Rotschild&Stiglitz (RS) Equilibrium In their pathbreaking paper, Rothschild and Stiglitz define the RS Equilibrium as a set of contracts such that,
i) each firm breaks even
ii) there exists no other contracts making non-negative profits when offered.

And they show that no pooling equilibrium exists and there may only be a separating equilibrium.

In this separating equilibrium, high risks get full coverage but low risks only partial coverage with premia equal to loss probabilities respectively.

What would you think about another separating contract bundle, in which high risks are cross-subsidised by low risks, as shown in the graph with red insurance lines and green indifference curves (G, H)?

When the new bundle is offered, high risks would prefer it to E and low risks to F (see the indifference curves). And high risks are indifferent between G and H, as in the usual separating equilibrium. Of course, this new bundle wouldn’t make an equilibrium as the yellow shaded area attracts low risks only and it might be the case similar to pooling.

However, wouldn’t the possibility of such bundles lead the usual separating contract to an instability? The new bundle making a non-negative profit seems to violate the second condition of Rothschild-Stiglitz equilibrium.

Numerical Example

Utility function: natural log
Initial wealth=100
Loss in case of accident=90
Loss probability of low risks=0.1
Loss probability of high risks=0.9

1) Incentive Compatibility Constraint for Original RS Eq.

log(100-0.9*90)=0.9*log(100-0.1*x-90+x)+0.1*log(100-0.1*x)

where x is the partial coverage amount that low risks receive in original RS eq.
Matlab calculates;

x=6.455383900159773
Utility(Low risks)= 4.424718438632044

2) Low risks subsidising high risks

Add 0.08 to the premium of low risks and subtract the same amount from high risks. (P.S: This is just an example, I also have the number maximising the utility)

new coverage=15.424194558495868
new utility(low risks)=4.452073664660258
utility of high risks=3.265759410767051 (same for full coverage at higher premium and partial coverage at lower premium)

P.S: There is no need for computing the utility of high risks because they obviously reach to a higher utility level by still receiving a full coverage with a cheaper premium.

Finally, we see that new bundle beats the original RS bundle, doesn't it?

Below are the Matlab codes I made use of:

y=0.08 %subsidy amount
ICC = @(x) log(100-(0.9-y)*90) - 0.9*log(100-90-(0.1+y)*x+x) - 0.1*log(100-(0.1+y)*x); %Incentive Compatibility Constraint
z = fzero(ICC,50); %coverage amount of low risks WRT ICC
z
u=0.1*log(100-90-z*y+z)+0.9*log(100-z*y) %utility of low risks

Edit: I also might have a general algebraic proof, but I need to verify it and will share it here afterwards.

It is important to remember that figures are not proofs. Minor mistakes in the figure are hard to detect but can change the outcome. Algebra is much more reliable.

Right now in your figure the low risk type consumers have indifference curves that are upward sloping somewhere. It seems though that is not critical here.

The visualisation of subsidies seems to be the bigger issue.  You need G and H to be the same distance away from their respective original zero profit lines. Thus the 'new' isoprofit lines should run parallel to the old ones, and at the exact same distance.

It seems to me that it is impossible to place GH on these lines while preserving incentive compatibility because of the relative slopes. However, this is also not proof. If you outline an exact GH pair and type up the equations that you think show that they constitute an improvement while being incentive compatible, you may get better answers (as well as a better understanding of the model).

EDIT

It seems to me, that in your numerical example the subsidies do not cancel each other out. It is not enough to change the rates by the same percentage, because the different types do not buy the same amount of insurance.

It costs $8\% \cdot 90 = 7.2$ to subsidize the high risk types, but only $8\% \cdot 15.4242 \approx 2.78$ is collected from additional fees on the low risk types. If there are the same number of people in both types, this will not work. But it seems to work if there are for example three times as many low risk people. I actually find this very surprising. I checked both IC constraints and all the utilities, and everything seems to check out.

Your new offers will definitely not be an equilibrium though. As long as one insurer offers the high risk insurance, other insurers can offer just the low risk one. With no high risk customers to subsidize, they will make a profit. By moving the high risk offer a bit more to the right they can siphon off all the low risk consumers from the cross-subsidizing insurance company, who will then suffer losses. So I guess the RS equilibrium is a kind of long run equilibrium.


Why would a digital Yuan (or some other national currency turned into a CBDC) "help fast-track international settlements" in that currency? I'm trying to understand the basis of this claim

As the main mode of transaction in the world's second biggest economy, China's currency should be a natural contender for the spot. However, the U.S. dollar still accounts for 88.3% of all international settlement transactions while the renminbi has only 4%. Digitization of the currency will help fast-track international settlements using the currency and help make it more popular among international traders.

I guess it might actually depend on the technical form the CBDC ('digital Yuan' or whatever) actually takes. Presumably, if it can be stored abroad without any bank account, i.e. just a digitally (and possibly chain-)signed certificate, then this (storage part) may be easier.
But it still raises the issue: how does e.g. a US consumer get their hands on digital Yuans with which to pay something that they buy directly from China (e.g. via Alibaba)? Any exchange that would sell those digital Yuans that would have to take dollars in, no?
I'm guessing that it has something to do with 3rd party countries, e.g. someone in the Eurozone could just exchange euros for digital Yuans, is that the gist of it?

How China Influences the U.S. Dollar
https://www.thebalance.com/how-does-china-influence-the-u-s-dollar-3970466

For export transactions to United States companies/consumers, Chinese firms typically receive dollars as payment for their exports. These firms then deposit the dollars into local banks, in exchange for yuan, which can be used to pay their domestic workers and vendors. The local banks then send the dollars to China's central bank, the People's Bank of China (PBOC). The PBOC holds the dollars in its foreign exchange reserves and regularly adjusts these reserves by buying or selling dollars via foreign currency markets in exchange for yuan.
By stockpiling dollars, the PBOC reduces the supply of dollars available for trade, putting upward pressure on the dollar and downward pressure on the yuan. By selling dollars, the PBOC effects the opposite result.

It is important to note that the PBOC does not sit on cash reserves. It uses the dollars it accumulates to buy U.S. Treasuries, which are safe-haven assets that provide some incremental return over cash.

The Same Problems Plaguing the Yuan Will Plague China's Digital Currency
https://qz.com/1899402/why-chinas-digital-yuan-wont-internationalize-the-renminbi/

Because the digital currency is little different from the yuan itself, it will on its own “not be a game changer that elevates the renminbi’s role in international finance,” wrote Eswar Prasad, a professor of trade policy at Cornell University, using the currency’s official name. “After all, the Chinese government still restricts capital inflows and outflows, and the People’s Bank of China still manages the renminbi’s exchange rate. Neither policy is likely to change significantly anytime soon.”

What investors are looking for in a reserve currency isn’t the technology—it’s a currency that’s stable, underpinned by a strong economy, freely convertible, and able to be used widely.


Stokey, Lucas (1989) p 11 FOC My Lagrangian is:

$L=\sum\limits_{t=0}^T \beta^tU(f(k_t)-k_{t+1})+\sum\limits_{t=0}^T\lambda_t(f(k_t)-k_{t+1}).$

My FOC for $[k_{t+1}]$ is:

$\beta^tU'(f(k_t)-k_{t_1}^*)(-1)-\lambda_t^*+\beta^{t+1}U'(f(k_{t+1}^*-k_{t+2})+\lambda^*_{t+1}f'(k_{t+1})^*=0$.

$\textbf{My Question:}$
I try to recover from their FOC and backtrack, but how did they get (5) as the Euler Equation on page 11?

Their EE is:

$\beta f'(k_t)U'[f(k_t-k_{t+1})]=U'[f(k_{t-1})-k_t]$ for $t=1,...,T$.

Reference:

Stokey, N., R. Lucas, and E. Prescott, Recursive Methods in Economic Dynamics, Harvard Univ. Press, 1989

Your FOC is correct. They took the FOC a step further and concluded that $\lambda_t=0$ for $t<T$. I quote from page 11:

  To obtain these conditions note that since $f(0)=0$ and $U'(0)=\infty$, it is clear that the inequality constraints in (4) do not bind and it is clear that $k_{T+1}=0$

The intuition is that the inequality constraints will only bind when you consume all capital today ($k_{t+1}=0$) or you invest all your capital and consume nothing today ($k_{t+1}=f(k_t))$. Both of these cases (save the former when $t=T$) imply that you will be consuming nothing for at least one period. Combining this with the MU being infinite at zero, obviously this cannot be optimal.


What do economist state as evidence that "structural reform" works? The typical recipe for countries to improve their growth rate and escape high indebtedness is to reduce expenditure and implement structural reform. What do people use as evidence of structural reform working?

A rather positive view of structural reforms

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

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

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

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

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

A rather negative view of structural reforms

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

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

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

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


Who gathers the production/use data that makes up GDP? And how? I know that GDP can be calculated from the national accounts using data which says how much of each industry's output was bought by each other industry.

But how is this kind of supply/use data gathered? Is it via questionnaire?

If so, is a random sample of businesses quizzed, or does every business have to submit data on what they bought and from whom?

Where is this data-gathering process documented? (For, e.g., the UK.)

In the UK, which uses what is know as "the output approach", the Office for National Statistics (ONS) sends out around 45,000 surveys to business, called the Monthly Business Survey (MBS).

The ONS claims that this covers "around 55% of the economy.".

In this survey, questions about turnover and purchases are asked which then allow the ONS to assign total outputs and intermediate expenditures to industry classifications. In the case of the ONS, they use Standard Industrial Classification (UK SIC 2007).

There is a separate survey for construction called the Construction Output Survey, and there are additional forecasts and models to fill in the gaps.

These output and expenditure estimates are then deflated by things like the Consumer Price Index (CPI) to produce an inflation-adjusted GDP figure.

More detail on this process along with many more links to data sources is available in the GDP quality and measurement information document.


Why does the inverted yield curve occur with US Treasury securities? I have always thought that US securities were considered nearly risk free as the government will almost always pay them back. So why is it the case that investors currently view short term securities as more risky than long term ones?

In general a longer maturity rate is practically speaking as equally profitable as a series of short maturity rates compounded. So for example, investing for 1 year thrice is as equally profitable as investing for 3 years. Investing for 6 months 16 times is as equally profitable as investing for 8 years. There is no certainty of equality and you can make money if you guess correctly. If there are many guessers then near equality will be realized.

The U.S. Treasury short maturity yield is higher than some of the long maturity yields (today), not because the short maturity investments are riskier. It is because the very short term rate (overnight) set by the Federal Reserve is high enough that the 3 month rate is higher than the 10 year rate. In other words, the expected mean rate for the next 3 months is higher than the expected mean for the next 10 years. Sometime in the future there might be a recession and the Federal Reserve might want to lower the overnight rate to help the economy. Sometime in the future the Federal Reserve might increase the overnight rate to reduce inflation. Both these considerations form expectations.

In addition to overnight rate setting, the Federal Reserve has engaged in other activity that influences Treasury rates. The Federal Reserve increased its holdings of Treasury bonds in the period from 2008 to 2014. These economists (Gagnon and others) in a paper titled "The Financial Market Effects of the Federal Reserve’s Large-Scale Asset Purchases" estimated that these purchases have been a downward force on long term rates.

The graph shows that those holdings have been decreasing since early 2018 so there might be reason to think that any downward force on long term rates is now not what it used to be so if we see a rise in those rates there is reason to think of causes that are apart from a change in risk perceptions.

Addendum: @Michael rightly points out that market participants bidding up long term bond prices is another force that lowers long term bond yields.


Specification form for household utility function What determines the choice of the functional form to be adopted for the household CRRA utility function, when utility depends on consumption and leisure?

For examples, we have these three functional specifications:

$U(c,l)=\left[\dfrac{c^φ l^{1-φ}}{1-θ}\right]^{1-θ}$
$U(c,l)=\dfrac{c^{1-φ}}{1-φ}+\alpha \dfrac{(1-N)^{1-γ}}{1-γ}$
$U(c,l)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$

c: consumption, l: leisure, N: labor supplied. Total time endowment: unity

In growth models, in order to obtain a steady-state in growth rates (i.e. a constant growth rate in long-run equilibrium), the CRRA utility function when labor-leisure choice is present must have a specific functional form.
The mathematical proof can be found in Barro & Sala-i-Martin book (2nd ed) , Appendix 9.4, pp 427-428.
It is shown that the CRRA utility function must take the form ($\ell$ represents labor here)
$$u(c,\ell) = \frac {c^{1-\theta}\cdot \exp\{(1-\theta)\cdot \omega(\ell)\}-1}{1-\theta}$$
where $\omega(\ell)$ is some function of labor, and $\omega'(\ell)<0$.
For $\theta=1$ this simplifies to $u(c,\ell)=\ln(c) + \omega(\ell)$.

After we impose additive separability, the exact functional form of the component related to leisure/labor is sometimes formed so as to have a constant Frisch elasticity of labor supply, For example
$$u(c,\ell) = \ln(c) +\alpha \frac {(1-\ell)^{1+1/v}}{1+1/v}$$
has Frisch elasticity equal to $v$, free of $\ell$. See this thread for more details on this.
The Frisch elasticity of labor supply is defined as the elasticity of labor while keeping the marginal utility of wealth constant. Its expression is derived given the solution to the utility maximization problem (i.e. by using the relations emerging from the optimaztion conditions).  After some manipulations this expression is
$$\eta_F = \frac {U_{\ell}\cdot U_{cc}}{\ell\cdot [U_{cc}\cdot U_{\ell \ell}-U^2_{c \ell}]}$$
Under separability between consumption and labor/leisure,
$$U^2_{c \ell} = 0 \implies \eta_F = \frac {U_{\ell}}{\ell\cdot U_{\ell \ell}}$$
To use one of the functional forms of the OP
$$U(c,N)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$$
we have
$$U_{N} = -\alpha N^{-\gamma},\;\;\; U_{NN} = \alpha \gamma N^{-\gamma-1}$$
so
$$\eta_F = \frac {-\alpha N^{-\gamma}}{N\cdot \alpha \gamma N^{-\gamma-1}} = -\frac 1 {\gamma}$$
So functional forms that lead to a constant Frisch elasticity, treat it as a "deep" preference parameter. Functional forms that lead to a Frisch elasticity that includes (the optimal) level of labor (as is the first functional form of the OP), treat it as a derived measure, dependent on the optimization framework also.


What would happen if the US defaults on its debt? As has been widely discussed already, the situation in Greece is considered more serious than in the US, as they have dollar-denominated debt and can only print Greek currency. The US, has the benefit of World Reserve Currency status. All of that does make sense, but I'm thinking there might be more to the story. So I would like to invite speculation on the consequences if the US one day defaulted on its debt.

A country going bankrupt is hardly a novel event, but few countries of the wealth and influence of the US are can be found in historical precedent. One of the only examples that came to mind is when Russia defaulted in 1998. That had global repercussions to North and South America and beyond. Also it destroyed the business model of Long Term Capital Management, an investment fund headed by a team of nobel-prize winning economists and mathematicians.

World Reserve Currency aside, is the Russian default a fair comparison? What different things might we expect if it were the US to default in the coming years?

The Russian default was an extraordinarily insignificant event compared with a current-day US default on all treasury securities.  There are no remotely similar events to compare it to.  This makes it hard to make guesses about all the ramifications.  Still, speculating is fun.

My guesses, based on a complete default (no money recovered) on all treasury securities:

Massive failure of important financial institutions all across the world.  LTCM was a hedge fund: a type of institution the is specifically expected to have significant failure risk.  On the other hand, treasury securities are held by institutions that are not well-equipped to bear risk.  Pension funds, banks, central banks, possibly whole governments would immediately fail.  There would be ripple effects as other institutions linked to them (or who they owe money to) faced an uncertain future.
Total freezing of most important financial transactions and many markets.  Treasury securities are used as currency and margin in many large and important financial transactions.  We witnessed a great deal of hobbling of financial institutions and markets when a few CDS's and mortgage-backed securities failed.  The loss of confidence in financial institutions and markets would be incomparably greater.
Drastic increases in all interest rates and a sharp decline in all stock prices.  The Russian default had consequences because it caused investors to dump risky investments in favor of safe ones.  A failure of the safest investment (treasuries) would cause utter loss of confidence in all investments and investors wouldn't know where to put money.  Cash maybe?  They would pull money out of both safe and risky investments, driving down both stock and bond prices.
Widespread unemployment. Cash available for lending would dry up.  Firms would be unable to borrow and have to forgo projects and lay people off.  These people would reduce their spending and default on some loans, causing amplified effects on interest rates and economic activity.  I would not be surprised if starvation and loss of social order ensued.
Irreparable harm to confidence in the US government.  The government would never again be able to borrow at low rates, it may not be able to borrow to fund its usual deficit.  This would be problematic since the depression and economic turmoil would require expanding the deficit.  It would be my guess that the US government as we know it would not continue.

I know this sounds a little alarmist, but wholesale default on all US government debt is truly an end-of-the-world-as-we-know-it scale event.    The world truly would change dramatically. Probably future generations would read about the golden age of humanity that ended with the great default.

Realistically, the US is not going to completely default on its debt any time soon.  If there was even a bit of worry about it, people and institutions would stop using it as such a critical component of their transactions and saving.  This would also be a dramatic change, but it would happen over time and reduce the effects, should a default ever happen.  I'm not sure what people would use as a substitute.  It would be uncharted territory.

A more likely, but still apocalyptic, scenario would be some partial default or sustained high inflation.  See the above consequences, but scaled down and including loss of confidence in the dollar.


How can interest rates be so low? Over and over again reputable economists and pundits like Alan Greenspan tell us that the 0% interest rates are "unsustainable", yet they are sustained.

How can interest rates be so low? Throughout all of history I could only find interest rates down to 3%, never lower, yet now we have 0% or even negative interest rates. How is this possible?

There is a demand for debt and a supply of debt. The demand for debt is determined by the extent of profitable investment opportunities that are around. The supply of debt is essentially people's saving multiplied by a factor depending on how much risk financial institutions are willing to carry.

Suppose we live in a world where there are no clearly profitable investment opportunities, but many projects which return the same money you put in on average. So nobody would take on debt at positive interest rates. However, even a a zero interest rate, people would still want to save money, for example to insure against bad times, or for retirement. So there's an equilibrium in which the interest rate is zero, but people still borrow and lend.

Another possibility is that people just expect deflation, so a zero nominal interest rate still goes along with positive real returns on your investment.

It is not clear to me what you mean by sustainable, but clearly such situations can arise, albeit rarely. Japan has nominal rates close to zero for a long time now.


Can changes in the value of money be measured via interest rates and future contracts? For most goods, we can compare the value of them by how people trade them. If in a free market, 10 apples can be traded for 20 oranges, then apples are twice as valuable as oranges. Likewise, if 10 apples can be traded for \$1, and 20 oranges can be traded for \$1, the same thing holds.

When I lend or borrow money, I am essentially buying or selling future money. If Alice lends Bob \$10 for 10 years with \$1 interest, then Bob is essentially selling Alice \$11 future dollars for a price of \$10 today. Additionally, if Bob buys \$10 of future corn today, and then corn is \$11 in the future, that means \$10 of todays money is worth \$10 of future money (on average since future corn today may be evaluated differently then future corn in the future, just due to unknown information. The free market is pretty smart in the long run though.)

Can this be used to track the changing value of money then, such as inflation and deflation?

(I guess its hard to use this, since both interest rates and the money supply are simultaneously manipulated by government backed institutions.)

Yes, but...

In theory what you suggest is pretty standard text book, and in fact Keynes makes exactly this point in his book The General Theory of Employment, Interest and Money as a way of getting a grip on the idea of interest and "Time-Value of Money" (don't have copy to hand, will update reference when I get a moment).

The problems come, as always, in the details - in fact in details that you have glossed over in your original set up.

If these 10 apples can be traded for those 20 oranges, then how can you be sure that another 10 apples can also be traded for a different 20 oranges.  Or even, are we sure 100 apples can be traded for 200 oranges?

Adding the dimension of time to this, creates even more problems, but putting those aside the relationship you are looking for is

$$F=S_0e^{rT}$$

That is the future price ($F$) is equal to the price now ($S_0$), inflated by the time value of money.

Now back to apples and oranges.  Are we certain that 10 apples now is the same as 10 apples in the future?  Or more pertinently does the 10:20 apple:orange exchange rate persist into the future?  Probably not.

Apples and oranges are both seasonal, and also suffer seasonal demand.  We could image that apples are easy to come by now, but hard to get in 6 months time, vs oranges.  Since you can't store them, this has to be reflected in the futures price.  This means you will get a different $e^{rT}$ value for apples and oranges

These main causes of these difference are usually broken down thus

Storage costs: can the thing be stored and delivered at the future date?  If so how much does it cost to store?
Carry: How much do you earn holding the thing for longer
Convenience yield:  How much premium are you willing to pay for the convenience of having it to hand?
Credit risk:  How much do you trust that the person will deliver (to the best of my knowledge there are no apple or orange futures, so we would be talking about a forward contract)

Take all these into account, and you have a commodity implied measure of future inflation.  Of course, inflation as usually considered also includes services and retail products, on which there are very few (no?) futures/forwards.  If you could find iPad futures and child care futures, then you would in theory be there.


Total amount of bank loans in a country I am looking for total amount of customer loans banks are giving in a country. I can find data for US at
http://research.stlouisfed.org/fred2/series/CLSACBW027SBOG

However I am looking for statistics for also other countries, Europe, China, Russia, Japan, India, etc.

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


Would a major wealth redistribution cause a major inflation? I've seen a video on wealth distribution on the US where perceived, "desired"(by the majority) and actual income distributions are shown. I've been thinking what would happen to the purchasing power if such a distribution was attained. How much could the supply increase to balance the huge amounts of money that were just being invested and accumulated and will now be spent on consumer goods? If an inflation would occur, is it possible to estimate it?

Inflation is only inflation if supply can't keep up, and if the money supply increases without growing the economy.

Massive wealth distribution (edit: on a long time scale) would be a huge boost to the economy. Why? All the purchases of consumer goods will increase production of all those goods, which will move through the entire economy causing expansion. Of course there will likely be some inflationary effects but as long as the wealth distribution didn't happen overnight whatever mechanism of distribution is probably going to redistribute slower than what the economy can handle anyway.

If you wanted to estimate potential inflationary effects you'd have to figure out the stickiness of production schedules of a basket of consumer goods that the poor would be likely to buy compared to the speed of redistribution.

Also, it's worth pointing out that a lot of money in equity is 'virtual money'. It doesn't actually exist (or rather, it exists, as long as you don't use too much of it at once). What I mean is, if for example Bill Gates wanted to liquidate half his stock, he would sell the stock at a massive loss to the current market rate and would get a lot less than you would think. The same applies to sales of any equity or land... In theory those assets have the value on paper but you can't just redistribute that form of wealth, it would end up destroying the paper wealth since the buyers wouldn't be able to afford it, and would require the seller to sell at a 'loss'... Forced distribution on a short timescale of large amounts of wealth would result in that wealth being destroyed without any real advantage, because the wealth isn't in actual money. So you could also argue that there would be some deflationary effects due to wealth distribution too!


Does the introduction of credit card affects Money Velocity? In my macroeconomics class, our professor showed an example of the introduction of the credit card. She says that this will decrease the money people are willing to hold, and thus increase the money velocity. But why is that true? I mean, although it seems like the credit card as a new mean of payment makes us feel we are not holding much money, but the credit card company are actually paying for us them using deposits. So the money people hold didn't actually change because while consumers hold less cash or deposit, banks that offer credit card will need to hold more cash or deposit to pay for it. So I think the claim that money velocity increases is false. Am I right?

Credit cards are a way of deferring payment in the form of inter-temporal substitution; it helps reduce the household's liquidity constraint that may prevent it from consumption smoothing as much as they would like if their transitory income was much more spread over the household's lifetime.

You are right that the amount of money doesn't change. Loans require giving away money, but credit cards are a way of creating debt, so banks do not need to hold more cash to give a line of credit like you suggest.


Is the government a producer, a consumer, or both? Example: A soldier is paid \$50,000 a year to serve in the army. So \$50,000 worth of defense services are produced and consumed.

In this example, who is the consumer (C) and who is the producer (P)?

Possible interpretations:

C: government. P: soldier (a private producer).
C: government. P: government (soldier is merely an employee of the government qua producer).
C: all individuals that make up the economy. P: government.

Please answer with reference to the above example. Please make use of economic logic and also how the above would actually be done in national accounting.

The government is a producer of goods/services that are then usually not subject to market transactions.

Some goods/services that the government produce can be said to increase directly the utility of the households/individuals in the economy.

Others can be said to function as intermediate goods, entering, usually as a positive externality, the production function(s) of the private sector.

Yet others, the literature contents, can be treated as both being utility-enhnacing and private production enhancing.

An example: national roads (without full-pricing tolls). They are used by individuals to go to recreational trips (utility-enhancing) but also by businesses to transport their goods (production function externality).

Another example: a police force. It can be said that it provides a sense of security to individuals (utility enhancing) but also, it reduces uncertainty for businesses and so increases the productivity of production inputs (in the sense that one does not have to divert scarce private resources to attain the desired degree of security , when a police force operates). Likewise for military (where instead of local outlaw, the danger comes from foreign aggressors).

Etc. In fact, for most goods/services provided by the government, I would say that the dual nature prevails.

In the Political Economy sub-field, in some models the concept of government self-consumption arises, in order to capture the phenomenon of corruption which leads to part of taxes being consumed by corrupted government officials and not used to produce the aforementioned goods and services.


Where is the money that was in the economy pre-lockdown? Countries are in lockdown, companies have no customers and thus no income.

They have to lay off staff which in turn means ex-staff have no income. Everyone simultaneously and quickly depletes any savings so they can no longer buy goods and services.

Where is the money that was circulating in the economy before lockdown?

A very insightful question. The thing I'll add here is the famous identity, MV=PQ. Essentially you're asking why Q dropped (real goods & services), while M (money supply) likely is the same. There are (at least) two answers:

V (Velocity) could have dropped - which has the same effect as shrinking M. $100 making 10 round trips is very different from $100 making only 1 round trip.
More nuancedly (sp?) - You could be seeing a collapse in the "composition" of the Money Supply. I.e. Maybe M2 collapsed back into M1, etc. If lots of people called in their loans (M2+) & converted to cash (M0), the same amount of money might be there...but it would have been "sucked" from the system, in a sense.


In Indonesia, reducing fuel subsidy strongly impacts the price of everyday goods. Why? In Indonesia, petrol/gasoline price is regulated by the government and historically it is heavily subsidized. I know, economists typically consider fuel subsidy to be a bad policy, for many reasons, including: it appears to disproportionately favor the wealthy who have more cars.

However, in Indonesia, fuel price seems to strongly affects other prices in general. I mean, when the government tries to reduce subsidy and increase the price, we will see prices of everyday stuff like food increase significantly. So, some people say increasing fuel price actually affect the poor the worst, because they barely have enough money to buy basic needs. The effect is so bad that such attempts to reduce subsidy are usually met with big protests by the general population.

Some source:

Jakarta Calm After Subsidized Fuel Prices Are Increased
Indonesia inflation jumps after fuel price hike
INDONESIA: Poor hit hard as fuel prices rise.

I've lived in other countries where fuel price is determined by market and fluctuate freely (like say the US), but I don't see such phenomenon (e.g. grocery prices don't noticeably increase as much when gas price increase). Why?

Consumers rarely have to pay the full price of petrol / gasoline. That's true in most of the world, and the US is no exception. There are explicit subsidies and favourable tax regimes in many extraction countries; and of course there are the implicit subsidies in the unpriced pollution - oil's externalities.

Those explicit and implicit subsidies have been there for decades: and consequently, oil products are hugely over-used, compared to the alternatives: that's a direct consequence of oil being a product with demand that's not perfectly inelastic - indeed, estimates of long-term price elasticity of demand for petrol are of a figure around -0.6 or so. That means that oil is embedded in large quantities in the supply chains for very many products.

The larger the historic subsidies, the bigger the impact of changes in subsidy levels to the price of other goods and services. And the poorer the population, the bigger the impact. That's why you see a bigger impact in Indonesia than in the US.

Now, although in the long run removing some of the subsidies will correct some of those imbalances, that will require capital reallocation and some industrial restructuring, all of which takes time.

And although in the long run, rising prices of some products, in the absence of rising wages, will choke off demand for other products, leaving inflation at no net change, the short-term effect of rising oil prices is to raise the costs in a significant part of the supply chain for a lot of goods and services, given an instant boost to inflation.


Why do banks charge a spread on credit? According to the doctrine of the money multiplier (https://en.wikipedia.org/wiki/Money_multiplier), a positive reserve ratio means a bank has LESS loans than deposits. See the table in the link. Therefore, if a bank gets a loan of 100, it can lend, say 80. Here I understand that the interest rate on loans must be higher than the interest rate on deposits, so that the bank can make a profit.

HOWEVER, there are other theories like endogenous money theory which state that actually, if a bank has 100 in deposits, it creates 900 of bank money (see https://en.wikipedia.org/wiki/Reserve_requirement). In this case, there is no need for the bank to charge a spread!

Can you guide me on this? I am confused!

Is this a proof that the money multiplier is correct?

You're getting two quite different concepts mixed up:

Banks earn a spread on loans for serving as an intermediary between savers and borrowers and for holding credit risk.
The process of borrowing and lending, of banking, leads people in an economy to have more liquid assets (money in a broader sense), than there is underlying cash.

When economists say banks "create money" they don't mean they're printing money off the presses and giving it to themselves as profits.

How banking expands the money supply

To understand banking clearly, its helpful (perhaps 100% required) to understand and use the fundamental equation of accounting:

$$ \mathrm{Assets} = \mathrm{Liabilities} + \mathrm{Shareholders Equity} $$

Let's say Warren Buffet decides to start a bank. He puts in \$10 billion of his own money. The balance sheet of the bank would look like:

$$ \begin{array}{r|rr}\text{Assets} & \text{Liabilities} & \text{Equity} \\ \hline \$10 \text{ cash}& & \$10\end{array} $$

Let's say he now collects \$90 billion worth of deposits from people. These bank deposits are assets for the people, but they are liabilities to the bank (the bank owes the depositors \$90 billion).
$$ \begin{array}{r|rr}\text{Assets} & \text{Liabilities} & \text{Equity} \\ \hline \$100 \text{ cash}& \$90 \text{ deposits} & \$10\end{array} $$

Now let's imagine that the bank lends out $80 billion. The balance sheet would then look like:

$$ \begin{array}{r|rr}\text{Assets} & \text{Liabilities} & \text{Equity} \\ \hline \$20 \text{ cash}& \$90 \text{ deposits} & \$10  \\ \$80 \text{ loans}\end{array} $$

Depositors have \$90 billion in deposits, and the borrowers have \$80 billion in cash. So the economy now has \$170 billion in money where there previously was only \$90 billion.

A key point to realize...

Across the whole economy, all these assets and liabilities cancel out! Borrowing and lending (on its own) is not creating any real wealth. The expansion of the banking system's balance sheet is simply creating more entries on the assets and liabilities side that cancel out. To the extent though that the liabilities of the banking system, deposits etc... are easily tradeable, this does create more money in the economy. More people have liquid assets in the form of bank deposits etc... (which are liabilities of the bank).


Identification strategy for measuring impact of employee turnover on project quality I am seeking help with identification strategy for figuring out causal direction in the following context:

There is a company where project managers (PM) rotate every 3-7 years across the country/globe. Naturally some PMs leave the company. I would like to measure the impact of PM rotation/turnover on project quality (as measured by time taken for completion ).

EDIT: I expect the PM rotation to be most likely independent of Project Quality. PM's must rotate between 3 to 7 years, with 7 being a hard limit with rare exceptions. PM's decisions are more likely to be driven by quality of life in a region, schooling, their own career aspirations, the type of opportunities that have opened up within or outside the organization. They cannot be fired (Govt) and don't get too much extra credit for a successful completion of a project and they don't get much flack if things are slow (unless it is a spectacular failure). Sometimes PMs move to a new region which is close to the old region so they are still able to hold on to their portfolio.

I would like help with identification strategy to distinguish between or tease out the separate impacts of the following scenarios:

Scenario 1: PM is rotated/changed --> Project quality suffers --> Project takes longer to complete.

Scenario 2: Project quality is good --> Project completes quickly --> PM didn't have the opportunity/time to leave

and

Scenario 3: Project quality is bad --> project takes forever --> PM rotates/leaves company due to natural career progression/rotation policy

I've spend the last many hours googling literature studying the impact of  project manager turnover on project quality but I haven't been able to find any good design. Would be grateful for any suggestions for identification.

This looks challenging, and the following would only work in circumstances where:

Reliable estimates can be made of how long each completed project should have taken to complete.
Good records are maintained of the progress of individual projects, enabling assessments to be made of the form "Project A was X% complete after n years".

Suppose two projects that were estimated to require 3 years were actually completed in 5 years, and in both cases a project manager had moved on after 2 years.  One might then focus on how complete each project was at the time when the project manager left.  If one project was 70% complete at that time, and so took a further 3 years on the last 30%, that suggests Scenario 1.  If the other project was only 40% complete after 2 years, and subsequently progressed at the same speed taking a further 3 years on the remaining 60%, that suggests Scenario 3.


Is there a study of world poverty rates over time pre 1800? There seems to be research showing that world poverty rates have declined since 1800, see eg this. Is there any research on world poverty rates before this period? I am guessing that during this period poverty in much of the world would have gone up as a result of European colonisation.

This is only for Europe unfortunately, but Bob Allen has a famous paper in which he presents reconstructed data for real wages in different European cities from 1350 all the way to 1799:

  Allen, Robert C. "The great divergence in European wages and prices from the Middle Ages to the First World War." Explorations in economic history 38.4 (2001): 411-447.

From real wages and price data, he then infers a measure of (extreme) poverty.

Look at Figure 5 in the paper. The scale of the y-axis is a measure of poverty in which 1 is the poverty threshold. Values above 1 on the y-axis correspond to real wages lower than survival rate (i.e. that do not allow a survival caloric intake), and values above 1 indicate a real wage above survival rate.


How to estimate Robot density in a country? I'm writing a thesis where a key element is the degree of automation (for Switzerland) through time.

I was counting on the IFR to get those data but I need to pay 2000k to get them...
Does anyone have an idea of how to estimate that? maybe a proxy? I cannot think of anything...

Thank you for your help

Daron Acemoglu has written a fair bit about the impacts of automation on labour markets. Bearing in mind that mechanization and "robotization" are different in important ways (with that importance increasing as AI and machine learning become more sophisticated and ubiquitous) you could use his work as a starting point.  Certainly, you'd be remiss not to cite it if you're writing a paper.

In https://www.nber.org/papers/w24196, Acemoglu outlines the challenges to the question you're asking and offers a framework of approach.  In https://www.aeaweb.org/articles?id=10.1257/jep.33.2.3, the framework is applied, and there are links to the data which you can look at for insight into how to measure this stuff.


Is Dollarization a good decision for a small open economy with an economic cycle that weakly correlates with the anchor country? I have been reading the work of Alesina http://scholar.harvard.edu/files/alesina/files/dollarization.pdf, about the pros and cons of dollarization (i.e. adopting the currency of other country, say the dollar). It is not the case of the European Union by the way, here the small open economy completely gives up on having its own currency.
Countries like Panama, El Salvador, and Ecuador have adopted such monetary policy and have done well during expansionary part of the cycle. Now, most commodities prices, of which these countries depends are down, and evidence will be available of the costs of dollarization. However, here I ask about theoretical reasons of why this extreme fixed exchange rate is better with a flexible scheme. The trade-off is between stable and credible monetary policies and flexible possibly manipulable ones in countries with weak institutions.

Like many questions in economics, optimal behavior depends on the set of alternatives. The alternatives of Panama, El Salvador, and Ecuador to dollarization are likely a heavily politicized monetary and exchange rate policy. And so for the hypothetical small open economy considered in the question, the key issue is "What's the institutional effectiveness and independence of the potential monetary authority?"

There are small, open economies with well run central banks (Australia and Switzerland) but also less well run ones (Turkey).  If Panama could have Australia's quality of central bank they'd probably prefer that to dollarization. That might explain why Chile and Costa Rica, with some of the best institutions in Central America choose not to peg their currency to the dollar. If Panama did manage its own monetary affairs without a peg they'd get likely get results more like Turkey. They likely prefer dollarization to the results they'd expect from independent monetary policy.

And indeed, when you peek into the past of many countries with pegged currencies, you often find an episode of poorly conducted monetary policy just before dollarization. Not Panama (never had its own currency since independence), nor El Salvador, but Ecuador introduced dollarization in response to a severe banking and financial crisis. Argentina also seemed to introduce dollarization in response to monetary crisis.

There are of course other considerations related to optimal currency areas. In particular, if you do a lot of trade with the USA dollarization is less of a sacrifice with respect to the best possible independent currency and monetary policy then if you do most of your trade with a neighboring country that uses a different currency. If Mexico adopts a pegged currency at some point the dollar makes a lot more sense than the Euro when 70% of their trade is with the USA and trade is a big part of their economy. But neighboring countries often have similar trading partners, and are in a similar position with respect to other optimal currency area criteria (no labor mobility to the reference
country, no fiscal transfer mechanism shared with the reference country, and probably similar business cycles) so institutions are going to play a important role in explaining cross-sectional differences in the decision to dollarize, and that is why I highlight that in this answer.


Graduate Level Textbook on Labor Economics and Macro-Labor What would be good textbooks at the graduate level that deal with Labor Economics and Labor-Macro (As in, Equilibrium Unemployment Theory, Shimer Puzzle etc)?

There's also Labor Economics by Pierre Cahuc, Stéphane Carcillo, and André Zylberberg. It's a broader labor econ book, but The "Unemployment and Inequality" fourth of the book covers these topics. I have not seen the second edition, but I expect that they did not alter that part for the worse.


Calculating the cost of a year without land-tilling The motivation to this question is the Biblical commandment of Sabbatical year, in which the land should be allowed to rest and should not be cultivated. I am trying to estimate how much would it cost to actually implement this commandment in an industrial economy.

For concreteness, I use data about the Israeli agricultural product; my main problem is the interpretation of the data, which is probably applicable in other countries.

Here are the relevant data for 2013 (rounded and in billions of ILS):

INPUT (excluding compensation for employee jobs) = 19.7. Of these: Input (purchased and from intermediate produce) = 17.5;  depreciation = 2.2.
OUTPUT: 30.0. Of these: crops = 18.1, animals = 11.9 (note that only land-cultivation is prohibited, while growing animals is permitted).
DOMESTIC PRODUCT: Gross = 12.5, Net = 10.3 (the difference between them is the depreciation.)
INSURANCE COMPENSATION AND OTHER RECEIPTS: 1.0.
INCOME ORIGINATING IN AGRICULTURE: 11.1.  Of these: Compensation for employee jobs: 5.3 (Q1: where do the other 5.8 go to?)

My assumptions are as follows:

The consumers should still eat fruits and vegetables. So, instead of growing fresh fruits and vegetables, some of these will be imported, while others will be preserved from the previous year (e.g. dried fruits and other products with long shelf life).
The farmers should be compensated on giving up the right to till their land, by giving them the same gross income that they have in a usual year.
However, other workers that deal with agricultural produce, such as: truck drivers, traders, workers in fruit markets, etc., should not be compensated because they can still work with imported produce (Q2: do the figures in the table include the income of these workers? Or do they belong to a different branch of economics?)

I did several calculations and got different results:

When there is no land-tilling, there is no depreciation and no insurance reception. Hence, only the 10.3 net domestic product is relevant, and from this we should extract the 1.0 insurance, for a total of 9.3. This should be multiplied by 60%, which is the relevant part of "crops" (since the other 40% are "animals" which can be grown as usual). We get: 5.6.
take only the compensation for employee jobs (5.3), since only employees should be compensated (should the insurance receptions be subtracted?). Again multiply by 60% and get 3.2.
in a different source, I found the fact that the total number of crop farmers is only about 7000. Assuming a gross annual income of 240,000, which a top decile income, I got to only 1.7 (billion). AFAIK, most farmers make less than that, so the actual figure should be even lower.

Since the numbers in each calculation are quite different, I am interested to know what causes the difference, and which calculation should be used?

The approach appears to be static. An intertemporal approach would recognize that part of the period's forgone production would have been used for investment, augmenting the capital base. So we also lose future consumption. But let's skip that.

In order to categorize what the country will lose (and who is going to lose it) I decompose output into certain categories and comment on each one

Output =
+ Imported materials  (we don't care what happens to them)
+ Services offered from abroad  (we don't care what happens to them)
+ Locally produced materials in other sectors  (will there be demand for their products from other sectors of the economy? Will they be able to export them? -possibly lost income here)
+ Services offered by locals (How much of them will be used to manage the compensating imports? -possibly lost income here)
+ Rents of land owners who do not engage in farming (lost income)
+ Wages of people working the land, i.e. "Direct Labor" (lost income)
+ Wages of other personnel in the sector (will they be needed to manage imports? -possibly lost income here)
+ Depreciation (equipment will erode by the simple passage of time, but land may even appreciate in terms of productivity-after all the biblical commandment has clear practical motives. So on the balance?)
+ Profits of farmers/entrepreneurs (lost income)

A basic conclusion from the above, is that by stopping production in one sector, we may also lose some production in other sectors.

...and think also about taxes.


Group contamination in Regression Discontinuity I'm an undergraduate student I have an empirical question regarding Regression Discontinuity and Diff-in-Diff methods. I'm currently evaluating the impact on fertility of a Conditional Cash Transfer in Brazil.

This is an oversimplified example of the problem we encountered.
A family will receive the benefit if their income per capita is, for example, $150, and will not receive it otherwise. We would use this fact to use a RD design to evaluate casual effects. The problem is, those above the threshold have incentives to get another child and reduce their per capita income to enter the "control" group.

Anyone has good references (articles, textbook), where I can find something that would help me? Helpful comments are greatly appreciated.

Are you looking for general references on Diff-in-Diff (DiD) and Regression Discontinuity (RD)?

If yes, a good starting reference at the undergraduate level is probably

J.  Angrist  and  J.S.  Pischke, Mastering' Metrics: The Path from Cause to  Effect, Princeton University Press, 2014. Look at chapters 4 and 5 for RD and DiD.

If who want to dig deeper:

J. Angrist and J.S. Pischke, Mostly Harmless Econometrics, Princeton University Press, 2009, chapters 5 and 6.

For recent references on RD

C. Carpenter and C. Dobkin, “The Effect of Alcohol Consumption on Mortality: Regression Discontinuity Evidence from the MLDA, American Economic Journal: Applied Economics 1 (2009), 164-182.
A. Abdulkadiroglu, et al., “The Elite Illusion: Achievement Effects at Boston and New York Exam Schools,” Econometrica, 2014.

On Did:

D. Card and A. Krueger, “Minimum Wages and Employment: A Case Study of the Fast Food Industry in New Jersey and Pennsylvania,” American Economic Review 90 (1994), 1397- 420.
D. Card, “Using Regional Variation to Measure the Effect of the Federal Wage,” Industrial and Labor Relations Review (1992) 46, 22-37.
C. Carpenter and C. Dobkin, “The Minimum Legal Drinking Age and Public Health,” The Journal of Economic Perspectives 25 (2011), 133-156.

Considering your example, I guess that it could be a contribution to show that a small increase in benefit per capita may increase fertility. Is the benefit permanent? If not, I doubt that people will have a strong incentive to get another child. Moreover, it takes time to get a new child! So, there is a time dimension issue. They cannot claim the benefit if they don't have the child! If your design allow for this, you can exploit the time dimension in case some families respond to the incentive.


vNM Expected Utility Theorem Proof I have been asked by my professor for Decisions Theory to refer to David M. Kreps' Notes on Theory of Choice for the proof of the vNM theorem of expected utility. Before the proof of the theorem actully starts, there are two Lemmas stated which are used in the proof of the theorem. I am unable to understand the proof of one part of one of the Lemmas. Specifically, the proof for part (c) of the below Lemma. I understand the intuition behind it. Its basically the indpendence axiom but with equivalence relation, however I do not get the proof at all. Any help is much apprciated.

And this is the proof for part (c). When the author talks about applying (5.2), he is referring to the independence axiom and when he talks about applying (5.3) he is referring to the Archemedean or the continuity axiom.

Let's write
\begin{align}
  x&=ap+(1-a)r\\
  y&=aq+(1-a)r
\end{align}
You want to show that $x\sim y$. But
\begin{equation}
x\sim y\quad\Leftrightarrow\quad x\not\succ y\;\text{ and }\;y\not\succ x.
\end{equation}
So the given proof tries to establish that $x\not\succ y$ (and in a similar way, $y\not\succ x$) is true.

To show this, use proof by contradiction. Suppose the contrary is true: $x\succ y$, and try to derive a contradiction. This is what the given proof is doing.


How does Russia have so much foreign exchange reserves? Source

Russia is under U.S. sanctions. How does Russia have so much foreign exchange reserves?

russia uses its oil revenue to setup russia stabilization fund in 2004. The money in that fund is used to pay for national budget deficit. The foreign exchange reserve of Bank of Russia is not affected by the US led sanctions.


Has US real median personal/household income been stagnant since the late 1970's/early 1980's? This is referring to an earlier question: Why are real median household incomes stagnant?

How do the following graphs actually tell you that US real median personal/household income has been stagnant since the late 1970's/early 1980's?

Real Median Personal Income in the United States

Real Median Household Income in the United States

How are these graphs in line with the following graph, which does look more like stagnant wages (when comparing wages in constant dollar):

For most workers, real wages have barely budged for decades

as well with the following graph which also seems to show stagnant wages:

Wage Stagnation in Nine Charts

As said in the comments, there is an important difference between household income and wages. Household income includes income from capital. As a higher share of total income is going to capital than to labour (see below), you would expect a divergence between household and wages.

Similarly, productivity increases are not being translated into higher wages, which means a higher distribution of gains toward capital (see below).

(Source of image here)

Another important issue is that the median can be a misleading figure when distributional changes are occurring. For instance, below is the percentile level of household income for the US:

(Source of image here)

As you can see, the median can mask quite an heterogeneous development for different percentiles. Since lower percentiles source their household income mainly from wages, a stable median wage but an increasing median income is entirely consistent with higher inequality plus lower labour share.


Effect of specific tax on demand and supply There are intersecting supply and demand curves. A tax is levied on suppliers. Say the tax is a fixed tax (for example, VAT) and not an ad valorem tax.

Why can we not just move along the demand curve by the value of the tax (as producers just increase the price by the value of the tax)? So for any given point on the demand curve, just increase the price by the given value of the tax and that is the new demand - why does it not work like this?

Whilst I agree a parallel shift of the supply curve by the value of the tax makes sense I do not understand where is the flaw in the earlier method?

The answer has to do with the fact that (under typical assumptions) producers experience increasing marginal costs as output rises. This makes the supply curve upward-sloping.

Suppose firms simply increased the price by the amount of the tax. Consumers would then buy fewer units of the good, resulting in less profits for producers. However, marginal costs are also lower at the lower level of production, meaning that firms could profitably produce at a lower price than that. So they lower the price and increase production a bit to partially offset the losses caused by the tax (note, the price remains above where it would be without the tax, and production still less than without the tax).


Government Spending vs. Investment Spending What is the difference between government spending on physical capital and investment spending by government on physical capital?

Government spending is the spending by the government on goods and services (non-capital goods). This means that the government buying physical capital would not be included in government spending, and would be included in investment spending.


Marginal Revenue Interpretation of Auctions I'm reading Klemperer's survey on auction theory (pages 17-18) which discusses the relation between traditional microeconomic price theory and the revenue equivalence result. Firstly, I don't quite understand the analogy: marginal revenue is the revenue due to additional quantity sold, but an auction might just sell a single non-divisible object.

  In particular, it is helpful to focus on bidders’ ‘‘marginal revenues’’.
  Imagine a firm whose demand curve is constructed from an arbitrarily large
  number of bidders whose values are independently drawn from a bidder’s
  value distribution. When bidders have independent private values, a bidder’s
  ‘‘marginal revenue’’ is defined as the marginal revenue of this firm at the price
  that equals the bidder’s actual value.

I'm not sure why "the marginal revenue of this firm at the price that equals the bidder’s actual value". Does this not assume that the bidder actually bids his actual value? This surely isn't the case for e.g. first-price auctions.

  Bulow and Roberts follow Myerson to
  show that under the assumptions of the revenue equivalence theorem the
  expected revenue from an auction equals the expected marginal revenue of
  the winning bidder(s).

I'm confused with this (which probably stems from my confusion with the earlier parts): if you sell 1 item, then trivially the marginal revenue equals the expected revenue. Surely I am missing something here.

This analogy is due to Bulow and Roberts.

Think of a diagram from Econ 101 with price on the vertical axis and quantity on the horizontal axis and consider a monopolist with constant marginal cost of zero. Given a demand curve, you can draw the marginal revenue and find the monopoly quantity where this function intersects the axis.

Now take that same diagram and interpret "quantity" as "probability that a buyer accepts price $p$", i.e., $q = 1- F(p)$. Now, maximize revenue $pq=q F^{-1}(1-q)$ wrt $q$ to reach some form of the well known marginal revenue = marginal cost. The derivative of the revenue above is
$$\frac{\partial pq}{\partial q} = F^{-1}(1-q) - \frac{q}{f(F^{-1}(1-q)} = v - \frac{1-F(v)}{f(v)}, $$
where I replaced $q=1- F(p)$ and set $p=v$. This is Myerson's virtual value. The expected revenue of an optimal auction is the expectation of the highest virtual value. The part $\frac{1-F(v)}{f(v)}$ is the information rent, which takes care of the fact that you don't bid your true value in a first-price auction.


Proper definition of extensive form game Consider the following game: Let $N := \{1,2,3,4\}$ denote a set of agents.
Going from 1 to 4 each agent can decide how many of the remaining agents he wants to integrate into a coalition.
The strategy sets are given by $S_1 := \{1,2,3,4\}$, $S_2 := \{1,2,3\}$, $S_3 := \{(1,1),(1,2),(2,1),(2,2)\}$ and $S_4 := \{(1,1,1)\}$.
I want to generalize the game for  $|N| = n$. So basically agent 1 picks a number $s_1 \in N$. Then agent $1 + s_1$ picks a number $s_{1+s_1} \in \{1,\ldots,n-s_1\}$ and so forth. Eventually there exists an agent who integrates all the remaining agents $\exists j \in N : s_j = 1+n-j$.

What's a proper definition of this simple game? What are the strategy sets?

Let the set of players be $N=\{1,\dots,n\}$.

According to my understand your description of the game, I take the following statements to be true:

Each player chooses a number of players to integrate, not players with specific identities. As a result, the player making the choice may or may not be included in the coalition.
Each player has to choose at least 1 player, if still available.
Each player moves exactly once.
The game is with perfect information.

Under the above interpretation, player $i$'s ($i\ge2$) action space would be $A_i=\{0,1,\dots,n+1-i\}$, as, in principle, $i$ could choose at most $n+1-i$ other players (when all players $j<i$ chose $1$) and at least $0$ (when all previous players have exhausted the player list). An exception applies to the first player, where the option $0$ is not feasible. Thus $A_1=\{1,\dots,n\}$.

Player 1's strategy space would be $S_1=\{1,\dots,n\}$. Player 2's would be $S_2=\{1,\dots,n-s_1\}$, since after $s_1$ players are picked by player 1, there are only $n-s_1$ players left to be picked by player 2. Carrying this argument forward, player $i$'s strategy space would be $S_i=\{1,\dots,n-s_1-\cdots-s_{i-1}\}$.

Generalizing from the above reasoning and taking into account boundary cases, let history at stage $i$ be $h^i=(s_0,s_1,\dots,s_{i-1})$, where we set $s_0=0$. (In general, though, stages should be indexed by a parameter different from the player index. But in your game, since each player moves exactly once, we can use the same index for stages as well as players to economize notation.) The history dependent strategy space for player $i$ would thus be
\begin{equation}
S_i(h^i)=
\begin{cases}
\{1,\dots,n-\sum_{j=0}^{i-1}s_j\} & \text{if } n-\sum_{j=0}^{i-1}s_j\ge 1 \\
\{0\} & \text{otherwise}.
\end{cases}
\end{equation}


How can austerity and quantitative easing happen simultaneously? In 2010 the UK government simultaneously started an austerity program which cut public spending, taking money out of the economy, while enacting a quantitative easing program which injected hundreds of billions of pounds into the economy.

How can these two seemingly conflicting programs co-exist?

The division comes from the entities guiding these sets of policies.

The austerity program falls under fiscal policy, which is controlled by Parliament. Quantitative easing falls under monetary policy, which is controlled by the Bank of England.  The policies enacted by these two entities are not necessarily coordinated, and can sometimes work opposite each other's objective.


Show that this income tax is effectively a lump sum tax This is a standard income, leisure tradeoff model.
$$ \max_{c,l}  \min\{c; l\} $$

$$s.t. \space c = w(1-t)(1-l)$$
$l$ is leisure (where total time is 1),
$w$ is wage, $c$ is consumption, and $t$ is the labor income tax.

I was asked to show that this income tax is effectively a lump sum tax.

The definition for lump sum tax is as follows: A tax whose value is independent of the individual's behavior.

Individuals can do nothing to avoid the lump sum tax, and there are no relative price changes. I guess that the utility function had something to do with it, but I do not see how the tax can be viewed as a lump sum tax.

A lump sum tax is a tax which only has an income effect, it has the equivalent effect of reducing the agent's wealth. In particular, it has no substitution effect, it does not incentivise the agent to switch between $c$ and $l$.

In your case the agent's optimal choice of $c$ and $l$ is given by:
$(c^*,l^*)=\left(\frac{w(1-t)}{1+w(1-t)},\frac{w(1-t)}{1+w(1-t)}\right)$
(show this)

It is easy to see that increasing $t$ is equivalent to decreasing $w$.


Literature on DSGE models featuring a financial sector I recently read a ECB primer on DSGE models featuring the financial sector and became interested in the field. I also started to read some of the earlier literature in that field (e.g. Gertler & Karadi, 2011) but was hoping to receive some hints on where this literature is currently evolving and if there are any more recent papers that have been widely recognized?

This is actually one of the more exciting development in DSGE modelling since financial crisis. Including financial sector or friction is definitely a growing area of research and interest.
Unfortunately its quite a new development so there are not that many papers yet out there (however if you consider doing research in this field that also means there is a lot of untapped opportunity to do work there). You will probably want to read most of the works of Gretler who is one of the pioneers in this area.  Gertler and Kiyotaki (2015), Gertler and Kardi (2015) and Gertler, Kiyotaki, and Prestipino (2016) are good texts.
Next papers such as Christiano, Motto, and Rostagno (2014) are looking at financial sector in a bit roundabout way but I think can be still considered to be part of this research area.
Lastly, this paper from Christiano, Eichenbaum, and Trabandt is a nice literature review on DSGE modeling since Great Recession. It covers the above mentioned papers plus also some others. It is a general review that looks also on other advances in DSGE modeling and also at their criticism but I would say its worth reading nonetheless even if you are only interested in the financial DSGE models.
You will also find more sources by looking at papers cited in the literature above.


Unemployment and the Minimum Wage---what are the main counter-arguments to Card and Krueger? Card and Krueger's paper (AER 1994, "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania") uses a difference-in-difference identification strategy to identify the causal effect of a minimum wage increase on unemployment. (A summary can be found here.) The main finding was that the increase in the minimum wage had a negligible or even non-existent effect of employment. There have been several criticisms to the paper---including criticizing the quality of the data or the fact that employers might have anticipated the change.

My question is, what are the main economic explanations for why employment did not significantly fall? What other evidence could be used to test these explanations?

Isaac Sorkin, a grad student at Michigan, has addressed this. Here is Miles Kimball blogging it, Isaac Sorkin: Don't Be Too Reassured by Small Short-Run Effects of the Minimum Wage, Jonathan Meer and Jeremy West: Effects of the Minimum Wage on Employment Dynamics, and The Economist—Destination Unknown: Large Increases in the Minimum Wage Could Have Severe Long-Term Effects. The main argument there is that previous work measures short run elasticities, which are less responsive than in the long run. You can surely find more by checking Sorkin's citations.


Repeated games with decreasing marginal returns The standard analysis of repeated games assumes that the payoff of a player from a repeated game is a sum (or arithmetic mean, or discounted sum) of the payoffs in the basic games.

But what if the players have decreasing marginal returns?

For example, suppose the basic game is Matching Pennies. In each basic game, a player can either win 1 or lose 1. However, the average utility of a player from winning e.g. 10 times, is not 1 - it may be less than 1 if the player has decreasing marginal returns.

Are there references that deal with such repeated games?

It seems to me that this will not make a difference if you only consider pure strategies. Consider that player $i$'s winnings is $x$ and from this he gets utility $U_i(x)$. As long as $U_i()$ is increasing in $x$ player $i$ will prefer outcomes with larger winnings to lower ones, which is exactly what he would do if he had constant marginal returns.

Another way of forming this argument is that as long as $U_i()$ is increasing another utility function representing the same preferences over outcomes would be $\hat{U}_i(x) = x$. (You can perform a monotone transformation that transfers $U_i()$ to this.)


STATA: Impulse Response Function after ARDL? I am working with an ARDL model in STATA but am unable to figure out how to run the impulse response functions with the estimated coefficients. Numerous papers talk about ARDL and IRF, but not a single one I've found actually outlines the process.

Can anyone point me in the right direction?

In order to compute IRF in stata you should check "varbasic" in Stata's manual: http://www.stata.com/manuals13/tsvarbasic.pdf


Constrainted optimization: merge two constraints into one Consider the following problem
\begin{align}
&\max_u F(x,u)\\
\text{s.t. }& u \in [0,\bar u].
\end{align}

Any idea how to merge the two constraints $u \geq 0$ and $\bar u - u \geq 0$ into one constraint $f(u,\bar u) \geq 0$?

$$0\leq u \leq \bar u  \implies -\frac {\bar u}{2} \leq u - \frac {\bar u}{2} \leq \frac {\bar u}{2}$$

$$\implies \left | u - \frac {\bar u}{2}\right| \leq \frac {\bar u}{2}$$

$$\implies \frac {\bar u}{2} - \left | u - \frac {\bar u}{2}\right| \geq 0$$

ADDENDUM
In a comment it was proposed that we could instead use the squared expression to achieve differentiability everywhere,

$$\frac {\bar{u}^2}{4} - \left ( u - \frac {\bar u}{2}\right)^2 \geq 0$$

Let's see: we then are allowed to decompose the square and write

$$\frac {\bar{u}^2}{4} - u^2 + u\bar u - \frac {\bar{u}^2}{4} \geq 0$$

$$\implies -u^2 + \bar u u \geq 0 \implies u(\bar u -u) \geq 0$$

which is nothing more than the multiplication of the two separate constraints.

ADDENDUM II
If we have $u \in [a,b]$ for $a<b$ arbitrary reals, then the general expression is

$$-u^2+(a+b)u-ab\geq 0$$


What are the factors that drive openness in developed vs developing countries My questions are:

What are the factors (both positive and negative) that drive openness in a country?
Do they differ between developed (e.g USA) and developing (e.g China) countries?
Will these factors remain a key factor to drive openness for the respective country in the future?

So far I researched on stage of development, specialisation of each country's skills, degree of trade liberalisation and the geographical size.

I assume you mean openness to international trade, and that you're interested in both policy choices (does the country want to be open) and physical constraints (can they be open).  The main factors that come to mind are:

Size: the smaller you are, the more you are forced to rely on international trade, as it simply isn't feasible to complete many economic operations entirely within your borders.  Lichtenstein and Andorra are much more open than the US or Russia primarily for this reason.
Remoteness and transport costs: if you're tiny, but in the middle of nowhere, like Tuvalu, the cost of trading internationally means that anything you can feasibly produce domestically will be cheaper than imported goods.  There is little point in trying to protect your domestic market because transport costs already provide more protection than you're likely to want.  This also applies to a lot of small landlocked countries, like Burundi.  Some dependencies of more advanced countries overcome this by massively subsidising transport, often linked to military bases.  E.g. consider the money the UK has spent on an airport in St Helena.
Taxation capacity: this matters very little to rich countries, but developing countries often have a small tax base, large informal sector, and find it hard to replace tariffs with alternatives (VAT, income tax, etc).  Some would like to encourage international trade, but are constrained in how quickly they can remove tariffs.
Proximity to the technological frontier: Ha Joon Chang would argue that the history of protectionism in countries like the UK and US make this point very clearly.  If you lag behind your competitors, you're more likely to protect industries you want to catch up.  If you're ahead, you are more likely to advocate for free trade, knowing that the more unconstrained international competition is, the more you will gain.
Relatedly, supply capacity is perhaps the equivalent for developing countries far from the technological frontier.  Some countries have very weak technology but so much cheap labour that they ought to be able to compete in global markets for simple manufactured goods, but lack critical elements necessary to develop successful firms in these areas.  These constraints come in many forms, a couple of the most common are poor physical infrastructure and weak financial services.  If you have supply capacity you will be keener to promote international trade.  This is not an issue for rich countries.
Comparative advantage will tell you what your country is likely to produce and export if very open.  As I explained in this answer, you will be more keen to rely on your comparative advantage if it is in an industry that is experiencing rapid improvements in productivity.  If your advantages are all in stagnant industries, you'll be more tempted to use protection and other state intervention to shift your comparative advantage, and then open your markets.  This is a bigger issue for poorer countries.
Political culture: many countries have attitudes to openness that are not entirely or directly driven by economics.  For instance, some would argue that European economic integration in the late 20th Century was primarily an attempt to reduce continental violent conflict.


How much is an asset worth that produces a constant income? Say you have an asset that produces for you £1k per year profit (adjusted for inflation) at a constant rate? (The proverbial immortal goose that lays golden eggs).

Logic would say it's potential is infinite.

But for an individual of age 30 it is only worth up to £50k in their lifetime. And they may care less about what happens to it after they die.

But to a company who buys it, companies can live essentially forever.

An asset that gives you a constant sum of nominal income every year is called a perpetuity. (The UK government used to issue such a product and called it a consol.)

Everyone "knows" that £1,000 received next year is worth less than £1,000 received this year. But how much less?

To try to formalize this notion, economists use the concepts of discount rate and present value. The discount rate is the rate at which you discount future returns.

If for example your discount rate is $5\%$, then this means you treat £1,000 received next year as equivalent to £950 received this year. (And an economist would say that the present value of £1,000 received next year is £950.)

We can now answer your specific numerical question:

For simplicity, suppose your discount rate is fixed as $\delta$.

Then the present value of this asset is: $$1000+(1 -\delta)1000+(1 -\delta)^21000+\dots = \frac{1000}{\delta}.$$

More generally, suppose your discount rate each year is $\delta_t$ ($t=0,1,2,\dots$).

Then the present value of this asset is: $$1000+(1 -\delta_1)1000+(1 -\delta_1)(1 -\delta_2)1000+\dots.$$

The brief discussion above is usually covered in greater depth in a course on Intermediate Microeconomics. See e.g. Varian (2014, 9e) Ch. 10.


Difference between equilibrium and k-rationalizability I'm reading a Structural Models of Nonequilibrium Strategic Thinking: Theory,
Evidence, and Applications by Crawford, Costa-Gomes and Iriberri. They write the following:

  In two-person games, a player can find his set
  of k-rationalizable strategies via k rounds of
  iterated strict dominance, without the need
  for fixed-point reasoning. Thus, k-rationalizability is cognitively less taxing than equilibrium, especially for small k.

K rationalizable strategies are those that survive k-rounds of iterated strict dominace (for n=2 player games). For n >2, k-rationalizable strategies are those that survive k-rounds of iterative never best response elimination.

What is meant by fixed point reasoning here? Is there a difference between fixed point reasoning and iterative never-best-response elimination?

Relevant section is Section 2.2 found on page 12.

What is meant by fixed point reasoning here?

Specifically, Nash equilibria. One way to define a Nash equilibrium in words is "a strategy profile from which no player can be made better-off by unilaterally deviating to a different action".  In other words, a Nash equilibrium is a "fixed point" because if the game ends up there, play stops because there are no more optimal moves.

  Is there a difference between fixed point reasoning and iterative never-best-response elimination?

Yes, they are different solution concepts.  The key is the last paragraph on page 12 of the article - basically, think of k-rationalizability as half-assing the game while still keeping the assumptions of full info, rationality, belief in mutual rationality, and so on.  Instead of stopping play when there are no optimal moves, play stops when some kind of decision-making resource is exhausted. It is interesting precisely to the extent that it may or may not model how real people, who face real information costs and processing time limitations, might approach games.


If my IV (denoted Z) equals 1, my endogenous var D always equals 1; is this a problem? I have a binary instrumental variable Z={0,1} and a binary endogenous variable D={0,1}. By construction, D=1 necessarily holds if Z=1. There are also cases where D=1 if Z=0, but there are no cases where D=0 if Z=1. (Side note: in the first stage of 2SLS, I get an F-value of around 3000.) Intuitively, I feel I should have more variation, i.e., that I should also have cases where D=0 if Z=1. Are my intuitive worries justified?

Let $\hat{\pi}_0$ and $\hat{\pi}_1$ be the first stage coefficients. That is, $\hat{D} = \hat\pi_0 + \hat\pi_1 Z$. When $Z_i D_i = Z_i$ (if $Z_i=1$ then $D_i=1$), we can show that $\hat\pi_0 + \hat\pi_1 = 1$ always and $\hat\pi_0 = (\bar{D}-\bar{Z})/(1-\bar{Z})$. (Proof is fun.) Unless $\bar{D} \simeq 1$, $\hat\pi_0$ is far from unity, which means $\hat\pi_1$ is far from zero. Not a surprise that the p-value is very small.

Another argument: Because $ZD=Z$, we have $Z(1-D)=0$. Thus, $Z$ and $1-D$ are strongly correlated unless $E(Z)=0$ or $E(D)=1$. By the way, try probit D Z to see an interesting result.

Whether you need more variability is another thing. The question is whether it is possible that $Z$ is exogenous and $D$ is endogenous. Need to think more.

EDIT: Thought about whether it is possible that $Z$ is exogenous, $D$ is endogenous, and $ZD=Z$ at the same time. It looks possible. Let $u$ be the regression error term. We can let $D=Z+(1-Z)\xi$, where $\xi$ is a Bernoulli random variable, so that $ZD=Z$. Let $E(u|Z)=0$ as we wish. Then $E(Du)=E(u|Z=1)P(Z=1) + E(\xi u|Z=0) P(Z=0) = E(\xi u|Z=0) P(Z=0)$, which can be nonzero. It is possible that $Z$ is uncorrelated with $u$, $D$ is correlated with $u$, and $ZD=Z$, I guess. So my answer is: No worries.

EDIT: But "$Z=1 \Rightarrow D=1$" means that $D$ is exogenous if $Z=1$, and is endogenous if $Z=0$. Whether this is OK depends on the context.


If government impose price control, what happens? I have this MCQ homework that I would like your opinion about

Suppose government chooses to impose a price control on a good. Which of the following is NOT a potential consequence of the restrictions on price?

Excess supply may occur.
Excess demand may occur.
No unexploited opportunities will exist.
People may spend a lot of time waiting in line to acquire the good.
Underground markets may emerge.

I think the answer is 3 because when black market exist with shortage and ppl will want to exploit the situation.

Those are all potential consequences of price controls. It would all depend on how the price controls were done, what measures were put in place alongside them, and all the context of the good itself.

If the price is set at too high a level, excess supply may occur. If too low, excess demand may occur. If it's a good with an extremely limited market that is met under the price control, then no unexploited opportunities may exist. If potential demand is greater than the rate at which it can be retailed, then people may spend a lot of time waiting in line to acquire the good. And underground markets may emerge in any case.


Regression of first differenced log-transformed model I am cosidering to transform a regression equation applying logarithms to the dependent and some of the independent variables (the ones I am actually interested in while leaving unchanged the others that I would like to use just as controls). This in order to interpret the coefficients of the transformed variables, say b, as "1% change in explanatory variable induces b% change in the dependent variable". Then I am thinking to first-differencing transform the model, that is, to introduce for both dependent and independent variables (again just the ones log-transformed of interest) the first lagged values (the values in the previous period) and the taking the differences between current and previous values as variables of the regression to be estimated. Wuold there be any issue in following this procedure?

What you describe is very standard in academic econometric practice. It essentially answers what is the elasticity of Y with respect to X identified from changes in X and Y.

Here are a few problems that can be introduced by this process

First-differencing is very similar to $Y_t = \alpha + \beta_0 * Y_{t-1} + \beta_1 * X_{t} + \beta_2 * X_{t-1} + \epsilon$ with the additional restrictions that $\beta_0 = 1$ and $\beta_1 = -\beta_2$
First-differencing can introduce serial correlation in standard errors


Pareto distribution and Lorenz curve I am currently studying Pareto distributions and their relation to Lorenz curves but I am having a hard time understanding the concept. If I am understanding correctly, Lorenz curve describes the wealth distribution and Lorenz curve of a Pareto distribution means that we are assuming that the income is Pareto distributed. Is this correct? But if Pareto distribution itself describes wealth distribution, why do we need to take a Lorenz curve of it?

Also, assume the cumulative distribution function for a Pareto distributed random variable $X$ with parameters $\alpha$ and $x_m$ is

$F_X(X)=\begin{cases} 1-(\frac{x_m}{x})^{\alpha}, \quad x\ge x_m \\ 0, \quad x<x_m \end{cases}$

and the Lorenz curve of it to be

$L(F)=1-(1-F)^{1-\frac{1}{\alpha}}$.

What do the parameters $\alpha$ and $x_m$ mean in the Lorenz curve? Is $\alpha$ the income level? What does the variable $x$ mean?

Apologies if this is a sort of stupid question but I cannot seem to understand it.

With my very limited knowledge of development economics:

$\left(\frac{x_m}{x}\right)^\alpha$ represents the proportion of the population that has an income larger or equal to $x$ where $x\geq x_m>0$ and $x_m$ is the minimum income amount.

Example 1:

Suppose $\alpha\rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $50,000$?

$$\lim_{\alpha \rightarrow 1} \left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{50,000}\right)^1=1$$

This suggests that $100\%$ of the population has an income larger than or equal to $50,000$.

Example 2:

Suppose $\alpha \rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?

$$\lim_{\alpha \rightarrow 1}\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^1=0.1$$

This suggest that $10\%$ of the population has an income larger than or equal to $500,000$.

The story does not end there. What about $\alpha$? It must be the case that $\left(\frac{x_m}{x}\right)^\alpha$ is bounded between 0 and 1 (it is a proportion). As such, it must be the case that $\alpha>1$. In the literature, $\alpha$ is refered to as the Pareto Index. Let us consider another example.

Example 3:

Suppose $\alpha=2$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?

$$\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^2=0.01$$

This suggest that $1\%$ of the population has an income larger than or equal to $500,000$. From this, a high $\alpha$, or a high Pareto Index, suggests a small proportion of high-income individuals.

What is a reasonable value of $\alpha$? We can normalize $x_m = 1$. Suppose we wanted to know the share of total income that is received by those individuals with an income above, say $\tilde{x}$. We take the following integrals:

$$\int_{x_m}^{\infty} x^{-\alpha} dx = \int_{1}^{\infty} x^{-\alpha} dx = \frac{1}{\alpha-1}$$

$$\int_{\tilde{x}}^{\infty} x^{-\alpha} dx = \frac{\tilde{x}^{1-\alpha}}{\alpha-1}$$

Therefore the share of total income that is received by those individuals with an income above $\tilde{x}$, call it $q$, is the ratio of the two integrals:

$$q = \tilde{x}^{1-\alpha}$$

We also want to define the proportion of the high-income population with an income larger than $\tilde{x}$ as $p$:

$$p = \tilde{x}^{-\alpha}$$

The next logical question would be what value of $\alpha$ allows us to reconcile the empirical observation that $20\%$ of the population has $80\%$ of the total income? This implies that $p=.2$ and $q=.8$.

$$q = p^{\frac{\alpha-1}{\alpha}}$$

Taking the log of both sides:

$$\ln(q) = \frac{\alpha-1}{\alpha}\ln(p)$$

Solving for $\alpha$:

$$\alpha = \frac{\ln(p)}{\ln(p)-\ln(q)} = \frac{\ln(.2)}{\ln(.2)-\ln(.8)} = 1.16$$

Therefore, an $\alpha=1.16$ is consistent with what is observed empirically. To answer the question, $x_m$ is the minimum income amount and is usually normalized to $1$. $\alpha$ is the Pareto Index which is a measure of the proportion of high-income individuals. A high $\alpha$ suggests a low number of high-income individuals.

The transformation $L(F)$ simply allows you to compute what % of the total wealth is controlled by those with an income below $\tilde{x}$.


Arbitrage Pricing Theory (APT), orthogonal factors Why in the Arbitrage Pricing Theory (APT), one of the assumptions is that the factors has to be orthonogal? what if not?

If you want to describe excess returns in terms of exposure to common risk factors, you want the risk factors to be orthogonal. However, if you have $k$ factors with no perfect collinearity, you can always orthogonalize them and use those. You then call the orthogonalized factors the risk factors.


Typical duration of bonds purchased in QE I know that QE involves the purchasing of government bonds by the central bank "printing money" but I am curious about what the typical time to maturity of those bonds are. I.e. the typical time before the central bank is faced with the choice of "un-printing" that money (taking it out of the economy) or doing some more QE.

This paper is a few years old. You might want to look at page 6.
Briefly, it varies by government and across time.
https://www.treasury.gov/resource-center/data-chart-center/quarterly-refunding/Documents/February2015TBACCharge1.pdf


Does a stock market crash "undo” the multiplier effect's money creation? The economy grows for multiple reasons, but one is the multiplier effect due to fractional reserve banking.

When there is a crash, is that artificially created money essentially destroyed, resetting the economy to how much money is "really" in the system?

Sorry, I know this may be phrased informally but my knowledge of economics is limited. I just know that a crash is regarded as an economic "reset", while "multiplying" money is artificial and depends on the market's confidence.

(originally posted here, I hope at least one of these forums is the correct place: https://money.stackexchange.com/questions/77099/is-a-market-crash-the-undoing-of-the-multiplier-effect)

All price changes in the stock or commodities markets impact the money supply to the extent it results in a change in margin credit. Do note that the amount of credit itself must change and not the price alone.  A crash only has an impact if it results in net changes in the amount of credit.

This is a second order effect of a crash and not a first order effect.  It also usually has no material impact on the money supply.  Although a crash may have a large impact on notional concepts of wealth, the effect on the supply of money itself is usually neutralized by the banking system.  It is normal, even in the 1929 crash, to see bankers announce they will keep open the brokerages' lines of credit and stand ready to purchase safe assets such as Treasuries in unlimited quantities from these brokers to hold the money supply roughly constant.  If brokers sold enough treasuries, it could even expand the money supply.

The size of the money stock (M2) is 13.4 trillion dollars.  The total size of all margin credit from all sources is 554 billion dollars.  Although this is four percent of the money supply, which is a lot, you are also forgetting that in a crash margin purchases will begin happening too.  Just as brokerages will be issuing margin calls, other investors smelling blood in the water will be purchasing on margin, though this will be a smaller effect.

The Fed normally drops the interest rate to expand the money supply after a crash to compensate for the loss of money created by the reduction in margin credit.  Remember that 554 billion would not be wiped out in a crash, it would just be reduced, at least initially, as brokers required repayment.


Issues if insurance is not compulsory for population So I remember this in my Econ 101 course thing.. eg. If all cars are legally required to be insured, insurers work off a population statistic to calculate cost of crashes/ individual issues (population statistics are public). If insurance is not compulsory, worse drivers insure, better ones don't, then insurers can't price off the population statistic because they only have a sample statistic (not public) of worse drivers on their books. I believe this was called the 'pooling problem' in my lecture, but sadly google comes up with entirely different ideas for that search! Please enlighten me! :) (if possible, some links to theory would be great, thanks!)

If insurance is not compulsory for all drivers there can be a case where the affected parties are unable to be compensated when an accident happens. In that case those affected by the accident wont have the means by which they can have their property replaced.

Insurance provides a service which guarantees a certain level of wealth for a party who is insured. Without insurance sub optimal transactions can take place making everyone worse off. This can be illustrated via an abstracted example.

Example
 1. Party A has wealth of \$50 and Party B has a wealth of \$100 and all their wealth is stored in their cars.
 2. Party A and Party B both get into a car accident.
 3. Party A destroys Party B's Car and legally owes Party B \$100
 4. without insurance Party A is unable to compensate Party B and loses all his income.

This whole issue can be avoided via the purchase of car insurance. So the law requiring all drivers to have insurance actually leads to a net increase in welfare for all involved  because it allows legally required transactions to take place.