How does lending money contribute to society? I essentially asked this question here, but IMO didn't get an answer from anyone who knew what they were talking about.

Normally someone gets paid because they work in a job, i.e. they contribute to society. I get paid by lending money to other people and charging interest. What exactly am I being paid for? It doesn't feel that I'm doing anything to better society.

It enables them to buy things they otherwise couldn't afford... in the near term. But ultimately they have to pay back the debt plus interest, so they can actually afford less in the long term. Is it just that I'm being paid to satisfy people's impatient desires? It doesn't seem that I'm contributing to long-term economic growth, and it seems that I may even be hurting it.

Someone suggested that the loans can be used to finance investment, so I thought about it for a while. I considered an economy consisting of three agents: BANK, person A, and person B.

Suppose person A is destitute and BANK lends £100 to person A in exchange for £110 later. Then person A pays person B the £100 to build a factory. Then person A manufactures goods with the factory and sells them to person B for £150. It seems that BANK has made £10 for nothing, as person A and person B could have come to an arrangement where person B would build the factory + pay person A £50 in exchange for the goods manufactured by person A directly. Maybe it's different with more than three agents. But I still come to the conclusion that the agents could collectively come to an agreement that wouldn't involve the BANK "middleman". Maybe that wouldn't work in practice. But in any case, it doesn't feel that lending money does anything to contribute to society.

In this simplified hypothetical world, yes, person B could give 100 currency to person A to build the factory directly. But what if person B does not want to take on that risk? How does person B know that person A will be contracted to make 150 worth of goods properly? Person B could just wait to see if someone else finances person A and then buy from A if the factory is successful. A single individual lending, or even small businesses lending out money can be a relatively large amount of money for that entity (so, high risk), whereas banks are able to diversify risks, by making many different loans and spending time evaluating how risky different people are.

Banks take on risk that other people/firms can't afford to or don't want to. Interest rate is very roughly, the price needed to be willing to take on these risks, or more precisely, the rental rate of money.


What is a substitute/complement in terms of mixed partial derivatives? I am trying to understand how substitutability relates to mixed partial derivatives. I thought the change in marginal utility with respect to a change in the amount of $x$ would correspond to $$\frac{\partial U}{\partial x}$$ so I got confused when I take the partial of that with respect to $y$. Does this measure the rate MU changes wrt $x$ as we change $y$? How is that related to being a substitute?

It is very important here to note that there are multiple, mutually inconsistent, possibilities for how to define a substitute/complement.

One way is to say that $x$ and $y$ are complements if an increase in $y$ raises the marginal utility of $x$ (or, given symmetry of mixed partials, vice versa):
$$\frac{\partial^2 U}{\partial x\partial y}>0\tag{1}$$This is the suggestion in foobar's answer.

Another way is to say that $x$ and $y$ are complements if a decrease in the price of $y$ raises the Hicksian (aka compensated) demand for $x$. Since Hicksian demand is the derivative of the cost (aka expenditure) function by Shephard's lemma, this can also be expressed as a condition on mixed partials:
$$\frac{\partial^2 C}{\partial p_x\partial p_y}<0\tag{2}$$
This is the suggestion in snoram's comment, and it is the notion more commonly taught in micro classes.

These definitions are not equivalent! Indeed, in any case with only two goods, those two goods must be substitutes according to (2), regardless of whether the cross-partial of $U$ in (1) is positive or not.

One can give fruitful labels to these concepts (though these labels are more common in the case of production rather than utility functions). Following Hicks, we can call complements by definition (1) q-complements: if $x$ and $y$ are q-complements, an increase in the quantity of $y$ leads to an increase in the marginal value of $x$. Meanwhile, we can call complements by definition (2) p-complements: if $x$ and $y$ are p-complements, a decrease in the price of $y$ leads to an increase in the demand for $x$. See, for instance, Seidman (1989) for a brief overview.

Both concepts are useful in different situations - it depends on what you're interested in!

More technical note: you might notice that (1) and (2) do not seem very similar to each other: (2) is a compensated concept, keeping us on the same indifference curve, while (1) is not. This is a valid criticism, and indeed there is an alternative notion of "q-complements" that is compensated, and a notion of "p-complements" that is not.

The compensated notion of q-complements, which is probably more relevant for most consumer theory applications than (1), asks whether the marginal return to $x$ increases as we increase $y$, while staying on the same indifference curve. (It's more relevant for consumer theory because it doesn't depend on the inherently ambiguous cardinality of $U$. Indeed, apparently Hicks introduced this as the consumer-theory definition of "q-complements" in his 1956 Revision of Demand Theory, though I don't have a copy of it myself.) This notion also has a mixed partial characterization, in terms of something called the distance function, which is a cool micro theory tool that no one learns anymore; the matrix of mixed partials of the distance function is called the Antonelli matrix, and it is a generalized inverse of the beloved Slutsky matrix.

If we wanted to think about other versions of p-complements, there are several options. One way is to hold income constant, and say that $x$ and $y$ are complementary if a decrease in the price of $y$ increases Marshallian demand for $x$. This is a valid notion (called "gross" complementarity rather than "net"), but it's not very nice because it's not symmetric (due to income effects) and hence doesn't have a mixed partial characterization.

Another, nicer way is to hold marginal utility of wealth constant (this is called "Frisch" demand, and is the consumer theory analog of profit maximization, which holds price of output constant), and then ask whether a decrease in the price of $y$ leads to an increase in the demand for $x$. This depends on entries in the inverse of the Hessian matrix of mixed partials of $U$, revealing an inverse relationship with (1) (which depends on the Hessian matrix itself) that parallels the inverse relationship noted above between the Antonelli and Slutsky matrices.


Must total world net exports be zero? On the Wall Street Journal Op-Ed page (Trump’s Real Trade Problem Is Money  ), John D. Mueller discusses the gold standard and claims something that I've either never heard before, don't understand, or both.

  Many economists wrongly assume that total world net exports must equal
  zero, but in fact countries participating in the international gold
  standard had combined net exports equal to the total increase in world
  gold reserves (which in turn approximated world gold exports). As a
  result, world monetary policy was countercyclical: When the prices of
  other goods fell, the profitability of gold mining rose.

First, I thought it was tautological that global net exports had to be zero. Is Mueller's claim correct? Does it turn on a technicality of the definition of net exports? Please help me understand his position.

Second, is it accurate to characterize gold standard monetary policy as counter-cyclical?  My understanding is that recessions in the gold standard era were marked by deflation where as in the fiat currency era recessions are merely dis-inflationary. If so, is it fair to call the gold standard substantially less counter-cyclical than fiat currency systems?

That depends on your model of the world, and on how you define "goods and services".

In a gold standard world, you essentially exchange goods and services with gold. Gold mining is therefore simply an expansion of the money supply. Adding gold to both the nominal and real side of the economy seems to me like today adding money as a good to GDP. If you do this, you get a positive trade balance. It makes a lot of sense to seperate monetary gold, which is on the monetary side, and commodity gold, on the goods side, as Henry mentioned in the comments.

In my opinion, it does not make sense to count gold twice. It seems reasonable to seperate the monetary and real side of the world, which then would yield a balanced world trade.

To the second question: I am no expert on gold mining, but I could imagine that it is not possible to "drive up" gold production in a recession. What is true is that a (nearly) constant money/gold supply leads to inflation being counter cyclical.


quasiconcave vs convex function I have been struggling trying to understand the difference between a quasiconcave and a convex utility function. As far as I understand a function can be both at a certain point, but is not clear to me why sometimes is said that the function is quasiconcave rather than just convex.

Can someone please tell me what are the implication of

Strict quasiconcavity implies single-peakedness, i.e. any strictly quasiconcave function has a unique supremum (or maximum if the domain is compact). Hence, any strictly increase convex function is also strictly quasiconcave.

Here are a couple figures to illustrate the difference. The function on the right is not quasiconcave because it has two local maxima.


What is the difference between Common Goods and Common Pool Resources? All this time I was thinking that common-pool resources are common goods, until I read this excerpt from a book.

Common pool resources can be depleted by overuse.

For example, an area of communal grazing land would be a common pool resource because grazing too many animals decreases the amount of grass available to each animal.

Wikipedia is a common good but are not a common pool resource - reading a lot of articles on Wikipedia does not reduce the number of articles available for everyone else to read.


In which cases prior to the Greek crisis did the IMF admit mistakes in it's economic policies? Has the IMF ever admitted mistakes on it's economic policies prior to the Greek crisis?

Yes.

  The International Monetary Fund yesterday admitted that its mistakes
  helped plunge Argentina deeper into the red during the currency crisis
  that crippled the country's economy three years ago.

  In a report published yesterday by its independent evaluation office,
  the IMF said it ought to have prevented the Argentine government from
  following poor economic policies.

  "IMF surveillance failed to highlight the growing vulnerabilities in
  the authorities' choice of policies and the IMF erred by supporting
  inadequate policies too long," it said.

IMF admits mistakes in Argentina crisis By Edmund Conway

The actual report is here: IMF's Independent Evaluation Office Announces Release of Report on the Role of the IMF in Argentina


Monetary policy takes medium run time to take effect. Why study it in short run? Well I'm reading a book on the IS-LM model.
At the end of the chapter, page 125 in my edition, the author presents one paper showing some graphs where the monetary policy (monetary contraction - increasing 1% of the federal funds rate) influencing output, but with its greatest impact only being felt after 4 quarters, and continues well into the 8th quarter.
At the same time, when the impact of the monetary policy is the greatest, is when the price level also begins to noticeably change.

According to the author, since in the IS-LM model we've assumed that the price level is given, the fact that the price level only begins changes at 4th quarter, is a good support for our assumption.

Well, my question then is, why should we study the monetary policy as if it had the same time to impact as fiscal policy in the short-run, when the graphs support more the medium-run impact? Or is it a matter of magnitude of the policy at hand?(The Central Bank can move the interest rate with a lot more leeway than just a 1%)

In an IS-LM model, expansionary monetary policy increases output and decreases the interest rate. Expansionary fiscal policy increases output and increases the interest rate, so the two types of policies are not equivalent.

Furthermore, monetary policy does not work in the long-run. See an AS-AD model for more details on that. The only reason monetary policy actually works, is because of the fixed prices and their slowly changing (increase) in the medium run. Otherwise inflation would rise immediately in response thereby making monetary policy not have any effect on real GDP.

To really get to the medium run you have to understand the short run first. Effects observed in the actual economy often are not as quick as in our models and take time. I wouldn't consider 4-8 quarters the medium run.

Hope this helps.


Why income elasticity of demand of luxury good in greater than 1? According to textbook and wikipedia, "if income elasticity of demand of a commodity is less than 1, it is a necessity good. If the elasticity of demand is greater than 1, it is a luxury good or a superior good."

Why was 1 chosen as the threshold?

I think that luxury good is consumed in larger proportion by relatively rich people. How does
income elasticity of demand is greater than 1 reflect this?

As for your first question: income elasticity of demand is just a percentage change in quantity demanded divided by a percentage change in demand. If you divide two things that are equal you get one: $\frac{a}{b}=1 \iff a=b$ (as long as $b \neq 0$). Same thing goes for income elasticity of demand, $1$ is not just some random value that was chosen to separate between goods. When $\%\Delta$ in quantity demanded is greater than $\%\Delta$ in income, then $\mathrm{YED}>1$ just as in terms of letters $a$ and $b$: $\frac{a}{b}>1 \iff a>b$ (as long as $b > 0$).

As for the second question: income elasticity of demand that is greater than one can be translated to "as the income grows, proportionally more is spent on luxuries". Which is exactly what you have in mind.


annualized quarterly growth rate calculating I was using R to calculate the annualized quarterly growth rate of real GDP by 400*diff(log(rgdp)).

If I perform in this way some of the percentage change is big.

However, if I calculate as 100*diff(log(rgdp)) than results seems to be accurate as what shows up on the most of the website.

Why is this happening? follow by the growth formula it should be multiplied by 400, not 100.

Perhaps quarterly growth should be multiplied by 400, but not annualized quarterly growth?


Why are most of open market orders so far from the mid price? Say there is an exchange with open buy and sell orders, why are there so many orders placed. So far away from the mid price. So for example is lowest sell order is say 1.01 and highest buy order is 1.00. It seems there is a trend that most of the open orders will reside away from the mid price. That is the total amount of orders at a given price is proportional to the distance from the mid price.

Three suggestions:

Spoofing: some people place buy and sell orders that they do not want executed (so away from the mid-price) to create a false impression of demand or supply , so that the market moves in their desired direction
Catching the flashcrash: occasionally the market has a blip with a temporary gross excess of supply or demand causing a volatile price shift which then returns to close to the original price when other players outside the market notice the unjustified price swing; leaving a bid some way away from the midprice can profit from such events providing that the price returns
Volatility survival: rather like opposing WWI trenches which got further apart over time as proximity became unsurvivable, short-term small-scale volatility will tend to pick off many of the bids close to the midprice, while leaving those further away untouched for longer periods of time

Market regulators tend to regard the first of these as bad (illegal in the US), the second as good, and the third as a natural though undesirable indicator of lack of liquidity


Why is matching function increasing in arguments? In matching theory, the matching function $m_t=m(u_t,v_t)$, where  $m_t$ is the number of matches(new hires) between unemployed looking for work($u_t$) and vacancies($v_t$).

This matching function is usually assumed to be increasing in its arguments, i.e. partial derivatives are positive.

What's the rationale for this?

Is there any data allowing us to conclude that it should be so?

Any help would be appreciated.

Think about it this way. The function is measuring the number of new matches being created. If we increase either the vacancies or the amount of unemployed, the function will generate more matches. Therefore, the partial derivatives are expected to be positive.

For example consider the matching function with something other than employment. If we consider a marriage game where we have a pool of single men and women, increasing either the amount of single men or the amount of single women, the amount of new matches created should increase. Note, that while increasing one of the two arguments might not actually increase the amount of matches, at worst it has no impact (such as in the edge case where we have only men and add another man). Since our function only considers new matches created in time T, increasing either argument should have a partial derivative of greater than or equal to zero. i.e. creating more matches.

Imagine now that instead of single men we have active job seekers (analogue to unemployment), increasing unemployment causes an increase in new matches.


Why are complementary goods defined as when a good's demand is inversely correlated with the price of another? Take a standard example: hamburger patties and hamburger buns. It is understandable that if the supply of hamburger patties increases (for whatever reason), the price of these patties will decrease, thus causing an increase in the demand for hamburger buns.

Now, let's say the price of hamburger patties decreases due to a decrease in demand (maybe doctors come out saying beef causes cancer). Why would the demand of hamburger buns increase when the demand of hamburger patties decreased? Standard definition of complementary goods (that I've seen so far) implies that the price decrease of hamburger patties would cause the demand for hamburger buns to increase, but that defies common sense.

Wouldn't a more sensible definition be that goods are complementary when the demand of one good is correlated with the quantity demanded of the other?

Why are complementary goods defined as when a good's demand is inversely correlated with the price of another?

We know that (in this model) the price of patties influences the demand of buns. This is because a cheaper patty makes the whole burger cheaper, which makes me demand more burger and therefore more bun .

  Why would the demand of hamburger buns increase when the demand of
  hamburger patties decreased?

Other things being equal, the price of patties influences the price of buns. The theory says nothing about how the price change came about. You implicitly assume that something changing patty demand would change bun demand, also. In the model world, there are only explicit assumptions.


Why $v_i=(X_i-\mu _X)u_i$ is i.i.d? I don't understand. Ok, we have $\beta_1-\hat{\beta }_1=\frac{\frac{1}{n}\sum_{i=1}^{n}v_i}{(\frac{n-1}{n}){s_{X}}^{2}}$. So, for the first OLS assumption results that $E(v_i)=E((X_i-\bar{X})u_i\overset{p}{\rightarrow}E((X_i-\mu_X)u_i)=E(X_i-\mu_X)E(u_i)=0$ (since $E(u_i|X=x_i)=0$). Then Stock and Watson says that for the second OLS assumption $v_i$ is i.i.d. and $Var(v_i)<\infty$. Surely this is for the casual sampling of variables but i can't prove it clearly.
After this, we have $E(v_i)=0$ and $Var(v_i)={{\sigma _{v}}^{2}}<\infty$ so we can apply CLT and saying that for $n\rightarrow \infty$ $v_i\sim N(0,{{\sigma _{v}}^{2}}/n)$ since $\frac{1}{n}\sum_{i=1}^{n}v_i=\bar{v}$.

Even after the comments exchange I am not totally sure what is the problem of the OP, but I will answer the question in the title.

Assume that we have a model

$$Y_i = \beta X_i + u_i,\;\;\; i=1,...,n$$

Assume that we obtain a "random sample" of observations from $\{Y_i, X_i\}$. This makes each observation $i$ identically and independently distributed from all other observations.

Now consider the random variable

$$v_i=(X_i-\mu _X)u_i = (X_i-\mu _X)(Y_i - \beta X_i) = X_iY_i - \beta X_i^2 - \mu_XY_i+\mu_x\beta X_i$$

while also

$$j\neq i : v_j=(X_j-\mu _X)u_j = (X_j-\mu _X)(Y_j - \beta X_j)=X_jY_j - \beta X_j^2 - \mu_XY_j+\mu_x\beta X_j$$

$v_i$ is a function of random variables $\{Y_i, X_i\}$ only, while $v_j$ is a function of random variables $\{Y_j, X_j\}$ only which are independent from  $\{Y_i, X_i\}$. So  $v_i$ is independent from $v_j$.

More over, since $X_i$ follows the same distribution as $X_j$, and $X_j$ follows the same distribution as $Y_j$, we have that $X_iY_i$ follows the same distribution as $X_jY_j$.Etc. So we also conclude that $v_i$ will have the same distribution as $v_j$.


How can we incentivize volunteer rescue forces like the Cajuan Navy which assisted in Louisiana? State-sanctioned militias in the state of Louisiana were an important part of the state's natural disaster response. Tiny boats went off into flooded areas to rescue people. These rescuers may have broken some laws but were allowed to do it because of the broader humanitarian interest they served.

Question: How to incentivize people to get a permit?
The incentives to get a license to become a member of a Cajun navy would be in times of peril---such as hurricanes--- to get to the disaster area quicker. But, this is not a great one. What could the state do to register people in times of non-crises? Tax-breaks? I am not sure

The United States Code provides for such a force and I believe Louisiana has one.  They are usually called "state defense forces."  They are a parallel state force with the state components of the National Guard.  They are not subject to call up by the President and serve in policing and emergency management roles in the states.  Not all states have them.  In some states they are both voluntary and require you to provide your own equipment, in others, they are fully funded.

I believe the Military Department of Louisiana has or had such a structure.  Nothing prevents the State of Louisiana from providing either pay or tax incentives to organize and manage the membership and to permit it to exist broadly.  Considering the frequency of natural disasters in Louisiana adding formal training and incentives to such a force would benefit all parties involved.

Imagine the effectiveness of the Cajun Navy if it had received formal training and worked within a planned response.  In most states, these are token forces primarily made up of ex-military who wish to continue service, but no longer wish to be in a potential combatant role.  They serve a small, but often important role, in mitigating the immediate effect of a disaster, but I don't think the states actually think of organizing any large number of people.  Most likely this is because they exist, but are so small that legislators forget they exist and don't expect much from them.


Is QE the equivalent of printing money? I heard somewhere that QE is not the same as printing money -- in fact as I know it, QE is the purchase of assets in exchange for more liquid and high-quality, better collaborated debt. In other words, it stimulates the exchange of liquid fiat, and indirectly increases the money supply in circulation (that is, money not in reserves) by inducing banks to lend, and therefore practice fractional reserve banking. So the analogy to "printing money" is a misnomer.
Is this explanation wrong?

It is correct that quantitative easing (QE) is not exactly the same as just printing money. However, I would hesitate to say equating it to printing money is complete misnomer (although its not accurate analogy either) and I also dont think that your description of QE is completely accurate.
QE actually begins with creation of new money. As explained by Fed St. Louis QE is defined as:

large-scale asset purchases—in the hundreds of billions of dollars range—of, for example, mortgage-backed securities and Treasury securities.

Moreover, this large-scale asset purchase is payed for by Fed creating new reserves. This creation of new reserves is in principle equivalent of actually physically printing out the bills even though it is done electronically.
The reason why the above should not be  equated to just printing new money (as explained here in this VOX article) is that:

asset purchases by the central bank are financed by money creation, but not money in the form of bank notes. The money is in the form of reserves held at the central bank. That form of money is a liability of the public sector – taking the central bank and the government together in an expanded definition of the public sector. It is a liability that pays the interest rate set by the central bank.

The above is why QE can be considered  an asset swap and why you can argue that the asset purchase just by itself is not money creation (although it indirectly encourages it) as it also reduces the money supply since Fed (or other central bank) is taking the money (in form of those bonds that can count as broad money) out of economy.
However, there is a caveat to this. The ultimate purpose of QE is to increase money supply which also happens when you print new money. The QE, if successful, does de facto increase the money supply when it encourages more lending. Even if the original swap of reserves for bonds might leave broad money supply unchanged the additional lending will increase it.
Thus this is essentially an argument about proper nomenclature/semantics. For example, lets say that in certain situation we know that if central bank would lower the interest rate by some percentage the corresponding increase in lending would make money supply expand by $\\\$ 1000$ and we know we could also do that directly by just printing out the $\\\$ 100$ bill which through multiplier (assuming $10\%$ reserve ratio) would also expand the money supply by $\\\$1000$. Hence, is it valid to call cutting interest rate printing money? Well certainly from narrow scientific semantic point of view that would be a misnomer as it would not follow proper economic nomenclature that makes distinction between the two actions. But this being said, since those actions would have equivalent effect I would not say that using such analogy is completely incorrect.
On a certain level this really is just an English language/semantics problem - If you go somewhere and later realize that you have to turn back and backtrack to the same spot would saying; 'well that is the same as staying still,'  be a good analogy or a misnomer? The actions have different implications in terms of effort etc. yet in both cases outcome is the same.  Moreover, before turning to economics I used to study physics which abounds with analogies that are far more outrageous yet commonly used in classroom (see this rant).
Hence to sum up, while calling QE printing money is not completely proper, the analogy has some merit to it.


Interpersonal comparison of utility What are the criteria which have been proposed to deal with the problem of interpersonal comparison of utility?

There are not just rules but whole theories. Namely cardinal utility allows you to make interpersonal utility comparisons (as opposed to ordinal utility). A common cardinal utility framework is the one based on  von Neumann-Morgenstern.
Von Neumann-Morgenstern cardinal utility requires that axioms of completeness, transitivity, continuity and independence are satisfied (see von Neumann-Morgenstern (1944) Theory of Games and Economic Behavior), so you can view them as 'conditions' for interpersonal utility comparison (although there are also other approaches to cardinal utility that might have different conditions).
Furthermore, note these rather necessary conditions for interpersonal utility comparisons not sufficient ones, as mentioned in the comments a good book that discusses this in greater detail is Roemer Theories of Distributive Justice.


Is there any pressure to keep currency exchange rates around 1? I notice that for any pair of currencies, the exchange rates seem to stay around 1 (as opposed to, say, 1,000,000). This is no longer the case if one country experiences an economic crisis that results in a very high inflation, but otherwise, rates don't seem to deviate very much from 1.
As you may have guessed I have very little knowledge in finance, so correct me if any of my assumptions are wrong, my aim here is to learn.
An example I'm thinking of is the Leu in Romania. It was revalued several times, the latest in 2005, when 10,000 old Leus were converted to 1 new Leu. That made the exchange rate with euros, for instance, closer to 1, given that 1 old Leu ≈ 0.00025 Euros, then 1 new Leu ≈ 0.25 Euros.
Why is this? Is this something most countries do? Does this principle have a name?

There are some counterexamples to the premise of your question:

A USD has been on the order of 100 Japanese Yen JPY for decades. By the way, 1/100 JPY used to be called "sen", and 1/1000 JPY used to be called "rin" (source: https://en.wikipedia.org/wiki/Japanese_yen#Introduction )

Bank notes:

Price tags:

A USD has been on the order of 1000 South Korean Won for some time.

Bank notes:

Price tags:

A USD is about 3500 Colombian Pesos (COP) now. Again, it's been in the thousands for decades.

Bank notes (note that they write "mil" in small letters instead of 000):

Price tags:

Note that $ denotes the peso, not U.S. dollars.

A USD is about 20,000 Vietnamese Dong (VND). Again, it's been in the thousands for decades.

Bank notes:

Price tags:

etc. These countries are not in any kind of economic crisis. The exchange rates are reasonably stable. The large numbers don't inconvenience anyone (except for some badly designed IT systems that can't handle very large or very small numbers accurately). These countries are all doing well.
Conversely, let us consider two countries at the opposite end from "doing well", whose currency is rapidly depreciating versus USD:

A USD is somewhere around 1,000,000 Venezuelan Bolivar (VES). Venezuela did redenominate its currency from "VEB" to "VEF" to "VES", over the last few years, dropping many zeroes. It did not help. The rapidly depreciating currency is one of the symptoms of their economic problems, not the cause.

A USD is somewhere around 100 Russian roubles (RUB). The rouble has been depreciating rapidly for years, especially after the West imposed painful sanctions to punish Russia for its aggression against Ukraine in 2014. Russia did already redenominate the rouble from "RUR" to "RUB" dropping 3 zeroes not long ago:

That did not help. Russia is still broke:

Russians looking for expired food.
Desperate Russian officials can't avert another imminent sovereign default, but are looking to delay the inevitable. They discuss redenominating their rouble again, dropping 2 more zeroes, and making 1 new rouble more valuable than 1 U.S. dollar, probably some variant of the Soviet "confiscatory redenomination" of 1947 (see https://en.wikipedia.org/wiki/Soviet_ruble#Fifth_Soviet_ruble,_1947%E2%80%931961 ). This won't help. Except for stroking Russian national pride, I fail to see the benefit of having a currency unit worth more than 1 USD.
Let us therefore look at some examples of currencies whose 1 unit is worth substantialy more than 1 USD.

A Kuwaiti dinars (KWD) is somewhere around 3 to 3.5 USD . This does not pose any inconvenience when a subunit of the currency is small enough. A 1/1000 fraction of KWD, worth about 1/3 of US cent, is called "fil". Here is a 1/2 KWD (500 fil) bank note:

Price tags:

This bottle of water costs 0.055 dinars, i.e. 55 fils, which is about 16 U.S. cents.

Chile widely uses an inflation-adjusted currency en.wikipedia.org/wiki/Unidad_de_Fomento - 1 unit of account is worth about 40 USD. It is not available as cash (bank notes or coins), but is primarily used to denominate real estate prices, stock prices, loans, etc.

1 Bitcoin (XBT) is worth thousands of dollars as of this writing. Conveniently, one hundred millionth of a bitcoin is called a satoshi and can be used in transactions.

Back when the U.S. dollar had a lot more purchasing power than it has today, prices commonly used "mill" (1/1000 USD). It's still perfectly legal (see https://en.wikipedia.org/wiki/Mill_(currency) ) and is often seen as the "9/10" on U.S. gasoline prices:

Contemporary gasoline prices.

Antique gasoline price sign: the gas costs 10.3 cents and the tax is 1.7 cents, i.e. 17 mills.

Also the Euro (EUR), the British pound (GBP), and the Swiss franc (CHF) are all worth more than 1 USD (but usually less than 2 USD).

A good (but not quite complete) list of recent redenominations can be found in Wikipedia https://en.wikipedia.org/wiki/Redenomination#List_of_currency_redenominations . As you see, just swapping 1,000 units of the old currency for 1 unit of the new currency changes nothing. However in the 1947 Soviet confiscatory redenomination cited above, if the state owed you 10 old roubles before, then the state now owed you only 1 new rouble. But if you owed the state 10 old roubles, then you still owed 10 new roubles to the state. This may be the kind of scheme that Russian officials are contemplating again.


Incentive compatibility I have a set of objects $A=\{a_1,a_2,a_3,\cdots,a_m\}$ and one player, who proposes a vector of valuations $t=(t_1~t_2~t_3~\cdots~t_m)$ for the $m$ objects. $T$ is the set of all such valuation types. Given $T$ I design an allocation function $$f:T\rightarrow [0,1]^m$$ and a payment function $$p:T\rightarrow \mathbb{R}$$
The mechanism $(f,p)$ is said to be Incentive compatible if $$t(f(t))-p(t)\geq t(f(t'))-p(t')~~\forall t,t'\in T~~~~~~~~~~~~(1)$$
One can easily derive from the above eqaution that incentive compatibility implies the following: $$t(f(t))+t'(f(t'))\geq t(f(t'))+t'(f(t))~~~~~~~~~~~~~~~~~~(2)$$
that is, $(1)\Rightarrow (2)$.

Is the converse ($(2)\Rightarrow (1)$) true?

First of all, the general form of the problem you got there is extremely demanding. In multidimensional screening problems, analytically often all hell breaks loose in a sense that it is just not tractable. One way out might be this recent approach by Gabriel Carrol: Robustness and Separation in Multidimensional Screening (also provides an introduction that hints at the intractibility of the problem).

Coming back to your question: $(2)$ does not imply $(1)$, because
$(2)$ only depends on $f$ and you can add a transfer rule that is not IC.
As a counterexample, consider a single good, $m=1$.
Take some strictly IC mechanism, $(f, p)$ and consider two types, $t$ and $t'$. So, $(1)$ is
$$U(t):= f(t)t - p(t) > f(t')t - p(t')\\
U(t'):= f(t')t' - p(t')> f(t)t' - p(t)$$
and by construction $(2)$ holds as well.
Let $t'>t$ and a lager allocation probability $f(t') > f(t)$ is accompanied by a larger expected transfer, $p(t') > p(t)$ - otherwise $t$ would love to fake being $t'$.

Now, construct $\widetilde{f}(s) = f(s)$ for all $s\in T$ and let $\widetilde p (t)= p(t')$ and $\widetilde p (t')= p(t)$. Since $(2)$ only depends on the allocation function $f$ and these functions are the same in both mechanisms, $(2)$ holds for $(\widetilde{f}, \widetilde{p})$.
Obviously, $(1)$ is violated for the reason named above:
$$\widetilde f (t') t - \widetilde p(t') > \widetilde f (t) t - \widetilde p(t).$$

To address your other problem (in your comment), use the integral formulation of expected utility.
Rewriting $(1)$ yields,
$$f( t)(t-t'))\geq  U(t) - U(t') \geq f (t' )(t-t')$$
implying that $U$ is Lipschitz continuous, implying $U$ is differentiable a.e., and equals the integral over its derivative:
$$U (t) = U(\underline t) + \int_{\underline t}^t f(s) ds$$
where $\underline t$ is the lowest possible type.
Then, you rewrite,
$$f (t) t - p(t) = U(\underline t) + \int_{\underline t}^t f(s) ds \\
f (t) t  - U(\underline t) - \int_{\underline t}^t f(s) ds = p(t).$$
Suppose $\underline t=0$, then
$$p(t) = p(0) + f(t)t - \int_{0}^t f(s) ds.$$
This implies the revenue equivalence theorem: If two auctions have the same allocation rule, payment functions (and thus revenue) can only differ by a constant. The same trick works for multi-unit auctions with multi-unit demand, see, e.g., Krishna, Ch 14.


Linear programming, shadow price range I'm not sure how to determine the range for which a shadow price is valid.

You might be able to skip straight to the question here.

I've been introduced to it using the following approach in 2D.

Context

Given an optimal solution which is at a corner, there are two intersecting
lines. Say that these lines represent the following inequalities

\begin{equation}
  \begin{aligned}
    2K + 3S & \leq 10 \\
    K + 2S & \leq 6 \\
    Z & = 30K  +  50S
  \end{aligned}
\end{equation}

Then the shadow price is the change in the objective function, $Z$, when the
right hand side of an inequality is changed by one unit.

Computing this for the second equation is done as follows:

\begin{equation}
  \begin{aligned}
    2K + 3S & \leq 10 \\
    K + 2S & \leq 6 + \Delta\\
  \end{aligned}
\end{equation}

Then subtracting the second equation from the first twice

\begin{equation}
  \begin{aligned}
    S & = 2 +  2 \Delta
  \end{aligned}
\end{equation}

Therefore

\begin{equation}
  \begin{aligned}
    K &= 2 - 3 \Delta
  \end{aligned}
\end{equation}

And the objective function can be written as

\begin{equation}
  \begin{aligned}
    30K + 50S &= 30(2 - 3 \Delta) + 50(2 + 2\Delta) \\
    &= 160 + 10 \Delta
  \end{aligned}
\end{equation}

And the shadow price is found from $z(1) - z(0)$, where $z(\Delta) = 160 + 10
\Delta$ which gives $10$.

Therefore the shadow price is 10.

But how do I compute the range for which this is valid? I can get through the
algebra here pretty easily, but there's little meaning to it.

Question

How do I determine the range for which I can adjust the second inequality such
that the difference made to the objective function is $10 \times
\text{adjustment}$.

The shadow price is formally not the increase in the objective function for relaxing a constraint by a single unit, but by an infinitesimal relaxation. In the world of linear programming it can be valid for up to a unit but it doesn't have to be. In your example problem it is actually not.  This becomes clearest when we use pictures.

I am going to assume that you were maximizing, not minimizing your objective function and furthermore that you have two additional constraints that you hadn't actually stated:

$K\geq0$ and
$S\geq0$

That would result in the following graphic with K on the y-axis and S on the x-axis, and the optimum at the red dot. The black dotted lines are isoquants of Z, for Z=100, Z=150 and Z=200.

The feasible region is made up of the x and y axis, the blue line from the y-axis up to the crossing, and the orange line from that point on to the x axis.

Let us now relax your constraint $K+2S\leq6$ by one unit, as you propose. In essence this means that the constraint becomes $K+2S\leq7$ and when we draw the new constraints we get:

Turns out that the new constraint no longer plays a role in determining the optimum! The optimum is now determined by the constraints: $S\geq0$ and $2K+3S\leq10$.

The answer to your question should now be clear: the shadow price is valid up to the point where that particular combination of constraints continues to determine the optimum. In this case that means that you can increase 6 to $6\frac{2}{3}$, for a corresponding increase in the objective function of $10*\frac{2}{3}=\frac{20}{3}$  After that point the combination of constraints no longer determines the optimum.

As a side point let me add that at least the Excel solver (and I expect therefore many other solvers too) report up to which point you can relax the constraints before the optimum is determined by another set of constraints.


Since 1980, in which years has the US debt-to-GDP declined and why? In other words, what were the factors in the reduction of the debt-to-GDP ratio?  Why did the ratio decrease during some booms and not others?

The time series for the debt-to-GDP ratio are available on the FRED wesbite: link to debt-to-GDP chart. I will let the reader download the data, rather than list the years here.

As for why it falls, it requires a good macroeconomic model. The reason being is that we need to explain both the fiscal deficit and the rate of nominal GDP growth.

Policymakers do not directly control the deficit. Faster growth means a greater tax take, less social spending. Meanwhile, capital gains taxes are determined by the hard-to-forecast changes in stock prices.

Nominal GDP (the denominator) can grow as a result of faster inflation, or faster real growth. The precise determinants of this have bee the subject of economic debates for decades. I obviously cannot answer it here.

One factor to keep in mind that the existing level of the ratio matters a lot. If nominal GDP growth is 5%:

you need a fiscal deficit of 5% of GDP to keep the ratio unchanged if the starting level is 100%,
you only need a deficit of 2.5% if the starting level is 50% of GDP.

If you look at the chart, the biggest fall in the ratio after 1980 was during the late 1990s. There was a brief fiscal surplus, and growth was relatively strong. There was a technology boom, along with the tech stock bull market/bubble that generated a lot of capital gains taxes. (The Clinton administration raised taxes earlier, which was also a factor in the surplus.)

(In a comment, you asked about the 2000s. During that period, the ratio rose, but that can be partly traced to the increased defense spending that happened. Once again, there’s too many moving parts in the budget and economic growth to give a simple explanation of how the ratio moved.)


What was the Summers memo of 1991 sarcastic about? I have read a twitter thread about Summers memo (wikipedia article) today, and trying to understand the last part (bolded).

  The problem with the arguments against all of these proposals for more pollution in LDCs [Least Developed Countries] (intrinsic rights to certain goods, moral reasons, social concerns, lack of adequate markets, etc.) could be turned around and used more or less effectively against every Bank proposal for liberalization.

So, if I understand, moving toxic production to LDC to decrease world-wide cost of healthcare is a bad thing due to some argument. But there are policies that are deemed to be good and accepted by World Bank, that can be struck down using the same argument?

Question: Which policies are those? Which "Bank proposal for liberalization" does the memo refer to?

The wiki article says also that

  Summers had only signed it, and that it was intended to be "sarcastic"

But I am not sure toward what policies or arguments this sarcasm was pointed.

The context of this memo was an emerging debate about structural adjustment and its environmental implications. The World Bank has attached a wide range of liberalization policies to many of its loans. These policies were meant to increase trade and foreign investment and to decrease fiscal deficits, all of which should help countries to pay back their loans. Critics argue that imposing these forms of liberalization on LDCs forces them to prioritize repayment over other policy goals, including environmental protection.

Here is a 2003 review on environmental impacts of structural adjustment by Bank researchers which briefly mentions and dismisses the "pollution haven... hypothesis that 'dirty' industries migrate to low-income countries after trade liberalization because of differences in costs of pollution abatement and looser environmental standards (p. 27)."


Why does anyone prepare their own food? I'm not an econ person, so this is a super naive question, but here goes: why, given the comparative advantage that specialization produces, does anyone prepare their own food rather than buying from restaurants all the time? It seems like there should be economies of scale with respect to food production (restaurants prepare food far more efficiently than I can do at home). Where I live, it's often cheaper to prepare your own food than ordering take out, but I can't understand why.

The weights used in the U.S. CPI (from 2016) are available here: link to BLS weights. Note that these weights are based on consumption data, so that they are meant to align with actual spending patterns.
If you look at them, you will see that the majority of the weight is either processed food or food away from home. Note that some categories explicitly state “processed,” but even things like salad dressing are put together in a factory.
Factories have a comparative advantage for processing food, but not for delivering hot food. Meanwhile, food is normally shipped in a form that can be stored. A box of pasta can be held in a store a lot longer than pasta that has been boiled and put into a container. Households do the final heating steps. As such, the premise of the question is somewhat flawed - processing is being done largely in an “efficient” fashion (although there are debates about food quality).
Restaurants are required to pay workers and rent and expect to make a profit, whereas food prepared (or just heated up) requires no payment. Cooking is labour intensive, and so it is hard to compete against \$0/hour if we look at the cost of the meal. Individuals might value their time at more than \$0/hour, but how they value their time does not matter if they cannot get more paid work and face a strict monetary budget constraint. Meanwhile, travelling to get a meal takes time. Finally, being reliant on food away from home poses risks, as seen in the pandemic.


How does capital investment (factory, plant, etc.) benefit the region they are located in? Generally, capital investment is always good for the economy, but I am not sure about the underlying mechanism. For example, if a car factory is built in a certain state, does it improve the economy solely by creating jobs and generating car sales thereby increasing the regional GDP?

I’ve also another related question. I often read that states in the U.S. frequently fight for large capital investment in their respective regions from a certain company, I’m assuming it would boost the economy tremendously. However, if a factory (lets say it’s located in state A) ONLY ships and sells its products to state B, would it improve state A’s economy as much as if sales were only made in state A?

My hypothesis is that the factory in would still improve the economy in state A due to the creation of jobs and its overall costs of production (purchasing from other businesses in the area), although it wouldn’t boost the economy as much as it would in the other scenario.

Please correct me if I happen to have said anything wrong.

There are more effects for economy then just what you mentioned.

  For example, if a car factory is built in a certain state, does it improve the economy solely by creating jobs and generating car sales thereby increasing the regional GDP?

Not only because of this there are also other effects. Depending on what kind of investment it is it can support technology transfer between the countries or also increase stock of human capital (through on job training/learning by doing), which according to endogenous growth theories can accelerate economic growth itself (although it is worth noting endogenous growth theory is still not completely accepted by profession even though Romer who developed it even received Nobel Prize for it because it’s hard to test).

Moreover, apart from such dynamic growth effects there are more beneficial macroeconomic effects on GDP. From macroeconomic perspective all spending including investment has a multiplier attached to it. For  when the factory increases GDP by the investment people will also have more money to spend increasing GDP even further and so on. From macroeconomic perspective your spending/investment is someone’s else income so if you increase your spending/investment other people will spend more as well.

Also aside from macroeconomic effects any new job openings put upward pressure on wages, however depending on how many jobs are offered and what is already situation in the market this effect might be very small.

  if a factory (lets say it’s located in state A) ONLY ships and sells its products to state B, would it improve state A’s economy as much as if sales were only made in state A?

Not really as measured by GDP. The GDP identity is given by:

$$GDP= C+I+G+X-M$$

Where C is consumption I investment G gov. spending, X export and M import.

So even just looking at the identity it is possible to see that if the cars are not consumed but exported GDP does not change.

However, there is even more to it. Trade theory in economics tells us that there are both static and dynamic gains from trade. Hence to the extent that the cars are exported because in state A they can be produced more efficiently then in state B, and in exchange state B trades to state A some of its products that they can produce more efficiently there will be both static one time increase in the level of their welfare from specialization, and also potentially some dynamic effect assuming that the longer the state specialize in the good they have comparative advantage in the better they become at its production.


Alternatives to avoid sanctions Recently we saw that the EU and Iran devised an instrument called INSTEX to avoid dealing in U.S. dollars and, hence, avoid U.S. sanctions. Similar thing is being planned between India and Russia (regarding the S-400 deal) to avoid U.S. sanctions under CAATSA.
I don't understand how changing the currency under the deal circumvents the problem. Isn't the whole problem the deal itself rather than the currency? The U.S. can still impose sanctions if it gets to know that a deal has happened. Then, how does this INSTEX work?
Sorry if I posted it on the wrong platform. I don't know where to go for such international, political and economics' questions.

You have most of the explanation here - the only missing piece is that as the de facto reserve currency of the world, almost all global trade is conducted in US dollars.  Typically what happens is that a buyer will convert local currency to \$USD, the \$USD will be exchanged in the transaction, and then the seller will convert that \$USD to their local currency.

There are two points in that chain where the United States can intervene - when the buyer converts to \$USD, and when the seller converts back to local currency.  The US' ability to impose credible sanctions rests on its ability to exert this influence, and historically it has had it by way of the \$USD being the reserve currency for the world.

In principle, nothing has ever stopped other nations from circumventing this mechanism.  What has made $USD desirable as a reserve currency has been the relative political and financial stability of the US economy, which allows the \$USD to act as a store of value.  The relative part is important - in a global exchange economy mediated by fiat currencies, there really are no non-currency safe-havens.  Every boat on the ocean rocks in a storm, but some rock less than others.  It is better to weather a bit of sea-sickness than to jump to a boat that may capsize.  US-imposed sanctions reflect a belief that nations would rather capitulate to US demands than skip over to the next boat.  For much of the globalization era this has been true for most nations.

However, there has been cause to question this relative stability in recent decades (especially since 2009) and so it's very telling that we are seeing these agreements.  They indicate that to some nations, whatever benefits have historically flowed from transacting in $USD are viewed as no longer being present.  Whether this is good or bad is open to interpretation - after all, it should be expected as other large economies "catch up" to the US in terms of economic activity, stability of institutions, etc.


Can we use hotelling's lemma as a rule of thumb when wage rate can vary? Disclaimer: I've been doing some simulation with excel to see how Hotelling's Lemma behaves when allowing another parameter to vary. I did this by producing 150 randomly generated profit functions (based on normal distributions) and allowed them to vary in the parameters of interest. The goal is to learn about how far I can apply use of Hotelling's lemma outside of ideal conditions (like when wage varies as well).

Recall that  what Hotelling's Lemma tells us (essentially) is that the change in profit from a change in price is proportional to the quantity produced,meaning that the profit function increases/decreases linearly when price changes holding all else fixed. This is exactly the  result I got  when simulating it.

.

I then asked If we allow wage rate to vary as well, I found some neat and messier structures of how Hotelling's Lemma changes. In this case the dermination of price and wage are sampled from a normal distribution such that $\mathcal{N}(5,2)$ and inputs and outputs are sampled from a normal distribution such that $\mathcal{N}(50,25)$. These are my results:

Though, this method is (clearly) unscientific and violates the strict definition of the lemma, can we use hotelling's lemma as a rule of thumb when variability in wage rate is small?.

Display of excel code/setup available upon request.

I don't quite know what it means to apply a mathematical result to a situation in which the assumptions are violated, but the same idea as in the classical version of Hotelling's lemma can be used to understand the marginal effect of wage changes and you can combine the two to understand what happens on the margin if you change both prices and wages.

The underlying logic of Hotelling's lemma was that for a profit-maximizing firm, the marginal effect of adjustment outputs (and therefore inputs) is zero. This tells us that the marginal effect of wage changes on profits should be proportional to inputs used, that is
$$\frac{\partial \pi(p,w)}{\partial w}=-x(p,w),$$
with $x(p,w)$ the optimal input bundle. This is indeed true and the proof is exactly the same as for the marginal effect of $p$, just replace the derivative with respect to $p$ by the derivative with respect to $w$. If $f$ is the production function, the profit function is given by
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$
Taking the derivative with respect to $w$ now gives us
$$\frac{\partial \pi(p,w)}{\partial w}=pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial w}- x(p,w)-w\frac{\partial x(p,w)}{\partial w}.$$
The first order condition for $x(p,w)$ being a maximizer of $pf(x)-wx$ implies
$$pf'\big(x(p,w)\big)- x(p,w)-w=0$$ and therefore
$$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial w}-w\frac{\partial x(p,w)}{\partial w}=0.$$
Substituting this in, we get
$$\frac{\partial \pi(p,w)}{\partial w}=-x(p,w).$$

The envelope theorem generalizes the underlying idea that marginal benefits of adjustments for optimizing agents are zero for more general parameterized optimization problems.


Finding Bayesian Nash Equilibrium I'm recently new to Game Theory and I've recently started teaching myself about Bayesian Nash Equilibirum. I've stumbled across a problem set that I can't seem to wrap my head around concerning Bayesian Nash Equilibrium.

The problem set is shown below:

For question 3, I initially tried to solve the first problem using Mixed Bayesian Nash Equilibrium but that doesn't make sense since both Player 1 and Player 2 have weakly dominated strategies, so why would they mix? Also when I combine the matrices I find no Pure Strategy Bayesian Equilibrium.

Is there something I'm missing here? How would I go about solving this question? Some example's would be very helpful

For question 4, I have no idea where to even start solving this question. I haven't come across any questions or tutorial on how to solve for Bayesian Nash Equilibria when BOTH players have don't know what game they're playing. Any and all help on how I would solve this problem would be greatly appreciated.

The Key to BNE is that players that know something (about the state of the world or their type) can condition their strategies with their information. That is, for example, in question 3, type A could choose strategy U, while type B could choose strategy D. Therefore from the second player's perspective, there are four possible pure strategies of his opponent. Namely UU, UD, DU, and DD. Which represents the actions of each type of player 1.

What you want to do is construct another table where player 1 has these four strategies and player 2 has strategies L and R whose payoffs are the expected payoff. For instance, in the cell (UU, L) the payoff to player 2 is $0.5(4)+0.5(2)=3$, and for (UU, R) it has a value of $0$. This table is only useful to asses player 2's payoffs since player 1 knows his type, so no need to fill a number for player 1. After constructing the table you realize that player 2 has a weakly dominant strategy (L). You have to go back and forth between the two tables (for type A and B) and the third table for player 2 to find all the fixed points.

For this game, there are several equilibria; so maybe that is throwing you off. For example, suppose player 2 plays her weakly dominant strategy, L,  in that case, both types are indifferent between U or D, so they both could choose U. Given the strategy UU, player 2's unique best response is L, so this is an equilibrium. I.e. (UU, L) is an equilibrium.

Try to convince yourself that (DD,R), (DD,L) are the other equilibria, and there are no more equilibria.

As for question 4, it is actually simpler. Since no player knows the state of the world, they cannot condition their actions on the state of the world. so the pure actions for player 1 are U or D and for player 2 are L and R.

You will have to construct another table where players average out the payoffs of each of the four possible combinations of actions. For example for the cell (U,L), the payoff for player 1 is 1 and for player 2 is zero. Similarly, the payoffs on cell (U,R) is $(0,0.5)$. After constructing the new table you can find the BNE by the usual algorithm to find standard simultaneous-move NE.

Try to convince yourself that there is a continuum of equilibria for this game. In all of them, player 2 chooses $R$ since it is a dominant strategy, and player 1 is indifferent between both actions so chooses any mixed or pure strategy (which technically speaking is a  type of mixed strategy too).


Change in money supply Suppose a bank makes a loan of $1M to someone taking out a mortgage to buy a house by crediting that person's bank account with a deposit equal to the size of the mortgage. Suppose the required reserve ratio is 10%.

a)How much has the money supply increased by?

b)Does reserve or high powered money change? Why or why not?

What I did: $\frac{$1M}{0.10}$ = $10M (this will be the increase in money supply)

Because the reserve ratio is 10%, the reserve money changes by 10% of 10,00,000 = 1,00,000

Is this correct?

The person taking out the 1 million dollar mortgage will take the funds and use them towards buying a 1 million dollar home. If the home is sold by Joe who keeps his money under the pillow then in this scenario money supply does not increase at all. If, on the other hand, the party selling the home will deposit the 1 million dollars in his/her bank account with bank A, assuming a 10% reserve ratio, bank A will keep 100,000 dollars as reserves and lend out the remaining 900,000. Bank A lends these 900,000 to Bob, who will take them and deposit them with Bank B. As long as Bank B will again keep 10%, 90,000 in this case, and lend out the remainder - 810,000 to some other customer and if this chain continues in this fashion with no one keeping money under the pillow, then the overall money supply will increase by 1,000,000 + 900,000 + 810,000 + ... = 1,000,000/0.1 = 10,000,000. The banks will end up increasing their reserves by 100,000 + 90,000 + 81,000 + ... = 100,000/0.1= 1,000,000.


CES Production Function with $\rho>1$ In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho} $, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho>1$, the production function won't be concave anymore (and hence production set will not be convex), but what does that imply about profit and cost functions?

The problem with $\rho>1$ is that it means the marginal product of factors is not decreasing ($\rho<1$) or constant ($\rho=1$) but increasing, which is an odd assumption. Such functions yield isoquants that are concave, and might lead to only one factor being used (as BKay said).

As in any generic CES, the marginal product of factor $x_i$ is

$$ MP_i = \left(\frac{y}{x_i}\right)^{1-\rho} $$

The derivative of this MP with respect to $x_i$ is, after some rearranging,

$$ (\rho-1) \left(\frac{y}{x_i}\right)^{1-\rho}\left(\frac{x_{-i}}{x_iy^{\rho}}\right) $$

For $\rho>1$, this expression is positive, which means that the productivity of a factor increases as more of that factor is used.

Regarding isoquants, you can find these by rewriting the production function as $x_2=g(y,x_1)$. In the generic CES, this is

$$ x_2 = \left(y^{\rho} - x_1^{\rho}\right)^\frac{1}{\rho} $$

These are linear in the case of $\rho=1$, convex in the case of Cobb-Douglas (where the function above is $x_2=\frac{y}{x_1}$, a hyperbole), and concave in the case of $\rho>1$. For example, select $\rho=2$ and you have:

$$ x_2^2 = y^2 - x_1^2 $$

which is the formula of a circle centered at $(0,0)$, with radius $y$. Normally, for production theory only $x_i \geq 0$ is interesting, which gives you the concave isoquants for different levels of $y$. The figure below shows an example, were for a given factor prices ratio, there is a corner solution (point A):

$\hskip3cm$

(Code for reproducing figure here)


Solving a HJB with a probability to transit to a new state I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time.
The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}=i_t-\delta k_t$. There is a probability $\rho$ per unit of time to transit to a new state where a tax is imposed, and where the profit becomes $f(k_t)-i_t-\tau k_t$. This is a partial equilibrium problem and we assume that $r$, $\delta$ and $\rho$ are exogenous.
I am trying to solve this problem using Hamiltonian-Jacobi-Bellman (HJB) functions:
$$\begin{align*}
rV_1 &= \max_{i} \{f(k_t)-i_t+\rho (V_2-V_1)+\dot{V}_1\} \\
rV_2 &= \max_{i} \{f(k_t)-i-\tau k_{t}+\dot{V}_2\}
\end{align*}$$
I know how to solve the second equation, following the method of Walde 2012 using dynamic programming. First, rewrite $$f(k_t)-i_t-\tau k_t + V'_2(k_t)\dot{k_t}=f(k_t)-i_t-\tau k_t+ V'_2(k_t)(i_t-\delta k_t)$$ Then, take the FOC with respect to $i_t$, which yields
$$V'_2(k_t)=1$$
Then, use the envelope condition to find
\begin{align*}
rV'_2=f'(k_t)-\tau+V''_2(k_t)(i_t-\delta k_t)-\delta V'_2(k_t)
\end{align*}
This can be simplified, using the FOC, as$$\begin{align*}
f'(k_t)=r+\delta+\tau
\end{align*}$$
which yields the same solution as a simple Hamiltonian would.
However, I am unsure how to proceed next and how to solve for the solution before the tax is imposed and the uncertainty resolved.
Additionally, if you have some references regarding dynamic control in continuous time, I would be very interested, especially if they treat the case of HJB with additional constraints.
Thank you in advance for your help!
EDIT: for clarification, once we reach the new state where the tax is imposed, there is no possibility to go back to the previous state. The only uncertainty is about when the tax will be imposed, i.e, when the uncertainty will be resolved.

I would leave this as a comment but I cant. You are on the right track.

Once you know $V_2(k)$ then you can plug that into to the first hjb and solve.

To solve for $V_2$ you need to find the optimal $i$ as a function of $k$. Then plug $i(k)$ into the 2nd HJB. That will give you a second order ode. Solving that will give you $V_2(k)$ and you go to 1.


Has there ever been a first world recession where inequality didn't fall? Gini index fell everywhere in great depression, great recession, the 80s recession, and so on. Has this ever not been the case? Have any countries not seen inequality fall in recession?

What you suggest can easily be rationalised, if those on fixed incomes (such as pensioners) tend to be poorer while those with economically sensitive incomes (such as workers and shareholders) tend to be richer and see their incomes more adversely affected by recessions.

But I am not convinced it is true in every case. The UK recession in 1980 may be an example, looking at this graph from the Office for National Statistics and I would expect there to be examples for other countries too.


Economic Overview of Pacific Island Nations What are the key drivers of growth for these small pacific island nations? I would expect tourism and natural resources to be pretty central to these economies. Is there anything else? What are the key barriers to growth here? I would imagine tyranny of distance and small populations couldn't help.

I am interested in answers which related both to macro and microeconomics.

I would be happy with any references pointing me in the right direction as well.

As a group, there are similarities. They are all small and remote island economies. None have a population over 1 million people. Fiji is the largest with a population of roughly 900k. It is important to remember that they are still different countries and it can be unhelpful and misleading treating them all as identical.

The Pacific island region has considerable potential for development, especially in the areas of tourism, fisheries, forestry, mining, and agriculture.

Challenges

Small size
limited natural resources (diversity and capacity to capitalise on)
narrowly based economies (small populations and isolation limit capacity to diversify production.)
large distances to major markets
vulnerability to exogenous shocks, can affect growth and have often led to a high degree of economic volatility.
They are some of the most vulnerable nations in the world to disaster. In particular, climate change and rising sea levels.

Long term viability of economies has been questioned. Fiji is the only one approaching independence from aid.. Many Pacific Island economies have found it difficult to achieve sustained economic growth and hence sustained growth in living standards

See this link for a brief overview by the world bank.


Why does a decrease in interest rate reduces the velocity of money circulation? I have seen sources that claim high interest rates increase the velocity of money circulation, but haven't really seen any concrete explanation for this. This is how I think it works: High interest rates -> high opportunity cost of holding money -> Demand for money decreases -> people dont want to hold money and will spend it -> higher velocity of income.

On the other hand,wouldn't a high interest rate make saving more attractive and thus reducing the velocity of income circulation?

EDIT: Sources that apparently claim this are referred to below:

This source does not explicitly state the relationship but somewhat explains the causes: http://thismatter.com/money/banking/money-demand-money-velocity.htm

A wikipedia page that seems reasonably credible states this relationship, but I am unable to fully understand their explanation:
https://en.wikipedia.org/wiki/Velocity_of_money

This exam question from Cambridge International Examinations that explicitly states the relationship (the correct answer is D)

You are absolutely right. Since higher interest rates increase the opportunity cost, you wan't to get rid of it faster (at the margin).

The core of the argument is that the cost stems not only from the amount of money holdings, but also from the duration. Say you are constrained about the amount of money holdings you need in your daily life, but you can still make sure the money is not idle for too long.

For the second part of your question, you are mixing up concepts. The interest rate influences the Consumption/Savings Decision, but it is the same decision problem as in a (model) world without money, i.e. it is a choice between real variables, whereas money velocity is a monetary question.


Why are firms taken to be profit-maximizing? Shouldn't that make them risk-neutral? Intro texts normally explain that insurance firms (casinos, etc.) "work" by diversifying risk from many clients. Unsaid, then, seems to be that risk is bad for both firm and client.

But why should a firm even need such diversification, if they merely seek to maximize profits? Likewise why would a firm ever itself be a purchaser of insurance, or spend resources trying to "manage" risk? This seems to be a rather deep oddity in the basic picture we're taught from square one.

Why would a [risk-neutral] firm need to diversify if all it wanted to do was maximise profit?

Suppose there is a risk-neutral firm that has two strategies it could follow: risky and safe. The safe strategy (e.g. holding a diversified portfolio) yields a profit of $1$ every year forever. The risky strategy (e.g. not diversifying) yields a payoff of $2$ with (independent) probability $x$ each year. With probability $1-x$ a risk-taking firm goes bankrupt and earns no more money ever again. The firm discounts the future at rate $\delta$.

The lifetime profit of the safe strategy is
$$\sum_{t=0}^\infty\delta^t=\frac{1}{1-\delta}.$$
The lifetime profit from the risky strategy
$$\sum_{t=0}^\infty 2x^t\delta^t=\frac{2}{1-x\delta}.$$
The safe strategy yields a higher lifetime profit if
$$x<\frac{2\delta-1}{\delta}\iff \delta>\frac{1}{2-x}.$$

In words, even though

the firm is risk-neutral and
the one-shot profit from the risky option (e.g. not diversifying) is higher,

the long-run profit from the safe option (e.g. diversification) can be greater over the lifetime of the firm.

In this example, the safe option maximises lifetime profit when (a) the probability of risky option leading to bankruptcy (and hence the loss of all future profit) is high or (b) the firm values the future highly relative to the present.


Difference between LMSR and Liquidity-Sensitive LMSR What is the difference between the LMSR (logarithmic market scoring rule) and the Liquidity-Sensitive LMSR (LS-LMSR) and what would cause prediction markets like Augur to focus on the later model?

The LMSR has a number of parameters which governs its function, but the LS-LMSR focuses, in particular, on the sensitivity function, AKA parameter b. This parameter determines how sensitive the market maker is to incoming orders. The higher the value, the less sensitive the market maker is, and the slower the price will change, because it requires more liquidity to increment the price.

It's up to market operators to set b appropriately, taking into account the amount of liquidity that's expected. However, this can be difficult to manage, as traders enter and leave the market. LS-LMSR outlines a programatic way to dynamically asses conditions and change the LS-LMSR accordingly, setting the value higher as the market thickens, and vice versa. Parameter b can be changed by the operator whenever the need arises, but LS-LMSR alleviates some of that management burden.

It probably would be a good idea for Augur and Gnosis to implement a dynamic way to manage liquidity, since relying on manual edits impedes scaling to large numbers of markets.

The LS-LMSR also describes a way to run the market maker at a profit. In many instances, operating at a loss isn't such a bad thing, at least to a pre-determined degree, since the initial liquidity is what attracts traders to enter the market. As with b, you can also edit this in time and bound losses to your liking (LMSR already supports this), but this paper provides a way to run profitably, or at least break-even.

As for whether to implement, this is a business model question. Exchanges will have to weigh the cost/benefit of sacrificing some money upfront to get participants to trade, with the desire to generate transaction revenue as the market intermediary. Both approaches are valid, but dependent what outcome you're trying to achieve.


What are all the countries that have a value added tax? I'm a web developer (not an economist) and i'm working on a program that deals with value-added tax (VAT), maybe it is a stupid question but, is the value added tax obligatory? I just want to know if this tax exists in every country all over the world?

No, many countries do not. For example. the United States does not have a VAT. According to the paper The Consequences of the Value-Added Tax on Inequality in Developing Countries (Alavuotunki and Pirttilä (2015)) here is an extensive list:

A4 – Countries and introduction year of VAT. (Source: IMF)
Asia-Pacific
1977    Korea, Republic of
1985    Indonesia
1986    New Zealand
1986    Taiwan Prov.of China
1988    Philippines
1989    Japan
1990    Pakistan
1991    Bangladesh
1992    Thailand
1994    China,P.R.: Mainland
1994    Singapore
1997    Nepal
1998    Mongolia
1998    Sri Lanka
1999    Cambodia
1999    Papua New Guinea
1999    Vietnam
2000    Australia
2005    India
no VAT  Brunei Darussalam
no VAT  China,P.R.:Hong Kong
no VAT  Lao People's Dem.Rep
no VAT  Malaysia
no VAT  Myanmar

Small Islands
1973    San Marino
1990    Iceland
1992    Cyprus
1992    Fiji
1994    Samoa
1995    Malta
1997    Barbados
1998    Vanuatu
2004    Cape Verde
2007    Antigua and Barbuda
2007    St. Vincent & Grens.
2010    St. Kitts and Nevis
2012    St. Lucia
2013    Seychelles
no VAT  Bahamas, The
no VAT  Comoros
no VAT  Grenada
no VAT  Maldives
no VAT  Solomon Islands
no VAT  São Tomé & Príncipe
no VAT  Timor-Leste
no VAT  Tonga
no VAT  Tuvalu

Americas
1967    Brazil
1968    Uruguay
1970    Ecuador
1973    Bolivia
1973    Peru
1975    Argentina
1975    Chile
1975    Colombia
1975    Costa Rica
1975    Nicaragua
1976    Honduras
1977    Panama
1980    Mexico
1982    Haiti
1983    Dominican Republic
1983    Guatemala
1990    Trinidad and Tobago
1991    Canada
1991    Jamaica
1992    El Salvador
1993    Paraguay
1993    Venezuela, Rep. Bol.
2006    Belize
2007    Guyana
no VAT  United States

North Africa and Middle East
1976    Israel
1985    Turkey
1986    Morocco
1988    Tunisia
1991    Egypt
1992    Algeria
2001    Jordan
2002    Lebanon
2008    Iran, I.R. of
2009    Djibouti
no VAT  Afghanistan, I.R. of
no VAT  Bahrain, Kingdom of
no VAT  Iraq
no VAT  Kuwait
no VAT  Libya
no VAT  Oman
no VAT  Saudi Arabia
no VAT  Syrian Arab Republic
no VAT  United Arab Emirates
no VAT  Yemen, Republic of

Sub-Saharan Africa
1960    Côte d'Ivoire
1980    Senegal
1986    Niger
1989    Malawi
1990    Kenya
1991    Benin
1991    Mali
1991    South Africa
1993    Burkina Faso
1994    Madagascar
1994    Nigeria
1995    Gabon
1995    Mauritania
1995    Togo
1995    Zambia
1996    Guinea
1996    Uganda
1997    Congo, Republic of
1998    Ghana
1998    Mauritius
1998    Tanzania
1999    Cameroon
1999    Mozambique
2000    Chad
2000    Namibia
2001    Central African Rep.
2001    Rwanda
2002    Botswana
2003    Ethiopia
2003    Lesotho
2004    Zimbabwe
2009    Burundi
2010    Eritrea
2013    Swaziland
no VAT  Angola
no VAT  Gambia, The
no VAT  Guinea-Bissau
no VAT  Liberia
no VAT  Sierra Leone
no VAT  South Sudan

Europe
1967    Denmark
1968    France
1968    Germany
1969    Netherlands
1969    Sweden
1970    Luxembourg
1970    Norway
1971    Belgium
1972    Ireland
1973    Austria
1973    Italy
1973    United Kingdom
1986    Portugal
1986    Spain
1987    Greece
1994    Finland
1995    Switzerland

Central Europe and the ex-Soviet
1988    Hungary
1992    Armenia
1992    Belarus
1992    Estonia
1992    Georgia
1992    Kazakhstan
1992    Kyrgyz Republic
1992    Latvia
1992    Lithuania
1992    Moldova
1992    Russian Federation
1992    Tajikistan
1992    Turkmenistan
1992    Ukraine
1992    Uzbekistan
1993    Czech Republic
1993    Poland
1993    Romania
1993    Slovak Republic
1994    Bulgaria
1996    Albania
1998    Croatia
1999    Slovenia
2000    Macedonia, FYR
2003    Montenegro
2003    Serbia, Republic of
2006    Bosnia & Herzegovina
no VAT  Kosovo, Republic of


How do reserves move between the 12 federal reserve banks? In single central banks systems (e.g. Bank of England) then the final ledger for electronic money transfer is the adjustment of reserve account balances.

In the Federal Reserve System, there are 12 banks providing reserve accounts responsible for the banking in their region.  How do these 12 ledgers operate such that money can transfer from one region to another?  Is there an overall ledger?

Edit to clarify in response to answer:

An electronic transfer from bank A to bank B in Sterling results in an RTGS transfer between the banks' reserve accounts.  The Bank of England debits one reserve account and credits the other - so that the total reserve balance remains constant.

In the federal reserve system, what happens when money transfers from bank A in region 1 to bank B in region 2?  Does the federal reserve bank in region 1 simply reduce the reserve balance of bank A without an offsetting transaction on its balance sheet anywhere?  Is there an overall ledger that manages the "total reserve", or is it simply a technical constraint of RTGS that the same amount of money is destroyed at one bank as is created at the other?

This is surprisingly subtle.

When, for instance, when bank A in the Richmond Federal Reserve district sends $1000 in reserves to bank B in the Minneapolis Federal Reserve district, reserves are taken out of bank A's account at the Richmond Fed and placed into bank B's account at the Minneapolis Fed.

Now, bank A's reserves are a liability on the books of the Richmond Fed, while bank B's reserves are a liability on the books of the Minneapolis Fed. Without any offsetting change, therefore, the process would result in the Richmond Fed discharging a liability and the Minneapolis Fed gaining a liability - and if this continued, regional Fed assets and liabilities could become highly mismatched.

The principle, then, is that there should be an offsetting swap of assets. It would be too complicated to swap actual assets every time there is a flow of reserves between banks in different districts. (There's over $3 trillion in transactions every day on Fedwire, the Fed's RTGS system - and if even a fraction of those are between different districts, the amounts are really enormous.) Instead, in the short run the regional Feds swap accounting entries in an "Interdistrict Settlement Account" (ISA). In the example above, the Minneapolis Fed's ISA position would increase by \$1000, while the Richmond Fed's ISA position would decrease by \$1000, to offset the transfer of liabilities.

So far, this is all very similar to the controversial TARGET2 system in the Euro area, in which large balances between national banks have recently been accumulating. The American system is different, however, because ISA entries are eventually settled via transfers of assets. Every April, the average ISA balance for each regional Fed over the past year is calculated, and this portion of the balance is settled via a transfer of assets in the System Open Market Account (the main pile of Fed assets, run by the New York Fed). Hence, if in April the Minneapolis Fed has an ISA balance of +\$500, but over the past year it had an average balance of +\$2000, its balance is decreased (by \$2000) to -\$1500, and it has an offsetting gain of \$2000 in SOMA assets.

As this example shows, since it is average balances over the past year that are settled, not the current balances, ISA balances do not necessarily go to zero every April. Historically, they were fairly tiny anyway, but since QE brought dramatic increases in reserves, these balances have sometimes been large and irregular. In the long run, though, the system prevents any persistent imbalances from accumulating.

(Note: the process in April is a little bit more complicated than I describe, since some minor transfers of gold certificate holdings are also involved. Basically, gold certificates are transferred between regional Feds to maintain a constant ratio of gold certificates to federal reserve notes; the transfers of SOMA assets are adjusted to account for this. Wolman's recent piece for the Richmond Fed is one of the few sources that describes the system in detail.)


How does the federal funds rate compare to the federal discount rate? The federal funds rate is the interest rate charged to banks when they borrow from each other overnight in the federal funds market to satisfy their reserve requirements. This rate is influenced by the Fed through open market operations. The federal discount rate is the interest rate charged to banks when they borrow from the Fed to satisfy their reserve requirements. This rate depends on how much money banks are in need of and how much the Fed is willing to provide.

It seems to me that both of these rates should be almost equal most of the time. If the discount rate is lower than the federal funds rate, then the depository institutions would borrow from the Fed, instead of from each other. Thus, this additional demand for the central bank's reserves drives up the discount rate such that it equals the federal funds rate. The opposite would occur if the federal funds rate is lower than the discount rate. Is this correct?

You are right the two series closely follow each other for the reasons you mention. During quite some time the discount rate actually used to be a ceiling for a funds rate. This is precisely, because if the federal funds rate was below
the discount rate, most banks adjusted their reserve positions in the federal funds market and when the federal funds rate rose to the level of the discount rate it was cheaper to borrow at discount rate so banks simply do that.

However, they are not always as close as you would expect, especially the discount rate does not anymore acts as a ceiling. There are several reasons for that in literature, primarily because borrowing at both of those rates has a different strings attached to it.

For example, Meulendyke (1998) argues that:

  Since the mid-1960s, the basic discount rate frequently has
  been below the prevailing Federal funds rate. The Federal Reserve
  has relied on administrative procedures to limit access to the window by restricting the frequency and amounts of borrowing.
   ... The Federal Reserve’s administrative
  restrictions used to be the primary factor that discouraged borrowing, but in the last decade banks themselves have been responsible
  for much of the limitation. Heavy borrowing in the 1980s by a few
  banks with financial difficulties caused others to avoid the window
  for fear depositors might conclude that they were also in trouble.
  Reluctance to borrow contributes to a seemingly contradictory
  result—namely, that increases in the amount of reserves in the banking system, when provided through the discount window, make
  reserve availability more restrictive on the margin because such
  increases put banks under pressure to find other sources of reserves
  to repay the loans.

and she further argues that:

  Until the mid-1960s, the Federal funds rate did
  not trade above the discount rate. During “tight money periods,” when the
  Trading Desk was fostering significant net borrowed reserve positions, funds
  generally traded at the discount rate, and the funds rate was not considered a
  useful indicator of money market conditions. When free reserves were high,
  funds often traded below the discount rate and showed some day-to-day
  variation. At such times, the funds rate received greater attention as an
  indicator of reserve availability.
  There was considerable surprise when the funds rate first rose above
  the discount rate, briefly in October 1964 and more persistently in 1965. As
  large banks became more active managers of the liability side of their balance sheets, they borrowed funds in the market in a sustained way. Banks
  had introduced large negotiable certificates of deposit (CDs) in 1961. But CD
  borrowings were (until 1991) subject to reserve requirements and (until
  1970) to interest rate ceilings under Regulation Q. Borrowings from other
  banks through the Federal funds market were free of reserve requirements
  and interest rate ceilings. Furthermore, they were not subject to the restrictions on prolonged use that were applied to the Federal Reserve’s discount
  window. The changes in liability management techniques meant that
  individual banks could expand credit even when they did not have free
  reserves if they were willing to bid aggressively for wholesale funding from
  other banks.

These and possibly also some other reasons will lead the two to diverge in some situations as borrowing at those rate might be subject to different conditions and have different implications. You can see on the graph below which plots of the federal funds rate in red and the discount rate in blue (based on FRED data) that they can sometimes differ quite substantially but indeed tend to follow each other closely.


Asset pricing vs Empirical asset pricing I get confused on what is the difference between asset pricing vs empirical asset pricing? Could you clarify your answer? Also, is asset pricing just assessing the current value of an asset?

Thanks,

As mentioned in Herr K's comment, asset pricing is the theory to price assets (such as equity, bonds, options, futures, swaps, etc). For this, you can use models like CAPM/Fama-French (returns), Black-Scholes (options), Swensonn (interest rates), and many many others. When you say empirical asset pricing, this means that you go to the data (and each model deals with a different type of data) and try to price an existing asset based on the information that you see on the market.

For example, in the Black-Scholes model, given the current stock price (S), the strike price for the option (K), the time until exercise (T-t), the risk-free rate (r) and the volatility of returns (sigma), you can use the formula directly and calculate the price of the option. Notice that this would be a theoretical price. In practice, you might observe something else in the market. You may want to do some arbitrage from here since there is mispricing (Don't go using this in the real market, but you get my point).

For your last question, asset pricing is not necessarily just about the current value of an asset. You probably would also want to estimate future values based on your models. In fact, it is really of no use a model that tells you today's price. For that, you just look at the price on the tape and ignore your model. The whole point of building models is to make predictions.


An increase in Japan's demand for US goods would cause the value of dollar to appreciate? An increase in Japan's demand for US goods would cause the value of dollar to appreciate? Why is that? The official answer is "Japan will buy more U.S. dollars." However, my thought was that isn't it usually the case that US companies get paid of Japanese yen rather than U.S. dollars? And hence Japan will not need to buy more U.S. goods. What is the problem in my thought?

You're just intermediating. The US companies would still have to pay their workers in USD, therefor, instead of the Japanese companies buying dollars in your scenario, the exporters are using the importers' Yen to buy dollars.

It's the same net effect.


Does the case of strategic complements always result in a situation where the Stackelberg leader is worse off? For example, in the case of Bertrand competition, is the Stackelberg leader worse off relative to the follower? If so, why? Is this because the first mover can always undercut until marginal cost is reached?

It depends on what you assume the firms know about each other.  Under full information / rationality assumptions, with identical firms, second-mover advantage disappears because the first-mover will recognize the credible threat of undercutting, and set P = MC and that will be that.

If the firms are not identical (e.g., different marginal costs) then turn order is irrelevant, because only one firm has the ability to make a credible threat of undercutting.  That firm will capture the market, whether it moves first or second.

Only if you drop the assumption of full information / rationality is it feasible that turn order would matter for Bertrand competition, because then the assessment of credible threat becomes a probabilistic one.


Variance in the Context of a AR (1) Model I wondered if someone could help me in terms of the required algebraic steps from expressions (3) - (4), for the the moving average representation of the AR (1) below?
Would be appreciated.
$$y_t=a +\theta_1y_{t-1}+u_t$$
$$var[y_t]= E([(u_t+\theta_1u_{t-1}+\theta^2u_{t-2}+...+\theta^{t-1}u_1)]^2)$$
$$var[y_t]= \sigma^2(1+\theta^2+\theta^4+...+\theta^{t-1})$$
$$var[y_t]= \sigma^2\frac{ 1-\theta_1^{2t}}{1-\theta_1^2}$$

$$Var[y_t] = E[(u_t+\theta_1u_{t-1}+\dots+\theta_1^{t-1}u_1)^2] = E[u_t^2]+\theta_1^2E[u_{t-1}^2]+\theta_1^4 E[u_{t-2}^2] + \dots +\theta_1^{2t-2}E[u_1^2]$$
the latter equality follows from the assumption that $u_t$ is not serially correlated (i.e. $E[u_{i}u_{j}] = 0\ \forall\ i \neq j$). Then, since $E[u_{t}^2] = \sigma^2\ \forall\ t$ it follows that
$$Var[y_t] = \sigma^2(1+\theta_1^2 + \theta_1^4+\dots+\theta_1^{2t-2})$$
it remains to find $(1+\theta_1^2 + \theta_1^4+\dots+\theta_1^{2t-2})$ which is equal to $\frac{1-\theta_1^{2t}}{1-\theta_1^2}$ according to geometric series sum formula.
Thus, $$Var[y_t] = \sigma^2\frac{1-\theta_1^{2t}}{1-\theta_1^2} $$


Expenditure function Let $u (x) = \prod_{{i\leq n}}$ $(x_i-m_i)^{a_i}$, where $m_i\geq$0 and $a_i\geq$0, Σ$_{{i\leq n}}a_i=1$ show that the expenditure function $e(p,u)$ is linear in $u$.
Based on the definition of the EF, my first guess would be to find the MRS in terms of $x_i$ and $m_i$ to we get $x_i^*$ and $m_i^*$. After we find these two in terms of each other, we substitute it back to the $u$ function to get now $x_i^*$ and $m_i^*$ in terms of $u$ as well. After this we substitute what we found to the budget constraint and prove that the expenditure function is linear in $u$ somehow.
I don't know if I should follow this logic or is there some other way to approach this problem? Any help is appreciated.

It is not quite true; the expenditure function is affine in utility, not linear. This is Stone-Geary utility, and I use the result that demand is given by
$$x_i^*=m_i+\frac{a_i}{p_i}\bigg(w-\sum_j m_jp_j\bigg).$$
One can interpret $\sum_j m_jp_j$ as "subsistence expenditure" and $w-\sum_j m_jp_j$ as "non-subsistence expenditure." Plugging this into the utility function, one can show that utility is an increasing linear function of non-subsistence expenditure. The inverse of a linear function is linear again, so we can write non-subsistence expenditure as $l(u)$ with $l$ linear (in the relevant range). This means then that total expenditure is
$$l(p)+\sum_j m_jp_j,$$
which is an affine function as the sum of a  linear function and a constant.


Dornbusch Sticky Price (Overshooting) model Stupid question: What do the phi and lambda from the equations above represent?

The equations you have are equilibrium conditions, while derivation (of course) starts with the demand equation and sets demand equal to supply. Then:

$\varphi$ ($0<\varphi<1$) is the income elasticity of money demand - the responsiveness of demand with respect to income
$-\lambda$ ($-\lambda <0$) is the interest rate semi-elasticity of money demand - the responsiveness of demand to the change in interest rates.

Semi-elasticity means that we care how much Y changes in percentages when we increase $X$ by one (increase by one, not by one percentage point).


How to test for invertibility in ARCH family models? I hope everyone is doing well.
Citing Enders (2014) in the book Applied Econometric Time Series: "the Box–Jenkins approach also necessitates that the model be invertible" while discussing necessary conditions for AR, MA and ARMA models. Both in algebra and in code (R), this is relatively simple to verify for smaller models.
How should one proceed when dealing with models of the ARCH family (ARCH, GARCH, GARCH-M)? Given one has two equations (that one describing the behavior of the mean and that one describing the behavior of the variance), must invertibility be verified for both, or must one only verify it for the mean equation?
Thank you!

It's not clear what you mean by "invertibility of the mean equation". Invertibility, in the Box-Jenkins context, is a property of ARMA processes. A characterization of invertibility is that an ARMA process is invertible if and only if the MA polynomial has no roots inside the unit disk in the complex plane. Empirically speaking, invertibility means the current (unobservable) innovation is a weighted sum of what has been observed, although the weights in the sum are unknown ex ante. An ARCH type model usually do not assume any type of ARMA specification for the mean.
As for the variance $\sigma_t^2$ term in, say, GARCH, by assumption it follows an ARMA process, and its invertibility can be checked like any ARMA model.
For "X"-GARCH extensions that leaves the ARMA linear process framework, it's not clear how you would define invertibility.


How to determine regressive tax I have a mock-up exam that my teacher gave me and one of the questions was the following:

  Given these four graphs, which one represents regressive tax?

I'm quite sure it's graph A, but I don't seem to be able to find a proper reason why. Any help and/or tips are welcome!

Thanks in advance

EDIT: As per request. How I interpret this: regressive means the higher the income, the lower the tax. Thus: tax drops with higher incomes. But I can't really determine to which graph this applies.

(T is tax, Y is income)

You're correct it is graph A. Regressive tax means that the lower income you have the higher %tax you have. If you imagine at the lowest level of income, you still have a flat tax(The intercept).

Since the slope is constant the %tax component is the same for all income levels.

Therefore this flat tax component is the part you need to look at. The flat tax represents a larger proportion for lower income earners. As a whole (%tax + Flat tax) the tax is then a higher % for lower income earners, hence regressive taxation.


Interpretation of coefficients in a regression with a lagged dependent variable I have estimated the following dynamic panel data model using GMM:
          $\ln Y_{it}=\beta_0+\beta_1\ln Y_{it-1}+\beta_2\ln X_{it-1}+\epsilon_{it} $
where $Y$ is employment and $X$ is productivity.
Suppose that the estimated coefficient on $\ln X_{it-1}$ is $\hat{\beta_2}=0.3$. If I understand correctly, because of the dynamic structure of the model, it can be considered that the dependent variable is the growth rate of $Y$. Hence, is the following interpretation of the estimated coefficient correct: any $1\%$ increase in productivity would be expected to increase the employment growth rate by $0.3$ percentage points?
If not, what would be the correct interpretation?

tldr If $\mathbb{E}[\varepsilon_{i,t}|X_{i,t-1}, Y_{i,t-1}] = 0$ then the coefficient $\beta_2$ is equal to:
$$
\frac{\partial \mathbb{E}[\ln Y_{i,t}|X_{i,t-1}= x_{i,t-1}, Y_{i,t-1} = y_{i,t-1}]}{\partial \ln x_{i,t-1}}
$$
If $\varepsilon_{i,t}$ is independent of $X_{i,t-1}$ and $Y_{i,t-1}$ (which is a stronger condition) then it is also equal to:
$$
\frac{\partial \ln \mathbb{E}[Y_{i,t}|X_{i,t-1}= x_{i,t-1}, Y_{i,t-1} = y_{i,t-1}]}{\partial \ln x_{i,t-1}}
$$
In both cases, they give an estimate for the elasticity of employment with respect to productivity yesterday (conditional on employment yesterday), i.e. what is the (expected) percentage point change in employment (today) due to a 1 percentage increase in productivity (yesterday) given the level of employment yesterday.
estimating elasticities
If
$$
\ln y = \alpha + \beta \ln x,
$$
Then the $x$-elasticity of $y$ is given by:
$$
\frac{\partial \ln y}{\partial \ln x} = \beta.
$$
It measures the percentage change in $y$ due to a 1 percent increase in $x$.
Now let's go to the stochastic framework:
$$
\ln Y = \alpha + \beta \ln X + \varepsilon
$$
And assume, as usual, that $\mathbb{E}[\varepsilon] = 0$.
The elasticity of $Y$ with respect ot $X$ is not really defined as $Y$ is not a function of $X$ (i.e. both are random variables).
There are two natural generalizations of elasticity in this setting:

$\dfrac{\partial \ln \mathbb{E}[Y|X = x]}{\partial \ln x}$, which is the elasticity of the conditional mean function $\mathbb{E}[Y|X = x]$.
$\dfrac{\partial \mathbb{E}[\ln Y|X = x]}{\partial \ln x}$, which is the derivative of the conditional mean function $\mathbb{E}[\ln Y|X = x]$.

Because the log is a non-linear transformation, the two are not necessarily the same.
However, if $X$ and $\varepsilon$ are independent then they are identical and equal to $\beta$. To see this, notice that for 1.
$$
Y = e^\alpha  X^\beta e^\varepsilon
$$
Then taking conditional expecations and using the independence between $\varepsilon$ and $X$:
$$
\mathbb{E}[Y|X = x] = e^\alpha x^\beta \mathbb{E}[e^\varepsilon|X = x] = e^\alpha X^\beta \mathbb{E}[e^\varepsilon],\\
\to \ln \mathbb{E}[Y|X = x] = \alpha + \beta \ln x + \ln \mathbb{E}[e^\varepsilon].
$$
Taking the partial derivative with respect to $\ln(x)$ gives:
$$
\frac{\partial \ln \mathbb{E}[Y|X = x]}{\partial \ln x} = \beta.
$$
For 2. we immediately have:
$$
\mathbb{E}[\ln Y|X = x] = \alpha + \beta \ln x + \mathbb{E}[\varepsilon|X = x] = \alpha + \beta \ln x.
$$
So:
$$
\frac{\partial E[\ln Y|X = x]}{\partial \ln x} = \beta.
$$
If only $\mathbb{E}[\varepsilon|X] = 0$ but $X$ and $\varepsilon$ are not independent, then only the derivation of 2 is valid and 1 and 2 are not necessarily equal.
If you want to estimate the elasticity of the growth rate, then the right regression is indeed:
$$
\ln(Y_{i,t}/Y_{i,t-1}) = \beta_0 + \beta_1 \ln X_{i,t} + \varepsilon_{i,t}.
$$