Money creation in textbooks since McLeay et al The paper "Money Creation in the Modern Economy" by McLeay et al was published in 2014 in the Bank of England Quarterly Bulletin and has had well over a thousand citations. The paper is damming about the description of the monetary system as given in many undergraduate level textbooks.
Here are some example quotes:

The reality of how money is created today differs from the description
found in some economics textbooks: • Rather than banks receiving
deposits when households save and then lending them out, bank lending
creates deposits. • In normal times, the central bank does not fix the
amount of money in circulation, nor is central bank money ‘multiplied
up’ into more loans and deposits.

rather than banks lending out deposits that are placed with them, the
act of lending creates deposits — the reverse of the sequence
typically described in textbooks

While the money multiplier theory can be a useful way of introducing
money and banking in economic textbooks, it is not an accurate
description of how money is created in reality

This description of the relationship between monetary policy and money
differs from the description in many introductory textbooks, where
central banks determine the quantity of broad money via a ‘money
multiplier’

And in contrast to descriptions found in some textbooks, the Bank of
England does not directly control the quantity of either base or broad
money

So my question now is - did the paper have any effect? Are the descriptions of the monetary systems in the latest editions of the popular textbooks (Mankiw and the like) any better? Are they now compatible with McLeay et al?
EDIT... I would suggest that there are two distinct issues that McLeay et al are complaining about:
A) Incorrectly suggesting that banks lend deposits rather than correctly stating that lending creates deposits.
B) Incorrectly suggesting that the money supply has a ceiling set by reserve requirements in practice.

Textbooks still include discussion of money multiplier together with more nuanced discussion of central banks setting interest rates (compatible with story in McLeay et al 2014). In fact they did so already before the paper was published in 2014.
I think in order to answer this question it will be important to first correct a misconception you seem to have about what the paper implies regarding to textbook story. You state:

The paper is damming about the description of the monetary system as given in many undergraduate level textbooks.

No. If you actually read McLeay et al 2014 they state:

While the money multiplier theory can be a useful way of
introducing money and banking in economic textbooks, it is
not an accurate description of how money is created in reality.
Rather than controlling the quantity of reserves, central banks
today typically implement monetary policy by setting the
price of reserves — that is, interest rates.

Consequently, the paper was certainly not damming of the textbook money multiplier story (and no fair reading of the paper can conclude they went as far as damming the description of whole monetary system in textbooks which goes way beyond just money creation), and going by this statement I doubt that the authors would even want to remove this story from textbooks as they themselves consider it useful. I also can't see anything else in the paper that would suggest that multiplier should not be included in the textbooks as a didactic tool (e.g. analogously most markets are not perfectly competitive,  yet I don't think that economic textbooks will ever be able to efficiently introduce more complex market structures before discussing case of perfect competition no matter how unrealistic such market structure might be).
Now onto your main question, one of the main point of McLeay et al (2014) is that:

Rather than controlling the quantity of reserves, central banks
today typically implement monetary policy by setting the
price of reserves — that is, interest rates.

Now this is something that was already discussed in textbooks quite long time ago. Let me give you some examples:
Blanchard et al. Macroeconomics an European Perspective. 2nd ed. Widely used mainstream textbook  with 2nd edition published in 2013 (year before the paper came out), states pp 69-70 (square brackets contain my remarks):

We have described the central bank as choosing the money supply and letting the interest rate be determined at the point where money supply equals money demand [i.e. the multiplier story]. Instead, we could have described the central bank as choosing the interest rate ... Why is it useful to think about the central bank as choosing the interest rate? Because this is what central banks, including the ECB, the Bank of England and the Fed, typically do...

the textbook then proceeds to give simplified down narration of what you could have read in the 2014 paper (but remember this is undergraduate textbook everything is simplified there).
Mankiw Macroeconomics 8th edition - originally published in 2012, states:

Although it is often convenient to make the simplifying assumption that the
Federal Reserve controls the money supply directly, in fact the Fed controls the
money supply indirectly using a variety of instruments. ...
There are various ways in which banks can borrow from the Fed. Traditionally,
banks have borrowed at the Fed’s so-called discount window; the discount rate
is the interest rate that the Fed charges on these loans. The lower the discount
rate, the cheaper are borrowed reserves, and the more banks borrow at the Fed’s
discount window. Hence, a reduction in the discount rate raises the monetary
base and the money supply.
In recent years, the Federal Reserve has set up new mechanisms for banks
to borrow from it. For example, under the Term Auction Facility, the Fed sets a
quantity of funds it wants to lend to banks, and eligible banks then bid to borrow
those funds. The loans go to the highest eligible bidders—that is, to the banks
that have acceptable collateral and are offering to pay the highest interest rate.
Unlike at the discount window, where the Fed sets the price of a loan and the
banks determine the quantity of borrowing, at the Term Auction Facility the Fed
sets the quantity of borrowing and a competitive bidding process among banks
determines the price. The more funds the Fed makes available through this and
similar facilities, the greater the monetary base and the money supply.

In fact the last paragraph was addition to the 8th edition that was not there in 7th edition (published in 2010) so you can see these textbooks usually catch pretty quickly.
Burda & Wyplosz Macroeconomics: A European Text 217-218 3rd ed, originally published in 2009:

Nowadays, ... most central banks choose to control the interest rate ... a central bank can set any interest rate it chooses, provided it stands ready to provide or withdraw liquidity in any amount needed to meet the derived demand corresponding to this rate.

the textbook then proceeds to discuss basically what is the point of the 2014 paper already in 2009.
The newer editions of the books that were printed post 2014 still contain this explanation. But given that they already addressed this issue prior 2014 I would not attribute it to McLeay et al paper per se. As clearly shown above textbooks addressed this issue already in 2009.
You can see these chapters being further expanded in the newer editions of the Mankiw and Blanchard et al Macro textbooks. Perhaps this further expansion can be attributed to influence of McLeay et al paper. Arguably Blanchard et al do this at much greater length then in Mankiw textbook.
In addition, it is worth to point out that I do not think that the above shows that McLeay et al were being duplicitous in their 2014 paper. In fact, if you read the paper carefully they never state that textbooks do not cover how money is being created by modern central bank. They always refer to money multiplier story (that admittedly is given perhaps too much space) in the textbooks that they are trying to 'debunk' but they actually never state that textbooks do not cover that modern central banks conduct monetary policy through interest rates.

Response to Edits:

I would suggest that there are two distinct issues that McLeay et al are complaining about:

A) Incorrectly suggesting that banks lend deposits rather than correctly stating that lending creates deposits.

I cant find a passage where McLeay et al state that textbooks got this point wrong, but if they state that somewhere it would be incorrect.  In fact, this is a picture that is from the above cited Blanchard et al Macro textbook pp 71 from 2013!

The picture clearly shows that demand for deposits creates demand for reserves which is exactly the point of McLeay paper. In addition there is nothing wrong in principle also arguing that from supply side the increase in supply of reserves would lead to increase in borrowing and deposit accounts. It is not the mode of behavior that modern central banks would choose but it is not wrong in itself as they could do that. The relationship in the diagram above has to hold but of course central bank has discretion in deciding whether to change interest rate (which affects the demand for money and then you get the effects coming from deposits) or to change supply of money directly where the effect would go opposite way. It is correct to state that modern central banks usually choose former rather than latter (and as proven above textbooks acknowledge that), but that does not mean it’s wrong to say that central banks could conduct monetary policy via direct changes in money supply.

B) Incorrectly suggesting that the money supply has a ceiling set by reserve requirements in practice.

Regarding, the B they nowhere say that reserve requirements do not create celling for lending. They state instead that banks will lend out money and then seek to raise their reserves by purchasing them on interbank market in order to maintain the mandatory reserve ratio - but they have to maintain that ratio.
First of all, as you can see from the statistics on excess reserves below (which is discontinued now as reserve requirements were abolished in 2020), prior 2009 banks held virtually no excess reserves. In such situation the amount of reserves supplied by central bank is hard cap for lending of commercial banks. If they would lend more they would literally break the law (as a tidbit in such situation the multiplier story still applies as the ratio to reserves and broad money should be approximately given by the multiplier). After 2009 market held excess reserves, in that situation you can think of every individual bank be unconstrained by reserve requirement because without central bank doing anything any individual bank can borrow excess reserves from any other bank to maintain their reserve requirement (this is what they mean in the paper when they say banks are unconstrained by reserves).   However, the banking system as a whole (all banks taken together) is still constrained by the reserve requirement and amounts of reserves. Even in this situation banks could not all at the same time lend more than they can given the reserves and reserve requirement. So this is not something that textbooks, which are written from the perspective of whole system rather than single bank, get wrong (and again I cant see them calming that textbooks do that in the paper). In fact here you could actually coitize the McLeay et al for not being careful enough about explaining this.


Relationship between forward rate, future spot rate, and liquidity preference I'm getting confused over the relationship between forward rates, spot rates, and liquidity preference.  I know that liquidity preference theory (i.e. that investors prefer shorter term investments because they are more liquid) states that the forward rate is greater than the future spot rate.  However, I am confused on what exactly the forward rate and future spot rate are.  My understanding is that:

The forward rate ($f_2$) is the hypothetical rate you would need to eliminate an arbitrage opportunity in the example below:

$(1 + r_1)(1 + f_2)$ = $(1 + r_2)^2$

The spot rate is the the expected yield on a zero-coupon bond.  But how does this play into the example above?

Moreover- just to make sure- am I correct in that longer maturity -> greater interest rate risk (because of inflation / liquidity pref)?

Thanks for your help.

In addition to comment given by @dismalscience, here you may find partial answer
(hope I got everything right below). Since many similar terms refer to concepts
that are close to each other, I'm also regularly fighting to get these somehow
in order. For example, there are forward and futures contracts that use similar
terms and are related to the yield curve concepts.

Yield curve premiums

On liquidity: lenders like to prefer short periods, because they are more easy
to convert to cash, while borrowers prefer long periods, since roll-over
terms involve risk. So lender wants compensation for which the borrower is ready
to pay. Longer you go, the premium is likely to increase but at decreasing rate.
This is different to the liquidity concept when the market for a bond is very
thin.

It is healthy to consider also other aspects with respect to risk-free bonds.
E.g. how good vs. bad times affect premiums. And why term premium? There is some
evidence of negative correlation between short bond return and economic growth,
and especially this correlation has larger magnitude than the correlation the
long term bonds have with economic growth.  This in turn, would mean that short
term bonds work out as better hedges against economic downturn leading to larger
premium requirement for the longer bonds.

It is possible, that we have positive liquidity premium at the same time when
other premiums are not, even so that the yield curve inverts, because the
spot rate
is geometric mean of forward rates. In this situation, the forward rate curve
would be below the spot yield curve. (This is not shown in the equations
below.)

Contracts

Forward rate (i.e. forward price) is the price agreed on the contract initiation
date. Pricing means a method to find out the forward rate "today" for the
contract.

Let us denote the value of the underlying asset $S_T$. Then, the value of
contract is $S_T - F_{0,T}$ at the "end" (time $T$) and we could write it as
$V_{0,T}(T) = S_T - F_{0,T}$. To get the value today, we discount with risk-free
curve, and is common to set the forward price $F_{0,T}$ such that neither party
does not have to pay to the other: $V_{0,T}(0) = S_0 - F_{0,T}/(1+r)^T = 0$,
where $r$ is the risk-free interest rate. So, $F_{0,T} = S_0(1+r)^T$.

Futures rate (i.e. futures price) is priced similarly, $f_{0,T} = S_0(1+r)^T$,
and it is said that "the futures price is the spot price compounded at the
risk-free rate". For the value, the marking-to-market makes the value zero in
each day, but just before that, the value is $f_{0,T}(t) - f_{0,T}(t-1)$.
If there is a carrying cost $c$, the price is  $f_{0,T} = S_0(1+r)^T +c_T$

Future rate and expected spot rate: consider carrying costs $c$ and risk $k$,
for which you don't want to pay, and that we don't know, what the spot price
will be at $T$, thus using expectation:
$$ S_0 = \frac{E(S_T) - c_T - k}{(1+r)^T}. $$

Put the future price into that and you will obtain $f_{0,T} = E(S_T) - k$, that
is, the futures price equals expected spot price less risk premium.

Yield curve

Forward rate $f$ is an interest rate today for a loan to be made in future.

The spot rate $r_T$ is the geometric mean of the forward rates, that is,
$$ (1+r_T)^T = (1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i) $$
or
$$ r_T = ((1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i))^{1/T} - 1.$$

Do forward rates imply future spot rate movements? Not necessarely, but find out
more about the trade called "rolling down the yield curve". However, the forward
rates do provide an estimate of the expected spot rate that is biased by
liquidity premium (premium is not shown in the above equation).


Basic question about why supply/demand should move to the equilibrium price? This is probably an extremely basic question, however I'm still unclear about the answer. In a typical supply and demand curve graph, such as

the equilibrium price is said to be

  the only price where the plans of consumers and the plans of producers agree—that is, where the amount of the product consumers want to buy (quantity demanded) is equal to the amount producers want to sell (quantity supplied)

and

  if a market is not at equilibrium, then economic pressures arise to move the market toward the equilibrium price and the equilibrium quantity.

But why should this be so? In the above figure, at the \$1.80 price the demand is 500, netting the supplier a 500*\$1.8=\$900 revenue. Which is higher than the $1.4*600=\$840 revenue it would get at the equilibrium price. And even though at the \$1.80 price the supply exceeds demand, the supplier could just choose not to produce more than the 500 demanded at that price. Or if it does produce it, it could store it somewhere for the future, for example. As a rational agent, the supplier wants to maximize its revenue, so dropping their price to the equilibrium price would be counterproductive to that goal.

What you write is true in a monopoly situation, but not when there are multiple firms on the supply side.

However, when equilibrium is discussed in this context people usually mean competitive equilibrium, that is equilibrium in a market where many firms compete. So at price \$1.80 several suppliers want to sell an aggregate amount of about 680 gallons. However demand is only 500 gallons. This would prompt the suppliers who are worried about not being able to unload all their stock to offer it at a slightly lower price, say \$1.79, undercutting their rivals' price. The idea is that by lowering the price somewhat you can greatly increase your market share, and unload your stock, to the general dissatisfaction of your rivals, who are now stuck with their own surplus gasoline.

Thus \$1.80 is not an equilibrium price, as suppliers have incentives to unilaterally push their own prices down.


Is there an equilibrium between privately owned and publicly listed companies? Private equity funds are increasingly taking formerly public companies off the exchanges. Is there an economic reason for any public companies to exist and what is the equilibrium share of public companies in the total corporate 'population'?

I don't know exactly how the numbers would work out, but you would expect an equilibrium but it will depend on many factors.

A complication is that you have public companies taken private and private companies that never go public.  It may be there is some equilibrium between these as they are unlikely to be equivalent. Ex public companies are often highly levered,  while never public are often debt free and cash rich.

Particular important factors would be interest rates and credit spreads.  This is because it boils down to the distribution of capital.  If there is lots of money available to lend then LBOs become attractive and companies can be taken private. If there is strong demand for equity then IPOs become easy.

I think an economy with a shortage of public companies would show overvalued equities and hence IPOs are encouraged (think dot com bubble - shortage of public web companies vs demand). If debt is cheap and there are too many public companies then you will see either take overs or LBOs.

I suspect the effect is most visible within sectors rather than the whole economy.


Envelope theorem for discrete choice sets? If we have a function
$$f(x)=\max_yg(x,y)$$

Then we can find the derivative $d/dx \ f(x)$ by realizing that
$$(1): \quad \frac {\partial }{\partial y}g(x,y^*)=0$$
because of the first order condition for maximization.

We can use this by recognizing that $$\frac d {dx}f(x)=\frac d {dx}g(x,y^*(x))=\frac \partial {\partial x}g(x,y^*)+\frac \partial {\partial y}g(x,y^*)\frac {dy^*(x)} {dx}=\frac \partial {\partial x}g(x,y^*)$$
Where the last equality follows because of result $(1)$.

However, my question is, what if $y$ can only take a discrete number of values? such as $0$ or $1$? We can of course say:

$$f'(x)=\begin{cases}
g_x(x,0) \quad \text { if } g(x,0) > g(x,1)\\
g_x(x,1) \quad \text { if } g(x,0) < g(x,1)
\end{cases}$$

However, I am wondering if there is a kind of "envelope theorem for discrete choice sets", that would allow us to simplify this (especially if the choice set is discrete but large).

There is an envelope theorem for the setting you describe.

Have a look at “Envelope Theorems for Arbitrary Choice Sets” by Milgrom and Segal (2002).


Elasticity of demand functions I have some question about the elasticity of the demand functions of two different traders/consumers. Suppose that $x_1$ and $x_2$ are the elasticities of the demand functions of these agents. Furthermore, they have linear demands, that are $d_1=a_1-x_1P$ and $d_2=a_2-x_2P$, where $a_1,a_2$ are the intercept points of the demand functions and $P$ is the price of the asset/commodity. What does it mean that the elasticity of trader $1$ is higher than that of trader $2$ i.e. $x_1>x_2$ or the opposite, $x_1<x_2$, in case they buy the asset/commodity or they sell it.

In the case of linear demand $d_i=a_i-x_iP$ (assuming $d_i$ is quantity demanded by individual $i$), the price elasticity of demand at point $(d_i,P)$ is
\begin{equation}
\epsilon_i(d_i,P)=x_i\cdot \frac{P}{d_i}.
\end{equation}
As @the_rainbox noted in their answer, price elasticity of demand varies along a linear demand curve. So in order to compare elasticities between different demand curves based only on the slope coefficients (the $x_i$'s), you need to fix $P$ and $d_i$; that is, assume that the demand curves of individuals $1$ and $2$ cross at some point $(Q_0,P_0)$.
Then, you can say things like
\begin{equation}
\epsilon_1(Q_0,P_0)\ge \epsilon_2(Q_0,P_0) \quad\Leftrightarrow\quad x_1\ge x_2.
\end{equation}
Or in words: $1$'s demand is more elastic than $2$'s at $(Q_0,P_0)$ if and only if $1$'s demand curve is flatter than $2$'s. [Note that since by convention demand curves are plotted in the $(Q,P)$-plane, a flat demand curve actually corresponds to a high $x_i$.]
Beware, however, that in contexts where a high degree of mathematical rigor is not required, it is sometimes taken as a rule-of-thumb that flat demand curves are "generally" more elastic than steep ones. Hence slope becomes a proxy for elasticity in those less rigorous discourses.


Is there an economic theory of why people like to hold on to wealth? It seems to me that many people work to become wealthy, but not because they want to consume all that wealth. Many people give it away, other's don't, but still they do not spend it. You hear about some rich 40-year old's throwing money away, but you seldom hear of a rich 80-year-old throwing anything away! What is the economic theory for "scrooge"? Are there simple preferences that make this the right strategy?
(I head somebody say that JP Morgan used to say it was a way of keeping score. Is there a theory of this?)

You could consider unspent money to be a call option. The set of future investment opportunities is unknown, so if you're optimistic about future opportunities it could be a valuable call option.

So saving money can be a profit/preference-maximizing behavior.


Why is capital income taxed differently than wage income? Why is income from capital gains taxed differently than wage income? There are perhaps historical, practical, and theoretical reasons. What are they?

(As far as historical reason go, I'm asking primarily about the United States. If anybody has any insight about this topic in other countries, I'd love to hear it as well.)

One reason is the inflationary gain problem.  Let me give an example with simple numbers.  I make \$100 in income and pay 20% tax of \$20.  I have \$80 left, which I invest in a stock.  The stock goes up in value at the same rate as inflation, about 3.5% a year.  After 20 years, it's worth about \$160, but \$160 has the same value now as \$80 did when I earned it.  So in effect, I have made no gain.  If I have to pay a 20% tax on the \$80 nominal gain, then I actually have \$8 less spending power than when I made the investment.  Just in order to keep pace, I would have needed returns of 125% inflation.

Another reason is to encourage investment.  A lower tax on capital should encourage savings, and the US is generally low on savings.

There is also a historical argument that if the income tax rate is 90% (which was the top rate starting in World War 2 and ending in the Kennedy administration), then it is impractical to tax investments at the same rate.  From 1921 to 1986, there was a lower rate on long term capital gains than on other income.  In 1991, capital gains tax was capped at 28%.  In 1997, long term rates were reduced again.

The current system is designed to encourage long term capital investments by decreasing the rate the longer something is held.  This does very little to help with nominal gains but does provide an incentive for long term savings.  It's unclear if it actually increases saving.  It certainly doesn't increase net savings, as the US tends to be fully leveraged, borrowing as much as is saved.

There have been proposals that focused more on the nominal gains issue.  For example, in the late 80s, early 90s there was a proposal to switch investments to working like traditional IRAs:  deductible at the time of investment but taxed the same as any other income at the time of sale or withdrawal.  Since this delayed the original tax to be paid at the same time as the capital gains, there is no separate tax on the nominal gain.


Models for inflation prediction I am looking for some statistical models that are accepted to best predict inflation rates. Anything that is more complicated than linear model will be appreciated. Or even recommend books that discuss inflation predictions.

I would also be interested in models that could relate inflation to GDP(or some form of feature extracted from gdp).

One approach that you might find useful/interesting would be Bayesian VARs.

An example of a Bayesian VAR being used to predict inflation can be found here.  In this paper, Tim Cogley, Sergui Morozov, and Tom Sargent motivate some methodology for Bayesian VARs and then compare their results to what the Bank of England uses in their "fan charts" (which they use to predict inflation).  Cogley and Sargent have two other papers on inflation that are related to this paper, "Drifts and Volatilities" and "Evolving Post-World War II U.S. Inflation Dynamics."  This series of papers is a good expose on some of the theory of inflation and how to statistically approach it.

I believe @FooBar is right that most banks are using large-scale VARs (similar to the aforementioned) and New Keynesian models.  This isn't directly related to the question, but there is a literature on model averaging in which they can combine the forecasts of VARs and N-K models to provide a "more informed" (I use the phrase more informed here lightly but I do think the model averaging is helpful) prediction as to what will happen.  An example of such a paper can be found here


Hypothetical Long-term Business Cycle I was looking through my textbook on Money, Banking, and the Economy and I came to the chart of the hypothetical short-term business cycle deviating around an upward sloping long-term cycle trend. My quarrel with this is does there necessarily have to be an upward sloping long-term trend?

I'm pondering that if we were to look at economies from different times (that are isolated in our analysis) couldn't there exist a possibility that in the more grandiose long-term an economy may exhibit flat, or perhaps negative long-term trends. Perhaps nations that are on the decline may exhibit these traits. Just a random question that I wanted to throw out there.

You are correct that there have in fact been periods of time when growth was flat, or even negative. For example, if one could observe the economy of the Cahokia over time, one would expect to see a period of negative growth. Further, the rapid growth we're to which we are accustomed is a very recent phenomenon.

It's impossible to know whether growth will continue at such a pace or slow dramatically; prominent economists such as Robert Gordon have argued that future growth may be much slower (though few argue that it will be flat or negative), a set of hypotheses known as "secular stagnation".

So while your textbook, showing cyclical variation around a positive long-term trend, correctly depicts modern business cycles, there's no reason to believe that such a trend must appear in all societies at all times.


What is the economic purpose of increasing the minimum wage? It is generally accepted among economists that minimum wage warps the equilibrium point between the supply and demand of labor by instituting a price floor and increases unemployment for unskilled laborers (while increasing the wage for the individuals who continue to work).
However, I have noticed that politicians continue to advocate an increase in the minimum wage.  Politics and political maneuvering aside, is there an economic reason as to why a higher minimum wage would benefit the economy?

Opinion of Economists

What makes you think that economists are so aligned against the minimum wage? Take a look at the IGM Forum that polls top academic economists. There is substantial disagreement about the effects and welfare implications of a minimum wage hike.

IGM Forum: Minimum Wage
NYTimes Blog Post about it

Theory about the minimum wage

Also, it's worth pointing out that besides empirical studies (like Card and Krueger), there are in fact theoretical reasons why a minimum wage might be welfare improving---see Lee and Saez (2012). (I'm just saying they exists, not commenting on the strength of the results.) In the Handbook of Public Economics, chapter 7, "Optimal Labor Income Taxation", they say this about that paper:

  Lee and Saez use the occupational model ... with endogeneous wages and prove two results. First, they show that a binding minimum wage is desirable
  under the strong assumption that unemployment induced by the minimum wage hits the
  lowest surplus workers first. ...

  Second, when labor supply responses are along the extensive margin only, which is
  the empirically relevant case, the co-existence of a minimum wage with a positive tax rate on low-skilled work is always (second-best) Pareto inefficient.A Pareto improving policy
  consists of reducing the pre-tax minimum wage while keeping constant the post-tax
  minimum wage by increasing transfers to low-skilled workers, and financing this reform
  by increasing taxes on higher paid workers. Importantly, this result is true whether or not
  rationing induced by the minimum wage is efficient or not.This result can also rationalize
  policies adopted in many OECD countries in recent decades that have decreased the
  minimum wage while reducing the implicit tax on low skill work through a combination
  of reduced payroll taxes for low skill workers and in-work benefits of the EITC type for
  low skill workers.


Solving differential equation in The Economics of Superstar (by Rosen) I am currently reading "The Economics of Superstar" written by Rosen (1981). I don't understand one differential equation he used in the paper. The equation is as follows:
$$\frac{dp}{dz} = (p+s)/z.$$

He says if we integrate this, we get $p(z) = vz-s$ for $v = (p+s)/z$. I think that it is a simple differential equation, but I am having trouble solving this. Could you give some help?

The differential equation is of the form

$$y' + f(x)y = q(x)$$

The correct answer in our case is

$$p = -s$$

so that you know what you are targeting.
Namely,  it does not depend on $z$. You can verify that it satisfies the differential equation.

Then the author just plays around like

$$p= - s \implies p -p = s-s \implies p - \frac p z z = \frac s z z - s $$

$$           p = \frac p z z + \frac s z z - s = \frac {p+s}{z}z - s$$

$$\implies p = vz -s $$

...and I suppose this playing around with identities is helpful to what he does in the paper.


What is the upper limit of circulating money on earth? I've come up with this question while discussing the topic of customer-perceived value with coworker.

Is there a theoretical upper limit to all circulating money in this world?

To avoid inflation or the simple answer "just print more money", let's use the current USD as the 'standard money' in this question. (I'm simply going to go off 'value of USD' from Google [if someone has a more precise way, then any suggestions would be much appreciated] where 1USD = .91 euro at the time of writing this).

According to Business Insider:

  according to the CIA, the total amount is $80 trillion if you include
  "broad money.

In my mind, assuming that all first-world counties' standard of living stay the same or get better, if third-world countries were to, all of a sudden, become first world countries in 2020, that would mean there is more money circulating in the world (more GDP per capita in the third-world country, thus more GDP per capita in the world).

And as the economies of third-world countries get better, they would create more value in their products/services that the people of their country produces.

So in an ideal world, if everyone lived as millionaires (aka, everyone drives $100,000+ cars, million-dollar homes etc.), due to the economy in every country flourishing, wouldn't that mean there is no upper limit to all circulating money in the world?

But as a counter, that's only in an ideal world - not reality.

Let's say you can buy a car for 50,000 USD. The manufacturer is  not going to create that car if it takes them $50,000 to make it. Where's the profit?

So then let's say there is a company that creates the rubber for the tire, another company creates the seats for the the car, so on and so forth. All those companies are not going to create their product without profit, because where's the incentive to do so without it?

If that's true, then someone at the bottom of the chain will be doing the work for minimal profit, which makes me think that the cars that are sold to people only exist because people at the bottom of the chain - which leads to, first world countries only exist due to third world countries (why do you think a t-shirt at GAP costs 3 dollars? Gap is selling you the shirt for 3 dollars because they only paid 1.50 dollars to make it, which someone did, most likely in a third world country), which means the upper limit of circulating money in the world would more or less look like y = ln(x) image of y = ln(x) because as first-world countries get wealtier, there is only so much they can grow because they're bounded to the work in third-world countries.

Disclaimer: My counter argument is simply a hypothesis and I have no grounds of proof. I'd be happy to debate/have someone shed some light if I am entirely wrong. The only reason why I mentioned it is because I wanted to provide a counter reason as to why there is an upper limit to all the circulating money in the world.

Clarification: What I am not asking is all the value in the world. I could create a painting and say it's worth 900 trillion dollars and that would 'increase' all the monetary value in the world to 980 trillion (assuming that we currently have $80 trillion in circulation). What I am asking is strictly circulating money.

What you are asking, dressed up in economics lingo, is as follows:

Is there a limit to real gross world product growth?

Short answer

A qualified no.

Longer answer

First of all, developed countries do indeed heavily benefit from the work of developing countries. Globalization has found a way to incorporate cheap labor from overseas for economic growth. However, incorporation of more labor is not the only way the economy grows. Innovation such as new ways to manufacture things more efficiently is another. Just search for "ways in which GDP grows" and most results should apply. But the bottom line is one could imagine a situation were autonomous robots produced expensive cars that were freely available to everyone without having to exploit human labor. Of course, whether the environment could sustain such intense resource usage or whether humans can bring automation/AI to such a point may put practical limits.


Completeness of the strict binary relation How do I show that considering the preference relation $\succsim$, then  $\succ$ is not complete?

I tried the following (which I don't know if it's right) but I'd also like to know if it's possible to generalize and say that $\succ$ will never be complete.

Suppose $X = \{a,b\}$ with preference relation $\{ a \succsim a, a \succsim b, b \succsim b\}$. Then, we $\succ$ can't be a complete relation on $X$ because neither $b \succsim a$ nor $b \not{\succsim} a$ are in $X$ and therefore it's impossible to conclude that either $a \succ b$ or $b \succ a$ is true.

Your counterexample looks correct.

But an even simpler example would be $X=\{a\}$. We have $a \succsim a$, so $a \not \succ a$.

More generally, we usually define

$\succsim$ to be reflexive (i.e. $a \succsim a$ for all $a$); and
$\succ$ by $a \succ b$ if $a \succsim b$ and $b \not \succsim a$.

By the above definitions, $\succ$ cannot be complete on any non-empty set $X$ because for any $a\in X$, we have $a \not \succ a$.


Why we use revealed preferred condition in GARP instead of directly revealed preferred? The definition of revealed preferred seems like coming out of nowhere. Why there's a string of direct revealed preferences lie between? Can we use directly revealed preferred in GARP condition ------GARP says that if $x \succsim_{D} y$, $y \nsucceq_{D} x$?

We need indirect comparisons because looking at direct comparisons is not sufficient to detect all types of inconsistencies. Suppose for instance that we observe $C(x,z)=\{x\},C(z,y)=\{z\}$ and $C(x,y)=\{y\}$. $C(x,y)=\{y\}$ implies that $y$ is "directly" revealed preferred to $x$. The reason why GARP is violated is that $C(x,z)=\{x\}$ and $C(z,y)=\{z\}$ imply that $x$ is "indirectly" revealed preferred to $y$, an inference that wouldn't be possible by looking at direct comparisons.


Is there any empirical evidence for the so-called "law" of diminishing returns The "law" of diminishing returns is usually "demonstrated" with made-up examples (programmers working on a software, farmers acquiring land...), but rarely on evidence. Is there any empirical research on actual data to acknowledge and quantify this law?

The above chart is from Agroecosystems, nitrogen-use efficiency, and nitrogen management by Cassman KG, Dobermann A, Walters DT, and it deals with this sort of question quite extensively for those interested in all the gory details.


A recursive production function I've been reading a book on a Kaleckian economics model, but there is a recursive function that has been bothering me. Can anyone provide some insight into why the equation why designed like this?

Quote from the book:

  $Y = \min\{aE, uK\} \quad (1.1)$

  Equation (1.1) denotes the fixed-coefficients production function, where $u$ denotes output; $E$, employment; $K$, capital stock; $a = Y/E$, labor productivity; and $u = Y/K$, the output-capital ratio. In the following analysis, we assume that the capital-potential output ratio is unity. From this, we can regard the output-capital ratio $u = Y/K$ as the capacity utilization rate.$^{4)}$
  ...

  $4)$ The capacity utilization rate $u$ is defined as $u = Y/Y^*$, where $Y$ denotes the actual output and $Y^*$ denotes the potential output. The capacity utilization rate is decomposed into $u = (Y/K)(K/Y^*)$, where $K/Y^*$ denotes the capital/potential output ratio and captures the production technology. If we assume that $K/Y^*$ is constant, then $u$ and $Y/K$ change in the same direction. From this, we can regard the output/capital ratio as the capacity utilization rate. In this chapter, for simplicity, we assume that $K/Y^* = 1$. Therefore, we obtain $u = Y/K.$

I assume $a$ and $u$ are the fixed coefficients. But even if we suppose $uK < aE $ then since $u = Y/K$, we should have $uK = aE$, or is that wrong? If correct, why do we use use the minimum operation?

Here is the digital book. Quote from page 20.

The min operation is known as a Leontief production function http://en.wikipedia.org/wiki/Leontief_production_function

You're right that with Leontief production functions, the equilibrium is always aE=uK, which is to say that production is linear, and factors are consumed in fixed proportion. There are some real world examples for this: to drive, for example, you need a car and wheels in an exact 1:4 proportion, and having 6 wheels per car doesn't let you produce anything more than 4 wheels per car. But for the most part the aim is just to motivate a linear approximation to a multi-sector economy, which we can easily plug into a computer and run economy-wide computations with. We didn't have high powered computers in those days (c. 1950), so computing multi-sector non-linear production technologies was simply out of the question.


Distribution of loans by types of interest Im looking for some kind of report that would inculde data on the distribution of loans by types of interest(fixed/floating), spesificaly global not just for one country(developed, undeveloped, developing world). If any of you have known to stumble upon such reports would be grand

Loans between corporations, that is what im looking for.

http://www.oecd.org/economy/surveys/

To complete the answer, I recommend reading the full economic studies country by country: banks, finance, debt, economic sectors, taxation, economic risk etc. Read these studies before making décisions !

I sincerely believe that these studies will help you in your economic analysis and banking.

We must understand why an interest rate is high or low, why the volume of business loans is strong or weak.

Only this type of study can explain and analyze the data rates between companies, and compare from one country to another.

For example, credit rates can reach 15% in an African country because the inflation rate is high and the rate of the central bank is very high! Without economic studies, it is impossible to compare.

French/Français :

Pour compléter la réponse, je conseille de lire les études économiques complètes pays par pays : l'écosystème bancaire, les financements, la dette publique, les études par secteurs économiques, la fiscalité, les risques économiques etc. Lire ces études avant de prendre des décisions !

Je pense sincèrement que ces études vont vous aider dans votre analyse économique et bancaire.

Il faut comprendre pourquoi un taux d'intérêt est fort ou faible, comprendre pourquoi le volume des prêts professionnels est fort ou faible.

Ces études peuvent vous expliquer et aider à analyser les données concernant les taux entre les entreprises, et les comparer d'un pays à un autre avec une approche globale.


What sort of impact would one expect on investment as a result of an approaching election? I am looking to observe and possibly quantify private investment during election periods.

I am interested in any studies, particularly empirical, which discuss the impacts elections can have on the economy in the lead up to them. From what I can tell, there is typically a suppressing impact on investment but I'd like to see some work on it.

There are many ideas to look into and some research:

IDEAS

Elections are presumably a time where the future becomes uncertain in the sense that there is a likelihood the new leader will change the course of a nation. From that point of view its about increased uncertainty.
Elections are often about corruption. The current political party will use corruption to remain in power. Often this means an increase in public expenditure or a loosening of monetary reins.
Potentially, some elections change the mood of a country because they absorb lots of people's energy, make them fight each other, as the candidates sometimes expose the least likable instincts in order to try and win.

EMPIRICS

Some of the work of Nick Bloom consists of structural estimation of a model of a firm that adjusts to changes in the amount of uncertainty in the environment. This kind of work suggests that for a fixed productivity level, investment (and disinvestment) rates fall in response to increases in uncertainty. The intuition is that of the threshold for exercising stock options: you are more likely to wait to exercise a call option if volatility is greater. Under the assumption that there is more uncertainty about future policy during elections, these models will tell you that investment will be lower during these periods.
Canes-Wrone and Park describe survey results in terms of people delaying purchases during election times.
Julio and Yook stuy cross- border capital flows in response to policy uncertainty, which is often the result of election periods.

SUGGESTIONS

Maybe, a way to go about this empirically is simply to look at the responsiveness of investment to q, as a function of how far elections are. If the coefficient of q in investment regressions is lower in the quarters right before presidential elections, you would have a reasonable and yet interesting result.


Is there a collection of Economics PhD prelims questions? Out of curiosity, do you know if there is a collection of Economics PhD prelims questions?

Exams from:

Harvard
UC Davis
MIT (only problem sets; go to graduates).

Although more cumbersome, you can also google something like "advanced macroeconomics exams pdf" or so, and you can discover personal pages of lecturers/professors that put up previous exams (or problem sets). For example, resources from a lecturer at Leipzig (go to bottom of page), Oslo, Dublin (one exam only), Helsinski.


Interpetation of coefficent in AR(1) model An AR(1) process is given as:

$$x_t=\rho_0+\rho_{t-1}x_{t-1}+\epsilon_t$$

This regression tells us that $x_{t}$ is a function of its value at time $t-1$.

My question is, how do you interpret its coefficient $\rho_{t-1}$? by comparison in a labor economics example (where only cross sectional data is used) for a case where you regress education on wage.
$$y_{wage}=\beta_0+\beta_{1} x_{educ}+u$$
If I were to take partials of this regression with respect to $x_{educ}$ I can interpret $\beta_1$ as the marginal returns to wage from education.

In the AR(1) process, following the same steps as before, I am not sure how to interpret the coefficient $\rho_{t-1}$.

What is its meaning?

For a second-order stationary series it is the correlation coefficient between the dependent value and its lag.
Specify
$$y_{t+1} = a+ \beta y_t + u_{t+1}\qquad u_{t+1}= \text{white noise}$$

The correlation coefficient between $y_{t+1}$ and $y_{t}$ is defined as usual

$$\rho_{(1)} = \frac{\text{Cov}(y_{t+1},y_{t})}{\sigma(y_{t+1})\sigma(y_t)}$$

$$\text{Cov}(y_{t+1},y_{t}) = E(y_{t+1}y_{t}) - E(y_{t+1})E(y_{t})$$

$$ = E\Big((a+\beta y_t+u_{t+1})y_{t}\Big) - E(y_{t+1})E(y_{t}) = aE(y_t)+\beta E\Big(y_t^2+u_{t+1}y_{t}\Big) - E(y_{t+1})E(y_{t})$$

We have $E(u_{t+1}y_{t}) =0$. Also, under first-order stationarity we have $E(y_t)=E(y_{t+1}) = \frac{a}{1-\beta}$

Using these we get

$$\text{Cov}(y_{t+1},y_{t}) = \frac{a^2}{1-\beta}+\beta E(y_t^2) - \frac{a^2}{(1-\beta)^2}$$

By definition the variance is

$$\text{Var}(y_t) = E(y_t^2) - [E(y_t)]^2 = E(y_t^2) -\frac{a^2}{(1-\beta)^2}$$

$$\implies E(y_t^2)  = \text{Var}(y_t) + \frac{a^2}{(1-\beta)^2}$$

Substituting,

$$\text{Cov}(y_{t+1},y_{t}) = \frac{a^2}{1-\beta}+\beta \text{Var}(y_t) + \beta \frac{a^2}{(1-\beta)^2}   - \frac{a^2}{(1-\beta)^2}$$

Things cancel out and we are left with

$$\text{Cov}(y_{t+1},y_{t}) = \beta\text{Var}(y_t) $$

Under the assumption of  2nd-order stationarity, $\text{Var}(y_t) = \text{Var}(y_{t+1}) = \text{Var}(y)$

Inserting all this back to the correlation coefficient

$$\rho_{(1)} = \frac{\beta\text{Var}(y)}{\sigma(y)\sigma(y)} = \frac{\beta\text{Var}(y)}{\text{Var}(y)} = \beta. $$

Note that the presence of the constant $a$ does not affect the correlation -it would be the same if $a=0$. This is because location parameters do not affect second-order statistics like the covariance and the variance.


Question on shifts in Supply and Demand Curves I have a super basic question on the shift in supply and demand curves ( I haven't seen this stuff since high school). More generally, this is about solving systems of simultaneous equations.

We have two curves, an IS curve and an LM curve. We know from theory
that the IS curve is downward sloping in $i-y$ space and the LM
curve is upward sloping in $i-y$ space. Let's take a basic example.
The IS curve is represented by
$$
i=-y
$$
and the $LM$ curve is represented by
$$
i=y
$$
 Equilibrium would imply values of $y$ and $i$ that solve these
equations simultaneously, or $i=y=0.$ Now suppose government activity
increased, shifting the $IS$ curve to the right. In this framework,
this would result in a horizontal translation of $2$ units of the
$IS$ curve.
$$
i=-y+2
$$
 while the $LM$ curve remains unchanged. The new equilibrium is given
by $y=1$ and $1.$ Notice that the initial increase of $2$ in the
IS curve resutlts in equilibrium increasing by only $1.$ Intuitively,
this is because as $y$ increases, so does $i$ according to $LM$
relation. And, as $i$ increases, $y$ decreases according to $IS$
relation. As both these relations have to be satisfied, some of the
initial increase is moderated. My question is: what ensures that $y$
in fact doesn't decrease? If the $LM$ curve determines interest rates
that are super sensitive to increases in $y,$ can it not be that
$i$ increases sufficiently that $y$ in equilibrium decreases? Graphically,
this is not possible- in the extreme case, the $LM$ curve is vertical,
$y$ does not change. But, what condition ensures that $y$ does not decrease?

Write $i=I(y,a)$ for the IS function (expressed as some function of $y$). Likewise, let $i=L(y)$ be the LM function. Assume the following properties:

$\frac{\partial I(y,a)}{\partial a}>0$ : $a$ is a parameter that causes the IS curve to shift up.
$\frac{\partial I(y,a)}{\partial y}<0$: the IS function slopes downwards
$\frac{\partial L(y)}{\partial y}>0$ : the LM function slopes upwards

For any $a$, we find the equilibrium $y^*(a)$ at the intersection of the two functions:
$$I(y^*(a),a)- L(y^*(a))=0.$$
Applying the implicit function theorem:
$$\frac{\partial I}{\partial a}+\frac{\partial y^*}{\partial a}\left(\frac{\partial I}{\partial y^*}-\frac{\partial L}{\partial y^*}\right)=0,$$
which implies
$$\frac{\partial y^*}{\partial a}=-\frac{\frac{\partial I}{\partial a}}{\frac{\partial I}{\partial y^*}-\frac{\partial L}{\partial y^*}}.$$

The numerator is positive by hypothesis. Thus, for $y^*$ to increase with $a$ we require that the denominator be negative: $\frac{\partial I}{\partial y^*}<\frac{\partial L}{\partial y^*}$. This is clearly true because $I$ slopes downward and $L$ slopes upward. But $y^*$ would also increase even if $I$ sloped upward—provided itdoes so more gradually than $L$.


General model of economics, economics as mathematical structure Contrary to the common view, the physics is all about models, there are no such thing as law of nature, because every physical law has its applicability range and therefore all of laws as models. But any physical law derives from one principle - https://en.wikipedia.org/wiki/Principle_of_least_action - any particle theroy (there are hundreds of them) is written as the model Lagrangian and then variational principle is applied to arrive at dynamical equations and observable quantities that can be corroborated in experiments.

My question is - is there similar approaches, theories, most general models in economics from which every other model or economic law can be applied. If there are such models, then the automation of economics can be achieved (assuming that the automation of mathematics will be completed, e.g. with project http://ai4reason.org/).

I have heard definition - the economics is a tuple <...> but I can not remember the book or article or the content of this tuple. Usually such tuples contain t]general mathematical objects: sets of agents with preference functions, set of interaction mechanisms (market system, competition patterns (monopoly etc.)), sets of external shocks (endogeneous factors) and so on...

I am seeking more for literature references, trends, keywords, not exhaustive answer. I can further research question by my own, but I should be informed from where to start my efforts.

Macroeconomics can (and should) be derived from the microeconomics, therefore I am seeking the model that states first principles - general microeconomic model (from which the general macro models could also be derived in the best case).

You may be interested in reading "Economic Models as Analogies", a 2014 article by Itzhak Gilboa, Andrew Postlewaite, Larry Samuelson and David Schmeidler, four well regarded economic theorists. They argue, somewhat contrary to your view of economic theorizing, that economic models are better viewed as "theoretical cases" rather than general rules akin to the laws of physics. Predictions are made by drawing analogies between the world in which the models live and reality. In Section 4 of the paper, they do propose a "model of economic models" that formalizes their argument. Although this model is probably not the one you were looking for, it may help clarify your (mis)perception of the methodology in theoretical economics.

The paper's abstract is as follows:

  People often wonder why economists analyse models whose assumptions are known to be false,
  while economists feel that they learn a lot from such exercises. We suggest that part of the
  knowledge generated by academic economists is case-based rather than rule-based. That is, instead
  of offering general rules or theories that should be contrasted with data, economists often analyse
  models that are ‘theoretical cases’, which help understand economic problems by drawing
  analogies between the model and the problem. Thus, economic models, empirical data,
  experimental results and other sources of knowledge are all on equal footing, that is, they all
  provide cases to which a given problem can be compared. We offer complexity arguments that
  explain why case-based reasoning may sometimes be the method of choice and why economists
  prefer simple cases.


Cost-push inflation and labour shortage Is it possible for there to be a shortage of labour in an economy when there is cost-push inflation?

The definition of cost-push inflation is an increase in aggregate prices driven by an increase in the cost of labour and/or production inputs.

A shortage of labour could put upward pressure on wages, yielding this type of inflation. So, yes, it is possible. However, it is not necessarily true that cost-push inflation is caused by a labour shortage and it is also not true that a labour shortage will always cause cost-push inflation.


Betting on the sum of scores Here is a game I like to play:

Every player gets a random number independently drawn from a fixed distribution. To fix ideas, suppose everyone draws a number from $\{2, ..., 10\}$ where all numbers are equally likely (that way you can do this with playing cards...)
The first player says: 'I think that the sum of the numbers is $k$ or higher' where the 'bet' $k$ is an integer that they need to choose.
The second player can either challenge or make a higher bet. If they challenge, the numbers are revealed and the sum computed. One of the two players is then knocked out depending on who was right -- and we move to the next round. If the second player doesn't want to challenge, they must then state 'I think that the sum of the numbers is $k'$ or higher' for some integer $k' > k$.
In that case, we move to the third player who can either challenge or make a still higher bet. And so the round continues.. until one player does inevitably challenge when the betting gets too high.
The above defines the rules in a particular round. And the players keep playing rounds until all but one is knocked out.

Despite having played this a few times, I really don't know what a sensible strategy (or alternately a Bayes-Nash equilibrium strategy) might be. Does anyone have any ideas? [If the game above is too complicated, I invite simplifications -- for instance, you might suppose that there are just 2 players.] For what it is worth, here are a couple of speculations on my part:

A very naive strategy is ignore the informational content of bets and stick with one's prior. In this example, the expected value of each number is $(2 + 10)/2 = 6$. Suppose then that my number is $x$ and that I face $n$ opponents. Then I might predict the sum to be $x + 6n$; and increase the betting until doing so would force me to bet higher than this (at which point I challenge).
Obviously, one can try to best respond to players who follow this kind of strategy. For example, if you see them betting some large amount $k$, you may infer that $x + 6n \geq k$ where $x$ is their number, or equivalently $x \geq k - 6n$. You might then compute the expected value of their number $x$ conditional on its being larger than $k - 6n$ (which is pretty easy in the uniform example).
I suspect that in a (Bayes-Nash) equilibrium, players must use mixed strategies. If your bet is a (strictly increasing) function of your number, then I can perfectly infer your number from your bet (if I know your strategy). This gives me a big edge -- and suggests that your strategy can't form part of an equilibrium.

Some thoughts:
Suppose there are two players $i,j$ and two numbers $1,2$. So type profiles are $\theta=(\theta_i,\theta_j) \in \{1,2\} \times \{1,2\}$, and let them be independet draws from the same distribution with $P(\theta_i=2)=p$ and $P(\theta_i=1)=1-p$ . On equilibrium path, this is a continuation game of any larger game because in the end there are always only two players left and in equilibrium they should be the ones with the highest types. This is the continuation game where the standing bid is so high that only the two highest possible numbers remain as plausible candidates.
Let $b$ be the current standing bid and solve backwards as player $i$:
First, $i$ overbids any $b\leq 1$ as challenging would mean losing for sure as for any $\theta, \theta_i+\theta_j\geq 2$.
Similarly, $i$ challenges any $b\geq 5$ as it means winning for sure as for any $\theta, \theta_i+\theta_j<5$ because 4 is the maximal sum.
$\theta_i=1$ challenges $b=4$ as $1+\theta_j\leq3<4$ to win for sure. $\theta_i=2$ challenges $b=4$ as overbidding to $b\geq5$ means player $j$ challenges the next round and wins. Challennging $b=4$ as $\theta_j=2$ means losing against type $\theta_j=2$, but winning against type $\theta_j=1$.
$\theta_i=1$ challenges $b= 3$ as overbidding means losing for sure as any type would challenge bid $b\geq 4$ for sure in the next round and the sum is at most 3.
$\theta_i=2$ overbids $b=3$ to 4 as challenging means losing for sure, given that $2+\theta_i\geq3=b$ for sure.
$\theta_i=2$ overbids $b=2$ as challenging means losing for sure, given that $2+\theta_i\geq3>b$ for sure.
$\theta_i=1$ overbids $b=2$ as challenging means losing for sure, given that $2+\theta_i\geq2=b$ for sure.
Hence, bidding 2 essentially means that the other one has the first move. You can construct multiple signaling equilibria here. Let's concentrate on the case in which nothing is inferred from such a move.
Consider the first round.
Suppose $p>1-p>\frac{1}{2}$.
As $\theta_i=1$, you could bid 4 in the first round. Both types must challenge, which means you lose against $\theta_j=2$ (sum is $3<4$) and lose against $\theta_j=1$ (sum is $2<4$). That is, you can only lose when bidding 4 as type 1 and you will never do this.
Alternatively, as $\theta_i=1$, you can bid $b=3$ in the first round, which means you lose against $\theta_j=1$ who challenges for sure (sum is $2<3$) and win against $\theta_j=2$ who would lose a challenge and loses from overbidding to 4, which you would challenge. Winning is more likely as you beat type 2 and $p>1-p$.
Alternatively, you can bid 2 in the first round, which means both types will overbid.
Type $\theta_j=1$ will bid 3 as bidding 4 or more means sure loss to them. You would challenge the 3 as type 1 and win (overbidding to 4 or more means sure loss to you).
That is you win at least with probability $1-p$.
What does type 2 do upon seeing $b=2$?
Given we assume no inference, they do the same as they would do if they started.
As $\theta_i=2$, you can bid 4 in the first round, which means you lose against $\theta_j=1$ (sum is $3<4$) and win against $\theta_j=2$ (sum is $4$). Both will optimally challenge you even if they learn your type - the high type knows they lost upon seeing 4, the low types knows they won. Thus winning is more likely, $p>1-p$.
Alternatively as $\theta_i=2$, you can bid 3 in the first round, which means you win against $\theta_j=1$ who challenges (sum is $3\geq3$, and them overbidding means you challenge in the next round and win) and lose against $\theta_j=2$ who overbids to 4 which you have to challenge and lose - $\theta_j=2$ would not challenge as this would mean sure loss to them. Thus losing is more likely, $p>1-p$.
Thus, back to type 1 bidding 2 in the first round. If $p>1-p$ and there was no inference, type $\theta_j=2$ bids 4 and you as $\theta_i=1$ win by challenging 4. But that would mean you always win as type $\theta_i=1$ when bidding 2. Then it must be that $j$ infers that you are type 1 when bidding 2. As a response, they would bid 3, which you can challenge or overbid and lose in any case. Hence, bidding 2 as type $\theta_i=1$ means, winning with probability $1-p$ and losing with probability $p$. Because $p>1-p$, this is worse than betting 3.
So $\theta_i=1$ bids 3 in the first round when $p>1-p$.
Now to type $\theta_i=2$ in the first round. Given the above, bidding 4 is better than bidding 3 given prior $p>1-p$. What about bidding 2?
Under the belief that type 1 bids 2, both player types would overbid to 3. Here, challenging would mean sure loss for type $\theta_i=2$. Hence, we overbid to 4 with the same outcome as bidding 4 in the first round. Under a belief that type 2 bids 2, a type 2 would overbid to 4 and win, and a type 3 would overbid to 3 and win challenge against 3 and also after a challenge against 4.
So $\theta_i=2$ bids 4 in the first round when $p>1-p$.
With $p<1-p$, type 1 would still not bid 4 in the first round, but also type 2 would not want to bid 4, instead they bid 3 and type 1 bids 2.


Asset pricing Coursera resources I am trying to learn John Cochrane's Asset Pricing. I notice there are Coursera resources (link). However, it is not available now. Did anyone try that class before? Does anyone know what's the next time it opens? Is it worth waiting for? Are there any good alternatives?

I have uploaded all the videos on YouTube as two separate playlists which cover asset pricing part 1 & 2 courses.

Asset Pricing part 1:  https://www.youtube.com/playlist?list=PLpccx1MwQZb7t7HJ6lBo4L3DXhknCD0K7&disable_polymer=true

Asset Pricing part 2:  https://www.youtube.com/playlist?list=PLpccx1MwQZb6XsKU3O81GrpqaxyyGoHpH

Regarding the class, I think Coursera moved to the new platform and thats why these courses and some others have disappeared. I heard that they are preparing the courses based on the new platform.


How Does Land Value Tax (e.g. Georgism) Decouple Lease Transfer From Improvement Transfer? In theory, a land value tax (e.g. Georgist single tax), works by the government replacing property taxes with leases on land, as an incentive to invest in property improvements.  This obviously works when the owner of the improvement is the holder of the lease on the improved land. However, if the land value increases, and the rent on the lease does not, the holder of the lease, in effect, privatizes the positive network externalities of others' improvements.  Therefore, it is necessary that the lease not fix the rent payments for the duration of the lease and, instead, dynamically adjust them to collect (thence distribute as a social good) the increased land value.

At some point a lease holder may be unable or unwilling to pay the rent, causing a lease transfer.  If his improvements are mobile, he can pull up stakes and move them.  However buildings are typically fixed improvements. Hence there can occur a decoupling of ownership of a fixed improvement from the ownership of the lease on the underlying land.  This would appear to be a theoretic problem with the LVT, since the new owner of the lease's right to improve the property would be constrained by the prior owner's use of the land in the form of fixed improvements.

How does LVT theory deal with this decoupling of land lease from improvement ownership with land lease transfer in the absence of improvement ownership transfer?

In practical terms the leaseholder will need a certain degree of certainty over her future rent payments on the lease to be able to make the investment. That’s why leaseholds often have long durations (in Germany typically 99 years). At the end of the lease the buildings become property of the freeholder. Depending on the jurisdiction the freeholder may have to compensate the leaseholder for the building. That is why leaseholds typically specify which type of building is permitted. Also the limited lease duration limits the ability of the leaseholder to actually benefit from the positive externality because her investment is already made; neither can she liquidate the increased land value at the end of the lease.

If the lease were permanent and subjected to real time land value taxes the leaseholder would have little incentive to invest. This would be equivalent to a one-year rental contract in which the tenant has a perpetual option to renew but the landlord can adjust the rent each year. It would be hard to convince someone to make a significant investment under these conditions. A solution may be for the leaseholder to buy insurance against excessive land value tax increases. But I have not heard of this in practice.


What is 'repo' in banking I keep hearing about repo rate etc. But this question is not what is the repo rate. It is: What is a definition of 'repo' ?

Repo is the short abbreviation for Repurchase agreement, as per Investopedia (https://www.investopedia.com/terms/r/repurchaseagreement.asp).


Why is the graph of unitary elastic demand a hyperbola? My teacher said that the graph of unitary elastic demand is a parabola:

But i fail to understand how in a hyperbola the percentage change of price and quantity demanded remains same. Can someone explain the same to me? First i thought that it remains the same rather in a straight line but a quick study proved that it is not the case in a line with a negative slope. But the question remains the same, how is it in a hyperbola the percentage change remains the same.

The price elasticity of demand is defined as:

$$E_P=\frac{dQ}{dP} \frac{P}{Q}$$

Although generally elasticity depends on price there is a special type of functions (isoelastic functions) for which elasticity remains the same along the whole function.

For example consider demand given by:

$$P=AQ^{1/e}$$

This demand function will always have the same price elasticity $E_P=e$ which you can verify by first solving the function above for Q and then applying the formula mentioned above. For $|e|=1$ the elasticity would always be unity. This is because the elasticity does not depend on anything else for example for a linear demand function the elasticity would depend on price as for $Q=a-P$ the $E_P=-P/(a-P)$ and hence as you correctly pointed out along line the elasticity constantly changes with price.

When you plot the function above you will get a picture that looks like hyperbola. See the picture below. However, not all functions that describe hyperbola have such property I don’t think your teacher meant to say all hyperbolas have constant elasticity.

.

Also the demonstration project from wolfram alpha where I got the picture has amazing utility that graphically shows how the elasticity remains same along every point of the curve. The utility even allows you to play with the parameters of the model. Here is the link for it if you want to explore it.

——————————————————-

Per request in the comment this is how you show that the elasticity is $e$ for $P=AQ^{1/e}$

Solve for $Q$ which gives you:

$$Q= \frac{1}{A} P^e$$

Apply the formula for elasticity which gives you:

$$E_p=e\frac{1}{A}P^{e-1} \frac{P}{Q}$$

Substitute for Q the original expression $Q= \frac{1}{A} P^e$

$$E_p=e\frac{1}{A}P^{e-1} \frac{P}{ \frac{1}{A} P^e}$$

Simplify the above expression and you will get:

$$E_P=e$$


Which currency do we consider when talking about trade balance ? The Marshall Lerner condition and the J curve made sense until one day I noticed that to make sense of all of this , we considered two different currencies. I was told for e.g. in the short run, export revenue would deteriorate since demand would be inelastic. But prices for exports only fell for the foreign buyer, not for domestic firms. If quantity being sold rose even by little, won't revenue earned by local producers have increased? Similarly, we considered effect on imports but this time considering domestic currency. My question is that when we look at trade balance , which currency do we use for analysis.

My reading of the Marshall-Lerner condition is that it assumes the price of exports is constant in the domestic currency and the price of imports is constant in the foreign currency.  Analysis is conventionally done in the domestic currency

But, initially after devaluation, volumes are assumed to stay the same (short-term inelasticity) and measured in the domestic currency the values of exports stays the same while the value of imports rises, leading to a deterioration in the measured trade balance; this is the immediate cost effect of devaluation. If you were to measure in the foreign currency, the short-term value of exports would fall while the short-term value of imports would stay the same, again leading to a deterioration in the measured trade balance


What impact would introducing 15 times more soy worldwide do to the price of soy? Soy is a commodity and the main export of Argentina. It is said Argentina has 40 million of hectares of arable land and it produces food for 400 million persons and that potentially could make food for 600 millions. I can't confirm this is true but it's said by the president of one of the main agrarian organizations of Argentina, probably with statistics made by them and/or data of the national institute of agrarian technology.

Africa it's said to have 600 million hectares of arable land, which is 15 times the arable land of Argentina. What would happen to the prices of soy related commodities if Africa proportionally introduces 15 times more soy related products to the world? Take any soy product you want for the answer, soy bean if you want for example. Can this be calculated? I've heard of prices simulations predictions for other commodities when introduced or retrieved from the market, such as oil, so I suppose this can be calculated.

Roberts and Schlenker (2013) estimate the price elasticity of demand for agricultural commodities to be about -0.08 to -0.05. These are elasticities estimated over much smaller changes in $p$ and $q$ than you are suggesting. Remember that the elasticity is effectively percent change in $q$ for a percent change in $p$, so we can ask what the percent change in $p$ is for a percent change in $q$.

If production is 15x more soy than there is currently (in total, so globally) then this is a 1500% ((1*15+1 - 1)/1) increase in soy. Using the high end (most elastic), we get that prices have to fall 18,750% (15/-0.08) to clear the market at this new higher quantity. This is impossible, so this tells us that our local elasticity measures won't do to solve this problem. It also suggests that there would be no reason to add so much soy bean production, because demand is likely unable to handle so much soy.


Do house prices increases affect inflation? In this book, page 202, the authors develop the idea that when the central bank increases the interest rates to get the CPI inflation back to target inflation, it will not necessarily cut off an eventual house price bubble, or the financial accelerator. And to justify this assertion, in a footnote they state: «The elements of housing costs included do not influence the overall price indices that enter the inflation target to a high degree. To see this plot the US Consumer Price Index against the US Consumer Price Index less Shelter(Data available from US Bureau of Labor Statistics)». (What I understand from this is that since the housing related costs that enter into CPI computation are negligible, the central banker will perceive an inflation different from what if it included housing prices directly. Did I understand correctly?)

So, I plotted the graph.

I included an extra series, "All items less Food". It behaves exactly like the CPI.

How can I deduce from the graph what the authors of the book are saying?

Any help would be appreciated.

Here's what the Bureau of Labor Statistics' FAQ has to say on CPI and housing costs:

Q: The CPI used to include the value of a house in calculating inflation and now they use an estimate of what each house would rent for -- doesn't this switch simply lower the official inflation rate?
A: No. Until 1983, the CPI measure of homeowner cost was based largely on house prices. The long-recognized flaw of that approach was that owner-occupied housing combines both consumption and investment elements, and the CPI is designed to exclude investment items. The approach now used in the CPI, called rental equivalence, measures the value of shelter to owner-occupants as the amount they forgo by not renting out their homes.
The rental equivalence approach is grounded in economic theory, receives broad support from academic economists and each of the prominent panels, and agencies that have reviewed the CPI, and is the most commonly used method by countries in the Organization for Economic Cooperation and Development (OECD). Critics often assume that the BLS adopted rental equivalence in order to lower the measured rate of inflation. It is certainly true that an index based on home prices would be more volatile, and might move differently from other CPI indexes over any given time period. However, when it was first introduced, rental equivalence actually increased the rate of change of the CPI shelter index, and in the long run there is no evidence that the CPI method yields lower inflation rates than some other alternatives. For example, according to the National Association of Realtors, between 1983 and 2007 the monthly principal and interest payment required to purchase a median-priced existing home in the United States rose by 79 percent, much less than the rental equivalence increase of 140 percent over that same period.

Ordinarily, because people can substitute between renting or owning a house, the price of a house should broadly reflect its rental value (e.g. if it were much cheaper to rent than to buy then people would stop buying and start renting until prices 'equalised'). However, if a bubble is being driven by speculation on future house price increases then it is likely that house prices will become decoupled from the long-run rental value of a house. You can see what this looked like during the last housing bubble in the below figure. It can clearly be seen that house prices (the blue line) were not well-reflected by the owner-equivalent rent included in CPI.


What does this graph mean? I know little about economics and am very confused on what this graph represents.
 I found it when I saw that the President had tweeted it and was wondering what it mean.  Specifics what "Jobs plentiful" and "jobs hard to get" are measured in.

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

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


Equivalence of Definitions of Continuity of Preferences We have two definitions of the continuity of preferences:

Def 1: $\succcurlyeq$ is continuous if for any sequences $\{x^n\} \subset X$ and $\{y^n\} \subset X$, then $n \in \mathbb{N}$ such that,

$\forall n, \quad x^n \succcurlyeq y^n$
$\lim_{n \rightarrow \infty} x^n = x, \quad \lim_{n \rightarrow \infty} y^n = y \quad (\text{where $x, y \in X)$}$

then $x \succcurlyeq y$

and

Def 2: $\succcurlyeq$ is continuous if whenever $x \succ y$, $\exists \ B_x, \ B_y$, open balls around $x, y$, such that $\forall \ x' \in B_x, \ y' \in B_y$, then we have $x' \succ y'$.

Show that the following definitions are equivalent to each other.

We want to show that for $\succcurlyeq$ on $X$, Def 1 $\iff$ Def 2

$\boxed \Longrightarrow$

Assume that $\succcurlyeq$ is continuous by Def 1.

Let us say $x \succ y$. Denote our open-balls as $B(x, r)$, an open ball around $x$ of radius $r$. Suppose $\forall n, \ \exists \ x^n \in B(x, \frac{1}{n}), \ y^n \in B(y, \frac{1}{n})$ such that $y^n \succcurlyeq x^n$. But then we have constructed $\{x^n\} \rightarrow x$ and $\{y^n\} \rightarrow y$, and by Def 1, $y \succcurlyeq x$, which is a contradiction.

$\boxed \Longleftarrow$

Assume that $\succcurlyeq$ is continuous by Def 2.

Let us take sequences ${x^n} \subset X$ and ${y^n} \subset X$ where $\forall n, \quad x^n \succcurlyeq y^n$ and $\lim_{n \rightarrow \infty} x^n = x, \quad \lim_{n \rightarrow \infty} y^n = y \quad (\text{where $x, y \in X)$}$ for $n \in \mathbb{N}$,

BUT $x \succcurlyeq y$ is false instead of true. We want to show this leads to a contradiction.

If $x \succcurlyeq y$ is false, (so $y \succ x$) then $\exists B_x, B_y$ such that $\forall y' \in B_y, x' \in B_x$, we have $y' \succ x'$. Because $\{x^n\} \rightarrow x, \{y^n\} \rightarrow y$, there exists $N$ large enough such that $\forall n>N$, we have $y^n \in B_y, x^n \in B_x$.

Thus $\forall n > N$, we have $y^n \succ x^n$, which contradicts $\forall n, \ x^n \succcurlyeq y^n$.


Tax Incidence and distribution Although the tax incidence is generally dependent on the nature of the elasticity of the demand or supply, in special scenarios can the producer take up the complete burden of the tax so that the price paid by the consumers does not change?
If this is possible does this mean that the supply curve is shifted in order to change the tax incidence?

There are two possible answers here, depending on whether you want to take a partial or general equilibrium setting. Note that the partial equilibrium analysis of tax incidence is only taught in undergraduate micro textbooks but is not considered a sensible approach in current research.

Partial Equilibrium: Suppose your supply is perfectly inelastic (for example a non-rival but excludable good which you produced last period) and demand is perfectly elastic (let's say another firm produces a perfect substitute with perfectly elastic supply sold at some price p). Then the producer will bear the full burden of the tax.

General Equilibrium: In a general equilibrium setting, anything can happen. In response to a tax, both supply and demand curves may shift. However, in this case the standard definition of tax incidence is no longer meaningful. In such cases, researchers either calculate some loss function for each consumer or simply look at price changes to analyze tax incidence.


Do the partial derivatives of the compensated demand have an interpretation? When obtaining the Marshallian demand from a utility maximization problem we have a classification of normal goods, inferior goods, Giffen-goods, etc.. These are related to the first derivative of the Marshallian demand function with respect to prices or income.

My question is whether a similar classification exists for compensated demand functions of the expenditure minimization problem or whether the (same) goods classification can be derived via the compensated demand.

To the question on whether the same classification can be achieved through the hicksian demand function: in general I believe the answer is no. To see this, relate the classifications to the Slutsky decomposition.

(Changes in) Hicksian demand is related to the substitution effect from the Slutsky decomposition. Any standard textbook should provide a reference to this.

There are two broad classifications:

1.Ordinary and Giffen goods: These refer to marshallian demand changes when prices change. This is what we call the Total Effect in the Slutsky
decomposition.

Normal and inferior goods: These refer to marshallian demand changes when income changes. This is by definition the Income Effect in the Slutsky decomposition.

As you see, the substitution effect, which is represented by changes in hicksian demand, is not a part of any of these classifications you mentioned.


Gapminder and OurWorldInData I have found two great resources to learn about the distribution of goods (of any kind) all over the world:

gapminder (Hans Rosling)
ourworldindata (Max Roser)

By reading their "about" sections they seem to be quite similar in approach: "Things are not going perfect but better than they seem".

For me it's hard to find out which other providers next to gapminder and ourworldindata are there around: providing similar access to similar data in a similar way (visual), with a similar (but possibly quite different) approach.

Any hints are welcome.

On income distribution, there is the World Inequality Database. On poverty, the World Bank provides lots of comparable figures, although I believe these are the same data as in gapminder.


Why does quantity supplied increase with price in economics? The Law of Supply is my worst enemy in economics because I could never truly understand it, and as a result, the stuff I learned after that was built on a weak foundation. The Law of Demand is totally different though, it makes perfect sense to me. I have spent hours thinking about this and I've figured out exactly what I don't understand.

The Law of Supply would make perfect sense to me if price was substituted with revenue. With each good supplied, revenue would increase in a linear manner. However, as I realized, it is not revenue we are dealing with. The way the Law of Supply works, revenue would increase in an exponential manner if we took a supply schedule and multiplied price and quantity for each price level.

I don't understand why price has to increase if quantity increases. Shouldn’t the increase in quantity supplied already generate more revenue to cover the extra costs of production? Why should the price be spiked up to further increase revenue? Why wasn’t the price that high in the first place if the good could be sold like that?

The Law of Supply seems counterintuitive to me in some ways. If you have only a small amount of something to supply, shouldn't you make the price HIGH so that those rare fools willing to pay for overpriced things (the ones at the top of the demand curve) would clear your stock and maximize your revenue?

One explanation that almost made sense is that the more you tried to produce, the higher the costs of production would get. A producer would have no choice but to raise prices if costs of production were that high. But this only makes sense if costs of production increase EXPONENTIALLY, which I don't understand why would happen with EVERY good! Why is it always assumed that marginal cost is increasing and not remaining constant?

On a side-note: how would the Law of Supply work in the digital realm, where stock is basically infinite? What would a supply and demand graph look like there?

On a side-side-note: why are the supply and demand graphs really considered CURVES when they are almost always represented by lines?

Maybe there's something fundamentally wrong about my understanding of this. It's supposed to be easy to understand, but for me it's not. I would be so grateful if someone could clear this up for me!

I believe that I have found the problem in this line:

  I don't understand why price has to increase if quantity increases.

The concept is actually the other way around.

Think of it in the same way as the Law of Demand.
If you go to the grocery store and you see a food that you like selling for \$.25/lb, you would buy a whole lot of it before the price rises.
Conversely, if you go to the grocery store and you see a food that you like selling for \$100/lb, you would probably wait to buy this item until it is cheaper or at least buy a small amount of it.
In economics, the price drives the quantity demanded by the consumer.

Now let us look at the Law of Supply.
Imagine that you are the owner of a company.  You go to the store, and you see that the item you are producing and the similar items produced by your competitors is selling for \$.25.  You would not necessarily want to produce a lot of the product because the margin between the selling price and the production costs (profit) is small.
Conversely, imaging going to the store and seeing that the item you are producing and the similar items produced by your competitors is selling for \$100.  You would want to produce a lot of the product because the margin between the selling price and the production costs is (presumably) large.
In this case, as in the other case, the price drives the quantity produced by the supplier.


Increase in the number of jobs in an economy: Central Bank raises interest rates? For example since the August job report came out, which showed hundreds of thousands ob jobs created. This will inevitable lead to the Fed to increase interest rates, but what is the motive behind doing this?

Would this not simply lead to a decreased shift in consumption and be an attempt at stagnating the economy?

When people find jobs, they start receiving salaries and their consumption increases. Increased demand for goods and services leads to increase in prices, or inflation. Fed is expected to respond with a rate hike to keep inflation in check.


features of Generalised Method of Moments estimation [

What does it mean that GMM is robust to distributional assumptions?

It is a fancy, confusing and indirect way to say that GMM, being a Method-of-Moments estimator, does not require distributional assumptions in order to estimate regression coefficients, just like Ordinary Least Squares, and in contrast to maximum likelihood that needs to make distributional assumptions.

The wording is confusing, because "robust to distributional assumptions" means that we get consistent estimates even if we have made the wrong distributional assumptions.


Is there book or research on economically sustainable design of town or village? I'm looking for a book which may help to design a self-sustainable small  settlement, which can maintain its own economy with limited connections to the outside world. Prefereably it should answer questions like:

What population is sufficient for such a town or village?
How to model microeconomics of such a settlment?

I would suggest the Handbook of Regional and Urban Economics. It currently has 5 volumes and collects papers on the theories and empirics of urban/city design. Selected chapters that may potentially address your questions include:

City size and place as policy issues
Financing urban public services
The theory of urban residential location
Therories of urban business location
Urban labour markets
Urban housing markets: Theory and policy
Micro-Foundations of Urban Agglomeration Economies


Basic Fundamentals question on regression analysis based on economic theory: In running any regression on demand based on parameters derived from theory, I'm having some confusion in some order of operations when the results come out inconsistent.

For instant in the case of multi-collinearity and having to drop parameters -- what is the case when parameters of equal importance to theory (like price's) have perfect collinearity? They can't be indexed or made into any ratio, so must be dropped.

My default is to discriminate between the parameters on their explanatory power -- R squared. But what about F-test of significance? If one is higher R than the other but may be nullified by the insignificant F test. Would the 'normal' protocol still be to include the parameter that results with that same higher R-squared despite the F-test being insignificant.
-- Or is it even not possible given the parameters are in the same model the F-test is being ran on?

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

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

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


Calculating rate of growth of per capita income Given this question:

National income is increasing by 1.5% a year and population by 2.5% a year. What is the rate of growth of per capita income?

Attempt:

Since per capita income is GDP/ population. I divided 1.5 by 2.5 and got 0.6. Is this right?
Thanks.

Consider: If national income is increasing at a slower rate than population growth, then intuitively per capita income will be falling. Here is a set-up for the rate of decline in per capital income.

$$\text{per capita income}_t = \frac{\text{GDP}(1.015)^t}{\text{population}(1.025)^t} \text{per capita income}_{t-1}$$


proof of uniqueness of Dominant strategy equilibrium This exercise is from book: Steve Tadelis, Intro to game theory, chapter 4 exercises.

Prove: If the game $\Gamma= \{N,\{S_i\}^n_{i=1},\{v\}^n_{i=1}\}$ has a strictly dominant strategy equilibrium $s^D$, then $s^D$ is the unique dominant strategy equilibrium.

My attempt:

Proof by contradiction: Suppose $s^D$ were not the unique DSE. Then $\exists s'^D\in S'^D$ such that $v(s'_i,s'_{-i})\geq v(s_i,s_{-i})$ where $s'_i,s'_{-i}\in S^D$ and $s_is_{-i}\in S^d$. But this contradicts the assumption that $s^D$ is the DSE.

Would this be correct?

You are along the right track but are making certain errors in your proof,

Your notation is confusing. You haven't defined what $S^{'D}$ is. I would suggest sticking with the defined strategy space and not creating unnecessary notation in the proof.
When you write $v(s'_{i},s'_{-i}) \geq v(s_{i},s_{-i})$, you are not only changing player i's strategy but also other players' strategies. However, you need to keep other players' strategies constant and then compare player i's options to choose the best strategy. Also, note which player's payoff function is being compared. So, you should instead write $v_{i}(s'_{i},s'_{-i}) \geq v_{i}(s_{i},s'_{-i})$.

Now, let's move on to how a proof by contradiction could look like.

Suppose there exists another dominant strategy equilibrium, $s^{A} \in S$. (Note that we are talking about both weakly and strictly dominant strategy equilibria here)
This means that $\exists i \in n$ such that $s^{A}_{i} \neq  s^{D}_{i}$ and $v_{i}(s^{A}_{i}, s^{A}_{-i}) \geq v_{i}(s^{D}_{i}, s^{A}_{-i})$.
However, $s^{D}$ is a strictly dominant strategy equilibrium, and so, $s^{D}_{i}$ is a strictly dominant strategy for player i. Inter alia, this implies that $v_{i}(s^{D}_{i}, s^{A}_{-i}) > v_{i}(s^{A}_{i}, s^{A}_{-i})$.
The inequalities in (2) and (3) contradict each other. Hence, the strictly dominant strategy equilibrium must be unique.


Is fisher equation a definition, identity? Or is it rather a very good estimate So let us agree that the fisher equation is $1 + i = (1 + r)(1 + \pi).$ Is it a definiton or good estimate. Intuition tells me this make so much sense and almost qualify as an equation that must hold. However, sources online tell me that it is a good approximation. If we accept that, can anyone suggest examples when the equation fails and why? Thank you!

The fisher equation has its basis in the fact that the real return on an asset is the nominal return divided by the inflation rate. If you hold a bond today, it gives you back $1+r_{t+1}$ tomorrow. This is basically $\frac{1+\iota_{t}}{1+\pi_{t+1}}$ such that the promised nominal rate is deflated by the inflation rate. Rearranging terms, you get your definition. This is exact and always holds conceptually. The approximation is that $i\approx r+\pi$  which is obtained via a Taylor series expansion.


Omitted variables in gravity model I'm trying to construct a gravity model for EU trade flows using a panel dataset, but am suffering from what appears to be a persistent omitted variable problem. My residuals display a queer log-like, almost polynomial type pattern when plotted against the log of trade volumes.

I have searched the literature and included every variable in the model that has appeared in previous studies, including the logs of;
- gdp per capita
- real and nominal gdp
- distance - multiple different measures.
- Real and nominal exchange rates.
- Price levels - various measures.
- Proxies for product differentiation.
- Exporter / importer fixed effects.
- Dummies for things like Schengen / Euro membership, home trade, etc.
Still, the curvilinear / log pattern remains.

Can anyone conjecture as to what the omitted variable may be? I have exhausted my knowledge of the theory.

Any help would be very gratefully appreciated.

Many thanks,

Robert.

The omitted variable bias in gravity model is an important issue given that some factors are unobserved or difficult to quantity. To solve this issue trade economists tend to rely on various fixed effect estimators. But, the question is what is your variable of interest?

Exporter-by-year and importer-by-year fixed effects

For instance, if you are interested in evaluating the effect of trade costs on trade, you can introduce exporter-by-year and importer-by-year fixed effects to absorb GDP, GDP per capita, price indexes etc.

Exporter-by-importer fixed effects

If you are interested in evaluating the effect of one country-specific variable on trade, you can introduce exporter-by-importer fixed effects to absorb distance or contiguity effects.

Exporter-by-year, importer-by-year + exporter-by-importer fixed effects

If you are interested in evaluating the effect of a time-varying bilateral factor, like free trade agreements, on trade (see Baier and Bergstrand, 2007), you can introduce exporter-by-year, importer-by-year + exporter-by-importer fixed effects to absorb distance or contiguity effects as well as GDP, GDP per capita, ....

References

Baier, S. L., & Bergstrand, J. H. (2007). Do free trade agreements actually increase members' international trade?. Journal of international Economics, 71(1), 72-95.
Baldwin, R., & Taglioni, D. (2006). Gravity for dummies and dummies for gravity equations (No. w12516). National Bureau of Economic Research.
Head, K., & Mayer, T. (2013). Gravity equations: Workhorse, toolkit, and cookbook. This last reference is a must to read to construct a gravity model.


real exchange rate formula doubt Real exchange rate formula is the exchange rate multiplied by the ratio of two prices.

what does this ratio actually imply? What is the rationale behind taking the ratio?

The ratio of relative prices $P^*/P$, where $P^*$ is foreign and $P$ home price level respectively,  is used in order to adjust the nominal exchange rate for changes in relative price levels in two countries. I think a good way of understanding it is to use analogy with real GDP. A real GDP $Y_r$ will be defined as: $Y_r = Y_n/P$ where $Y_n$ is nominal GDP where we would divide by $P$ in order to adjust for the effect of inflation (change in price level).
The real exchange rate is basically the same concept you are trying to adjust the nominal exchange rate for changes in price level, but you cannot just divide by $P$ because then you would be ignoring the changes in foreign price level.
You can see this more clearly if you look at the equation in terms of inflation (as in Copeland's Exchange Rates and International Finance textbook). Start with the definition for real exchange rate:
$$Q=S\frac{P^*}{P}$$
where, $Q$ is the real exchange rate and $S$ nominal exchange. Now take logs of the variables and to clean up the expression I will be using lowercase letters to denote logs (i.e. $\ln X = x$):
$$q = s + p^* - p$$
Now the above should hold across multiple time periods so lets examine the changes between $t-1$ and $t$ where $\Delta x_t = x_{t}-x_{t-1}$:
$$\Delta q_t = \Delta s_t + \Delta p_t^* - \Delta p_t$$.
Now finally by definition inflation ($\pi_t$) is the change in price level so $\pi_t\approx \Delta p_t$ so lets substitute that in:
$$\Delta q_t = \Delta s_t + \pi_t^* - \pi_t$$.
Hence the change in real exchange rate reflects the change in nominal exchange rate adjusted for inflation in foreign country and home country.