consequences of an inheritance tax on family owned businesses Whenever I see the pros and cons of an inheritance tax discusses one common argument against it is as follows:

If an inheritance tax is imposed on transferring a family owned business, then this tax will have to be taken from the cash flow of the business. This will often bankrupt the business and therefore also destroy all the jobs associated with it. Hence an inheritance tax is (on businesses) is a very bad idea.

However I think this argument is based on a simply fallacy, namely that the business has to pay the inheritance tax, whereas in fact the new owner has to pay it.

So suppose the inheritance consists of exactly the family business, previously owned to 100% by the patriarch, and there is a 30% (of the value of the business) inheritance tax. Now the successor has multiple options:

take the tax from the cash flow of the company, this is usually very destructive to the business and hence a bad idea
sell a 30% stake of the business for cash to some other investor and use the cash to pay the inheritance tax
get a credit for 30% of the company value with the business as security from some bank and use that to pay the inheritance tax

Neither of the other two options takes any cash out of the business directly, so there is no danger for the business or the jobs attached. All that changed is that the son only has about 70% of the wealth of the patriarch.

Question: Is the initially stated argument a valid argument against an inheritance tax and I overlooked something? Or is it just a plausible sounding straw man that doesn't hold upon closer inspection?

Note that I'm not interested on whether an inheritance tax would be a good or bad idea in general, that would be opinion based and not suitable as a question here.

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

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

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

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


Nash Equilibrium for n-shops Location Game So if two ice cream shops were to be placed in the location $[0,1]$, inorder to maximize their own pay offs, they both would finally come to the location $[\frac{1}{2}, \frac{1}{2}]$. This is also the Nash Equilibrium of the problem and does not require a lot of mathematics to understand.
Now my problem was when we are playing this game for 3 different shops, like there is no intuitive answer for the problem. Every possible answer I thought of ($[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}]$, $[\frac{1}{4}, \frac{1}{2}, \frac{3}{4}]$ etc.) is not a Nash Equilibrium.
So like is there a mathematical way that I can use to find the Nash equilibrium location for this game, or prove that it doesn't exist.
Also can this be generalized to $4,5 ...n$ games?

The following two claims hold in the general $n$-shop case.
Claim 1. In equilibrium a shop closest to an edge (0 or 1) cannot be alone.
Proof. Such a shop could gain customers by moving slightly inward.
Claim 2. In equilibrium at most two shops can be in any location.
Proof. Assume there is an equilibrium where there are three or more shops in a location. Denote the number of customers coming to this location from the left and right by $c_l$ and $c_r$ respectively. (In the 3-shop case $c_l + c_r = 1$.) All shops in the location have a payoff of $(c_l + c_r)/n$. By moving slightly to the left, a shop can attain a payoff that is arbitrarily close to $c_l$, and by moving slightly to the right a shop can attain a payoff that is arbitrarily close to $c_r$, therefore it can attain (almost) $\max(c_l;c_r)$. It is easy to show that if $n >2$ then
$$
\frac{c_l + c_r}{n} < \max(c_l;c_r).
$$

Using these two properties it is very easy to show that in the 3-shop case there is no pure Nash-equilibrium, so this is left as an exercise.
I happen to have a Desmos for this, if you'd like you can use it to validate the claims and reason about the proof.

Note: a mixed equilibrium does exist, see Shaked, A. (1982): Existence and Computation of Mixed Strategy Nash Equilibrium for 3-Firms Location Problem.

Interestingly, $n = 3$ is the only case without a pure equilibrium, $n \in \left\{2,4,5\right\}$ all have one, and for other $n$ values there are infinitely many equilibria. For a formal characterization one needs some additional properties, but no advanced mathematics. For a detailed discussion see
Eaton, B.C., and R.G. Lipsey (1975): The Principle of Minimum Differentiation Reconsidered: Some New Developments in the Theory of Spatial Competition.


Intertemporal budget constraint notion Let $W$ denote the investor's wealth $B$ is the the investment in the risk free asset (suppose that the risk free rate is taken to be zero) and $X$ is the investment on the risky security which has price $P$. Se the investors total wealth is:
$$W=B+PX$$
The capital gain that is generated in some some time interval $[t-1,t]$ is about $X_{t-1}(P_t-P_{t-1})$ and the portfolio rebalancing then occurring to the intertemporal budget constrained is:
$$B_t+P_tX_t=B_{t-1}+P_tX_{t-1}$$
I wonder why does the last equation hold? Does this come from the intertemporal budget constraint and how?

In the sequel it refers to the wealth change that is:
$$W_{t}-W_{t-1}=X_{t-1}(P_{t}-P_{t-1})$$

$\underline{Note:}$ It is from Kerry Back 1992 paper

At time $(t-1)$, the investor buys some risk free bond, $B_{t-1}$ and some risky asset $X_{t-1}$ at price $P_{t-1}$, such that the budget constraint holds, i.e.
$$
W_{t-1} = B_{t-1} + P_{t-1}X_{t-1}
$$
At period $t$, one unit of risk free bond pays off one unit, so $B_{t-1}$ units of risk free bonds pays off $B_{t-1}$ units of wealth at $t$.

For a risky asset, one unit of $X$ is now worth $P_t$. Since the agent holds $X_{t-1}$ units of the risky asset, she effectively has wealth $P_tX_{t-1}$ from risky assets.

Adding the two gives us the budget at period $t$, i.e.
$$
W_{t} = B_{t-1} + P_t X_{t-1}
$$
And period $t$ portfolio choice satisfies
$$
W_t = B_t + P_t X_t
$$
Combining the two gives us the desired equation.


What is "inter-temporal smoothing"? In a recent article about the OECD latest report it reads as follows:

  Mr White said QE and easy money policies by the US Federal Reserve and
  its peers have had the effect of bringing spending forward from the
  future in what is known as "inter-temporal smoothing". It becomes a
  toxic addiction over time and ultimately loses traction. In the end,
  the future catches up with you. "By definition, this means you cannot
  spend the money tomorrow," he said.

What does he mean by "inter-temporal smoothing" and "bringing spending forward". How do you "bring spending forward"?

Intertemporal consumption smoothing is using savings or borrowing to make consumption less volatile than income (or sometimes wealth). The intertemporal consumption smoothing  motive can arise in many settings, but an early and famous example is the Milton Friedman’s permanent income hypothesis. The idea is that if income contains permanent and temporary shocks then maximizing the NPV of lifetime utility will make you ignore temporary shocks and consume a fixed fraction of your permanent income.

Essentially, you know temporary shocks are temporary. If you cut back on consumption in the face of these temporary shocks you'd end up with consumption that is higher than average in better times. Because of declining marginal utility of consumption this is a worse consumption plan than keeping your consumption fixed in good times and bad.

In contrast, if you experience a permanent negative shock to consumption there won't (in expectation) be a good time to make up for the lost income. Leaving your consumption unchanged in the present would therefore require even larger cutbacks in the future. This is a worse consumption plan than cutting your consumption immediately and using that new and lower plan going forward.


Decline in the Value of the Domestic Currency I am using Mishkin's textbook "Economics of Money, Banking and Financial Markets." On the chapter on financial crises, the authors list a number of factors that could trigger a crises.

One factor is An Unanticipated Decline in the Value of the Domestic Currency. I quote "Because of uncertainty about the future value of the domestic currency in developing countries (and in some industrialized countries), many nonfinancial firms, banks, and governments in developing countries find it easier to issue debt denominated in foreign currencies rather than their own currency. This can lead to a financial crisis in a similar fasahion to an unanticipated decline in the price level.  With debt contracts denominated in foreign currency when there is an unanticipated decline in the value of the domestic currency, the debt burden of domestic firms increases.• Since assets are typically denominated in domestic currency, there is a resulting deterioration in firms' balance sheets and a decline in net worth, which then increases adverse selection and moral hazard problems along the lines just described."

I don't get the logic. The authors are a bit vague: a decline in the value of the domestic currency perhaps means a depreciation relative to other currencies. Now, if that is correct, if there is uncertainty about the future value of the domestic currency, why would firms issue debt denominated in foreign currencies? If domestic currency is expected to appreciate, then only does it make sense (forgetting uncovered interest parity for now). Am I thinking correctly about this?

Thanks!

Your question appears to be just: why would they issue foreign currency debt?

If an emerging country is growing rapidly, it often attracts capital flows. This is typically associated with a strengthening currency. Firms and investors extrapolate this tendency, and so they lend to the country in the currency of a developed economy (typically the U.S. dollar). If the local currency continues to appreciate against the U.S. dollar, it should be easier to cover the payments (since local currency earnings are worth more). Things break down if this extrapolation fails, as discussed in the question.

In terms of an example, you could look at the developing Asian countries ahead of the Asian Crisis of 1997. Borrowing in U.S. dollars by local forms was the root of the problem. The post-1997 tendency to accumulate large foreign currency reserves by central banks in countries pursuing an export-led growth strategy was a reaction to that experience.


Do rising inequalities create inflation? When the "richs"* get richers, the "poors"* get poorer... assuming:

h1) a constant volume of money,

h2) a constant volume of people,

h3) a close separation between the two groups of people,

h4) a constant influx of money from rich to poors (salaries) c1, and a constant influx of money from poors to rich (profits from stuff sold to poors) c2, such that c2 > c1

this means that the volume of money the richs have access to gets bigger over time and the volume of money the poors have access two gets smaller, therefore, also assuming:

h5) constant prices,

this means each person in this poor group has individually access to a smaller amount of money. Therefore despite a constant salary, they can afford to buy less things.

Could we therefore says that richs experiment a deflation and poors an inflation? If not, is there a more befitting name to describe this phenomenon?

Are the assumption realistics, or is the reasonment flawed?

Example:

100 persons, divided in R and P, with respectively 20 and 80 persons
1000 euros total volume of money
time t1: R has 500 euros, P has 500 euros
time t2: R has 700 euros, P has 300 euros
the prices remain constant at t1 and t2, e.g. 1 euros bread, 2 euros meat, 3 euros gas

at t1: people in P and R have on average respectively 25 and 6.25 euros

at t2: people in P and R have on average respectively 35 and 3.75 euros

therefore, despite constant salaries for people in P:

at t1 they can afford both bread, meat and gas

at t2 they can afford only bread and meat

*: I put quotation marks, because being rich or poor is subjective, but within a closed social group we can assume that such a distinction can be agreed upon.

This is not an answer that will state exactly which assumptions are valid/invalid but rather a guide to find answers to your question.

Here are various scientific articles about the topic:

These first two papers take what is called a Political Economy perspective on the matter. They assume that the central bank controls inflation, but the central bank is to some extent controlled by the population. If the population has higher income inequality, it prefers higher inflation and thus countries with higher inequality will face higher inflation.

Albanesi, Stefania. "Inflation and inequality." Journal of Monetary Economics 54.4 (2007): 1088-1114.

Dolmas, Jim, Gregory W. Huffman, and Mark A. Wynne. "Inequality, inflation, and central bank independence." Canadian Journal of Economics/Revue canadienne d'économique 33.1 (2000): 271-287.

This paper is also interesting, since it considers the different inflation faced by different income classes. It is somewhat orthogonal to your question, however.

Hobijn, Bart, and David Lagakos. "Inflation inequality in the United States." review of income and Wealth 51.4 (2005): 581-606.

However, to my knowledge, there are no papers written on income inequality as a direct cause of inflation without the involvement of the Political Economy argument above. This is mainly because in Economics our models usually tell us that the central bank has almost perfect control over inflation (except for negative inflation rates).

PS: If you find the readings above too difficult, consider reading a textbook on monetary economics:
Mishkin, Frederic S. The economics of money, banking, and financial markets. Pearson education, 2007.


Beginner level books on market economics Which books can be recommended for a beginner with a strong math background, who wishes to learn how markets work, what is a stock, what is a bond, how to trade e.t.c. and has a minimal knowledge of economics?

For introductory learning nearly any introductory economics textbook would do. For example, Mankiw's 7th edition has a section on "How Markets Work" and "Saving, Investment, and the Financial System". The latter discusses stock and bonds.


How does the pie get larger? I understand the concept of creation of wealth through trade, innovation, labor and capital. New value is being added to the economy. Let's imagine for a second that there are no central banks and the money supply is constant. This would imply that even though one can create wealth and add value to the economy the money would just be changing hands, meaning in order for one individual to get richer another would have to get poorer. We all know that in the real world central banks print money (I am ignoring inflation, nevertheless that's not the point) but that isn't the problem as far as I can tell. The current rebuttal of my example is that even though no new money is being created somehow the pie gets larger so everyone can get a larger slice. But how exactly does this process of expanding the pie happen? I can't grasp my head around the idea that the pie can enlarge while the money supply is fixed. I guess that's the same problem mercantilists had back in the 16th and 17th centuries.

(Mostly) ignore money for this growth issue; it's by and large a red herring that's distracting you.

Instead just think of technological progress for instance. Assume everyone is washing their clothes by hand. That takes a fair bit of time. Now someone invents a washing machine. Everyone (who can get a washing machine) will then have more time on their hand to do other stuff, like (say) invent and produce faster means of transportation, saving people time in yet another way and so on. All this is leads to improvements in efficiency (productivity).

If you apply this process of improvements to (say) agriculture, you get to produce more food with the same amount of resources, like with the same manpower and even the land being the same. Now there is a literal pie (of food) that gets bigger, which can feed more people. (The inventions would have to be appropriate, of course, like irrigation, mechanization [tractors etc.], or genetic selection/modification of crops.)

The additional people you can feed in turn can produce more stuff [until you run into limits on natural resources]. This is true even if you don't have technological progress. That's why demographic growth also matters for economic growth.

Of course, money becomes useful as a means of exchanging all this different stuff that people invent and produce, but it's not money that grows the economy, by and large, although financial services (including the basic presence of money) are also a way to make some processes more efficient. As an example of the latter, a loan allows someone to get machinery or even housing up-front, before they've produced enough stuff to pay for the item. That in turn buys them time (in their life) to produce stuff faster (using the machinery sooner rather than later) or even make babies, which would have housing where to live. I.e. financial services can speed up things which otherwise would happen slower or maybe not at all. (Do note that loans are money in the broad sense, in the modern economy; I'm not getting into details on that here; the first link explains it pretty well.)


Game Theory: finding nash equilibria of an extensive form game (game tree) How do you find the pure Nash equilibria from this commitment game?

The SPNE is (U, ps)

Do you find the NE by finding the best response (BR) of a player to a specific strategy of the other player?

e.g.

BR(p)={D}
BR(q)={U}
BR(r)={D}
BR(r)={U}

BR(U)={p}
BR(D)={s}

For the SPNE, you want to proceed by backward induction as explained by Lee Sin. For all other NE you want to construct the normal form representation (the usual table for simultaneous games) and solve for NE as if players were choosing their actions simultaneously. Note that the actions of player 1 are $\{U,D\}$, while for player 2 are $\{pp, pq, pr, ps, qp, qq, qr, qs, rp, rq, rr, rs, sp, sq, sr, ss\}$. This will indeed be a very large table.

Another way to do it is simply to look for fixed points directly. For example, assume that Player 2 plays $pp$, then the best response for player 1 is $D$ but if player one chooses $D$, the best response for player 2 is either of the following actions $\{ps, qs, rs, ss\}$ since $pp$ is not in this set, it is not a fixed point. iterate that process for all the actions of player 2.  Try to convince yourself that there is no other NE, except for the SPNE. And think about how would you change the payoffs so that the SPNE is the same, but now there are other NE's. (Hint: would changing the payoff of the terminal node after U,p from (2,1) to (1,1) work?)


What's holding the U.S. from decreasing unemployment by cutting H1B limits? I don't understand why doesn't the U.S. decrease the number of H1B visa holders (foreign workers) during the current economic crisis? The unemployment is skyrocketing, wouldn't this measure decrease unemployment? I'm sure there are many other economic implications, but I don't see any harm in opening up jobs to the U.S. citizens.
Recently there was a suspension of H1B applications, but I'm talking about cancelling all currently active H1B visas. Is this possible?
Or maybe this is wrong. Decreasing the number H1B workers will impact other economic factors?

The economy is not a zero sum game with a fixed number of jobs available. If it were, then restricting immigration or at least the access immigrants have to jobs would unambiguously benefit domestic workers. However, H1B immigrants, in particular, are frequently skilled workers that increase the productivity of the companies they work for, creating additional job opportunities, some of which are filled by domestic workers.
The direct effect of H1B workers is to drive down the wages and opportunities for domestic workers they compete with directly, but the indirect effect is the creation of complementary jobs, which may be skilled or unskilled. Similarly, increased immigration by low-skill workers drives down wages of competing low-skill domestic workers but increases the productivity of the economy, creating jobs (and goods/services) for skilled and unskilled workers.
A flow of new labor will benefit some domestic workers and harm others, irrespective of the current state of unemployment. Immigration is a political decision, and like other political decisions is made with an eye to who benefits from it and how influential they are on the political process, rather than a more abstract notion of what is good for "the country" or "unemployment." In other words, since some domestic workers are harmed by immigration and others are helped, you must define "good" before you can say whether H1B or any other type of immigration is good for the economy or its workers. The definition politicians use is "good for re-election," and since they have different constituencies, they disagree with each other on whether H1B or any immigration is good.


What is the different between a Sales Tax and a Value Added Tax? The USA has a sales tax which is applied at the point of sale, in states which have a sales tax. During the Democratic Party primaries, one of the candidates, Andrew Yang, proposed moving to a Value Added Tax (VAT), which his platform claims to be in place in 160 countries.
It terms of process, it seems that with a VAT, companies must charge the end tax even when selling to other companies, but only pay the tax collection agency the difference in what was collected and what was owed. With a sale tax, it seems that companies do not collect any tax when selling to other companies
Hence, what is the difference between a sales tax and a VAT in terms of the amount of taxes collected? It seems that in both cases, the taxes are only applied to the end consumer.

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


Why are my 2SLS estimates much larger than my OLS estimates? When performing IV regression, it is often taken as a sign of unbiasedness if the IV estimates are similar to the OLS estimates of the same model.  My question is:  If the estimates are very different, what possible explanations are there for the difference?

Say for example, with a strong instrument, that one finds that the IV estimates of some coefficient are 10 times larger than the OLS estimates.  Is this most likely a sign of severe bias in the OLS estimates?  Or is it just as likely that the exclusion restriction is not satisfied, and so the IV estimator is possibly even more biased than OLS?

I understand that this is an open question, but if anyone could just give me the basic range of possible explanations, and perhaps point me towards some literature on interpreting IV results when they are significantly different from OLS, that would be much appreciated?

Best regards,

Let me take an example based on the estimation of returns to education, which has been a well-studied problem. The usual result is that researchers find the 2SLS estimate to be larger than the OLS estimate by approximately 25%-50%, e.g. Card (1999, 2001).

Three reasons :

An omitted variable that could be negatively correlated with the amount of education. This omitted variable would lead to an downward bias in the OLS estimate of the schooling coefficient.
Errors in variables that arise because years of schooling are a noisy measure of education that leads to higher earnings. This measurement error in education biases the OLS estimate of the treatment effect toward zero. OLS estimates are thus too small. Since the IV estimate is unaffected by the measurement error, they tend to be larger than the OLS estimates.
It's possible that the IV estimate to be larger than the OLS estimate because IV is estimating the local average treatment effect (ATE). OLS is estimating the ATE over the entire population. By contrast, IV is estimating the local ATE: the instrument shifts the behavior of a subgroup of individuals for whom the returns to education are larger than average. In other words, the IV estimate is the effect of increasing education only for the population whose choice of the treatment was affected by the instrument, while the OLS estimate describes the average difference in earnings for those whose education differs by one year. Then IV estimates will be larger than OLS estimates because of heterogeneity in the studied population.

References

Card, D. (1999), The causal effect of education on earnings. In: Ashenfelter, O. C., and D. Card (eds). Handbook of Labor Economics Vol. 3A. Amsterdam: Elsevier, 1801–1863.
Card, D. (2001), Estimating the Return to Schooling, Econometrica, 1127-1152.


How is labor disutility modeled in Arrow-Debreu model? In DSGE models, utility function contains labor disutility components. But I cannot not see how labor disutility is incorporated in Arrow-Debreu-Mackenzie model.

In the Arrow-Debreu model, households are endowed with some commodities, producers can use feasible production plans to transform commodities, and households ultimately consume commodities - which they choose optimally given their preferences subject to budget constraints.

These commodities can include (potentially time and state-contingent) labor services provided by households. In this framework, provision of labor can simply be viewed as a form of negative consumption, which decreases utility but provides earnings that can be spent elsewhere. Indeed, the original Arrow-Debreu paper mentions this interpretation several times. For instance, on page 268 when they describe households in the model (which they call "consumption units"):


What is the economical advantage coming from the ability to speak English? I'm very much interested in the difference between anglophonic and non-anglophonic countries stemming just from the language skills. I'll be grateful for any references regarding the topic.

What is the approximate annual economical gain of a country stemming purely from investments in English language classes?

One of the best ways to study the benefit of speaking English, at least within an English speaking country, is by looking at the wage discrimination literature. Some, but not all, of wage discrimination turns out can be explained away by the ability to speak English. These studies are good to look at because of the controls for other factors. For example, it looks like the ability to speak English was a factor in this study, looking at before and after the impact of the IRCA in the US. The references I give of course are not generalizable to the whole world, because of their focus on the US. So I suggest looking more into the wage literature yourself.


PMI - Purhasing Managers Index I need the PMI time series for an econometic project but it seems I can't get it for free on the internet.

Does anyone know where can I download it for free?
Also a proxy for this series would be great.
The sample size: monthly, from 1960-70 to 2016

Such data can be found on Quandl, for example:

https://www.quandl.com/data/ISM/MAN_PMI-PMI-Composite-Index


eBay economics - total revenue impact from expanding ecosystem by 1 seller While I think my question is straight forward, I'm guessing the answer to it is anything but.  Thanks in advance for any insight you may be able to offer, even if it's just a directional assist.  Here's the question.

What is the impact on total revenue when eBay has 1 additional seller join its ecosystem with X dollars in product inventory?  How does this differ in the short vs. long term?

Acknowledgements, thoughts, and considerations:

The type of product the seller carries, number of bidders (buyers), and their respective bidder budgets all certainly play a factor in calculating the additional sellers impact on revenue.  Feel free to consider the question with the seller selling "average" product, constant bidder budgets, and number of bidders before and after the seller joins. In short, please feel free to simplify these (and any other) assumptions.
Seasonality should play a factor in the short term.  If the seller were to join in December, the impact would be different than if that same seller with the same X dollars in product joins in May.  In the long term, this shouldn't matter.
Playing with some supply and demand curves on paper tells me that the steepness/slope of the curves has a dramatic impact on the expected total revenue when multiplying P2 and Q2.  The revenue could be incremental to the existing revenue, decrease it, or keep it flat depending on these slopes.

Again, any light you can shine on the above would be greatly appreciated.  In case you're up for it, please find a final follow-up question below.

Say the X dollars in product inventory were 5% of the total pre-existing available eBay inventory.  If we increase X to represent 10% of total pre-existing available eBay inventory, how would that change its impact on total revenue?

eBay seems to think that additional sellers increase its profit (not just revenue), because it advertises the option to sell on its website and tries to recruit more sellers. That's the empirical answer.

Theoretically, eBay could refuse to list the additional seller's product and not charge the seller any fees, so by revealed preference, eBay cannot be worse off with an additional seller. If eBay was forced to list every seller who comes along, then the revealed preference reasoning would fail, so then eBay may or may not be better off with an extra seller.


"If $\lambda$ is greater than than 1, the system explodes." Why does the system explode? David Romer in his textbook Advanced Macroeconomics (Third Edition) writes regarding the speed of convergence of the Diamond model the following:
(Pg. 83)

Equation (2.60) [$k_{t+1}={1\over{(1+n)(1+g)}}{1\over{2+\rho}}(1-\alpha)k_t^\alpha$] gives $k_{t+1}$ as a function of $k_t$. The economy is on its balanced growth path when these two are equal, that is $k^*$ is defined by
$k^*={1\over{(1+n)(1+g)}}{1\over{2+\rho}}(1-\alpha)k^{* \alpha}\space\space\space\space\space\space\space\space\space\space\space\space$ (2.61)
Solving this expression for $k^*$ yields
$k^*=\left[{(1-\alpha)\over{(1+n)(1+g)(2+\rho)}}\right]^{1 \over{1-\alpha}}\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$ (2.62)
[...] We can also find out how quickly the economy converges to the balanced growth path. To do this, we again linearize around the balanced growth path. [...] Thus:
$k_{t+1} \simeq k^*+\left( {{dk_{t+1}}\over{dk_t}}\Big{|}_{k_t=k^*}\right) \left(k_t-k^*\right)\space\space\space\space$ (2.64)
Let $\lambda$ denote $dk_{t+1}/dk_t$ evaluated at $k_t=k^*$. With this definition, we can rewrite (2.64) [...]
$k_t-k^* \simeq \lambda^t (k_0-k^*)\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$ (2.65)
Where $k_0$ is the initial value of $k$

My Question
(Pg 84)

[...] if $\lambda$ is greater than 1, the system explodes

What does it mean the system explodes when $\lambda$ is greater than 1? Why does it explode when $\lambda$ is greater than 1?

Eq. (2.64) can be written as (at first order)

$$
k_{t+1} - k^* = \lambda (k_t - k^*) \tag{1a}
$$

Define the quantity $\kappa_t$ as

$$
\kappa_t \stackrel{\rm def}{=} k_t - k^* \tag{2}
$$

So that

$$
\kappa_{t+1} = \lambda \kappa_t \tag{1b}
$$

This can be solved very easy by realizing that

\begin{eqnarray}
\kappa_1 &=&\lambda \kappa_0 \\
\kappa_2 &=&\lambda \kappa_1 = \lambda(\lambda\kappa_0) = \lambda^2\kappa_0 \\
&\vdots& \\
\kappa_t &=& \lambda^t \kappa_0 \tag{3}
\end{eqnarray}

So the dynamics is $\kappa$ is driven by the value $\lambda^t$. You can see that if $\lambda = 1$ then $\lambda^t=\lambda$ for each $t$ and $\kappa$ remains constant. However, if $\lambda > 1$ the values $k^t$ will become increasingly large (explodes) as $t$ increases, that is

$$
\lim_{t\to \infty} \kappa_t \stackrel{\lambda > 1}{=} \infty
$$

Similarly

$$
\lim_{t\to \infty} \kappa_t \stackrel{0< \lambda < 1}{=} 0
$$


Using the sample mean to test hypotheses Suppose that we have some data-points $x_1, ..., x_n$ and want to test a model that specifies a stochastic process which supposedly generated the data points. For example, we might want to test the hypothesis that every $x_i$ was generated by a normal distribution with some mean and variance. In such cases, textbooks tell us to

Derive the distribution of the sample mean under the assumption that the probabilistic model is correct.
Compute the probability of observing the observed sample mean or one that is 'more extreme'.

This process gives us a p-value which we can use to evaluate the probabilistic model. Calculating p-values for different probabilistic models allows us to compare the models.

My question is simple: when calculating p-values, why do we consider the distribution of the sample mean (i.e. why is the sample mean our 'test statistic')? Why not use the median, mode, lower quartile, or any other function of the data? For that matter, why not do the natural thing and just compute the joint density $f(x_1, ..., x_n)$?

To be clear, I understand why we might focus on the sample mean if one's hypothesis does not specify a probabilistic process that generated the data. The reason is that, regardless of the distribution that generated the data, one might hope that the sample mean is distributed approximately normally (the central limit theorem). However, often one's hypothesis directly specifies the probabilistic process that generated one's data. I am asking whether there is any good reason to focus on the sample mean in this instance.

Edit: since an example has been requested, I will explain the particular problem that motivated the question. I am testing the predictions of Nash Equilibrium using some experimental data. In this case, the unique BNE is in mixed strategies, so the model I seek to test is probabilistic. I am wondering how to proceed when assessing the model's 'fit' (and comparing its fit to that of other theoretical models).

My question is simple: when calculating p-values, why do we consider the distribution of the sample mean (i.e., why is the sample mean our 'test statistic')? Why not use the median, mode, lower quartile, or any other function of the data? For that matter, why not do the natural thing and just compute the joint density $f(x1,...,x_n)$?

So, you have entered into two deep questions.  The first is “why the mean?”  The second is implicit in your joint distribution question, “why use the sampling distribution?”

I will answer the second before the first because the answer of the first depends somewhat upon the second.

There are two broad families of statistical methodologies, the Bayesian methods, and the null hypothesis methods.  Null hypothesis methods are sampling based methods.  Bayesian methods are generative methods.

Because of the nature of your question, I am going to overgeneralize a little bit to make the answer shorter than a book.

There are several reasons to use one class of problem over the other, and there are circumstances where you would only use one specific class of solution.  They are not interchangeable.

If one computed the joint density, $f(x_1\dots{x_n})$, one would fail because a probability can only exist with respect to a model.  Implicitly, you have a model in mind with the joint density, but it does matter to call it out.  In a Bayesian framework, that joint density would be the density averaged over all possible models and all possible parameters.

There are a couple of specific reasons to use Bayesian methods.  The first is if you have prior knowledge.  For example, let us assume that you know $\mu=1,000,000$ is almost impossible, but $\mu=5$ is quite probable.  That is the definition of prior information.  You know something.  It might be vague.  It might be unbelievably vague, but it is also not true ignorance either.

If you have prior knowledge, then non-Bayesian methods are inadmissible solutions.  A solution is considered admissible if the estimator cannot be stochastically dominated by another estimator.  Now admissibility is only one possible criterion.  It implies that for a given data set, the Bayesian method will be intrinsically the least risky to use and the most accurate when averaged over the sample space.

A second reason to use a Bayesian method is that they generate coherent strategies.  A strategy is coherent if a bookie that is taking all finite bets at the bookie’s stated prices, cannot be gamed into a sure loss by a crafty actor or set of actors.  Null hypothesis methods are not coherent.  If you use a null hypothesis method in a game, they can create sure losses for a bookie.

Consider the case of an inverse roulette wheel, where the ball is dropped, and the wheel spun.  Instead of betting what number it will land on, the croupier sees where the ball lands, but the players cannot see it.  Instead, the croupier tosses two fair coins.  If a coin comes up heads, then the croupier reveals the value one place to the left of the true value; otherwise, the croupier reveals one value to the right.

The sample space is $\{\{L,L\},\{R,R\},\{R,L\},\{L,R\}\}$.  To make this concrete, consider a wheel where the number came up 17.  They will either be given the set, $\{16,16\}$,$\{16,18\}$, or $\{18,18\}$.  Obviously, if you receive $\{16,18\}$ you should bet 17, but what should you do otherwise?

If, for example, you should receive $\{16,16\}$, Bayesian decision theory says you should toss a fair coin and either bid on 15 or 17.  You will win 75% of the time.  Frequentist decision theory says that the minimum variance unbiased estimator is 16.  You can check that for yourself.  You should bid 16.  Of course, sixteen is an impossible value, known with probability one.  You will win 50% of the time.

If the bookie placed odds using Frequentist decision theory but you played as a Bayesian, you would rake in the cash.  The Frequentist odds are not coherent.

Although not obvious yet, this is the start of the answer to the first question on the mean.  And, in fact, you do want to be careful.  If you have a Bayesian model, you do not want to test it with a null hypothesis method.  Conversely, if you have a Frequentist model, you do not want to test it with Bayesian methods.  They would produce different predictions.
So far, we have two reasons to use the joint distribution via Bayes theorem.  The first is accuracy.  The second is coherence.  So why use a null hypothesis method?

The first reason to choose a null hypothesis method is that you have no prior information.  The nice thing about null hypothesis methods is that they work equally well regardless of the true value of the parameter.  That is not true for a Bayesian method.  They do not work equally well for all possible values of a parameter.

In the example above, where you felt you could all but rule out $\mu=1,000,000$ would be problematic if $\mu=1,000,000$ as you would have underweighted the possibility of the case.  Bayesian methods weight the sample based on prior knowledge.

As well, they minimize the maximum possible loss you could face from drawing an unfortunate sample.  The minimum variance unbiased estimator and the maximum likelihood estimator generate sampling distributions that are worst-case distributions.

Bayesian methods, generally speaking, minimize the average loss you would expect to receive from drawing an unfortunate sample.  In the absence of prior information, it is challenging to minimize an average loss when you have nothing to compare to the sample.

The second reason is that you truly have a sharp null hypothesis.  Bayesian methods have no clean or nice way to handle a hypothesis of the form $\mu=k$.  This is because samples are not random in Bayesian thinking.  You saw the sample.  It is a fact.  It cannot be random as there is no uncertainty in it.  Parameters are uncertain in Bayesian thought.  You cannot see them; you cannot know what they are.  The $\Pr(\mu=k)$ is intrinsically zero.  It is one point over an infinite number of points.  The measure of a single point from an infinite number is zero.  On the other hand, Bayesian methods handle solutions over regions fine.

The third reason is that you need an unbiased estimator for some reason.  Assuring an estimator is unbiased is expensive.  There can be a huge loss of information to assure it is unbiased.  You may need a sample size many times larger than a biased Bayesian estimator to get the same result.  All Bayesian estimators are biased estimators.  Bayesian methods optimize the trade-off between bias and accuracy, though as the sample size becomes arbitrarily large, the Bayesian method will converge to the true parameters, so it becomes a just barely biased estimator.

These differences answer the question of using the joint distribution of the data to determine the probability versus using the sampling distribution of the model to determine a probability.

If you decide you want to use a Bayesian method, then you solve $\Pr(\mu\le{k}|{X})$, for example, instead of $\Pr({X}|\mu\le{k})$.  The converse is true for the null hypothesis method.

Now as to using the mean for a sampling distribution.  There are several reasons for this.  The first is that in most standard cases, the mean is the most efficient estimator of the parameter.  Secondly, all unbiased estimation has quadratic loss buried inside of it somewhere.  Even if you are making an estimate based on the median, that is the mean rank.  While the median minimizes absolute linear loss, it minimizes quadratic loss in rank space.

That takes us back to stochastic dominance.  If you have two unbiased estimators, say the sample mean, and the sample median, but the sampling distribution of the sample mean stochastically dominates the sampling distribution of the sample median, then why would you use the median?

You see the median begin to be used, generally, when robustness is a concern rather than power.

If you want to know the probability your model is true, given the data that you saw, then you use a Bayesian method.  If you want to know the probability of seeing data as extreme or more extreme than likely under your hypothesized value, then you use a null hypothesis method.  If you use a Bayesian method, then it depends on the joint density of the actual sample.  If you use a null hypothesis  method, then you will use the sampling distribution of the model to condition the data.  The mean will tend to be used due to issues of efficiency and power and not any other statistic.

EDIT
If your hypothesis concern is model selection, then you should use a Bayesian method.

Bayesian methods treat a model as a parameter.  It uses the data to determine which model is most probable.  You can have any number of models you like and any number of parameters.

Model selection by information criterion maps to stylized Bayesian posterior densities, but do not make their assumptions explicit.  Bayesian methods allow you to let the data select the model.

EDIT
For clarity, let us imagine that we have two possible models that data are drawn from.  Model one is $$\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}.$$  Model two is $$\lambda\exp{[-\lambda{x}]}.$$

$$\Pr(M_1;\mu|X)=\frac{\prod_{i=1}^n\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}\Pr(\mu|M_1)\Pr(M_1)}{\int_{\mu\in\Theta}\prod_{i=1}^n\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}\Pr(\mu|M_1)\Pr(M_1)\mathrm{d}\mu+\int_{\lambda\in\Theta}\prod_{i=1}^n\lambda\exp{[-\lambda{x}]}\Pr(\lambda|M_2)\Pr(M_2)\mathrm{d}\lambda}$$

$$\Pr(M_2;\lambda|X)=\frac{\prod_{i=1}^n\lambda\exp{[-\lambda{x}]}\Pr(\lambda|M_2)\Pr(M_2)}{\int_{\mu\in\Theta}\prod_{i=1}^n\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}\Pr(\mu|M_1)\Pr(M_1)\mathrm{d}\mu+\int_{\lambda\in\Theta}\prod_{i=1}^n\lambda\exp{[-\lambda{x}]}\Pr(\lambda|M_2)\Pr(M_2)\mathrm{d}\lambda}$$

The first probability statement $\Pr(M_1;\mu|X)$ can be reduced to $\Pr(M_1|X)$ by marginalization so that $$\Pr(M_1|X)=\int_{\mu\in\Theta}\Pr(M_1;\mu|X)\mathrm{d}\mu.$$

The same can be done for model two so that you can find the probability of model two as $$\Pr(M_2|X)=\int_{\lambda\in\Theta}\Pr(M_2;\lambda|X)\mathrm{d}\lambda.$$

Since $\Pr(M_1|X)+\Pr(M_2|X)=1$, it follows that these probability statements are the probability that these are the true models in nature given the data you have seen.

In the first model $\mu$ which is the median and the mode as this distribution has no mean or variance vanishes from the probability statement in the last line.  A point value is never calculated because it is marginalized out.   Likewise, $\lambda$ which is $1/mean$ is also never calculated as a point statistic.

These extra parameters are called nuisance parameters and their integral has to be solved, but only to make the integral vanish.

When solving Bayesian model selection, the parameters are nuisances and you need them as placeholders for calculation purposes, but you have to make them vanish.  If you didn't, then you could have points where model one was more likely than model two, but other points where that relationship is reversed.

Imagine that $\Pr(M_1|X)=99.99\%$, it would follow that $\Pr(M_2|X)=.01\%$ and you could ignore model 2.  I once tested seventy-eight models of bankruptcy.  One model had a posterior probability of about 57%.  the second model had a posterior probability of about 43%.  The remaming 76 models had a combined probability of $1/10000$ of one percent.  I could exclude the 76 models, but I couldn't exclude two models as both were approximately equally likely to be the data generating function.

Although I bypassed the problem using model averaging, that is I averaged the two regressions together by weighting one result by .57 and the other by .43, what it really meant was that I was close, but not quite there and that some similar convex combination of the two sets of variables were what mattered.


Does demand side economics only apply to manufacturing For example, if investment falls then manufacturing falls. but there is no investment in secondary markets.

Wouldn't recessions cause factories to collapse but why would they affect realtors, used cars, secondary retialers etc?

When factories collapse, workers in those factories are out of jobs... no money means they can't afford to buy a used car, etc.

Update: To go back to the question in the title, consumption is also a part of the demand, and it includes consumption in services as well.


Clarifying foreign direct investment and imports on International Transactions release I would like to measure US corporations' investments in property, plants, and equipment abroad versus their spending on labor abroad. However, I'm unsure of the categorizations on the International Transactions report.

BEA report for reference:

Suppose I'm American and I want to build a factory in Mexico. I have $1 million dollars.

Scenario A - I convert \$1 million to pesos and put the pesos in a Mexican bank. I use the pesos to purchase a factory and pay my workers every two weeks. Does this $1 million show up entirely as an outflow in reserve assets (page 9 line 75)?

Scenario B - I don't convert my money. I build the Mexico factory and pay my workers entirely in USD. Does the factory show up entirely in outflow of direct investment assets (page 9 line 62)? Does my periodic worker comp show up entirely in primary income payments compensation of employees (page 9 line 57)?

Now suppose at the end of the year, the workers convert their dollars to pesos. Does this conversion show up in currency and deposits (page 9 line 94)?

Edit: It seems that construction of the factory itself is can be considered as a import of construction services for America according to the IMF's treatment for compensation of employees.

I think it does not really work that way

Going through your references:

(Table 1 line 75) Reserve Assets: These are foreign assets held by the Federal Reserve or US Treasury for intervening in currency markets.  Since the US does not intervene, the small numbers are largely miscellaneous transactions related to IMF Special Drawing Rights
(Table 1 line 62) Direct Investment Assets: This is the closest as to what is going on.  Amounts sent out to buy or build the factory are counted here, as are later investments from non-repatriated income
(Table 1 line 57) Imports and income payments - compensation of employees: this is payments for short term workers coming into the country and then going home, such as seasonal agricultural workers. If you had built your factory by sending short-term construction workers abroad to build it then their income might have counted as (Table 1 line 29) Exports and income receipts - compensation of employees
(Table 1 line 94) Other investment liabilities - currency and deposits: assuming your workers abroad are resident there and you pay them in US dollars (why?), then this increases this line, but if they then change this into their currency by transacting with you in the US it reduces it back again

I would have thought that a more natural set of transactions would be something like:

You start with some US dollars for the investment which you change into Mexican pesos (increasing line 71 for you and either increasing line 94 or decreasing line 71 for somebody else depending on whether they are foreign)
You use the pesos to buy some land in Mexico, buy construction supplies and hire construction workers (reducing line 71 but increasing line 62) so you end up with a factory in Mexico
Now you have a factory in Mexico, your factory pays in pesos for workers resident and working in Mexico and for supplies, and receives revenues from sales.  Your factory's net profit or loss appears as Direct Investment Income (line 25), which if you do not repatriate it then also goes into further Direct Investment in your Mexican factory (line 62)


Sustainability of Collusion in a finite Bertrand competition with N > 2 firms In a infinitely repeated Bertrand competition, collusion is sustainable if, and only, if, the following inequality is satisfied,
$\frac{\pi}{N(1-\delta)}\geq\pi$
Where $\pi$ is equilibrium profits, and $0 < \delta < 1$ is the discount factor. If the game is finite, I make the following claim:
Collusion is not sustainable, as each firm has an incentive to deviate from collusion in the last stage of the game. However, how would I go ahead and prove this claim?
My initial attempt, was to setup the following inequality,
$\frac{1-\delta^T}{1-\delta}$ $\frac{\pi}{N}$+$\pi_{\varepsilon}$ $\delta^{T-1}$ $\geq$ $\pi$,
where $\pi_{\varepsilon} < \pi$, and is given to the deviating firm. Is this a valid approach, or should I change my approach?

You're already on the right track. Just need to reorganize your argument a little bit.
Hint: Prove by contradiction. Suppose that collusion is sustainable in an equilibrium. Show that the collusive strategy (i.e. choosing the monopoly price in every stage) is not optimal given that everybody else is doing the same. Then conclude that collusion cannot be an equilibrium, hence a contradiction.


Who looses Forex if someone gains? My understanding of forex exchange is that when there are two currencies cur1 and cur2, they are balanced in the world with a certain rate. People who buy cur1 at a high rate and sell in a lower rate effectively end up paying for the profit which someone has made by selling cur1 when the rate was high and buying when the rate was lower.

In a hypothetical world where no one sells their reserve of cur2 but people having cur1 reserves keep on buying cur2, how will the system be balanced? Who will gain or loose to compensate for the imbalance created? Please correct me if you think the question is incorrect or missing something.

The foreign exchange market is a huge over-the-counter market, primarily intermediated by large dealer banks with subsidiaries around the world (sometimes referred to as “money center” banks).

A buyer or seller of a currency is almost certainly at or trading with one of these banks. The dealer banks make money via a “spread,” where every buyer or seller of a currency effectively pays the dealer they’re trading with. For example, if 1 cur1 = 1 cur2, a buyer of cur2 might have to pay 1.01 cur1 for it, and a seller of cur2 might receive 0.99 cur1 for it, with the dealer taking .01 on both sides.

The forex market is large and highly competitive, and balance between the prices of any two currencies is just determined by supply and demand— as people trade cur1 for cur2, the supply of cur1 at dealer banks will rise, and the supply of cur2 will fall, so the price of cur2 will rise relative to cur1.  From a practical standpoint, as more people trade cur1 for cur2, a dealer bank will have to replenish its reserves of cur2, which it will often do by borrowing it.

In my opinion, the simplest way to think about all this is to remember that the primary source of demand for any currency is people who want the stuff denominated in that currency (be it consumer goods, financial or real assets, etc.). When more people want that stuff, relative to the stuff in other currency areas, the price of that stuff (i.e., the value of the currency) rises. For example, Canada is a huge oil exporter, and if you want Canadian oil, you first have to get Canadian dollars. As a result, when demand for oil rises, the value of the Canadian dollar rises.


Pure exchange economy with free and non-free disposal Question is as follows:

My answers are

(I)

Let k is constant utility level

Then

$$k=\sqrt{x_A^1x_A^2}$$

$$k^2/x_A^1=x_A^2$$

The first derivative is negative. So the indifference curve is decreasing.
The second derivative is positive.

Since utility function is quasiconcave, then preferences are convex because his prefers averages to extremes.

For B

Similarly

For constant utility level k

$$k=x_B^1/(1+x_B^2)$$

Then $$x_B^2=x_B^1/k-1$$

This is straight line.

Due to the linear indifference curve he is indifferent between averages and extremes. Thus not strictly convex preference.

(II)

In order to analyze whether preferences are nonstaiated and strictly monotonic, we need to look at its MU’s.

For A,

since good 1 and 2 may decrease the marginal utility, preferences are not monotonic.  But preferences are non-satiated.

For B,

Since marginal utility is increasing, the preferences are monotonic. But preferences not strictly monotonic because getting more x2 decreases utility. However preferences are locally nonsatiated because by giving the consumer a bit more $x_B^1$ holding $x_B^2$ constant, he strictly better off.

(III)

I did first three parts in this way. However, I couldn’t proceed the last three parts that I marked with red pen. Please help me to do this. Thank you.

Consider a pure exchange economy with two players - A and B.

A's utility function is

\begin{eqnarray*} u_A = x_A^{\frac{1}{2}}y_A^{\frac{1}{2}}\end{eqnarray*}

B's utility function is

\begin{eqnarray*} u_B = \frac{x_B}{1 + y_B}\end{eqnarray*}

Endowments of A and B are respectively

\begin{eqnarray*} \omega_A = (2,5), \ \omega_B =(5,2)\end{eqnarray*}

  Find the set of Pareto efficient allocations in this economy.

Set of Pareto efficient allocations consists of those feasible allocations in which B consumes $0$ unit of commodity $Y$.

$$\text{Pareto Set} = \left\{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right) \in \mathbb{R}_+^2 \times \mathbb{R}_+^2 | x_A + x_B = y_A + y_B = 7, y_B = 0\right\}$$

  Assume there is no free disposal. Player A can make a take-it-or-leave-it offer to B. Player B can either accept or reject. If player B rejects, then each player stays with his or her current endowment. If player B accepts, A's proposal is implemented. Find the equilibrium allocation outcome $\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right)$.

In a subgame perfect equilibrium, B's strategy will be to accept an allocation proposal $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ if $u_B(x_B, y_B) \geq u_B(5, 2) = \frac{5}{3}$, and reject otherwise. Given B's strategy, A will choose a proposal by solving the following maximization problem :
\begin{eqnarray*} \max_{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)\in \mathbb{R}_+^2 \times \mathbb{R}_+^2} & \ x_A^\frac{1}{2}y_A^\frac{1}{2} \\ \text{s.t.} & \ x_A+ x_B = 7 \\ & \ y_A + y_B = 7 \\  & \ \frac{x_B}{1+y_B} \geq \frac{5}{3}\end{eqnarray*}

Solving this problem we get the following proposal :
$\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right) =  \left(\left(\frac{16}{3}, 7\right), \left(\frac{5}{3}, 0\right)\right)$

  Assume there is now free disposal. Redo the last part allowing for free disposal.

Allowing for free disposal, in a subgame perfect equilibrium, B's strategy will be to accept an allocation proposal $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ if $u_B(x_B, y_B) \geq u_B(5, 0) = 5$, and reject otherwise. Given B's strategy, A will choose a proposal by solving the following maximization problem :
\begin{eqnarray*} \max_{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)\in \mathbb{R}_+^2 \times \mathbb{R}_+^2} & \ x_A^\frac{1}{2}y_A^\frac{1}{2} \\ \text{s.t.} & \ x_A+ x_B = 7 \\ & \ y_A + y_B = 7 \\  & \ x_B \geq 5\end{eqnarray*}

Solving this problem we get the following proposal :
$\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right) =  \left(\left(2, 7\right), \left(5, 0\right)\right)$

  Find competitive equilibrium (if it exists) in both the cases : (a) no free disposal, (b) free disposal

Competitive equilibrium consists of an allocation $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ and prices $(p_x, p_y = 1)$ such that

$x_A = \dfrac{2p_x + 5}{2p_x}$, $y_A = \dfrac{2p_x + 5}{2}$
$x_B = \dfrac{5p_x + 2}{p_x}$, $y_B = 0$
$x_A+x_B = 7$, $y_A+y_B=7$

Solving it we get, $p_x = \dfrac{9}{2}$.
Therefore, competitive equilibrium allocation is $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)=\left(\left(\dfrac{14}{9}, 7\right), \left(\dfrac{49}{9}, 0\right)\right)$ and supporting prices are $(p_x, p_y) = \left(\dfrac{9}{2}, 1\right)$. Equilibrium is the same in both cases - with or without free disposal.


What does poorest country by population mean? I understand poorest countries based on GDP but what does poorest countries by population mean? Below is the link to the information I was looking at.
http://worldpopulationreview.com/countries/poorest-countries-in-africa/

"Poorest country by population" doesn't really mean anything: I think this is just sloppy writing by the website. The table on the website is actually ranking countries by

$$\text{GDP Per Capita}=\frac{\text{GDP}}{\text{population}}.$$

This is a measure of the value of goods or services that the average person in a country produces.


State of general equilibrium theories Post Blanchard 2008 This question arose from a useful comment I received upon answering a question on this site, and a recent question from a doctoral student.

The so-called theory of general equilibrium (including the many variants) is a cornerstone of economic theory, occupying economists' minds since (at least) the 1800s. Although, like other concepts in economics, different schools of economic thought remain polarised about its usefulness (both empirically and theoretically) and its falsifiability, [see Sonnenschein–Mantel–Debreu theorem (1976), positive corner-case results of Brown and Matzkin (1996) and any one of a number of further negative results Nachbar (2004), Kirman (1992), Carvajal (2003),...].

Post Blanchard's now famous 2008 declaration that "the state of macro is....good" and the characterization of DSGEs as state-of-the-art. What is the state of GE theories today?

To understand why macroeconomists use DSGE as a tool, in general, it's a good idea to read up on the Lucas critique.

More colloquially, DSGE models provide macroeconomists with a laboratory that allows quantitative comparison (and ranking) of different economic policies.  Further, the process of writing a model disciplines thinking: if one cannot write down a minimally specified model that replicates (to varying degrees of success) some real world dynamics of interest, then do we really understand what is happening to begin with?  It's easy to play a casual story telling game that explains economic phenomena (with critical errors).  We unfortunately see this all the time in politics, or even on CNBC.

Also, DSGE models do not all share the same objective.  There are some medium and large scale models employed at central banks that are used to guide monetary policy.  Other models are much smaller and stylized, and seek to answer questions about how a government should optimally finance itself, or how to tax different factors of production, or how informational frictions affect prices.

Finally, to answer your question on the state of dynamic general equilibrium theory, researchers continue to apply this mode of analysis to deepen our understanding of economic dynamics.  This entails writing models of economic agents incentivized by objective functions, facing resource and informational constraints, interacting with one another and nature.

Every DSGE model has some fatal flaw which cripples it from perfectly explaining real world dynamics.  Perfection is not the point of DSGE models, so when macroeconomists say that DSGE models are "useful" or "good", it's important to ask questions about the context of those descriptive terms.  It's also important to remember that macroeconomists are, at the same time, some of the harshest critics of DSGE models.


What economics theory explains changes in purchasing different types of goods? I am thinking about the first of the month. I spend more of my money on superior goods and, because of that decision, less on normal and inferior goods. Are income and substitution effects enough to explain this or not?

I am thinking about a hypothesis for why people change their types of goods given that budgets change according to pre-decisions.

I will not misguide you: this is not only a goods decision or money constraint; it is also related to the time constraint and time-involved work.

For example, an agent has a given time constraint, T, to work on his given projects over a time period, t, which requires only T-r amounts of time for the period t+1. Hence he spends the time r to do other things according to opportunity cost theory. Therefore, he spends his time working on this project only at amounts equivalent to T-r and the opportunity cost of this project decreases as time approaches t+1.

So at this point, if he is an expert at this job, everything will go smoothly but, if not, this is a false prediction and the agent has to face his false decision and survive on his present condition. The hypothesis starts here: 1. Will he increase his purchasing power (i.e., sleep less to increase the time constraint)? 2. Will he bargain with his boss to increase his time constraint? Why?

If you research "discount rates", "time preference" or similar, you'll find all sorts of stuff about seemingly inconsistent decisions, some of which can in fact be explained quite well and some of which show some ways that people can be dumb sometimes. You'll find lots of approaches to modelling comparisons involving things like someone revealing an implicit interest rate of over 100% per annum in one situation while accepting 5% as a good return in another situation.

I wouldn't focus too much on any specific paper. Check the bibliographies and see the variety of what's out there, and maybe narrow in on a handful of most-promising conceptualizations afterwards.

As a point of contrast, you could introduce the lifetime income hypothesis of Friedman as discussed in many 1st and 2nd year textbooks.


Why are Portuguese migrating out of Portugal? Where the world's population is growing and shrinking

Apart from a low fertility rate, why is the Portuguese population declining?

As you write, fertility rate explains most of it, and immigration explains the rest.

People tend to migrate to richer countries, and Portugal is below average EU income. (Portugal is an EU member. The EU has freedom of movement, the citizens of any member state can relocate to other member states and work there.) Real per capita GDP is 30% less than in its neighbor Spain, and almost 50% less than in Germany.

Eurostat Real GDP per capita table


Fixed Effects Variation Suppose I wish to run the following gravity regression:
$$
Y_{ijt}=\boldsymbol{X_{ijt}'\beta+}ij+i_{t}+j_{t}+\epsilon_{ijt}
$$
where the LHS is bilateral trade between $i$ and $j$ in year $t,$
the first part of the RHS denotes control variables, specific to $i$ and $j$. The term $ij$
is the pair fixed effects term, which is a fixed effect for a particular
term. The third term is the importer-time fixed effect, and the fourth
term is the exporter fixed effect. Importantly, within each year,
I have bilatereal information between various possible combinations
of importers and exporters. My questions are as follows: 1) Are the
coefficients on $ij$, $i_{t}$ and $j_{t}$ identified? My first
inkling is yes. That is because, assuming a balanced panel, the total
number of observations are:
$$
T\times I\times J
$$
 whereas the number of coefficients to be identified as:
$$
IJ+IT+JT
$$
 which is less than $TIJ$ if:
$$
IJ\left(1-\frac{1}{T}\right)>I+J
$$
 which should generally be the case.

2) On what basis is $\beta$ identified? What variation is being used?
The $ij$ pair fixed effects would imply this is being identified
from within country pair- over time variation in $\boldsymbol{X.}$
What role does $i_{t}$ and $j_{t}$ play?

Thanks!

It isn't enough to have a large number of observations. You need multiple observations of each $i$ and $j$ for multiple time periods and doing trade with multiple countries. If, for example, if countries $k$ and $l$ only trade with each other, it wouldn't be possible to distinguish between $k$, $l$, and $lk$. In additions, you need multiple observations because you want to identify a time subscript on $i$ and $j$. If you have many countries, many periods, and many trade relationships, and the data are not sparse then this may not matter much.

You may find this paper of use:
Gravity or dummies? the limits of identification in gravity estimations (Hornok 2011)

  This paper argues that identification of trade policy effects with a
  gravity equation that includes country-time dummies to control for the
  theoretical Multilateral Trade Resistances (MTR) is severely limited.
  In most cases heterogeneous policy effects, i.e. more than one policy
  dummies, cannot be identified separately, because the policy dummies
  and the country-time dummies are perfectly collinear. Although a
  single policy dummy can b e identified, the estimate may not b e
  meaningful, because country-time dummies absorb to o much of the
  useful variation of the data. Standard estimation techniques often do
  not reveal these problems. The pap er demonstrates these arguments by
  taking four typical research questions on the effect of a trade
  policy, checking the
  identifiability of the corresponding policy effect and deriving the
  estimates. Empirical exercise on estimating the trade effects of EU
  enlargement complements the analytical findings.

And Estimating the gravity model without gravity using panel data
(Westerlund and  Wilhelmsson 2009)

  "...$\alpha_{ij}$ which is unidentified in the fixed effects
  formulation of the model. In order to identify $\alpha_{ij}$, a random
  effects assumption is needed. But such assumptions are generally not
  satisfied in practice..."

@ChinG, here is the basic gravity idea:
Newton's law of universal gravitation tells us that the force of gravity is
$$F = \frac{G M_1 M_2}{R^{2}_{1,2}}$$
which implies
$$f = g + m_1 + m_2 - 2 \cdot r_{1,2} $$
(where lowercase letters are the logs of upper case ones). In the trade equation, the force of gravity is replaced with the volume of trade. Usually, $M_1$ represents the economic size of economy one (say measured by $GDP_1$ and in your regression analogous to $I_t$), $M_2$ represents the economic size of economy two ($GDP_2$ and analogous to $J_t$), and $r_{1,2}$ is the geographic or effective economic distance between the two countries ($\alpha_{i,j}$). GDP isn't exactly the right measure because countries differ in fraction of the economy devoted to trad-able output. Think of it as analogous to the importance of that country in global trade (in an absolute and not relative sense).


Confusion about the convexity of the best response correspondence I am recently reading the proof of the existence of the Nash Equilibrium. As a math student, I do understand the use of Berge's maximum theorem and Kakutani's fixed point theorem, but I am not sure why the mixed-strategy space and the image of the best response correspondence are convex.

For example, if I have two mixed-strategies $\sigma_i,\sigma_i'\in BR_i(\sigma_{-i})$, why is the mix of these two strategies, $\lambda\sigma_i+(1-\lambda)\sigma_i'$ ($0\leqslant\lambda\leqslant1$), also a best response? Since $\sigma_i,\sigma_i'\in BR_i(\sigma_{-i})$ suggests $u_i(\sigma_i,\sigma_{-i})=u_i(\sigma’_i,\sigma_{-i})=\max\{u_i(x_i,\sigma_{-i})\}$, how to prove that $u_i(\lambda\sigma_i+(1-\lambda)\sigma_i',\sigma_{-i})$ also equals to maximum?

I would have thought that the if the opponent strategy $\sigma_{-i}$ is given, even if it is a mixed strategy, then $$u_i(\lambda\sigma_i+(1-\lambda)\sigma_i', \sigma_{-i})= \lambda u_i(\sigma_i, \sigma_{-i})+(1-\lambda)u_i(\sigma_i', \sigma_{-i})$$ and so if  $u_i(\sigma_i, \sigma_{-i})=u_i(\sigma_i', \sigma_{-i})$ then $u_i(\lambda\sigma_i+(1-\lambda)\sigma_i', \sigma_{-i})$ is equal to both of them. If they are best responses then so too is the linear combination strategy


Why would investment banks not care about their risky exposure to subprime mortgages? At the end of The Big Short, Steve Carell's character asserts that the banks weren't acting stupid, and that they knew they were getting a bailout so they just didn't care. Why wouldn't they care? From what I understand the bailout was just a bunch of big loans. Why would you not care about losing a bunch of money and having to take out huge loans?

The movie over-simplifies the large banks' attitudes somewhat. In reality, they differed:

some banks limited their exposure to junk debt (I will use this term because there was junk other than subprime mortgages involved). They'd buy up a little junk, repackage it into new instruments, and re-sell their instruments to clients. So their own exposure to the junk was limited, which is commendable. Their losses came a little later, when the clients who bought the repackaged junk sued them for billions of dollars and eventually won.

some banks were buying up more junk than they could use right away for repackaging - holding an inventory, intending to re-sell it eventually, and hoping to earn more money by holding the high-yielding junk meanwhile. This bet did not go well. (And they got sued too.)

a couple of banks thought they did not have junk exposure, but they were mistaken. One very prominent bank CEO went on TV to tell the public that his bank had no junk exposure. He might have believed it, but a few days later it turned out that all the repackaged junk that his bank sold to clients had a "par put" clause, meaning that the client could require ("put") the bank to buy back the repackaged junk at face value ("par"). When the repackaged junk was trading at about 20 cents on a dollar, the clients massively exercised the option, causing the bank to lose tens of billions of dollars. (And they got sued too.)


Still don’t understand why a trade deficit must be financed by foreign borrowing (even after reading other answers) Maybe my understanding of how the real world works is now clouding my ability to understand the theoretical because I aced this stuff in college, but now can’t wrap my head around it.  Yes, I’ve read all the similar questions but I still don’t see a clear answer to the question.  I’m guessing I’m getting tripped up the semantics but could use some help in figuring this out.
Why exactly does a trade deficit require foreign borrowing?  If I decide to buy some French wine and go to my banker for a loan, who in turn creates a demand deposit, increasing the money supply, why does my purchase of French wine require foreign borrowing?  A bottle of French wine to anyone who can help me understand this!  Thanks.

The assertion that a trade deficit must be financed by foreign borrowing is technically incorrect (although it is often approximately true).

The accounting identity for international transactions has a few extra terms, one of which is grants between governments (foreign aid). A country could run a trade deficit financed by foreign aid.
Equity financing also shows up in international transactions. Instead of emitting debt, a country can emit equity.

The reason why the statement is approximately correct is that the  largest new flow across borders is the transfer of debt instruments, noting that currency notes are a liability of government. Since net imports had to be paid for with some instruments (possibly bank deposits), there has to have been an offsetting flow of instruments out of the country. Since those instruments are debt instruments, people loosely refer to this as “borrowing,” although it could be the transfer of a bond issued years earlier. Most people do not refer to selling a bond out of their portfolio as “borrowing,” so one needs to be careful of this terminology.


For what demand function is a monopoly most harmful? Consider a firm with zero marginal cost. If it gives the product for free, then all the demand is satisfied and the social welfare increases by the maximum possible amount; call this increase $W$.

But because the firm is a monopoly, it reduces the demand and increases the price in order to optimize its revenue. Now the social welfare increases by a smaller amount, say, $V$.

Define the relative loss of welfare (deadweight loss) as: $W/V$. This ratio depends on the shape of the demand function. So my question is: is this ratio bounded, or can it be arbitrarily large? In particular:

If $W/V$ is bounded, then for what demand function is it maximized?
If $W/V$ is unbounded, then for what family of demand functions can it become arbitrarily large?

Here is what I tried so far. Let $u(x)$ be the consumers' marginal utility function (which is also the inverse demand function). Assume that it is finite, smooth, monotonically decreasing, and scaled to the domain $x\in[0,1]$. Let $U(x)$ be its anti-derivative. Then:

$W = U(1)-U(0)$, the total area under $u$.
$V = U(x_m)-U(0)$, where $x_m$ is the amount produced by the monopoly. This is the area under $u$ except the "deadweight loss" part.
$x_m = \arg \max (x \cdot u(x))$ = the quantity which maximizes the producer's revenue (the marked rectangle).
$x_m$ can usually be calculated using the first-order condition: $u(x_m) = -x_m u'(x_m)$.

To get some feeling of how $W/V$ behaves, I tried some function families.

  Let $u(x)=(1-x)^{t-1}$, where $t>1$ is a parameter. Then:

  $U(x)=-(1-x)^{t}/t$.
  The first-order condition gives: $x_m=1/t$.
  $W=U(1)-U(0) = 1/t$
  $V=U(x_m)-U(0)=(1-(\frac{t-1}{t})^{t})/t$
  $W/V=1/[1-(\frac{t-1}{t})^{t}]$

  When $t\to\infty$,  $W/V \to 1/(1-1/e)\approx 1.58$, so for this family, $W/V$ is bounded.

But what happens with other families? Here is another example:

  Let $u(x)=e^{-t x}$, where $t>0$ is a parameter. Then:

  $U(x)=-e^{-t x}/t$.
  The first-order condition gives: $x_m=1/t$.
  $W=U(1)-U(0) = (1-e^{-t})/t$
  $V=U(x_m)-U(0)=(1-e^{-1})/t$
  $W/V=(1-e^{-t})/(1-e^{-1})$

  When $t\to\infty$, again $W/V \to 1/(1-1/e)\approx 1.58$, so here again $W/V$ is bounded.

And a third example, which I had to solve numerically:

  Let $u(x)=\ln(a-x)$, where $a>2$ is a parameter. Then:

  $U(x)=-(a-x)log(a-x)-x$.
  The first-order condition gives: $x_m=(a-x_m)\ln(a-x_m)$. Using this desmos graph, I found out that $x_m \approx 0.55(a-1)$. Of course this solution is only valid when $0.55(a-1)\leq 1$; otherwise we get $x_m=1$ and there is no deadweight loss.
  Using the same graph, I found out that $W/V$ is decreasing with $a$, so its supremum value is when $a=2$, and it is approximately 1.3.

Is there another family of finite functions for which $W/V$ can grow infinitely?

An arbitrarily large ratio should occur with demand curve

$P=\begin{cases}
\frac{1}{Q} & \text{if } Q>1 \\
2-Q & \text{if } Q\leq 1 \\
\end{cases}$.

The monopolist prices at $P=1$, but the consumers' surplus if $P=0$ is infinite, because the area under the demand curve contains $\int_1^\infty \frac{1}{Q}dQ=\infty$.


Price elasticity when relationship between sales, price and other factors is not linear For commercial deployment, price elasticity is calculated through linear regression which assumes that there is a linear relationship between price and sales. I have a)price and b)social media ratings for my product. I know that social media ratings influence consumer decision to purchase or not purchase something. As per assumption of linear regression if social media rating is low we need to reduce price to improve sales. However, beyond a point price cuts will not help in increasing sales. This is a violation of linearity.

So my question is what if

1) Relationship is not linear even after log transformation of the data?

2) How can i know about the point at which i need to stop from cutting price further.

To give a background, i have already read this Econometrics: Is elasticity meaningful in my, or any, regression? However, it didn't seem to answer my second question, which is will my sales continue to increase if i keep dropping price although i have a low social media rating?

My background: I am a statistician and my economics knowledge is all i read in college. I am working a revenue optimization problem

There is a confusion between a "linear relationship between two variables" and an "econometric equation that is linear in the unknown parameters to be estimated".

The first has to do with what happens in reality, and it implies that the marginal relation is constant. The second may be obtained even if the actual relation is not linear, but non-linear in specific ways that permit to obtain it by a suitable transformation of the data.

To illustrate this, for the OP's case, the actual (deterministic part of the) relation may be

$$X_d = AR^aP^{-b} \tag{1}$$

Demand $X_d$ for the product is a positive non-linear function of social ratings $R$ and a negative non-linear function of its own price $P$. The marginal effect of price for this relation is not-constant

$$ \frac {\partial X_d}{\partial P} = -\frac{b}{P}X_d <0$$

By assuming $(1)$ we have already made a series of assumptions about the interplay of the variables involved. This specification permits us to obtain "an econometric equation that is linear in the unknown parameters to be estimated" since by taking logs we have

$$\ln X_d =\ln A +a\ln R +(-b)\ln P \tag{2}$$

So while the marginal effect of price on demand is not linear and not constant, the elasticity of demand with respect to price is constant, and equal to $-b$ (the sign indicating direction of influence).

But is $(1)$ an adequate way to represent the actual relation?

So the proper way to proceed here is to

1) To the best of our knowledge, using evidence and logical arguments, we determine the qualitative interrelations between the variables involved: is the effect positive/negative? Is the relation of their levels linear/non-linear? Is it monotonic or, say "inverted-U", etc.

2) We construct a mathematical form that reflects qualitatively the conclusions/assumptions arrived at in step 1. For example, if we believe that an "inverted-U" relationship exists between levels of $Y$ and $Z$, this could be modeled by $Y = a + bZ + cZ^2$ with $c<0$

3) If the mathematical expression we obtain in step 2 is not linear in the unknown parameters of interest, we check whether it can be transformed into one that it is. Of course, there are estimation methods for non-linear relationships, non-linear least-squares being the easy example. But experience has taught us that our estimation techniques are better when they estimate equations linear in the unknown parameters, this is why we always try to arrive at such a specification, even if we may accept in the process certain approximating steps to what we have obtained in step 2 (and not just exact transformations).


How did Brexit wipe $2 trillion off world markets? I read here (https://www.theguardian.com/business/live/2016/jun/24/global-markets-ftse-pound-uk-leave-eu-brexit-live-updates) and elsewhere that Brexit wiped \$2tr off world markets.

How did that happen?

Is this equivalent to \$2tr worth of a certain good, sinking to the bottom of the ocean?

Think about it this way:

You own 100 houses on a private island in Italy. These houses could be sold for $1 million each.

All of a sudden, the Italian prime minister outlaws all boats. There is now no way to get onto the island apart from swimming.

Since you can no longer get onto the island by boat, and have to swim, no one wants to live in your houses, and thus no one wants to buy them for \$1 million. Your houses are still quite nice, though, so people instead will offer to buy them for $200,000.

Your houses were originally worth \$100 million in total (that's how much you would have got if you sold them) but are now only worth $20 million. You have 'lost' \$80 million.

In Brexit, instead of houses, we have stocks. These stocks were originally worth a certain amount of money. When Brexit occurred, people no longer wanted to buy or own British things, because they thought that they would be worth less. The total value of British things dropped by $2 trillion. It's important to note that the things are still there (the stocks didn't disappear!) but are simply valued less to investors.