in choosing a model.  $r^2$ is non-decreasing in the number of parameters.  You can improve $r^2$ quite often by adding a set of unrelated variables, such as the rainfall in Quebec to estimates of the price of diamonds in Singapore.  Significance assumes you believe the null is true.  If you have multiple models, then you obviously do not believe one to be true.

When using Frequentist methods, model selection should be done with something similar to the Akaike Information Criterion or the Bayesian Information Criterion.  Both are summary point estimates of a stylized Bayesian posterior density over the model space.  Only use one, don't use both.  There are also a few other similar methods.  The information criterion are essentially Bayesian solutions with slightly different prior mass functions.

You should write software to go through each combination of variable, excluding variables that are collinear, and run either the AIC or the BIC.  The AIC and the BIC measure how close a model is to the data generating function.  This is not like significance an doesn't imply significance.


Is marginal cost different from extra cost?

In this graph they use marginal cost to describe a sort of linear approximation of the change in the cost function $C(x)$ at some point $x_0$, that is
$$
MC_{x_0}(x) = (x-x_0) \cdot \frac{\partial C(x_0)}{\partial x}.
$$
This is indeed not the usual way the term marginal cost is used.


Regarding the role of the Fed in controlling interest rates

It is exceedingly rare for commentators to discuss the discount rate. As such, referring to “rate cuts” typically refers to Fed Funds.

In any event, under normal circumstances, the discount rate is kept at a constant spread to the Fed Funds rate, so they move at the same time.


Are supply and demand always (with arbitrary dimensions but still) linear?

Absolutely not, linear curves are a simplification made for (many) basic economics courses. When considering the basic mechanisms of Supply and Demand (supply rises when the price rises, demand drops when the price rises, etc.) there is no need to use more complicated curves, but as soon as you introduce elements like optimization, it is in many cases necessary to use non-linear functions.

In real life we don't generally observe curves as such, but rather different quantities sold with different prices. We can estimate curves based on this kind of data, but to me that's more of an econometric exercise.


Are these products substitutes or independent?

The goods are perfect substitutes because the marginal rate of substition between them is constant for all values of $x$, $y$ and $m$, i.e. there is $c \in \mathbb{R}$ such that
$$
\forall (x,y,m) \in \mathbb{R}_+^3: \ MRS_{x,y}(x,y,m) = - \frac{MU_x(x,y,m)}{MU_y(x,y,m)} = c.
$$
This essentially means that the goods $x$ and $y$ are so much alike that the rate of substitution does not diminish.

The total quantities may be limited, but (I think) perfect substitution has nothing to do with set of feasible allocations or the budget set, only with the preference ordering.

Varian defines perfect substitutes in subchapters 3.4 and 4.3. Neither subchapter considers feasibility.


Can measures of "efficiency" be objective?

Efficiency can be a very "objective" concept because it is conditional on whatever "non-economic" environment production will take place. Consider the "production possibilities frontier" of a society, that says "given the resources we have, we can produce $100$ of X only, or $70$ of Y only, or some combination of the two, in a year".  As long as we are on the production possibility frontier, we are efficient.
Right. How do we measure how much we can produce? It looks like a totally objective "engineering" exercise (however complicated). "We have available $1000$ labor hours and with them we can produce $100$ units of $X$, it is simple as that".

Hmmm... and why do we have $1000$ labor hours available, and not, say, $2000$? How did we determine that? We determined it by considering perhaps some "average" number of hours worked per person times the working population of the society. Ok, and how did this "average" number of labor hours per working person cοme about? In practice it is strongly affected by the institutional environment in the specific economy. But his institutional environment is taken as given in determining whether we are efficient or not.

So this concept of efficiency is conditional on technology and socio-economic institutions (market structure is included here).

But can the socio-economic institutions be assessed as to whether they cause inefficiency themselves? They certainly can, but we just moved one step higher than the previous concept of efficiency. And here things get cloudy, because the socio-economic institutions are never free of ideology, values and the like.

So to say "the structure causes inefficiency" has the possible rebound "given your values which is not my values". In theory we could even quantitatively assess that -but we would have to perform real-world experiments, something like "let's introduce slavery, and see whether in a long term perspective we produce more or not". Ok. Are you willing to play slave in this scenario (that will last many-many years), so that the world will benefit from the objective knowledge we will gain?


Does this cartoon about the transpacific partnership contain errors?

Small mistake #1: What capital is

The author identifies "bonds, real estate deeds, stock certificates" with "capital" (p. 5). But in mainstream economics, capital refers not to these, but to durable goods that can be used to produce other goods and services.

It may be argued that this is a mere quibble, since the author is communicating to a mass audience and in everyday parlance, that is what people think of as capital. Unfortunately, "capital" recurs throughout the comic and is, indeed, a key component of his arguments against free trade. He frequently conflates "capital" in the fuzzy, ill-defined layman sense and "capital" in the more precisely defined sense that is used in academic economics.

For example:

On pp. 5-6, capital is depicted as pieces of paper ("bonds, real estate deeds, stock certificates") that China owns.

Then on p. 12, capital is depicted as businesses that are shipped off on boats from Iceland to Guatemala.

Then on p. 13, "stocks, bonds, and real estate" (which he defined earlier to be capital) are given to some Chinese-looking man on the left, who then gives back in turn $ \$ $ to the grouchy, jowly man in the middle, who then distributes this $ \$ $ either as savings or investment. Investment is depicted as some excavators constructing some buildings. The text below then claims, "People who 'buy capital' are investing their money." This is all terribly wrong and confusing: So what exactly is capital? Is it something that corresponds to the $ \$ $ symbol? Is it "stocks, bonds, and real estate"? Is it excavators or buildings? What is its relation to savings and investment?

As already said above, capital refers to durable goods used to produce other goods and services. Investment in turn refers simply to the purchase of capital.

On p. 21, capital is depicted instead as boxes being shipped off on a boat by some corporate fat-cat displaying his middle finger. It is unclear what is in the boxes (pieces of paper? excavators? office buildings?).

P.S. I am not an expert on trade. This small mistake is merely one I managed to spot. In my opinion there are many more mistakes. But this post is long enough, so I hope someone who is an expert on trade will find it her duty to explain the other mistakes, especially since this comic has for some reason received undeserved attention.


How to have same utility function for two persons?

Strictly speaking, infinitely different utility functions can give the same preferences over lotteries. If person A has utility function $f(x)$ then person B with utility function $g(x)=A+B\cdot f(x)$, $B>0$ will make identical decisions over all lotteries. There would be no way to know which individual was actually using utility function $f(x)$ and which was using $g(x)$ because they would always make the same decisions.


Direct revelation mechanism's sets of strategies and types

In mechanism design you are free to choose the rules of the game. The designer can determine $(S, g)$, i.e., what players can do and what happens when players played some strategy profile $s \in S := \times S_i$.

In a direct mechanism, players are simply asked to report their type. Hence, every player $i$ must have a strategy that corresponds to "I am type $\theta_i$" for all $\theta_i \in \Theta_i$ such that we can, without loss of generality, set $S_i = \Theta_i$ and each strategy corresponds to a type report.

Together with the revelation principle, direct mechanisms are very powerfull, see my post here.


Different futures contracts have different delivery/final settlement machanisms. You need to look up each contract’s specifications.

The U.S. Treasury bond/note futures are an example of a complicated delivery mechanism. There are rules that set the basket of eligible bonds to delivery, as well as the factor relating the futures price to the price paid for the actual bond. That is, the price depends on what bond you deliver.

In the comments, you added more information. The key to understanding futures contracts is that holders get a daily profit/loss each day based on the change in the final settlement price. This means that even though the price for final delivery will likely be different than the price at which you entered the futures contract, you will get a gain/loss from the settlement process that matches the difference. The net result is that you effectively buy or sell at the fixed futures price you entered the position at; you just do not know the sequence of payments that occur as part of the settlement process.

For example, assume that you want to buy a commodity that is traded on a futures exchange. You enter into a long futures position at price of \$1000. If the futures price fell to 900 by the delivery date, you would pay \$900 for the delivery. However, you woukd have lost \$100 as a result of the daily settlement process on your long futures position. Your total cost is \$1000 - where you entered the position. The fact that there are up front gains/losses creates some complexity for fixed income, but that is a second order effect.

If you search around, you should be able to find materials explaining the difference between using futures to hedge versus a forward sale/purchase. That might help clear up your undertanding.


What caused "excess liquidity" in equity market, when CPI was deflationary at the same time period, which caused 'market crash in 1929' in USA?

The inflation data provided are not useful in this context. The average inflation shown is a mixture of the outcome of a post-WWI deflation, recovery, and price stability. One would need to look at more specific periods. FRED CPI data

Equity prices are supposed to equal the discounted value of future earnings/dividends (the dividend discount model). This means that fair value is highly sensitive to expectations about the future. The 1920s was a period of economic dynamism, where many large firms using new technologies became dominant. It is completely unsurprising that stock prices rose during the expansion.

The linkage between monetary tightening and stock prices is a standard feature of Austrian economics and/or financial market commentary, but it is not widely respected among other economic theories.


Here's my take, which could be wrong.  We assume a closed economy, so net exports, imports and exports themselves are all zero.

We have Y = C + I + G
Suppose we have the simple consumption function C= k +c*(Y-T).  We are only allowed to consume out of disposable income or have autonomous consumption.

Then Y = k+c*(Y-T) + I + G
This goes to

Y*(1-c) = k - c*T + I + G

You take T = t*Y.  So this is

Y*(1-c) = k - ctY + I + G

Subtract ctY from both sides.  Then we have

Y*(1-c*(1-t)) = k + I + G

The solution in the absence of the balanced budget would be

Y = [k + I + G]/(1 - c*(1-t))

Now have G = t*Y - the balanced budget

Y*(1 - c*(1-t)) = k + I + t*Y  or

Y*((1-t) - c*(1-t)) = k+ I or

Y*((1-t)*(1-c)) = k + I

or

Y = [k + I]/((1-t)*(1-c))

So my take is that the multiplier is right, in both cases, but that there is or will have to be some autonomous consumption expenditure in the economy, and that's what's left out.  You can say, "I don't need autonomous expenditure!"  But just try coming up with a Keynesian cross to settle output in nominal terms without one!


The problem of transformation: what is it all about?

According to Samuelson (1971) the transformation problem is to find some general formula by which transform the value of labor into market prices. This is not strictly speaking marxian problem as it already features in the classical works, especially Ricardo wrestled with this problem a lot, since labor theory of value is a classical theory but became associated with marxians later on because they derive profit from the ‘surplus value’, however empirically even industries with various different capital to labor ratios tend to have equal return on capital.
Samuelson argues there is no general solution to the problem. Since the labor theory of value fell out of favor in economics I am not sure if there are any more recent successful attempts, if there are any scholars that specialize in that here they might give additional comments on that.
The labor theory of value is not used by contemporary mainstream economists so the non-marxian solution is simply not to use the labor theory of value but rather the subjective theory of value which does not have this problem and has proven to provide better explanations than the labor theory of value.
Regarding the questions in PS:

Value is not fixed after the job is done. For example, a fashionable purse might have high value in a current year because it is in fashion but as soon as fashion changes its value might became very low. Or another example is given by 'diamond paradox'. In a normal situation diamond is more valuable than water because people already have access to high quantities of water so their marginal utility of consuming another glass of water is low, but they do not have access to many diamonds so their value is high. In a desert however the value of water would suddenly jump above that of diamonds.

No value is not generally equal to price. Assuming that the exchange is voluntary we can say that the price is either equal or greater than the value of money that buyer gives for it and the value for seller is either equal or lower. For buyers who refuse to partake in the transaction the value will be lower than price and for sellers that refuse to sell the value will be higher than price. Only in special cases is the price exactly equal to value.


The investment ($I$) equals savings ($S$) result is derived by Keynes just from national identity, as a result it just hold by definition. Keynes in the passage starts with the simplified version of national identity that omits government. That is:
$$Y=C+I \implies I=Y-C$$
And simply solves it for investment. The solution of investment tells us that private investment is equal to difference between income and consumption. But a difference between income and consumption is also by definition savings $(S)$. Hence keynes derives this result just using accounting. In the end accounting always has to hold because its just an identity based on definitions - in itself it has no economic meaning (In a proper version of identity that would include government this would still hold true but then investment would be equal to private and public saving $I =Y-C-G$).
Furthermore, later when it comes to the paradox of thrift Keynes does not abandon the above interpretation. Rather Keynes would say that during recession $I$ is fixed at some level $\bar{I}$ so any increase in savings $S=Y-C$, which can happen only by lowering consumption $C$ will be just corresponded equal fall in income $Y$. The argument is that your spending is an income for someone else - you cut your spending you will also cut their income - however by cutting their income you also force them to spend less and their spending is your income (this holds in liquidity trap where $I$ is fixed).  At least this is the contemporary interpretation of what Keynes meant as a standards of rigor in economics were not as high in the  past as they are nowadays always leaving some room for interpretation  (see for example Blanchard et al. Macroeconomics an European Perspective - the box about paradox of thrift).
Even when we go beyond mere accounting identity to the IS-LM AS-AD model which is currently the dominant model used to explain the macroeconomy the level of investment and effective demand will be determined first by peoples exogenously given expectations and preferences and the level of savings will be afterwards determined to match this (see Romer's Advanced Macroeconomics for more complex treatment of IS-LM models).

A  side note: you should also note that precautionary savings (i.e. money households set aside to have some reserve just in case) will still normally be transformed into investment provided it is held in some account even if it is at normal deposit account even though you as an account holder might never know about it. The above still applies though - I am just mentioning this because from reading your Q I thought you might be having this misconception.


The money market rate is the interest rate at money market which is market where highly liquid assets are traded (see more about this at Investopedia). It is called money market because in economics very liquid assets can also be considered to be 'broad money.'
The rate at money market rate and central bank policy rate will be similar because the policy rate influences all other interest rates in economy and has very large influence on money market since the price at which banks can borrow from central bank will directly affect the price of funds at money market.
Regarding your question:

If I would have used money market rate instead of policy rate to compare different interest rates of different countries would that be sound?

Well that depends on what is your research question. Depending on specific of your research using central bank interest rate might be more appropriate or less appropriate or you might consider using one as a proxy for other.
For example, in some research about transmission of monetary policy it will be important to distinguish between them because money market interest rate is one of the channels through which the central bank's policy (set with its policy rate) is transmitted to economy (e.g. like in Abbassi & Linzert, 2012).
In some other research where this transmission mechanism is of no interest you might use as a simplification only money market rate, or vice versa skip the money market rate and use policy rate directly as money market rate will be some function of policy rate.
Whether such simplification is justified is research specific. For example, in economics risk free rate and government bonds rate are completely distinct rates but if you can justify saying that it is virtually impossible for Germany to default so its bond rate will have very little risk and will be almost as good as risk free rate then you can use it in a research instead of risk free rate. However, while that justification might fly now, it would be dismissed in first half of 20centruy when Germany had large default risk.  The point is that although they are not same you can depending on context justify one as approximation of other (science is in the end all based on approximation - even in physics you don't deal with exact results and infinite precision).


Second Price Auction - adjusting PDF for reservation price

To find expected revenue of the seller in a second price auction with reserved price consisting of two bidders who bid their valuations in equilibrium, we do the following :

Given that valuations are i.i.d with pdf $f$ and CDF $F$ over $[0, \hat{v}]$, the revenue that the seller gets for different realizations of players' valuations are as indicated in the graph below:

Therefore, expected revenue of the seller is :
\begin{eqnarray*}  &&\int_r^\hat{v}\int_r^{v_2} v_1 f(v_1)f(v_2)dv_1dv_2 + \int_r^\hat{v}\int_{v_2}^\hat{v} v_2 f(v_1)f(v_2)dv_1dv_2  \\ &&+\int_0^r\int_r^\hat{v} r f(v_1)f(v_2)dv_1dv_2  + \int_r^\hat{v}\int_0^r r f(v_1)f(v_2)dv_1dv_2 \\ &=&  \int_r^\hat{v}v_1 (1-F(v_1))f(v_1)dv_1 + \int_r^\hat{v} v_2 (1-F(v_2))f(v_2)dv_2 + 2rF(r)(1-F(r)) \\ &=& 2\int_r^\hat{v} v_2 (1-F(v_2))f(v_2)dv_2 + 2rF(r)(1-F(r))\end{eqnarray*}


Question about the Ellsberg Paradox in Expected Utility Theory

The short answer seems to be yes your example violates expected utility... It mostly seems to me like a simple transformation of the first example you gave (but you got rid of the red balls).

As mentioned in other answers expected utility is not equipped to handle uncertainty because it deals with taking expectations and expectations cannot be computed when you don't know probabilities. For that reason, I think it is relevant to my answer to specify the distinction between what risk and uncertainty are.

Risk: This deals with facing a set of probabilities over outcomes in which the agent knows/understands the possible outcomes that he is facing
Uncertainty: This deals with an agent facing an unknown set of probabilities over possible outcomes

In a previous version of my answer, I wasn't careful in how I thought about and explained these ideas and I have clarified a few ideas in my head since then which hopefully translates to a clarified answer. Now to answer your question:

We have an urn with 60 balls. We are uncertain about the number of black balls and yellow balls in this urn. Imagine we present this urn with the above specified lotteries to an individual (with a monotonic utility function $u$) and ask them to choose between $L_{A'}$ and $L_{B'}$. Without loss of generality, imagine they choose to take $L_{A'}$.

This tells us by revealed preference $L_{A'} \succ L_{B'}$ which then implies

$$u(30) > u(B)$$

where $B$ is the number of black balls that are believed to be in the urn (no probabilities here, this is some number the decision maker arrived at and at this point we don't know how). This implies

$$B < 30$$

Now imagine that you give the agent the choice between $L_{C'}$ and $L_{D'}$ and the agent chooses to face lottery $D'$. Then

$$u(60) > u(30 + (60-B))$$

where $(60-B)$ is the number of yellow balls that the agent must think are in the urn. This implies

$$60 > 30 + (60-B) \Rightarrow B > 30$$

This means that an agent cannot prefer $L_{A'}$ to $L_{B'}$ and prefer $L_{C'}$ to $L_{D'}$ because that would imply $B > 30$ and $B < 30$. It is difficult to talk about where $B$ comes from because it is a hard concept to think about when we are used to working in expected value terms, this agent cannot compute any type of expectation over the sets of values that he thinks $B$ lies in. As an example of this, let me use a max-min approach.

Imagine that when he is presented the urn and lotteries that he is told that the number of black balls is either 15 or 45, but the agent doesn't know with what probability it is 15 and what probability it is 45.

Lottery $A'$ will give him $u(30)$ utility units for sure
Lottery $B'$ will give him either $u(15)$ utility units or $u(45)$ utility units.
Lottery $C'$ will give him either $u(75)$ utility units or $u(45)$ utility units.
Lottery $D'$ will give him $u(60)$

The agent is worried about uncertainty (an interpretation of the max-min is that he thinks it is a magic urn that tries its best to pay the least amount as possible, so it will choose the value that leaves him worse off if given a choice). Then using this in his decision process when comparing $L_{A'}$ and $L_{B'}$ he thinks these lotteries will pay the following amounts

$L_{A'}$ will pay $u(30)$
$L_{B'}$ will pay $\min(u(15), u(45)) = u(15)$

thus he chooses lottery $A'$. Now consider $L_{C'}$ and $L_{D'}$. He thinks these lotteries will pay

$L_{C'}$ will pay $\min(u(45), u(75)) = u(45)$
$L_{D'}$ will pay $u(60)$

thus he would choose lottery $D'$. This is one example of how uncertainty can be treated in a problem like this, but by no means is the only.

Note: Earlier I thought had a piece of my answer that talked about priors over the value of $B$, but this wasn't carefully thought through. You can have multiple values that you might imagine $B$ takes, but as soon as you assign a probability distribution to any of these values then you have left the realm of uncertainty and moved to the realm of risk.


If you're willing to pay \$10 to reduce your risk of death by 1 in 1M, then your VSL is \$10M.†

Example. Say there are two bike helmets. One costs \$20 while the other costs \$30. The two helmets are exactly identical, except that the more expensive one has been proven to reduce your risk of death by an additional 1 in 1M.

If you're willing to buy the more expensive helmet, then your VSL is (at least) \$10M. (If you're exactly indifferent about buying the more expensive helmet, then your VSL is exactly \$10M.)

I'm not sure what you mean by "the risk of death reduces the value of a statistical life."

†At least under the simplest approach, which ignores other important considerations like risk aversion and the discrepancy between willingness-to-pay and willingness-to-accept.


Oliver Hart said that financing deficits by printing money can lead to hyperinflation "once the economy is close to full capacity". What does he mean?

Full capacity means utilization of all resources in economy. That is all labor, capital and other resources are already used at their best uses.   Economy will not always work at its full potential for various reasons. For example, due to fall in aggregate demand during recession, which can cause unemployment to be above natural rate.
Monetary policy or fiscal policy can stimulate economy but only to the point economy reaches full capacity (i.e. all resources are put to their best uses). After that point as Hart mentions monetary stimulus will likely lead just to inflation or potentially hyperinflation.


Every country has a trade balance, which is defined as exports minus imports. The United States is a typical observing state, which means their imports are higher than their exports. I do not know the current numbers but USD 1m of imports and USD 600k might be right. In this sense you import USD 400k than you export.
Now as you already pointed out you have to pay you imports in USD. Now the point NickJ mentioned is insofar important  as to understand what you are actually entitled when holding USD. And all you are entitled to is to buy products within the United States. True, you as a single person could change it into another currency, but this does not change the fact that some else is holding USD and is sooner or later going to demand goods in USD.

So it is correct that in some sense running a trade deficit really brings debt to a country.

However, one has to understand that running a trade deficit is not only indebting. In some way it is a credit for future production and consequently employment. The debt you are causing when running a trade deficit can also be seen as in some way storing demand for US products which in some point in the future will be carried out.


Obj function yielding independent goods demand functions

So if we take a simple example with 2 goods and consider the consumer's problem:

$L=u(x,y)+\lambda(I-p_xx-p_yy)$

This yields the ratio of FOC:
$u_1/u_2=p_x/p_y$

Now plugging these guys into the budget:

$I=p_xu_1^{-1}(u_2p_x/p_y)+p_yy$

A sufficient condition is $u_1^{-1}$ be hod -1 to make $p_x$ drop out. But I don't think it's necessary.

Now I'm not exactly sure what that means (the inverse of the marginal utility of a good is homogeneous of degree -1), but a great thing to think about.


Econometrics - Need help interpreting results

You seem to have $H_0: \beta_1 \le 0$ and $H_1: \beta_1 > 0$. In this case, you reject $H_0$ in favor of $H_1$ if your t-value (not the absolute t-value, but the t-value itself) is bigger than the one-tailed critical value. In your case, $\hat\beta_1 < 0$ so your t-value is negative and so you do not reject $H_0$ in favor of $H_1$.

Edit

Regarding significant* variables with the wrong signs (negative when expecting positive in your case), Jeffrey Wooldridge (Introductory Econometrics: A Modern Approach, 7e, end of Section 4-2f, p. 134) writes:

"A significant variable that has the unexpected sign and a practically large effect is much more troubling and difficult to resolve. One must usually think more about the model and the nature of the data to solve such problems. Often, a counterintuitive, significant estimate results from the omission of a key variable or from one of the important problems we will discuss in Chapters 9 and 15."

Note *: "Significance" means rejection of the null in favor of the two-sided alternative. "Significance" does not necessary imply the rejection  of the null in favor of a one-sided alternative.


Have you tried the Bank for International Settlements? The stats are pretty comprehensive: https://www.bis.org/statistics/rppb1501.htm


Solving Constrained Optimization Problem with Two-Period Model of Human Capital

The problem is that you are ignoring the division $ \frac {0} {0} $, which is in $ \frac {\partial L} {\partial v} $. Before looking at the solution, and seeing that indeed when $ \theta_ {2} = 0 \ \Rightarrow \ v ^ {*} = 0 $, I want to note that the first constraint $ c_ {1} = w - (1 -v) \theta_ {1} h_ {1} ^ {a1} $ can be more realistic and logical. It is easy to see that $ \frac {\partial c_ {1}} {\partial v}> 0 $ in the restriction, which implies that by saving to invest in human capital, your income increases in the first period. This implies that there is no trade-off between investing human capital and consumption (if the interest rate were less than one it would reduce the problem a little, but it would still be inherently wrong). And second $ c_ {1} = w - (1-v) \theta_ {1} h_ {1} ^ {a1}$ does not allow saving wealth, only income. A more reasonable restriction would be this $ c_ {1} = (1-v) (w - \theta_ {1} h_ {1} ^ {a1}) $
\
I'm going to ignore the second observation and go on to answer your question, I just thought it relevant to point it out. We are going to work with this restriction $ c_ {1} = w + (1-v) \theta_ {1} h_ {1} ^ {a1} $. This does not affect $ \frac {\partial L} {\partial c_ {1}}, \frac {\partial L} {\partial c_ {2}} $, but it does $ \frac {\partial L} {\partial v} $. The third first order condition would be:
\begin{align}
\frac {\partial L} {\partial v} = \lambda_ {1} \theta_ {1} h_ {1} ^ a - - \theta_ {2} h_ {1} ^ a) (1 + v) ^ {1-a} a \lambda_ {2} \beta = 0 \\
\frac {\partial L} {\partial v} = \lambda_ {1} \theta_ {1} - \theta_ {2} (1 + v) ^ {1-a} a \lambda_ {2} \beta = 0
\end{align}
We derive the euler equation by introducing the constraints at $ \frac {\partial L} {\partial v} = 0 $:
\begin{align}
\frac {\theta_ {1}} {w- (1 + v) h_ {1} ^ {a}} = \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {\theta_ {2} h_ {1} ^ {a} (1 + v) ^ {a} }
\end{align}
It seems that you can safely say that $ \frac {\theta_ {2}} {\theta_ {2}} = 1 \ \forall \theta_ {2} $, but this is not true when $ \theta_ {2} = 0 $, but this is not the case because when this occurs $ \frac {\partial L} {\partial v} = 0 \ \forall c_ {1}, c_ {2}, v $. This is made clear by multiplying $ \frac {\partial L} {\partial v} = 0 $ by $ \theta_ {2} $:
\begin{align}
\frac {\partial L} {\partial v} = \frac {\theta_ {1}} {w- (1 + v) h_ {1} ^ {a}} + \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {\theta_ {2} h_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = \frac {\theta_ {1} \theta_ {2}} {w- (1 + v) h_ {1} ^ {a}} + \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {h_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = 0 \ \forall c_ {1}, c_ {2}, v
\end{align}
Therefore the solution of the euler's equation for $ v $ is valid if only if $ \theta_ {2} \neq 0 $. What is the optimal value of $ v $ if $ \theta_ {2} = 0 $? Since we have three unknowns and $ v $ only appears in 2 of them, we cannot derive a solution by substitution. You have to see what the effect of v is on the utility function. To achieve this, we obtain the optimal values of $ c_ {1} $ and $ c_ {2} $ by solving for the euler equation (without substituting the constraints and we obtain the following:
\begin{align}
c_ {1} ^ {*} = \frac {\theta_ {1}} {(1 + v) a} \\
c_ {2} ^ {*} = \frac {\theta_ {2}} {\theta_ {1}} (w (1 + v) ^ {1-a} - (1 + v) ^ {- a} h_ {1} ^ {a} \theta_ {1} = 0
\end{align}
But this implies that $ c_ {2} ^ {*} $ is a corner solution, therefore the constraint $ c_{2} = \theta_ {2} h^ {a} $ is not binding, so which is not valid. I'm not going to go into much detail why, but you can learn more in Chapter 18 of Simon and Blume's  mathematics for economists  book. The idea is that if  $c_ {2} $ is 0 the constraint of this variable cannot be fulfilled, it remains as an inequality and is irrelevant to the optimization problem. So it becomes an optimization problem in 2 veritable $ v, c_ {1} $ but how $ \frac {\partial c_ {2} ^ {*}} {\partial v} <0 $. So the optimal level of $ v $ is 0? Without any restriction on the value of $ v $ the solution is $ v = - \infty $ !!!. With restriction its optimal value is equal to 0. Note: this is only true if you make a modification to the utility function, otherwise the optimization problem is not defined. See explanation in "edit".
Edit
I made some corrections to what I had previously written. On that $ v $ does not depend on $ \theta_ {2} $, it does; only in a similar way to what happens when you use quasilinear utility functions. In these types of functions, the good that appears linearly in the utility function does not depend on income if you derive first-order conditions. The problem with this is that the first order conditions are only valid, in this case, when the quantities consumed of both goods are positive. This comes in any intermediate microeconomics book.
It turns out this is general. The first order conditions of an optimization problem in which it is not explicitly specified that the values of the endogenous variables must be greater than or equal to 0. This implies that there will also be parameter values that make the optimal solution let 0 be the endogenous variables. In this case, $ \theta_ {2} $.
The problem is that when $ \theta_ {2} $ equals 0, the objective function is undefined $ \theta_ {2} = 0 \ \Rightarrow \ c_ {2} = 0 \ \Rightarrow \ log (c_ {2 }) = - \infty $, so the value of $ theta_ {2} = 0 $ is not possible. But with slight modifications like changing in the utility function $ log (c_ {2}) $ for $ log (c_ {2} +1) $, the problem is defined and by substituting the constraint of $ c_ {2} $ in the utility function, it becomes independent of $ c_ {2} $ and the optimization problem becomes 2 variables and since $ v $ is only a cost, the optimal solution is $ 0 $ with restriction and $ - \infty $ Without restrictions. So when $ \theta_{2} = 0 \ \Rightarrow \ v^{*} = 0$, and when $ \theta_ {2} \neq
 0$ the optimal value of $v$ is a constant that comes out of the problem of optimization. So there is a dependency, but it breaks when $ \theta_ {2}> 0 $.
Now why is there no dependency when $ \theta_ {2}> 0 $ ?. As @Bertrand commented, this could be changed if there were disutility of accumulation of human capital in the utility function. But it is not the only way, you can also allow to save wealth to invest in human capital, this will create the expected dependence for $\theta_{2}>0$. I think that change is very reasonable; why the agent would be not allowed to save wealth to invest in human capital?
Regarding what they told you about what happens when $ w = 0 $, again you will see that this puts more restrictions on the parameters, in fact it has to be true that $ a \beta = 1 $, otherwise the optimization problem is not definite.


Why does the definition of MRS follow from the implicit function theorem?

It's actually pretty straight forward. The implicit function theorem for two variables is given as follows (as long as some regularity conditions hold):

For $F(x, y) = 0$,

$
\frac{dy}{dx} = -\frac{\partial F / \partial x}{\partial F / \partial y}
$

In the case of MRS, we want the marginal change in $x$ associated with a marginal change in $y$ required to maintain a certain level of utility, $c$, such as (conveniently) $c=0$. So, starting with

$U = U(x, y) = 0$,

we have

$\frac{dy}{dx} = -\frac{U_x}{U_y}$

Note that $c=0$ is just a simplification for exposition. For a general $c$, you can just subtract it from either side of the equation and you get the same result since $c$ disappears in the derivative.


Some basic questions on infromation asymmetry

You want to look at a previous post: Imperfect vs incomplete information .

Basically complete information relates to the assumption that players have common knowledge about the game and the utility function of all players. However, they may not know the cards that a player was dealt with, or there might be an action taken by some player that they do not observe.

Perfect information relates to observing all moves and strategies available to a player, but perhaps not knowing the utility or being uncertain about the game. Examples of perfect but incomplete information are a bit harder to imagine, but the examples I like to think of are board games where players have secret goals or missions.

Asymmetric information intersects with both of these classifications. A game has asymmetric information if one player has more information than another player. That is, perhaps one player has complete information and the other one has incomplete information, or perhaps one player has perfect information and the other one has imperfect information. There are many examples here, perhaps one of the most famous ones is the market for used cars, where the seller usually knows better the condition of the car than the buyer. I.e. the information is asymmetric.

Your question about the bid-ask spread is a bit too general. I think that people have used economic theory to rationalize the existence and size of the bid-ask spread. To my knowledge, there are many reasons why this spread exists, and some of these mechanisms have nothing to do with the information.

For example, consider the market for currencies. If in a small town there is a single bank that exchanges currency X for currency Y and vice versa. The bank has market power and can get away with a large bid-ask spread. Even if there is complete, perfect and symmetric information, people might prefer to exchange currencies at a relatively large ask and a relatively low bid, instead of traveling to the next town to find a  different bank.

Pertaining information, I can think of traders in the financial market exchanging currencies. If there is a dealer with better information as to where to find more favorable exchange rates, it will have more leeway to increase their own bid-ask spread. Since the their clients might not know there are better options out there in the market.


How does protectionism harm the producers and workers in export sectors in the country implementing the protectionist policies?

Several channels I could think of:

As you said, protectionist policies tend to raise the cost of goods that could previously be imported. If these goods are intermediaries for the export sector, the firms in those sectors would be affected. This is the most direct channel I could think of.
You could also argue that the other countries will respond to the protectionist policy by imposing their own protectionist policies, which would eventually hurt the exporters. Think of US agricultural product exporters who got hurt as a result of the US' first imposing tariffs on Chinese exports through China's retaliatory tariffs.
In the longer run, protectionist policies keep firms without comparative advantage in business. Of course, there's an argument of "infant industry" who needs protection... but aside from that, without such policies, the resources employed by these "sunset industries" could be diverted and used more productively elsewhere.


Effect of the Bank Home Loans of Property Prices

This is a big question. I will just comment on one aspect.

  Are the prices rising so high as a result of increasing amount of
  affordable loans or vice versa, the demand for property has driven the
  supply of cheap loans?

We know that people can pay for a house in a few ways:

Use proceeds of the house that they are selling.
Use savings from wages and salary.
Sell other assets (e.g., equities) and use proceeds.
Get a mortgage.

Unless the other sources of cash (1-3) rise enough to cover higher house prices, mortgage lending has to rise. One can try to do a lot of statistical analysis, but the usual relationship is that mortgage borrowing rises at the same time as house prices going up.

Returning to the part of the question I highlighted above, one could argue either way.

One argument is that house prices rose (for some reason), and so mortgage borrowing has to rise in response. In this view, the amount of mortgage loans is not a driver of anything, it is just a mechanical response to higher house prices.

Going the other way is possible. However, we need to remember that mortgages are inanimate objects - in what sense can they drive behaviour? It is safer to say that some humans have changed their behaviour, and the rising loans is a consequence of that behavioural change. The driver is the behaviour change (a possibility not covered in the question.)

The likely suspect is that the behaviour of bankers has changed. Their "effective lending standards" have changed for any number of reasons, and so households can borrow more. (One possible factor is lower interest rates; although formal lending standards may not have changed, the same household can qualify to borrow more under the existing standard.) This extra mortgage borrowing capacity allows households to bid up the price of houses.

I do not think I have fully answered the question, but this discussion may help you rethink how to pose the question.  This website has a difficult time with extremely broad questions, so you may get more responses if you ask a few very specific questions.


Suppose you face a single buyer whose willingness to pay, $v$, is distributed according to $F(v)$. If you charge a price $p$, he will buy if and only if $v>p$, leaving you with expected revenue of $$r(p)=\Pr(v>p)p=[1-F(p)]p.$$

Let's maximise revenue by computing an FOC:

$$r'(p)=1-F(p)-F'(p)p=0.$$

We can rearrange this as

$$\phi(p)\equiv p-\frac{1-F(p)}{F'(p)}=0$$
i.e., the 'virtual valuation' should be zero.

If we return to $r'(p)$ and think about what the individal terms mean, we can see where the "cost" you ask about comes from: a unit increase in price causes an extra unit of revenue in the fraciton $1-F$ of the time that the buyer is willing to buy, but reduces the liklihood of him buying by $F'$. This is the fundamental trade-off that a seller who doesn't know the buyer's willingness to pay must make.

If you know anything about standard monopoly theory then this setup should be quite familiar. Usually, when we look at a profit-maximising monopolist with demand $D(p)$ and zero marginal cost we solve
$$\max_p D(p)p\iff \underbrace{D'(p)p-D(p)}_{\text{marginal revenue}}=0.$$
In an auction-like setting, the 'demand' $1-F(p)$ is simply the probability of the buyer being willing to pay $p$.


Firstly find the marginal utility of $x_1$ and $x_2$. Hint: Use the chain rule.
$\frac{\partial U}{\partial x_1} = \frac{na(aln(x_1)+ bln(x_2))^{n-1}}{x_1}$
$\frac{\partial U}{\partial x_1} = \frac{nb(aln(x_1)+ bln(x_2))^{n-1}}{x_1}$
The consumer maximises their utility when $\frac{MU_{x_1}}{P_{x_1}}= \frac{MU_{x_2}}{P_{x_2}}$.
From that we simply need to solve for $x_1$ and $x_2$.
Plug each expression into the budget constraint which in this case is $w =P_{x_1}x_1 + P_{x_2}x_2$. Then solve for $x_1$ and $x_2$, giving each demand function.


What oil price will minimise new resource extraction?

An oil price of 0 will make any extraction of oil unprofitable and hence imply no further oil extraction.

You should think about how prices carry signals. A low oil price (even if not 0) carries the signal that we, as society, do not value that oil as much. The conclusion is imminent.


Check the article:

Olley, G. Steven, and Ariel Pakes. The dynamics of productivity in the telecommunications equipment industry.

This is an Econometrica paper (1996). It is the standard method to estimate productivity nowadays.


Is weak sustainability (Hartwick's rule) possible on a global scale?

First of all Hartwick's rule has originally been derived under rather specific assumptions, among others a Cobb-Douglas production function and constant population and technology. Therefore copying it one on one to the real world would require a huge leap of faith.
Although it has later been generalized to broader classes of production functions, people have also pointed out that instead of a prescriptive rule (invest resource rents in man-made capital to realize constant consumption), it is in fact a descriptive result. If we observe constant utility then resource rents must have been reinvested.
Asheim, Buchholz and Withagen (2003) show among others that it is perfectly possible in a world with constant technology and population not to have zero net investment, and still constant utility over time, as well as situations where we follow the Hartwick rule and have non-constant consumption (and hence non constant utility).
Blindly following the Hartwick rule, on a global scale or on a regional scale for that matter would therefore not be a good idea. That notwithstanding several papers have shown strong links between savings in the past allowing for constant or even growing consumption in the (far) future (e.g. here, here and here).
Intuitively it makes sense that saving rents from non-renewable resources to make-up for lost production capacity is a good policy measure. Whether or not it will be enough is a different matter, and very much depends on degree of substitutability and technological progress.


The data plotted in the graph are based on a consumer survey conducted by Conference Board. They ask consumers how hard it to get a job. If 34.1% of people say that "jobs are plentiful" and 18% of people say "jobs are hard to get" then the figure would plot a value of $34.1-18=16.1$.

The figure originally comes from this Bloomberg article, but you can read a little more about the underlying data here.


(1) The limited liability constraint ensures that the agent does not pay out of her own pocket the investors dividends. The investor can only take as much as the company made, and cannot charge the agent money she did not produce.

(2) It is reasonable in a sense. In the real world, we believe investors cannot force a manager to pay for bad performance from her own bank account (at least I believe it).

Relatedly, on a technical note, limited liability is necessary to make problems "interesting" in Principal-Agent models. It is well known that is both agent and principal are risk-neutral, the principal "sells the firm" to the agent, and the agent does the optimal thing, and we actually do not have a moral hazard problem, in the sense that we can achieve first best.

To make the problem interesting, we usually assume either:

(i) the agent is risk averse or

(ii) we have limited liability.


Bernoulli transformation in " A Close Form Solution to the Ramsey model" by Smith (2006)

If $z = k^{1-\alpha}$ then

\begin{eqnarray}
\dot{z} &=& (1-\alpha)k^{1-\alpha-1}\dot{k} = (1-\alpha)k^{-\alpha} \dot{k} \\ &=& (1-\alpha) k^{-\alpha} [k^{\alpha} - \delta k - c] \\
&=& (1-\alpha) [1 - \delta k^{1-\alpha} - c k^{-\alpha}\color{blue}{(k/k)}] \\
&=& (1-\alpha) [1 - \delta z - c(k^{1-\alpha})/k] \\
&=& (1-\alpha) [1 - \delta z - x z]
\end{eqnarray}

I will leave $\dot{c}/c$ for you to work out


Collusion model with imperfect public monitoring

You'll get different answers depending on assumptions, so the answer is maybe. Two papers come to mind.

For a continuous time model, see Sannikov and Skrzypacz (2007). They show that collusion is impossible with imperfect monitoring if new information arrives continuously and actions can respond quickly in accordance with new information.

A well known model using discrete time periods is Green and Porter (1984). They show that collusion is possible. Firms can sustain collusion by allowing for the possibility of price wars.


Based on the 2010 US Census, Wikipedia has pretty much exactly what you seem to be looking for.

A sample graph:

though they of course also have tables and links to more data.

Or if you prefer raw data, the IRS SOI is available for 1996-2012.

And of course there's Piketty and Saez seminal paper on the subject (See Table 5 for their data analogous to the included Wikipedia picture)

And of course there's the BLS which publishes these things regularly by demographic, hourly only, or occupation/industry.


Continuity of Prices for Constant Elasticity of Substitution Production Functions

I think so. I didn't work out the proof myself but found in Wikipedia another version of the CES production function that has the same shape as your cost production function by replacing $1-\sigma$ with $r$. Here is the link. The exponents in the parameters $\alpha_i$ should cause no further complication.


"Rules of origin" cover this. Try

Wikipedia https://en.wikipedia.org/wiki/Rules_of_origin or
WTO: https://www.wto.org/English/tratop_e/roi_e/roi_info_e.htm  or
EU: https://ec.europa.eu/taxation_customs/business/calculation-customs-duties/rules-origin_en or
USA: http://2016.export.gov/FTA/nafta/eg_main_017791.asp

Essentially the rules tell someone trading between two places with a free trade agreement in which circumstances something partly originally imported from a third country (whether under another free trade agreement or having paid an original tariff) can be freely traded under the agreement


Logic Fallacy in a single government purchase creates infinite GDP?

It seems that you misunderstood the concept of marginal propensity to consume. The marginal propensity is only valid for an infinitesimal increase in income. As soon as you move away from the original starting point where the marginal propensity was 1, there may be a different marginal propensity. The multiplier however assumes that the marginal propensity to consume remains the same at all income levels (since it is essentially an infinitely repeated application of the marginal propensity to consume).

The above problem with such analyses is why economists switched to models with so-called microfoundations in which the firm and consumer behavior is explicitly modeled via profit/utility maximization. In a model with fixed resources it would quickly become clear that a marginal propensity to consume of 1 for all income levels would violate market clearing conditions.