How does the russian interest rate raise will defends the Ruble from devaluation? This strategy (interest raise to minimize currency devaluation) is being adopted in countries like Brazil and Russia:

Brazil interest rate http://www.tradingeconomics.com/brazil/interest-rate

Russia interest rate http://www.tradingeconomics.com/russia/interest-rate

How does the interest rate rise protect the currency value?

I'm used to see news about governments raising the interest rate with the aim of protect the currency value. But I never saw an explanation about how these two things are related.

An interest rate is the return that a lender earns on money lent to someone else. If the interest rate is higher then it makes lending money more attractive because the return is higher.

In particular, if the interest rate in, say, Russia increases relative to that in the USA then American lenders will switch from lending their money in America to lending their money in Russia where the return is greater. But in order to lend money in Russia, these investors are first going to have to convert their dollars into roubles. This increased demand for roubles causes the price of the rouble (i.e. the exchange rate) to increase.


Equivalence of producer surplus areas I really can't understand how the sum of the rectangles leads to the sum of the triangle.
My book's(Chapter 12, page 430)  explanation is:

Equivalence of these areas can be shown by recognizing that each point
in the supply curve in Figure 12.11d represents minimum average cost
for some firm. For each such firm, P - AC represents profits per unit
of output. Total longrun profits can then be computed by summing over
all units of output.

The author doesn't explain how the bold part is valid and its not obvious from the graph as well.

The profit of a firm $i$ is given by:
$$
\pi_i(p) = p q_i - C_i(q_i)
$$
where $p$ is the price, $q$_i is the output of firm $i$ and $C_i(.)$ is the cost function which differs across firms.
The first order condition gives:
$$
p = \frac{\partial C_i(q_i)}{\partial q_i} = MC_i(q_i^\ast)
$$
This shows how to obtain the optimal supply of firm $i$, i.e. where $MC(q_i^\ast)$ is equal to $p$ (see figures a,b and c in the question).
Total profits for the firm is then:
$$
\begin{align*}
\pi_i(p) &= pq_i^\ast - C_i(q_i^\ast),\\
&= \left(p - \frac{C_i(q_i^\ast)}{q_i^\ast}\right)q_i^\ast,\\
&= \left(MC(q_i^\ast) - AC(q_i^\ast)\right) q_i^\ast.
\end{align*}
$$
This corresponds to the shaded areas in figures a,b and c in the question: the areas equal the difference between $MC(q_i^\ast)$ and $AC(q_i^\ast)$ multiplied by $q_i^\ast$.
Now, consider the first order condition:
$$
p = MC_i(q_i^\ast).
$$
We can invert this function to obtain the supply curve of firm $i$:
$$
q_i(p) = q \text{ whenever } p = MC_i(q).
$$
The total supply in the market at price $p$ is determined by adding the supply curves of over all firms.
$$
Q(p) = \sum_i q_i(p).
$$
Notice that, as usual in economics, figure d in the question draws the supply and demand curves wrong, as they put $p$ on the vertical axis and $q$ on the horizontal. Mathematically, the two should be switched.
If we would do it correctly, it should look something like this:

If $p^\ast$ is the equilibrium price, then the producer surplus is given by:
$$
PS = \int_0^{p^\ast} Q(p) dp = \sum_i \int_0^{p^\ast} q_i(p) dp
$$
Now let us make a change of variables $p \to q$, where $p = MC_i(q)$. Then $dp = \frac{\partial MC_i(q)}{\partial q} dq$  so:
$$
\int_0^{p^\ast} q_i(p) dp = \int_0^{q_i^\ast} q \frac{\partial MC_i(q)}{\partial q} dq.
$$
Then use integration by parts to get:
$$
\begin{align*}
\int_0^{q_i^\ast} q \frac{\partial MC_i(q)}{\partial q} dq &= \left[q MC_i(q)\right]^{q_i^\ast}_0 - \int_0^{q_i^\ast} MC_i(q) dq,\\
&= q_i^\ast MC_i(q_i^\ast) - C_i(q_i^\ast),\\
&= p^\ast q_i^\ast - C_i(q_i^\ast) = \pi_i^\ast.
\end{align*}
$$
Here we assume $C_i(0) = 0$ and we used the first order condition to substitute $MC_i(q_i^\ast) = p^\ast$. We also used $\pi_i^\ast$ to denote the profit of firm $i$ at price $p^\ast$. From this:
$$
CS = \sum_i \int_0^{p^\ast} q_i(p) dp = \sum_i \pi_i^\ast,
$$
which shows that the consumer surplus indeed equals the sum of all profits in the industry.


How can I show whether the following function is Increasing, Decreasing, or Constant returns to scale? I need to show whether or not the following function is increasing, decreasing, or constant returns to scale, but I can't figure out how to extract the t from it when it is put in the form $y(tX_1, tX_2)$.
$$
y = X_2\sin{\left(\frac{X_1}{X_2}-\pi \right)}
$$
Is there any other way of showing IRS, DRS, and CRS? Any help would be appreciated!

You have $$F(x_1,x_2)=x_2\sin\bigg(\frac{x_1}{x_2}-\pi\bigg).$$
If you multiply all factors by $t$ you get
$$F(tx_1,tx_2)=t x_2\sin\bigg(\frac{tx_1}{tx_2}-\pi\bigg).$$
Simplify the fraction and you are almost done.


What is "Stated-capital"? What is Stated Capital? I have found some definitions on the Google, but I have not been able to comprehend it from this definitions. Could somebody explain it to me?

"Stated Capital" is the nominal value (or "par" value) of all the outstanding shares of a company.

When a company issues shares each has a nominal price, say EUR $1.00$. But of course they may be sold for more (as far as I know, it is forbidden to issue a share and receive an amount less than its nominal value. Afterwards of course, the share of a company may be re-sold for less than its nominal value -but not when it is issued originally).

If it is sold for more, say EUR $1.50$, then the EUR $1.00$ is designated and presented as "Shareholder's Capital" in a specific line in the financial statements (usually the first line of Equity), while the "above par" value, the remaining EUR $0.50$, goes into another Equity line.


Solow-Swan production function- What is '$L$' If $L$ represents the amount of labour in the Solow-Swan production function:

$Y/L$=$f(K/L)$

how can $Y/L$ equal 'per-capita income' if $K/L$ is the capital/labour ratio?

I thought per-capita income was income divided by the entire population.

Thanks.

$L$ is labor, which is equal to population in Solow-Swan model. Solow did not actually believe the workforce and population had the same size, but what's important is that labor is supposed to be in direct relationship with population i.e. there is a number w between 0 and 1 such that

$L = w *$ population

Direct proportionality between population and workforce is one of the assumptions of the Solow model.


Extensive form of multiplayer games I want to model an interaction between a government and its citizens using the game theory extensive form, but I want to express the posibility that the citizens don't act as a unique agent. For example, government imposes a new tax, some of the citizens pay it without problems, but others don't do it because socioeconomical problems.

Obviously, both situations can happen at the same time so in the extensive form there wouldn't be a unique selected leaf node. ¿Does this mean that I can't use extensive form in this case?

I'm looking for an example in the game theory literature, but I can't find anything.

Thank you

There is no possibility to model a

  "single" player [that] can simultaneously choose different options.

Neither in the normal nor in the extensive game form. Different types of a player can be only be used if they are mutually exclusive. So suppose the citizens (one player!) can either be poor or rich and this may affect their pay-offs and thus actions. There is no possibility for the citizens to be poor and rich at the same time (while represented by the same player).

Your only option is to follow Sub-Optimal's solution and model different types of citizens as different players.

As Bayesian pointed out, you can then model simultaneous decisions by citizen 1, 2 and so forth by setting their information sets accordingly.


Can anyone explain the connection between complete market and perfect risk sharing? I understand that in complete market, ex post consumption growth is equal for all investors. Why is that perfect risk sharing? The book also states that

  only aggregate shocks should matter for risk prices. Any idiosyncratic risk will be equally shared.

Why is this case in complete market?
What does it have to do with financial innovation?

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


How do central banks create money to pay interest on reserves Referring to this answer (https://economics.stackexchange.com/a/15321/12013) for the technical process of money creation.

How do central banks create the money to pay interest on its reserves (assuming positive interest rate)? I don't see the process being backed by any asset. And if the money is fiat-ly created with no asset backing, would raising interest rate not have the opposite effect of driving inflation and devaluing the currency (rather than slowing inflation and appreciation)?

Yes I am aware of the PQ=MV formula of conventional central bank wisdom, that raising interest rate has the effect of restricting money supply and slowing inflation. I would like to know if the factor mentioned above exists, and if yes, when will each factor/force dominate net inflation and exchange rate.

Since there are many types of interests, the answer can be limited to IOER or this.

The payment of interest is straightforward. The central bank is a bank, and pays interests on deposits in exactly the same way as any other bank.

If the central bank wants to pay $1 in interest to a depositor, it:

increases the amount on deposit by \$1 and
reduces the central bank's equity by $1.

That is, it just reshuffles the right hand side of the central bank's balance sheet; the asset side is unchanged. (On the income statement, there is a correpsonding $1 interest expense.)

All the central bank needs to do is ensure that the aggregate interest paid on reserves is less than the aggregate interest received from its lending and bond/bill holdings to prevent equity from going negative.

Since 100% of a central bank's assets pay interest, and its currency (dollar bills) do not, keeping the net interest flow positive is not too difficult (in a positive interest rate environment).

You asked what backs the currency. The answer is the taxing power of government. There's a few theoretical justifications for that statement, One is provided by Fiscal Theory of the Price Level (edge of mainstream), and another is referred to as "neo-Chartalism" (post-Keynesian).

As for the relationship between inflation and the rate of interest, the answer depends upon which school of economics you follow. Increasing interest rates increases the interest income of the private sector (which should increase effective demand, as you note), which is traded off by the effect on behaviour of the higher rate of interest. The conventional answer is that the behavioural effects dominate; that is, higher interest rates slow the economy and inflation.


Count panel models for endogenous regressors I have a count variable as the dependent variable and several continuous regressors on the RHS. The dependent variable and some regressors are supposed to be endogenous to unobserved factors. I have a set of IVs for the endogenous regressors.

I have two questions:

Is there any model for count panel data (especially for fixed effect) that addresses endogenous regressors?

I have been looking into the literature and have not found anything. It is either count models for panel data, or count models for endogenous regressors, not both.

Some books mention that it is difficult to address endogeneity in count panel data, but they do not really explain why. Anyone knows why?

Thanks!

Windmeijer (2000, Economics Letters) presents a treatment of the estimation of count-data models with fixed effects and endogeneity.

http://www.sciencedirect.com/science/article/pii/S0165176500002287

See this slideshow by Wooldridge for a pedagogic and progressive presentation of panel data models with endogeneity. Count-data models are introduced slide 54. He clearly explains the possible solutions and relate the count-data case to other nonlinear models.

http://www.ifs.org.uk/docs/wooldridge%20session%204.pdf


Would a fair distribution of wealth from the super rich increase the purchasing power/life quality of the average person? I will start by saying that I do not have any background in finance and that the following question might be very much naive, but it has been bugging me for a while now. The general narrative seems to be  that redistribution of wealth from the super rich (top 1%) would benefit the other 99% and increase their quality of life. In the answer to this question, it is calculated that in case of redistribution from the top 1%, everybody among the bottom 99% would receive around $111,000. However, this answer does not really go into detail on what would happen in terms of purchasing power if everyone in the bottom 99% would get said amount. Specifically, my question is the following (and is, to first degree, purely of abstract nature). Let us consider a closed economic system in which most of the population is similarly wealthy, but a very tiny fraction owns a large amount of money. For simplicity, let us assume that we have a city with 1,000,0000 inhabitants from which 999,999 inhabitants are similarly wealthy, while there is 1 inhabitant who owns 90% of the wealth. Let us assume the wealth this one person owns is purely in the form of cash and he basically just hoards his money, so that this money is virtually non-existent in our system. Now assume that his money gets equally distributed between the inhabitants. Would this really make any difference to the purchasing power of the inhabitants and thus to their life quality? Wouldn't prices just simply rise so that purchasing power would stay roughly constant? The question boils down to two main assumptions:

The number of super rich people is much smaller than the number of average people
The money of the super rich is hoarded and virtually non-existent in our closed economic system

If my reasoning is correct with those assumptions, how are these assumption distorted in the real world?

tl;dr: In the hypothetical you set in the body of your question redistribution cannot help poor. However, this is not because redistribution could not significantly raise the welfare of the poor but rather because in your question 'the rich' actually dont have any resources to share with the rest. In fact the way how you set up your hypothetical example what you call 'redistribution' is equivalent to monetary expansion.
However, in real life redistribution, even redistribution of money, is not equivalent to monetary expansion and can help to raise material welfare of people especially of the poor and to the extent quality of life depends on material welfare their quality of life as well.

Full Answer:
The full answer will be structured in three parts. I will first try to answer your question in body of the text, then explain the problems with your assumptions and finally try to answer question in your title with small changes that will be still true to the spirit of the question and make it actually answerable in objective fashion (I do this because I think that most of the attention this question is driving comes from the question in its title).
PART I: Answer to the Question in the Body:
In order to answer your Q lets try to model it. In your example you talk about hypothetical economy when nobody works there are just people with some given wealth so we are in some endowment economy where people are endowed with some output $Y$ which constitutes their 'wealth', but rich people have only cash/money $M_r$. What even more you assume that the cash was not used in anyway so for all practical purposes it simply is not even part of the economy. Moreover, as per the question I will assume the money are equally distributed among everyone.
Well taking assumptions above seriously we can just analyze the effect of such money transfer the same way as just an expansion of money supply using simple equation of exchange (this is not normally appropriate model for evaluating redistribution but in your case redistribution is just monetary expansion in disguise and this model is a simple textbook example of a model that can be used to analyse effect of monetary expansion):
$$MV=PY$$
where, $V$ and $P$ which were not yet introduced  are velocity of money and price level respectively. In this case $Y$ is fixed as there is no production just everyone has some certain level of wealth. Hence when $M$ increases by adding the $M_r$ either $P$ increases or $V$ must drop (which would happen if the money people get from the rich end up 'buried' and not used), or some combination of thereof.
In either case whether prices or velocity changes people cant be made better off by the transfer of cash in your hypothetical scenario as in your scenario the rich $1\%$ really has nothing of value to give to the $99\%$. Their endowments $Y$ remain fixed. Again, I would normally never recommend using this model for analyzing redistribution but in your case you are not really talking about redistribution.
PART II: Problems With Your Question:
The main problem to your question is that if really taken to its logical conclusion it boils down to the following. Does creating more money make people any richer? Well answer to that is no. What makes people rich is the amounts of goods and services they have access to. In fact from an economic perspective the $1\%$ of population in your question must be miserable bunch because by your assumption they only have money while the other $99\%$ actually have some other 'wealth' which presumably in your question implies some goods and services to enjoy.
However, the above situation is clearly absurd and does not correspond to reality. Rich people do not sit upon piles of sterile cash that is not used in economy. If their money circulates in the economy either because the rich spend it, or put it in bank into their accounts or invest it into some assets they will already be part of the $M$ and redistribution, even redistribution of money, will not expand $M$.
Redistribution might decrease output due to the fact that it distorts incentives, however at the same time there are arguments that it might boost output as high levels of inequality can make harder for poor people to become entrepreneurs or realize their full potential in different ways. Empirical studies actually show that the relationship between economic growth and intensity of redistribution is mixed, suggesting that on net it has no impact on economic growth save for cases of extreme redistribution that would be well above what we typically see in developed nations nowadays (see Ostry, Berg, and Tsangarides 2014 and sources cited therein). When it comes to velocity redistribution can slightly increase it as some research shows that people with lower savings rate contribute to higher velocity (see Wang & Ding 2005), however the effects are nowhere near strong enough so any transfer will have no net effect on welfare of recipients.  As a matter of fact it is generally agreed by the profession that over long time periods it is an increase in $M$ which leads to inflation (see pretty much any conventional textbook for example Blanchard et al Macroeconomcis an European Perspective or Mankiw Principles of Economics).
As a consequence of the above, redistribution can be more or less viewed not just as a transfer of money but as a transfer of resources and in fact this is how redistribution is treated in wast majority of the literature in public economics (see models presented in any public economics textbook). Many redistribution models will not even explicitly include money at all and this is not due to lack of oversight but deliberate simplification (the same way as physicists often may assume that some part of space contains perfect vacuum even if it has some particles) reflecting the view that redistribution is not simply some money creation especially when financed by taxes (even though relative prices might change which can still affect welfare but this is accounted for in the models).
Part III: Answer to the Question in the Title:
The question in your title:

Would a fair distribution of wealth from the super rich increase the purchasing power/life quality of the average person?

is actually far more sensible but I will still make 3 changes that I think are true to the spirit of what you are actually interested in.
First I am going to change the word fair distribution to redistribution to get rid of the word 'fair'. Well what is fair or fair distribution? That's a question that human race pondered ever since civilization emerged from the fertile crescent and perhaps even in prehistory without any generally agreed upon answer and discussion of fairness belongs to moral philosophy not economics.
Second I will change wealth to wealth/income. The reason for this is that I feel that like many other non-economist you do not distinguish between wealth and income and treat them jointly. For example, who is in your opinion a richer person? A old retiree whose net assets are worth $\\\$1,000,000$ because his house happened to be in gentrified area and went up in value but with measly salary of $\\\$500$ working as shop greeter, or a superstar which might have no net assets (i.e. living in fancy hotel instead of owning house) but with monthly paycheck of $\\\$70,000$? The retired person has more wealth but I think many would actually consider person in that situation poor whereas the second person has no wealth but I think most reasonable people would consider such person rich.
Third I will replace average person with low income people. The reason for this is that in any right skewed income distribution (an income distributions are world are right skewed) an average person will actually be better off than more than half of the population and optimal redistribution systems are designed to help people at the bottom not people who are better off than most (even though unfortunately in practice governments often engage in such 'perverse' redistribution as well).
Hence I will try to answer the following question:
Would a redistribution of wealth/income from the super rich increase the purchasing power/life quality of the low income people?
the answer here when it comes to income is resounding yes, when it comes to wealth the answer is not clear or rather the answer would be maybe.
Redistribution via Income Taxes
The literature on optimal income tax shows that optimal top marginal tax rates for the richest can be somewhere in rage of $50-75\%$ in the US (see Saez 2001) an generally similar estimates are also found for other developed countries.
The simulations for these tax rates are already made with an explicit goal of maximizing actual welfare of the recipients and society overall  - not to maximize any monetary transfer per se - but to maximize their underlying utility. They are based on the following optimal non-linear tax function which comes from the seminal works of Mirrlees (1971) - who in fact got Nobel Prize in Economics for this contribution, Diamond (1998) and Saez (2001).
Furthermore, the optimal tax rate even takes into account all the labor supply response and other factors in fact the formula is given by:
$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_n0)}$$,
with $b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $.
I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part $\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$ is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part $\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b_n$ which depends on both utility of consumers and the societal utility function,  and finally $\frac{1-F(z_n)}{z_nf(z_n0)}$ is the part that captures the relative magnitude of distortions created by this taxation. Again since we are talking about improvements in actual underlying utility what the aggregate price level is does not matter.
If redistribution would not be able to improve welfare in poor this formula would give you zero marginal tax rates. Since simulations based on real world parameters (Saez 2001) show that top marginal tax rates could be as high as almost $80\%$ clearly redistribution can improve the welfare of poor.
An important caveat is that this optimal taxing formula does not take into account general equilibrium effects and general equilibrium effects generally (no pun intended) result in lower taxes but no reasonable estimate of the magnitude of general equilibrium effects would push the top marginal tax rates to zero. Even if we would say that in general equilibrium these taxes would be half of what partial equilibrium analysis suggests they would still result in redistribution that would be able to significantly help the low income individuals.
Redistribution via Wealth Taxes
When it comes to wealth taxes the literature is much smaller as studying wealth taxes is exponentially more difficult than income taxes due to the fact that they are rare and we also don't have good data. As a consequence the discussion on wealth taxes to some extent takes a form of twitter fights between various economists (I am referring to Summers-Saez & Zucman infamous twitter battle) rather than in some proper research.  This being said some academic work was already done on this topic but it remains highly inconclusive. I will try to present both views fairly starting with the no wealth tax view and then presenting the pro wealth tax view.
Anti wealth tax view:
Some economists argue that optimal wealth tax is simply $0$. If this is so then obviously wealth tax cannot help improve welfare of the poor. There are several reasons why this would be so which are all summed nicely in this article from Larry Summers. Here is the digest:

Wealth taxation also raises practical concerns — for example, issues of valuation: Is a partnership in a law firm wealth? How will illiquid assets — like football teams or newspapers — be valued? And issues of liquidity: If someone owns 1 percent of Uber — still a private company — she will owe roughly $20 million in taxes each year, but it’s unclear where she can get this money. She can’t sell shares and, if involved with the operation of the company, is likely to be barred from borrowing against the value of her stock.

There are also family unit issues: If a couple files separately or gets divorced, do they get two $50 million exemptions? And issues of gaming: There will be incentives to use legal structures to relinquish direct ownership of assets while maintaining control of them. For example, owning assets in a trust or a nonprofit to benefit from wealth while avoiding tax liability. Granting that capital income should be taxed more heavily than it now is, and that unrealized capital gains going untaxed is a serious problem, there is also a question of just how a punitive tax is appropriate.

It is important not to be misled by the 2 percent annual rate: A 50-year old who has accumulated a substantial fortune can expect to pay more than half of it in taxes before she dies.

Imagine that a wealthy person invests in 10-year treasury bonds, with a 2.4 percent return. The wealth tax would extract 2 of the 2.4 percent return. Combined with income taxes levied at a 40 percent rate, the wealth tax could make the effective tax rate on capital income well over 100 percent. And then at the end of life would come the estate tax. While we are not aware of formal estimates of the loss in economic efficiency from wealth taxes, we suspect that if levied without concomitant reductions in income tax rates or estate tax rates, the ratio of burden on the economy to revenue raised would be far higher than with the base-broadening measures we advocate.

....

The Organization for Economic Cooperation and Development recently assessed wealth taxation and concluded that “from both an efficiency and equity perspective, there are limited arguments for having a net wealth tax.” Of the three countries with a wealth tax, two — Norway and Spain — raise an average of 0.305 percent of GDP. These taxes generate less than one-third of what the wealth tax estimates despite having a much broader base: While precise data are hard to come by, we suspect that less than 10 percent of this revenue — or 0.03 percent of GDP — comes from those in the top 0.1 percent of the wealth distribution.

Pro wealth tax view
However, Saez and Zucman the main proponents of wealth tax  would argue that the worries mentioned above are overstated (see here).
According to Saez and Zucman the main reason why the European wealth taxes failed is that Europe tolerates (and arguably even encourages) tax competition. In this article they argue that:

The specific form of wealth taxation applied in a number of European countries had three main weaknesses. First, European
countries were exposed to tax competition and tax evasion through offshore accounts, in a
context where until recently there was no cross-border information sharing. Second, European
wealth taxes had low exemption thresholds, creating liquidity problems for some moderately
wealthy taxpayers with few liquid assets and limited cash incomes. Third, European wealth
taxes, many of which had been designed in the early 20th century, had not been modernized,
perhaps reflecting ideological and political opposition to wealth taxation in recent decades.
These wealth taxes relied on self-assessments rather than systematic information reporting.
These three weaknesses led to reforms that gradually undermined the integrity of the wealth
tax: the exemption of some asset classes such as business assets, a preferential treatment of
others such as real estate, or a repeal of wealth taxation altogether.
A modern wealth tax can overcome these three weaknesses. First, offshore tax evasion can
be fought more effectively today than in the past, thanks to recent breakthrough in cross-border
information exchange, and wealth taxes could be applied to expatriates (for at least some years),
mitigating concerns about tax competition. The United States, moreoever, has a citizenshipbased tax system, making it much less vulnerable than other countries to mobility threats.
Second, a comprehensive wealth tax base with a high exemption threshold and no preferential
treatment for any asset classes can dramatically reduce avoidance possibilities. Third, leveraging
modern information technology, it is possible for tax authorities to collect data on the marketvalue of most forms of household wealth and use this information to pre-populate wealth tax
returns, reducing evasion possibilities to a minimum. We also discuss how missing market values
could be obtained by creating markets. In brief, the specific way in which wealth was taxed in a
number of European countries is not the only possible way and it is possible to do much better
today

This being said even the most extreme serious wealth tax proposals are somewhere in ballpark of $5\%$ - this is much less then the top marginal taxes that are optimal for income redistribution. It can still be argued to be somewhat well-being enhancing but clearly it is debatable of how significant would this effect be.
Nonetheless as correctly pointed by Brian in his +1 answer/comment the argument can be made that wealth tax will help to erode the political power of the rich which will ultimately help the poor. There are also counterarguments, in fact Summers argues that wealth tax might actually increase the political power of the rich but this debate is something that is outside the field of economics so feel free to read their papers and articles I linked here in full to make up your mind about this. I personally am agnostic about this power argument as there is not much research done on this issue. I suspect that even Summers and Saez and Zucman would agree that on both sides of this argument evidence is mixed.
Hence when it comes to wealth taxes it is hard to say if they can improve the lives of low income people significantly, but this is not because of a high inflationary effect redistribution via these taxes would have but rather due to the fact that optimal wealth taxes are either zero or low.
The only exception of the above is land or similar type of property which can still be counted as wealth and which is actually easy to be taxed even at high rates without generating some of the above issues (although the liquidity and difficulty of valuation argument would remain) . However, this being said most of the wealth around the world does not really consist of land so this is more or less side note.
PS: The above is just a tip of an iceberg. I did not discuss how factoring in the effect on incentives for education change the result, what is the role of capital, inheritance, consumption taxes in all of this, I also did not discussed in great lengths general equilibrium analysis, or fully factored in all dynamic effects, but all I have written still holds in spirit even if the issues mentioned here could change the actual numbers presented. If your intent is to actually learn about the field then in addition to sources already presented here and sources cited therein I wholeheartedly recommend reading the Mirrlees review which is probably the best literature review on optimal taxation there is and also the book by Inequality Atkinson who before his death was one of the top experts in the area of inequality and redistribution.


Depreciation vs. obsolescence in RBC and New Keynesian DSGE models A common calibration for depreciation rates within RBC models is to assume 10% depreciation rate (based on NIPA stats, for instance). This implies a half-life of about 6.5 years. But this estimate seems a little low for advanced economies where capital is often in the form of computerized technology etc, which would seem to depreciate at a much faster rate (say 25% pa, implying a half-life of more like 2.5 yrs). Further, with technological advance you have the problem of obsolescence - the useful life of a capital good may end long before it depreciates away fully using the 10% pa rate. There is an element of unpredictability around obsolescence as well.

How can this be accounted for within RBC and DSGE models? Would obsolescence pose a problem for calibrating a model assuming delta is approx. 10%?

Thanks in advance,

R.

On the empirical side, there might be answers for you in the national accounts. I only know about the french case : the french statistical institute (INSEE) has different depreciation data for different types of capital (e.g. buildings, machines, patents) and for different sectors. These data are supposed to reflect both physical depreciation and "normal obsolescence". The only reference I can give you is in french : Jean-François Baron, "Les comptes de patrimoine et de variation de patrimoine, base 2000", INSEE, 2008, http://www.insee.fr/fr/indicateurs/cnat_annu/base_2000/documentation/methodologie/nb10.pdf , but I suspect you can find similar data in other national accounts.

On the theoretical side, if I understand you, you mean that a capital good can become economically obsolete before it is physically out of use. I see three possible reasons (possibly overlapping) for this :

the price of inputs (e.g. the wage rate) rises so that this capital good is not profitable any more
the price of output falls but not the price of inputs, so that it is not profitable
production is demand constrained and there other more productive ("new") capital goods available, so that our capital good will remain idle (although it could be profitable if there were a higher demand)

To account for this problem, you have to drop the representation of capital as a single homogeneous magnitude (the number "K") and have different capital goods with different properties (productivity and input requirements). This is called a "vintage capital" model. I don't know if this has been done in an RBC or DSGE framework, but a quite complete neoclassical paper dealing with these issues is :
Solow, Yaari, Tobin, Weiszäcker (1966), "Neoclassical growth with fixed factor proportions"
http://sites-final.uclouvain.be/econ/DW/DOCTORALWS2004/bruno/vintage/solow%20yaari%20tobin.pdf


Significance of circular flow diagram? A staple of intro economics textbooks is the circular flow diagram. Example from Frank et al. (2019, Principles of Economics)

What is the significance of the above diagram?

One thing it can illustrate is that Production = Consumption = Income. But it seems to me that this point could be stated separately without resort to some diagram such as the above.

Is there anything else the above diagram can usefully illustrate?

Thoughts: It seems to me that these circular flow diagrams are just some cutesy device that some textbook writer long ago first came up with, and which other textbook writers have simply and mindlessly stuck with.

These diagrams seem "cool" because they're analogous to similar diagrams in the natural sciences (e.g. the water cycle). But they have not anywhere near as much significance (or indeed any significance at all). Of course, I could very well be wrong--hence this question.

As you pointed out the diagram is used to illustrate that Production (or more appropriately output) = Income. In this case it equates it to consumption but that is not generally true of all circular flow diagrams.  However, besides that it has many other uses as well as it is an easy way of provide a bird’s eye view of all important flows in the economy.

Also you cant judge a tool only based on its most simplified version. A more involved versions of circular flow diagram as for example the one taken from Krugman's macro textbook that you can find below the text also include financial markets (where people can save), government, international sector and so on.

The point of these is to provide birds eye view of the economy, to show students that in economics everything is interconnected (and you would be amazed by how many people do not realize that), and also it also highlights the specific interactions that occur between different sectors of an economy. The circular flow also is a good and easy way how to introduce 101 econ students to the idea that in macroeconomics income is endogenous, I seen it also be used as illustration of how money multiplier works.

  Thoughts: It seems to me that these circular flow diagrams are just some cutesy device that some textbook writer long ago first came up with, and which other textbook writers have simply and mindlessly stuck with.

In a sense yes (except for the mindless part) because circular flow is just a didactic tool - you compare it to a water cycle and thats actually a good comparison. You can also describe water cycle in terms of physical processes in a mathematical model and you can do the same with circular flow which is ultimately based on the national income identity and related macro general equilibrium models but that might be too intimidating for highschool/undergraduate 101 students. The tool does not get included in textbook because professors mindlessly copy it but because its genuinely useful classroom device.

Arguably the circular flow does not have much use outside classroom or self-study, but  that does not mean that it is not a useful or significant tool as it helps students who's math skills might not be yet ready for mathematical presentation understand important concepts in macroeconomics. In most fields you will find that there are tools that outside classroom have no much use yet I would argue thats enough significance in itself. I would also not say it has any less significance than water cycle which is again mostly just teaching/visualization tool. You can compare it to other tools, for example when I was studying math and we were covering topology our professor would bring us examples of Klein bottle, 3D printed tesseract and other 'cutsy toys' - ultimately they were literary just cool looking paperweights but I found them useful when learning topology and I am sure other students did as well.  The circular flow just serves similar purposes and makes teaching as well as learning easier.


Fed must hold collateral equal in value to notes in circulation The Fed's website has a FAQ "Is U.S. currency still backed by gold?". Part of the answer writes:

  The Congress has specified that Federal Reserve Banks must hold collateral equal in value to the Federal Reserve notes that the Federal Reserve Bank puts in to circulation. This collateral is chiefly held in the form of U.S. Treasury, federal agency, and government-sponsored enterprise securities.

In 2017, there were \$41.6B in notes in circulation (source). (Of course, the Fed's total assets vastly dwarf this figure -- they are around \$4T now, but were over \$700B even in the years before the financial crisis.)

Three questions about the above statement:

The statement presumably refers to some specific law passed by Congress. What was this law and when was it passed?
Does the Fed specifically earmark some portion of its assets as "collateral" for the notes in circulation? For example, in 2017, did the Fed have \$41.6B in assets specially earmarked as "collateral" for the \$41.6B in circulating notes?
Does the Fed actually keep any record of this? (This was one thing I was looking for but unable to find.)

Section 16 of the Federal Reserve Act, 1913.
No, the Fed’s accounting generally follows the standard convention, where assets=liabilities+equity. All of its assets count as collateral; no particular assets are “earmarked” as collateral. By construction, the Fed holds assets against all its liabilities (mostly deposits), not just Federal Reserve Notes.
Yes, you can see the assets and liabilities of the Fed in its H.4.1 release.


Transitional Dynamics of consumption per capita in solow model I can write the transitional dynamics of output per capita as follows
$$y=f(k)$$
Take its derivative with respect to time t
$$ \dot{y} =f’(k) \dot{k}$$
Divide it by $k/k$
$$ \dot{y} =f’(k) \frac{\dot{k}}{k} k$$
And finally divide both side by $y=f(k)$
$$\frac{\dot{y}}{y} = \frac{f’(k)}{f{k}}\frac{\dot{k}}{k}k$$
Now I want to derive this for consumption for per capita
I guess
$$c= (1-s)y$$
$$\frac{\dot{c}}{c} =\frac{\dot{y}}{y} $$
So is this transition dynamics for consumption per capita correct? How can I obtain this?

You're right, since in basic Solow model (with population growth and no technological progress) macroeconomic closure condition (in aggregate terms) is:
$$Y(t) = C(t) + I(t)$$
where $$I(t) = sY(t)$$
Now replacing the second in the first equation:
$$Y(t) = C(t) + sY(t)$$
Factorizing we arrive at the equation you stated:
$$C(t) = (1-s)Y(t)$$
Taking the first equation in per worker terms (multiplying both sides by $1/L_t$) and differencing with respect to $t$:
$$\dot c = (1-s)\dot y$$
Using the identity $c(t) =(1-s)y(t)$, dividing the previous expression (as you did) we get:
$$\frac{\dot c}{c} = \frac{\dot y}{y}$$
I hope this is what you are looking for.


How is a country's interest rate and inflation linked? India has just dropped its lending rates. Is the drop in interest rates linked to the country's inflation?

Link to article - http://in.reuters.com/article/2015/09/14/india-economy-inflation-idINKCN0RE04V20150914

You're probably not looking for a complex scientifical discussion on the subject, so here's a very simplified description of the mechanism:

Lower interest rate => Loaning money is cheaper => More money in the system => Higher inflation.

So the change in inflation is one of the products of government/central bank changing interest rates.


Was the 2008 financial crisis caused more by the US subprime mortgage lending or more by an international energy and oil pricing crisis? I recently read The Big Short by Michael Lewis in which the depiction of the role of subprime mortgage lending in the US and then run away speculation on CDOs are shown to be the central and primary cause of the 2008 financial crisis.  Then today I watched a lecture called The Third Industrial Revolution (https://www.youtube.com/watch?v=QX3M8Ka9vUA) by Jeremy Rifkin in which he portrays the 2008 subprime mortgage crash as being an "aftershock" of the real financial "earthquake" which was international oil prices hitting $147/barrel 6 months prior to the 2008 crisis.  He continues this analogy by insisting global policy makers are, when they bail out the banks who invested in subprime mortgages or change investing laws, "trying to address the aftershock rather than the earthquake."  I have never heard this view of the situation and am ignorant as to how international oil pricing functions and affects the world economy.  How do people in the know square the relative significance of the US subprime mortgage investment bubble with this international oil price situation when trying to label the root causes of the 2008 global financial crisis?

It's worth differentiating between the financial crisis of 2007-2009, and the recession that began in December 2007.

The financial crisis was indeed caused by a run on wholesale funding that was precipitated by significant losses on real estate, both subprime and otherwise. As noted in the first link, stresses in wholesale funding first appeared in August 2007. At that time, oil prices (WTI) were still \$70-\$80/barrel.

Losses to financial institutions during the crisis that can reasonably be linked to oil are limited, and the timing is all wrong, so it's not credible to suggest that the increase in oil prices would have led to a financial crisis absent the actual mechanism (run on wholesale funding, significant losses on real estate assets) by which the financial crisis occurred.

That said, it's certainly likely that oil prices contributed to the recession that began in December 2007, as documented by James Hamilton, though the severity of this recession was in large part due to the fact that there was a financial crisis:

  Recessions associated with financial crises have been more severe and
  longer lasting than recessions associated with other shocks.
  Recoveries from such recessions have been typically slower, associated
  with weak domestic demand and tight credit conditions.

Thus, oil prices are likely to have contributed to pushing the economy into recession, which may have worsened credit losses, fueling the stresses in the financial sector, but it's not credible to claim that they were a primary factor in causing the financial crisis, as compared to the structural vulnerabilities relating to wholesale funding and securitized assets.


Is there really an "efficient equilibrium *price*" with externalities? In the "Intro to Micro" approach to externalities, I've always felt uneasy as a teacher with the idea that there is something like an "efficient equilibrium" with a single "efficient price".

To keep things simple, I will focus on external costs, but everything applies symmetrically to external benefits.

The graphs you often see are similar to those in the Wikipedia article on externalities. These graphs use the term "ideal equilibrium" instead of "efficient equilibrium", but the rest is standard (I include the captions from Wikipedia):

 "Demand curve with external costs; if social costs are not accounted for price is too low to cover all costs and hence quantity produced is unnecessarily high (because the producers of the good and their customers are essentially underpaying the total, real factors of production.)" (https://en.wikipedia.org/wiki/Externality)

I have of course no problem with recognizing and explaining that there is a unique efficient quantity.

My main issue is with the idea that there is a unique "ideal price" above the actual equilibrium price, and secondarily with the fact that this price is dubbed an "equilibrium" price (however "ideal" or "efficient").

At first sight, it seems ok to explain that, in the presence of an external cost, the good is under-priced because the actual equilibrium price only takes private costs into account (as opposed to social cost).

However, this only makes sense if we are talking about the price for consumers. For producers, a higher price would mean a higher quantity supplied. So in the presence of an external cost, the good is actually overpriced for producers, and underpriced for consumers only.

This is, in fact, the reason why a Pigouvian tax is a good instrument to solve the external cost problem: Because it increases the price for consumers and decreases the price for producers. Under a Pigouvian tax, the efficient quantity is indeed the equilibrium quantity. However, we now have two equilibrium prices (these are real equilibrium prices this time), one for consumers --- higher than the old equilibrium price --- and one for producers --- lower than the old equilibrium price.

This issue becomes even more salient if, after discussing external costs in the context of the above graph, you attempt to make sense of external benefit in the context of the symmetric graph (https://en.wikipedia.org/wiki/File:Positive_externality.svg) which again suggests that the good is underpriced at the actual equilibrium (whereas it is really underpriced for producers, and overpriced for consumers).

My questions are:

Am I missing something? Is there really a sense in which the price $P_S$ in the above graph is the "correct efficient price" across the board, both for producers and consumers?
Presentations of externalities that rely on graphs like the one above are typically agnostic about the "source" of the externality (whether it is a consumption or production externality) and only care about its sign (i.e., whether it is an external benefit or an external cost). I guess if we assumed that the external costs came from a consumption externality, then $P_S$ would indeed an efficient price (because it would reduce consumption to the efficient quantity, which is all we would care about). But then again, isn't it a stretch to call $P_S$ an "equilibrium" price when, clearly, imposing $P_S$ across the board for both consumers and producers would induce a shortage?
If you were teaching externalities out of a textbook using similar graphs and terminology, how would you try to make sense of all this for your students? Depart from the textbook and explain you don't agree with the idea of an "efficient equilibrium" with a single "efficient price"? Or try to work around the issue without opposing the textbook?

The blue lines are labelled as being costs, not prices. The line labelled "Private Cost" represents quantity supplied for a particular cost to the consumer. The line labelled "Social Cost" represents the quantity supplied for a particular cost to society. While, for the former, the cost is equal to the price received by the supplier, for the latter the cost is equal to the price received by the supplier plus the externality. $P_S$ therefore represents the price paid by the consumer, not the price received by the supplier.

In the case of steel smelting causing air pollution, the free market price represents the equilibrium if steel consumers are required to compensate steel suppliers for the cost that suppliers incur in smelting steel. The ideal price represents the equilibrium if steel consumers are required to compensate steel suppliers for the cost the suppliers incur, and also compensate the general population for the cost that they incur through pollution.


Premises on which the Efficient Market Hypothesis is built upon The Efficient Market Hypothesis tells us that stocks are "fairly priced" (i.e. not systematically over- or under-valued). That is, stock prices reflect all information that is already known about them. This is because any information from current or past events are already "absorbed" by the market. What is this "asborption" of information by the market supposed to mean? The market is comprised of its participants, buyers and sellers. I would also like to ask if the Efficient Market Hypothesis is built upon the following premises:

All market participants have perfect information at all times, i.e. they know the operations of the companies listed on the stock exchange fully well and that they are also well-equipped with the financial knowledge required to interpret any economic news that may affect the performance of the listed companies.
All market participants respond rationally and make decisions using the marginalist principle.
Market participants are able to assess the circumstances readily at any one time and are able to respond to any new economic information (e.g. government macroeconomic policy changes, changes to corporate earnings etc.) instantly.

I just learnt about this concept and so I would just wish to seek clarification on it.

The efficient market hypothesis does not require everyone being omniscient. It works through the supply/demand and price mechanism.

For example, imagine that you have private information that a company X is doing bad (without being a manager or other person banned from insider trading). If company X is doing bad then sooner or later it’s stock price will have to go down, so if you own it’s stock you would sell it now while the price is still high or if you don’t own the stock you would short sell it.

However, as you start selling the stock you are pushing the price down. Since it’s profitable for you to sell or short sell as long as the market price is above the actual value of the company, you would do that so much that the price would eventually be driven to the point where the price is exactly equal it’s value.

The central assumption of the efficient market hypothesis is the perfect market assumption. In a perfect market there are no transactions costs, information is costless, investors are rational. In the real world however, these assumptions do not completely hold and the market is not fully efficient.

This being said, there is evidence that despite these imperfections markets are efficient to a certain extent as prices do adjust in response to new information. In fact it’s astonishingly hard to beat the market, and difficult to find deviations from the efficient market hypothesis empirically but now we know they exist.


Authoritative Books on Subprime Mortgage Crisis I'm looking to learn more in detail about the situations and actions that setup the sub-prime mortgage crisis. In particular I'm looking for an authoritative book or books that provides a multifaceted approach such as

overview of the tools that were used (such as mortgage backed securities (MBS), collateralized debt obligations (CDO)).
Role of investment banks and predatory lending techniques.
Role of derivatives of the securities.
Role of  government regulations and government entities (Fannie mae freedie mac).
Role of crediting agencies such as Moodys S&P
Role of regulators (SEC) and how they failed to see the bubble.
Role of federal reserve.
What were the economic indicators that were picked up by people who saw the bubble early on?

I would appreciate if someone could tell me about books that provide a generic overview and books that go into great depth

Trying to think of what's on my bookshelf... this is an incomplete list, but it should get you started.

The Financial Crisis Inquiry Commission Report is the authoritative overview of the housing crisis and its role in the financial crisis. It's unique in that the staff of the FCIC was able to interview essentially all the major players and had access to documents that were (and many of which still are) unavailable to the public. It's also free.

The American Mortgage System: Crisis and Reform by Susan Wachter (editor) is a collection of essays that provide a good overview of many of the main issues.

For a lighter read, All the Devils Are Here: The Hidden History of the Financial Crisis by Bethany McLean and Joe Nocera is good, but flawed.

On Fannie Mae and Freddie Mac, I'd recommend Guaranteed to Fail: Fannie Mae, Freddie Mac, and the Debacle of Mortgage Finance by Acharya, et. al.

House of Debt (Mian and Sufi) isn't bad on some of the causality, but I find that it goes a bit off the rails when it comes to policy prescriptions.


«public-sector debt must now also include the central bank’s sharply rising debts». It didn't before? In this text by Carmen Reinhardt, there's the following sentence:

  For Greece, Italy, Portugal, and Spain, public-sector debt must now also include the central bank’s sharply rising debts.

What does she mean by this? Isn't all public debt usually considered?


What is the difference between monopoly profit and monopoly rent? I understand that monopoly profit is the return on capital (=profit) of the monopolist, which is larger than the normal profit in a competitive market.
As monopoly rent I understand the income in excess of the factor cost (=rent) of the monopolist.
In this sense both would refer to the same thing.
Is this correct?

Economic (not just monopoly) rent is what is otherwise known as produces surplus. And producer surplus is related to profit via

$$\text{Producer Surplus}=\text{Profit}+\text{Fixed Cost}$$

assuming that fixed cost is sunk.


Set of utilitarian weights I am wondering if there are literature which studies specific properties of the set of utilitarian weights?

To be more specific, when we have some axioms that characterize the preference of a social planner (and  gives a utility representation), we can usually say the set of utilitarian weights is convex and closed. But are there previous papers studying more properties about the set?

I am grateful if some one can point me out some related paper regarding properties of set of utilitarian weights.

One is Relative Utilitarianism (RU). Under the axioms below, society's preference can be represented by the simple sum of individuals' vNM utilities (each normalized to between $0$ and $1$). That is, every individual is given equal weight.

Pareto Axiom. (This is just the usual: If everyone prefers lottery $p$ to $q$, then so too does society. And if everyone strictly prefers lottery $p$ to $q$, then so too does society.)
Separability.
Invariance.
Anonymity. (Again, this is just the usual, though a small technical adjustment may be necessary.)

The original papers on RU are Dhillon (1998) and Dhillon & Mertens (1999). However, the first contains fatal mistakes and the second is indecipherable. See Börgers & Choo (2017) for a simple and correct exposition/proof of RU.

(Of course, the literature on utilitarianism and what you ask is vast. This is just one small example.)


Book recommendations on empirical methods in economic research and econometrics? This question is more so geared toward anyone with research experience within economics, though of course, anyone is welcome to respond.

I'm an undergraduate math major with a minor in economics. I'd like to know which textbooks are decent for introducing one to research methods in economics? Graduate level is fine, perhaps even preferred. If a student was to take his research to "the next level" (publishing worthy), which textbooks would provide the rigor and insight to develop said research?

Also, any introductory recommendations for econometrics?

Thank you in advance.

ANNENDUM:
From the limit experience I have in economics, I've found the following areas to fascinate me on multiple occasions (in no particular order):

Game theory
Macroeconomics (monetary policy)
Microeconomics (i.e., commodity pricing)
Pricing theory

Firstly, for basic introductory econometrics, the following is quite good:

  "Introduction to Econometrics" by Stock and Watson.

It is light on technical details, and heavy on intuition. This might not appeal to you if you are a math major, but it's the most important thing to get straight if you are interested in applying the empirical methods rather than developing new econometric methods.

Stock and Watson is very basic, so you will probably want to supplement it. Good choices are likely to be

  "Mostly Harmless Econometrics" by Angrist and Pischke

(this is another book that puts intuition first, but now with more formal detail), and for an advanced graduate treatment

  "Econometric Analysis of Cross Section and Panel Data" by Wooldridge.

I don't know Wooldridge very well, but I know that his book is popular and well-loved among econometrics grad students.

These books have a bias in favour of microeconometrics and deal less, if at all, with time series methods. More comprehensive coverage can be found in

  "Econometrics" by Hayashi

or

  "Econometric Analysis" by Greene.

The book by Hayashi is probably better for self-study; Greene is more of a comprehensive reference book.


What does pilot price mean in the context of marketing? In my lecture slide, it is said to be one of the competition-based price determination.

When I searched online, I was unable to find anything relevant.

If the phrase pilot experiment
or more famously television pilot is any indication, then pilot price is probably an initial price, one that the company experiments with when trying to find the ideal price of their product.


Treatment of natural disasters and wars in consumption of fixed capital Assets destroyed by natural disasters and wars are not included in consumption of fixed capital (CFC) in the US NIPA or the UN SNA.
Suppose though that the US decided to start including such asset destruction in CFC, and revised all past NIPA data to reflect this.
Certainly CFC will be revised upwards. My question is the following. Which of the following is correct?

A: Nominal GDP and nominal gross investment will stay the same, but nominal NDP and nominal net operating surplus will fall by the increase in CFC.
B: Nominal GDP and nominal gross investment will increase by the increase in CFC. Nominal NDP and nominal net operating surplus will stay the same.
C: Some mix of A and B.
D: Something else entirely.

In other words, is the same period investment undertaken to undo the natural disaster's destruction already included in gross investment (option A), or is it counted as intermediate consumption (option B)?
(It does not matter if the investment to undo the destruction does not happen instantly. In the second case, gross investment can end up being negative in the period of the disaster.)
I would have assumed it was obviously option A, but this sentence on the CFC Wikipedia page led me to doubt myself:

Ultimately, the distinction drawn between that portion of business operating expenditure included in CFC beyond depreciation, in contrast to that portion of operating expenditure included in Intermediate consumption may appear rather capricious, rather than theoretically justified.

A reference giving a definite anwer to this question would be particularly appreciated!

I am answering this both as a PhD in economics and as a Certified Accountant (I am both).
This quote for the wikipedia page is garbage, because it comes after the article has correctly informed the reader that CFC is an estimate that attempts to capture the economic reality rather than the accounting convention as regards consumption of capital for productive purposes.
If we characterize such estimates as "capricious", then all economic theory and econometric practice is "capricious". Certainly there are people who believe that, but one could imagine that we don't think highly of such opinions in this forum. I don't think such across-the-board unsupported criticism can have, or deserves to have, a "definite answer" from some "respectable source". In other words, I do not have to prove that I am not an elephant. Let the other side prove (not just argue) that I am.
More over, the sentence seems to imply that accounting/tax data are somehow "true to the essence of things". My objection to this is not related to the possibility of falsified records to avoid taxation or boost profits etc. This phenomenon exists but all past experience says that it does not exist to such extend as to make accounting/tax data unreliable on the aggregate.
So assume perfectly "honest" accounting data. In what sense are these data true to the essence of things? What is the essence of things here? Is the historical cost (which is a certain piece of information but referring to the Past), relevant for the Present?
Also, the passage appears to ignore that modern official accounting standards (both FASB and IFRS) advocate more and more for using "fair values"/estimates related to balance sheet assets than historical costs, exactly in an attempt to capture the economic reality of things, and in full knowledge that this will lead us to estimated Financial Statements, with a degree of uncertainty, inaccuracy, etc... So what? The purpose of Financial Statements is not perform or reflect account-settling, but to be the base of financial and economic analysis.
We "settle accounts" with the government through the filing of tax returns forms. We settle accounts with customers, suppliers, personnel, banks, etc by using the accounting ledgers, journals, the detailed bookkeeping and records of financial transactions. But we adjust all these data to produce Financial Statements, to the degree that the transactional reality differs from the economic reality. The fact that we may disagree as to what is the economic reality after all, should not surprise us, nor hinder us in the pursuit of relevant knowledge. It will not be the first "reality" that we disagree on.


First order condition of log functions in general and interpretation In the following, a first order condition for a log function is calculated.
I know how the left part was calculated, however, I am a little bit confused where the gamma on the right site comes from without further knowledge about the concrete function of N. Could someone explain?
And why can one interpret these findings as elasticity?
Info: R(a) is the ad revenue for a platform i dependent on ad level a. N is the number of viewers which depends on the ad nuisance of platform i and the other platforms -i. The maximisation problem is to choose optimal ad level a.

It just comes from the derivative of profit function. I assume that $a_i$ is the choice variable here so the derivative of $\pi$ wrt $a_i$ is (step by step):
$$\frac{\partial \pi}{ \partial a_i} = \frac{\partial \pi}{ \partial a_i} [ \ln R(a_i) ]  +  \frac{\partial \pi}{ \partial a_i} [ \ln N_i(\gamma a_i, \gamma a_{-i}) ]  \\
= \frac{1}{ R(a_i)}   R'(a_i)  +  \frac{1}{ N_i(\gamma a_i, \gamma a_{-i})}  N_i'(\gamma a_i, \gamma a_{-i}) \gamma $$
The gamma at the end pops out because of the chain rule. If you have $G(H(F(x)))$ then $\frac{dG}{dx} = \frac{dG}{dH} \frac{dH}{dF}\frac{dF}{dx}$. You just simply have to realize that $\gamma a_i = F(a_i) $.
Lastly to derive FOC you just set the last equation above to zero and you are there:
$$\frac{1}{ R(a_i)}   R'(a_i)  +  \frac{1}{ N_i(\gamma a_i, \gamma a_{-i})}  N_i'(\gamma a_i, \gamma a_{-i}) \gamma = 0 \\ \frac{ R'(a_i)}{ R(a_i)}    = - \gamma \frac{ N_i'(\gamma a_i, \gamma a_{-i})}{ N_i(\gamma a_i, \gamma a_{-i})} $$
To sum up, the $-\gamma$ is there because of the chain rule, nothing special aside of taking derivative is going on here.
PS: I just realized that I forgot to explain why you can interpret them as elasticities but great +1 answer of Herr K. already did that so I wont unnecessarily repeat it here.


How does lowering a corporate tax rate benefit a nation? When governments talk about about lowering corporate tax rates they claim there is benefit in one form or another. The problem is the mechanism of how the benefit can work is never explained.

The closest I've seen is that it will result in more investment but often the corporate tax is only after an organisation has made a profit.

Some sort of worked example of how this may affect decision processes or some analysis of changes that have occurred when companies have lowered corporate tax rates will also be nice.

I. The case for government revenue

The rationale behind this argument is that it it possible to increase the tax revenue by lowering the tax rate. A. Laffer draw a curve which represent this:

The idea is that if your tax rate is above $t^*$, then taxpayers try to evade taxes, develop the shadow economy, avoid taxes, or simply reduce the amount of work that they do because their is a lesser incentive to do more when the state takes a bigger share of the reward.

So the debate is therefore to know whether we are already beyond this optimal tax rate $t^*$ or below. Emmanuel Saez and Thomas Piketty argue that this optimal tax rate is about 70%. Others, such as Arthur Laffer, believe that it is much lower, and that the government revenue would actually increase if the tax rates were lower.

II. The case for growth

Lower tax rates can also be adopted in order to increase the attractiveness of the country to foreign investors, and hopefully develop the economic activity. Many countries have been known recently to struck tax deals in order to attract business: The UK with Google, Ireland with everybody, Luxembourg with Amazon and others, etc.

Similarly, developing countries tend to create special economic zones where foreign firms can develop their activities and build factories in return for lower taxation.


Can a complex interaction term mean more than what it's composed of? I'm cross-posting this question on both Economics and Cross Validated to get answers from a different perspective on each field. It is generally accepted to cross-post if the question is tailored to fit better in each community. See: https://meta.stackexchange.com/a/64069/510233

Cross-posted in Cross Validated: https://stats.stackexchange.com/q/403213/243829

I'll try to be as detailed as possible to best deliver what I want to know in this question.

In economic data, there are some variables that are composed of many other variables. Mostly indices or multiples. I can think of some examples like CPI, Consumer Sentiment Index, or even EV/EBITDA, PER.

Better and more general example would be something like Wind chill Temperature Index which is

where:

WCI = wind chill index, kcal/m2/h
v = wind velocity, m/s
Ta = air temperature, °C

What I want to know is whether a variable like WCI can have more meanings(effects) than v and Ta seperately combined.

Based on what I learned in regression analysis from Ecnonometrics class, a simple interaction term like  would be interpreted as the difference in marginal effect between a control group (D=0) and a treatment group (D=1), when expressed like:

Also, more general interaction term like VariableA * VariableB would be interpreted as an impact of VariableA on the coefficient of VariableB, in some examples like

PollutionLevel = B0 + B1*Population + B2*NumberOfCars + B3*Population*NumberOfCars
= B0 + B1*Population + (B2 + B3*Population)*NumberOfCars

(originally from a question posted on Cross Validated)

But I think these interpretations are too simple to capture the whole meaning(effect) of a new variable derived from other variables, which leads to other questions like:

What would be the interpretation of a super complex interaction term like WCI in regression perspective? (and other data science methods like Random Forest or Deep Neural Network, etc if possible.)
Would it still be valid and preferable to include an explanatory variable like WCI in the model when I already have v and Ta? Would it make the model more accurate?
Intuitively, I feel like people would care more about important indices but not necessarily the variables it is composed of. Can a complex interaction term derived from other variables have more effect than a simple combination of the other variables?

The third one is related to my original question of whether one variable that is composed of other variables in a dataset can have more effect on the dependent variable.

One example I can think of is the steepness of the yield curve that is considered very important in the bond market, which is a mere slope of the interest rates of bonds with different YTM.

People seem to care about the slope of the yield curve more than individual rates of each bond so I think it is a legit explanatory variable but it is unclear whether it is justified to introduce this new variable from existing variables or how it should be interpreted.

What you describe as "complex interaction term" is typically referred to as composite variable.  Interaction terms are products of two or more variables.  You also use terms like "effect" and "meaning", which in more technical sense, seem to refer to issues of unbiasedness, accuracy, and interpretation.

If a composite variable is a linear combination of other variables (e.g., their sum), then including that variable over its components will not change the predictive power of a linear regression model, though it will change the interpretation of regression coefficient(s).  If it is a non-linear combination, such as the case with your WCI variable, then you will obtain different predictions in a linear regression model.  If you are interested in interpreting the effect of WCI on your dependent/outcome variable (or if you believe it lies in the causal pathway and affects the interpretation of another variable of interest), then you should include WCI in the model.  The interpretation will be standard for regression models, i.e., by how much the value of dependent variable is expected to change for a one-unit increase in the value of WCI, holding other variables constant.  If, however, you are interested in the effect of wind velocity or temperature on the outcome, then you will have a hard time interpreting the results with WCI used as explanatory variable.  Including interaction terms or composite variables can also lessen the precision of your coefficient estimates, just like it does when you have multicollinearity.

In purely predictive (machine learning) models, you are most interested in the predictive power of your model, not in the interpretation of variables.  Those models readily accept "engineered" variables (hence, "feature engineering" is super important for models like random forests, though not for deep learning).  Accuracy will depend on a lot of factors, but using composite variables can certainly help.


R's felm() fixed effects vs. factor() within lm() I am wondering why using felm's fixed effect option versus using the factor() function is different? I was under the impression they were one in the same?
I am running a panel regression where i is the a unique FIPS code and t is every quarter. The variable of interest, a binary of whether the FIPS flooded or not, is constant across time. However, my y, housing prices, varies each with each t. When I use felm()'s fixed effect option for FIPS, flooded is NA. However, using factor(FIPS), gives an estimate for the variable of interest.
I'm under the impression felm()'s fixed effect function is working correctly. Since being flooded is time constant and has no variation within a given FIPS, the fixed effect is absorbing the effect of flooding. However, I'm not sure why factor(FIPS) within the regression would return an estimate since a fixed effect essentially the same thing?

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

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

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

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

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

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

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

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


Are government bonds internal or external debt? I was watching this video and since it's a bit oversimplified I wanted more information.
I was curious because internal debt is in the local currency from local lenders, while external debt is in a widely-used currency from foreign lenders.
Technically the government bonds market has foreign and local lenders so are govt. bonds both external debt and internal debt? At the same time, the bonds are traded in local currency, I assume, so are govt. bonds internal debt then?
If someone could clarify it would be much appreciated. Thanks so much for your time!

Here’s a website from the IMF that has guides on their definition of external debt: link to IMF webpage.
In the 2013 guide, on page 5, the definition is debt owned by foreign entities. So foreign-owned bonds count as external debt.
From a practical perspective, this definition has problems. Tax avoidance schemes typically rely on using international transactions, so determining the true locale of the beneficial owner of securities is difficult. The other major issue is that for developed country government bonds, being a foreigner offers no special treatment - they have exactly the same economic rights with respect to the bond as locales. As such, some people do not consider domestic currency bonds as being “external debt”, but that is using a new definition.


Return on two different ways to construct a portfolio I run the following regression:

$$r_i-r_f=a+b(r_m-r_f)+e
$$
Now I see that the alpha is not zero and I construct two different zero cost portfolios:

1) 100% long in ($r_i-r_f$) and -b short in ($r_m-r_f$)

2) 100% long in $r_i$ -b short in r_m and borrow/short (1-b) on$ r_f$

I have problems determining the expected returns of the two portfolios and determining if they are the same.

I would appreciate your help a lot!

I'm not sure I follow, but do the two portfolios mean the following?

For portfolio (1),
$$
r_i - r_f - b(r_m - r_f) = r_i + (b-1) r_f - b r_m
$$
and for portfolios (2)
$$
r_i - b r_m - (1-b) r_f = r_i + (b-1) r_f - b r_m.
$$
So, writing them out like this, they look the same.


Are Microeconomics limited to one country? For my IB Econ internal assessment I want to comment on the falling oil prices across the globe.

Would this be covered under Micro because it's 1 specific market? or is it international because it comes after OPEC lost it's influence on the market?

It is my impression that this would  fall in Intrernational Microeconomics, which is one of the two main subfields of Interrnational Economics. Essentially it is the modern "International Trade", where the prevailing features of the modern era (less trade barriers so less room for state policy, larger trade flows so increased degree of activity at firm-level, large private global players) make it acquire more "micro" aspects compared to the older International Trade field with its preoccupation with customs and tariffs (although these still exist of course).

It cannot be categorized as a "macroeconomic" subject, because macroeconomics always looks things from a general equilibrium (and sometimes disequilibrium) perspective -but a general one, not partial. And a cross-section of markets of the same good is a fragmented but single market of the world economy.

You may be talking about whole countries, like OPEC countries, but you will essentially consider them as producers of a specific product, not as national economies.


Structural Modelling of Expectations/Beliefs I'm trying to build a structural model explaining the job search process by the job-seeker. One of the crucial component of job search, that i want to model, is the expectation about the employer. So things like a worker will not apply for job to the employer if he thinks/believe (correctly or incorrectly) that employer is racist, or if we thinks/believe that this job won't pay him well. I want to model these expectations and don't know where to start.

I will really appreciate if someone can help highlight few references in this regard and offer any useful suggestion that can help me build the structural model and eventually estimate parameters of interest.

Check for example Postel-Vinay and Robin (2002) and references therein to see how people go about structurally modeling the labour market.

The general framework is a dynamic one, where the job-search process is a Poisson arrival process for each worker, and the results are derived by finding the market equilibrium conditions.

The interesting point in this literature is that indeed labour market decisions are based on expectations about the employers, but here specifically the expectation about how career trajectories diverge and what this means for reservation wages for the employee, and how this depends on the productivity of firms.
In the cited paper, the productivity determines what a firm may be able to pay eventually. It raises wages when competing offers arrive by firms that may theoretically also employ the worker.

The worker forms expectations about his future trajectory and this implies that he is willing to work for a high productivity firm for less money, simply because he expects it can raise wages in the future if competing offers arrive.

So you see that expectations are indeed important for the job-search process. For your idea, you have to be specific as to what the worker expects if a firm is racist. An interesting idea would be to think about blacklists and stigma, which would mean that the worker could be locked-in the current firm. This would require the firm to offer higher wages compared to others.
It's a straightforward mechanism, and interesting I think, but you need to show basically how this impacts the workers choices based on preference (since you are doing structural modeling). It may be a good idea to start with reduced form analysis, and also to check out the sociology literature as I am sure this sort of idea has been used before. Your contribution would be to model it and get exact parameter values, run counterfactuals and hopefully discover some surprising results.

What worries me a bit about the idea is that you are somewhat missing an objective for the firm to be racist, given that it optimizes everything else in equilibrium in this setup. Of course you may assume it the same way one assumed productivity, and while it may be econometrically sound, it is not really identified in a theoretical sense. What I mean is that racism would just be a parameter akin to pushing down what a worker expects from a firm in terms of future productivity. And since it is outside of the explanatory power of a structural model, it may be that structural modeling is not really the necessary approach - a reduced form will tell you as much.
Perhaps you can make it bite, by making the stigma structurally different from lower productivity, and perhaps exactly this matters. This would be where the result would be interesting to me.


What benefits does Bitcoin (i.e. cryptocurrency) offer? I really, really like moderate inflation. I live in Sweden where the interest rates just went negative because we are on the brink of deflation, and the rate of inflation we have now is far too low. We have huge problems attracting foreign attention and the swedes don't want to take out loans to invest since paying them back ten years down the road hurts just as bad as paying them now (and there's a real risk it might take more to pay them back later, if we go into deflation). It's a real head ache that's all the talk in our financial world.

Getting a hold of Bitcoin is very, very hard and expensive. The value of it has deflated dramatically from when it was introduced, and despite occasional crashes, it could very well just keep climbing in value. It also has a cap, which to me looks like it would unleash a deflationary spiral that would only end once we find a new currency (like the Great Depression was alleviated when we stopped using the gold standard).

But, I'm a computer scientist and I just can't stop finding bit coin supporters everywhere. It has thousands of evangelical missionaries using blogs and magazines to tout it as a means to liberate us from the bankers. I don't know, can't computer scientists be just as corrupt? It looks like dot coms with access to good cryptographic algorithms and server farms who can work in parallel could game the system even more, and once they've swept up all the bit coins they would wield more power than nation states.

Am I missing something? What does bit coin bring to the table that is worth disregarding deflation? Does it bring positive GDP growth? Ease of investment? Monetary power to the people? How? And, I keep coming back to this, wouldn't all these positives be completely wiped out once the 30s come knocking deflation style?

Why does this currency have so many supporters? Are everyone mad?

I'd say that some benefits of Bitcoin and other cryptocurrencies (i.e. "altcoins"), include:

Decentralization -- which is advantageous for those who are skeptical of monetary policy, runs on banks, etc.
Anonymity, or at least the perception thereof -- with increasingly comprehensive monitoring of transactions (particularly non-cash transactions), those seeking relatively anonymous methods of exchange have traditionally been out of luck in the realm of digital transactions. Bitcoin has widely been regarded as a "digital cash", providing some measure of relative anonymity in online transactions, though this is a decreasingly accurate perception as government databases cataloging transactions grow in sophistication (and some of these databases, such as that of the FBI's, have been opened up to public developers via coding marketplaces like Kaggle). Some altcoins, such as Monero (XMR), add measures to enhance the benefit of relative anonymity.
Portability -- they can be easily stored and transported offline (paper-based hashes or QR codes), on portable media, over the Internet, with or without the aid of a "wallet" or bank-like service.
Region Neutrality -- unlike fiat currency or Eastern-only currencies such as Perfect Money, cryptocurrencies seem to have less bias based on location or language.


Can I rename a utility function based on its properties? A big name researcher gives a name to a specific utility function 30 years ago. Now I am writing a paper and feel that a new name might be more suited because of the properties associated with the utility function.
That name was not heavily cited.
Can I give it a new name, or will the readers find the new name weird, overwhelming, and confusing?

You can do whatever you want, it's your paper. Will it make it more difficult to publish? Yeah. Referees are fickle and easily annoyed.
You would not be the first person to go down this route. For example, the term Maskin monotonicity is common place in implementation theory. However, some authors use another term, I believe Maskin invariance. The claim is that invariance is a more appropriate term for the definition. Its worth noting that all of these authors had the same advisor who is a giant in the field.


What are good books about financial markets? I found answers on this question online but people always recommend books for beginner investors who want to become rich. I just want book that is about history and role of financial markets and book that explains how financial markets work.I don't want to read how to become a millionaire by investing.

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

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


Does any international organization (such as IMF) provide data on capital control / financial openness? I've read some literature on measuring capital control / financial openness, but I am wondering if there is any data directly collected by international organizations like IMF that measure capital control.

Indeed the IMF reports on this. A recent review-overview paper is

Quinn, D., Schindler, M., & Toyoda, A. M. (2011). Assessing measures of financial openness and integration. IMF Economic Review, 59(3), 488-522.

In the introduction, the authors write

  "The aim of this article is to help researchers better understand the range of choices they have in measuring financial integration and
  globalization, the pros and cons associated with each, and some of the
  reasons behind the divergence in findings in the literature. In
  particular, it describes de jure, de facto, and “hybrid” indicators,
  and comparatively analyzes their data properties and how these
  measures relate to one another"

The main "De jure" (as the authors characterize it) resource (on which many scholars have built various indexes using wildly different methodologies) is the IMF's report Annual Report on Exchange
Arrangements and Exchange Restrictions (AREAER)

As the authors write, after presenting the various "De jure" indices,

  "De jure indices of financial globalization do not reflect the extent to which actual capital flows evolve in response to legal
  restrictions, either because of a lack of enforcement, or because
  controls in one area may induce a response in other asset flows. Also,
  even the more disaggregated indices may not capture subtle, but
  possibly important differences between countries’ capital control
  regimes. De jure measures, therefore, do not necessarily reflect a
  country’s actual degree of financial integration, highlighted by the
  fact that even countries with relatively closed capital accounts
  became substantially more financially integrated over the past
  decades... Thus, "de facto", or in some cases “blended,” measures
  present an alternative way to measure a country’s integration into
  global finance markets. These can be divided into three categories:
  quantity-based, price-based, and hybrid measures."

They then move on to a detailed study of many of all the above indices, and they conclude (they have also conveniently summarized qualitative assessments of each index in tables):

  "A key result is that most of the measures provide information that is linked in a meaningful way to economic outcomes. The exception is
  IF-Heritage, an indicator that we found not to be correlated with
  other indices, to have no measurable impact on economic growth, and to
  be linked in the factor analysis to unique dynamics that are not
  easily interpretable. Correlation and factor analyses suggest that
  investigators using de facto indicators of financial globalization
  will find differing identifying variances from those found in the de
  jure measures. In part, this is because de facto measures likely
  reflect the influences of many political and economic factors, of
  which legal restrictions of the capital accounts, as indicated by the
  de jure measures, are but one."

and

  "The bottom line of this paper is that researchers in this field have an unusually large choice of indicators, most of which are valid, with
  unique advantages and disadvantages."


Does quantity demanded refer to all consumers or individual consumers? When looking at points along the demand curve, is each quantity corresponding with each point referring to the total quantity demanded by all consumers in a market or the quantity demanded by individual consumers?

I am looking at the market supply and demand graph and the graph for each individual firm in perfect competition (the horizontal demand).

If it’s a “market demand curve,” that means the quantity demanded is that of the entire market. If it’s an “individual demand curve,” it’s the quantity demanded of an individual consumer. If you open an Econ 101 textbook and look at the supply–demand equilibrium diagram, they’re generally talking about market demand and supply curves.


Could you recommend a book or lecture notes about time series that is easy to understand? What are some time series texts that you would recommend to start studying? Some easy textbooks/lecture notes or step by step for undergraduate level.
I'm not so good at math/statistics.

The book I currently use for my time series class is "Time Series Analysis and Its Applications With R Examples" by Shumway and Stoffer, 4th edition. It's served me fine, and gives lots of example codes that you can plug into R and see what happens with example data.


expected payoff of an American put option I am reading about the American option and sources got me confused in the part where the American put option is considered on a non-dividend paying asset.

I understand that the payoff of an American put option at maturity is given by $$\max\left\{K-S(T),0\right\}$$ where $K$ is the exercise price and $S(T)$ is the price of the underlying at maturity.

Some sources state that considering n-arbitrage, the American put option problem aims to find the option value $$P(S,t)=\sup_\tau EG_\tau$$ where $G_\tau$ is the gain function and E is expectation.

In my understanding, the gain function is equivalent to the payoff function, hence $$V(S,t)=\sup_\tau E[max\left\{K-S(\tau),0\right\}]$$ for all stopping times $\tau$.

However, I also stumbled upon sources that say that $$(S,t)=\sup_\tau E[e^{-r(\tau-t)}\max\left\{K-S,0\right\}]$$ for all stopping times $\tau$.

What could be an explanation behind this?

An explanation behind this could be that, as you mention

  [...] the payoff of an American put option at maturity is given by
  $$\max\left\{K-S(T),0\right\}$$

and

  [...] sources [...] say that $$(S,t)=\sup_\tau E[e^{-r(\tau-t)}\max\left\{K-S(\tau),0\right\}]$$

Thus the difference between the two appears to be the discounting factor considering how far (in time) we are from the exercising/stopping date (which is the same as the maturity date if $\tau=T$), which implies to compute (continuously) the present value of this futur payoff. If  $(\tau-t \rightarrow0) \land (\tau=T \lor t \rightarrow T)$, both equations become equal.

If one decides (for some reasons) that the option will be exercised at maturity, i.e. $\tau=T$ and that "now", $t$, is "turning into this futur date $T$", i.e. $t \rightarrow T$, one gets

$T-t \rightarrow 0 \implies \sup_T e^{-r(T-t)} \rightarrow 1$

$\implies \sup_T E[e^{-r(T-t)}\max\left\{K-S(T),0\right\}]$

$\rightarrow  E[\max\left\{K-S(T),0\right\}]\rightarrow\max\left\{K-S(T),0\right\}$

Where the last above row shows that the expected payoff becomes certain.