Why goods prices are normalized in each period in growth models? As it is known, in simple economic growth models such as the Solow or the Ramsey model, all goods prices are set equal to 1 in each period.

Also, as we known from general equilibrium theory, homogeneity of degree zero of demand implies that only relative prices matter. Therefore, we can always set one price equal to 1 (or do other types of "normalization").

However, in economic growth models (which are general equilibrium models) we have an infinite number of good prices and all are set equal to 1. Why do we always do that? Doesn't it affect the analysis of the model?

I have search for an explanation, but I've been unable to find one. I have a hypothesis, though. Maybe we can normalize goods prices in each period because we assume the economy is in equilibrium at each instant. Then, by the same argument we use in static models, we can normalize goods price in each period. Is this explanation true? Or there is some other reason? And, if it is true, how do models that feature inflation are made?

In fact, in growth models, people usually assume that there exists a one generic good for which the price is equal to one and not an infinite number of goods. Then, it is possible to set the price equal to one.

You can look at Acemoglu's Introduction to Modern Economic Growth textbook. In this book, you can see in a decentralized economy (where there are different sectors, final good producer, intermediate good producer etc.) you can see that the price of the final good is normalized to one and there exists a price $p$ for the intermediate good.

Therefore, to sum up, in simple canonical solow model with only one sector, there exists a one generic good and for more complicated growth models (endogenous growth models), where there are different sectors, there are different goods, then this implies different prices.

Hope that is a clear answer for you !


Probit with panel data - assumptions violated with endogenous covariate? I am trying to understand how does the fact that a regressor is endogenous (as a result of reverse causality) in a dynamic probit model violate any assumptions that I may need to make for this model. E.g. for an OLS model, it is pretty straightforward - the orthogonality condition for the regressor being uncorrelated with the error term will not hold. Can similar logic be extended to probit models with panel data? I would appreciate simple explanations and/or references that explain this.

Yes it does. According to the Verbeeks guide to modern econometrics (pp418) standard panel fixed effects binary model assumes that error has “a symmetric distribution with distribution function $F(.)$, i.i.d. across individuals and time and independent of all $x_{is}$” [with the model being  $y_{it}^*=x_{it}’ \beta+ \alpha_i+u_{it}$].

Just a two pages later Verbeek also writes about Random effects probit model $y_{it}^*=x_{it}’ \beta +\epsilon_{it}$ that $\epsilon_{it}$ should also besides other things be independent of $x$.

Now both of the above panel models are technically static since no lags are involved but I don’t see any reason why this assumption could be relaxed just by making model dynamic. In fact there is a such thing as panel IV logit/probit, probably people would not invest time into development of those if this assumption could be relaxed by just adding lags.


Approximating common knowledge with common beliefs (Monderer and Samet, 1989) I'm trying to understand Monderer and Samet's 1989 paper on approximate common knowledge.

I'm stuck in the last part of the proof of the "agreeing to disagree" Theorem A, where the upper bound of the posterior is established. The lower bound seems clear but I just cannot understand how one of the terms is written as (1-p). Am pasting a picture below of the part where I'm lost:

The last line which establishes the upper bound of r_i is unclear to me. Any help would be greatly appreciated. The link of the paper is :

https://ie.technion.ac.il/~dov/cpb_monderer_samet.pdf

(Theorem A is in page 180-181)

Showing the left-hand inequality

Let's rearrange the equation,
$$
\begin{align*}
x &= r_i \frac{\mu\left( B^p_i\left( E \right) \right)}{\mu\left( E \right)} - \frac{\mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right)}{\mu\left( E \right)} \\
& \iff \underbrace{x \mu\left( E \right) + \mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right)}_{\text{LHS}} = r_i \mu\left( B^p_i\left( E \right) \right).
\end{align*}
$$
Since all terms in $\text{LHS}$ are weakly positive, we can omit $\mu( X \cap [ B^p_i( E ) \setminus E ] )$ from $\text{LHS}$, giving
$$
x \mu\left( E \right) \leq r_i \mu\left( B^p_i\left( E \right) \right).
$$
Since $\mu( E ) \geq p \mu( B^p_i( E ) )$, this directly implies $x p \leq r_i$.

Showing the right-hand inequality

Since $\mu( E ) \leq \mu( B^p_i( E ) )$ and $x \geq 0$, we have
$$
\begin{align*}
&\frac{1}{\mu\left( E \right)} \left[ r_i \mu\left( B^p_i\left( E \right) \right) - \mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right) \right] \\
&\geq \frac{1}{\mu\left( B^p_i\left( E \right) \right)}\left[ r_i \mu\left( B^p_i\left( E \right) \right) - \mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right) \right] \\
&= \underbrace{r_i - \frac{\mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right)}{\mu\left( B^p_i\left( E \right) \right)}}_{\text{RHS}}.
\end{align*}
$$
It must be that $X \cap [ B^p_i( E ) \setminus E ] \subseteq B^p_i( E ) \setminus E$, so
$$
\text{RHS} \geq r_i - \frac{\mu\left( B^p_i\left( E \right) \setminus E \right)}{\mu\left( B^p_i\left( E \right) \right)} = r_i - \frac{\mu\left( B^p_i\left( E \right) \right) - \mu\left( E \right)}{\mu\left( B^p_i\left( E \right) \right)} = r_i - \left( 1 - \mu\left( \left. E \right| B^p_i\left( E \right) \right) \right).
$$
By definition (and this is inequality (9) in the text) $\mu( E | B^p_i( E ) ) \geq p$, thus
$$
\text{RHS} \geq r_i - \left( 1 - p \right).
$$
Noting that the original equality was $x = \text{RHS}$, we have
$$
x \geq r_i - \left( 1 - p \right) \;\; \iff \;\; r_i \leq x + 1 - p.
$$


The growth rate in the Solow model Consider a very simple version of the Solow model with discrete time, a fixed population size and no technological progress (a fuller description of the assumptions is outlined here). Capital is at 'steady state' when the total amount of depreciation equals the total amount of saving; graphically, this occurs at the intersection of the blue and purple functions below:

Consider the adjustment process to the steady state. I have often heard it claimed that the lower capital starts, the faster growth will be. Hence, the model predicts that poor countries will grow faster than rich countries, assuming that these countries are the same in all other respects ('conditional convergence'). However, looking at the graph here, it seems that at a very low level of capital, growth will actually be very slow. The reason is that the savings function (blue) is barely above the depreciation line (purple), so the increase in capital is small. As capital increases, the gap between these functions gets larger, so the growth rate should increase. Eventually, however, the gap begins to get smaller, and growth does indeed slow down (as commonly claimed).

Of course, I have been discussing the growth rate in capital, while we are interested in the growth rate in output. And indeed the marginal product is capital monotonically falls as capital increases. Is this what ensures that the growth rate in capital is always falling? Or is there still a period when the growth rate in output is increasing?

It seems you are confounding growth and growth rate.  The difference between the blue and purple curves, i. e. growth,  may be small when capital is small, but the ratio of the difference to  capital, i. e. the growth rate will indeed be larger the closer capital is to zero.


Buy-or-sell deal when there is cash shortage A company is owned by two partners: partner $A$ holds a fraction $a$ and partner $B$ holds a fraction $b$ (with $a+b=1$). Partner $A$ wants to break the partnership. A common procedure for this is that $A$ should suggest a price $P$ for the entire company and let $B$ choose either of two options:

Buy $A$'s share for $aP$, or -
Sell his share to $A$ for $bP$.

This procedure incentivizes $A$ to suggest a fair price. Moreover, if $B$ thinks that $A$'s price is unfair, he can profit from this - if he thinks that the price is too low he can opt to buy and if he thinks that the price is too high he can opt to sell. The advantage of this procedure is that it guarantees fairness without requiring for an external arbitrator.

A possible problem with this procedure is that it requires $B$ to have enough cash for buying $A$'s share. If $B$ doesn't have enough cash, and cannot get credit, he might have to select the "sell" option even when the price is low. Moreover, if $A$ knows that $B$ is short in cash, he might exploit this fact and deliberately suggest a low price in order to force $B$ to sell his share.

MY QUESTION IS: is there an improved division mechanism that handles this problem?

Let's assume $A$ stated that the company is worth $P$.

I propose the following mechanism:

$B$ has to choose one of two alternatives:

Sell his share for $bP$.
Pay an amount of his choice, $x\leq aP$, and then with probability $\frac{bP+x}{P}$ get $A$'s share and with probability $1-\frac{bP+x}{P}=\frac{aP-x}{P}$ lose his own share (in addition to paying $x$).

If $B$ thinks that the company is worth less than $P$, he should go with the first option, just like in the original scheme.

Otherwise he should pick the second alternative, paying as much as he can to maximize his profit expectation. Should $B$ have enough money, he would pay $aP$ and get $A$'s share w.p. 1, if not, there is no way of allowing $B$ to get the company without ripping $A$, unless this is done randomly..


Derive the Cost Function TC(Q) Suppose $F(K,L)= 50L^{\frac{1}{2}}K^{\frac{1}{2}}$, the wage is $w = 5$
   (euros) and rent is $r = 20$ (euros). What is the cost of producing
  $1000$ units? Derive the cost function $TC(Q)$.

I know how to find out the cost.

$L = 40$ and $K = 10$. So the total cost is $400$.

But now I am stuck. The solution for the cost function should be: $$TC(Q) = 5\cdot \dfrac{4Q}{100} + 20\cdot \dfrac{Q}{100} = \dfrac{2}{5}Q$$
How do I get to this function?

You are trying to find the solution of $K$ and $L$ in terms of $Q$ to derive the total cost function with respect to quantity.

Page 13 of this lecture has the exact question you are interested in. (Google is your friend.)

From the Lagrangian you'll have $\frac{K}{L}=\frac{w}{r}$
From the production constraint you'll have $50K^{\frac{1}{2}}L^{\frac{1}{2}} = 1000$ and you have already derived the solution to this.

The first implies that $4K = L$ given your solution.

Then since,
$50K^{\frac{1}{2}}L^{\frac{1}{2}} = Q$

$$\implies 50K^{\frac{1}{2}}(4K)^{\frac{1}{2}} = Q \\
\implies 100K = Q \\
\implies K(Q) = \frac{Q}{100}$$

A similar solution can be found for $L(Q)$, and then it should be smooth sailing for you from there.


The Savage sure thing principle and Subjective utility representation I have tried reading and understanding Savage's proof of the subjective utility representation, it is too complicated. Is anyone aware of a shorter/more elegant proof of this?
It is not a problem if we assume a finite prices set.

The original is in Savage, L.J. 1954. The Foundations of Statistics. New York: John
   Wiley and Sons.

A good summary can be found at
http://www.econ2.jhu.edu/people/Karni/savageseu.pdf.

The Savage proof is known to be very elaborate, and long. It uses the sure thing principle as its main axiom. I was wondering if there is a more "modern" proof, that is both elegant and shorter. Or a nice challenge would be to try to prove collaboratively using some modern mathematics, like mixture spaces, (I am aware of Anscombe-Aumann).

In Kreps' (1988) book "Notes on the Theory of Choice", the issue is dealt with in chapter 9 "Savage's Theory of Choice Under Uncertainty", after discussing subjective probability in chapter 8. As usual, Kreps' style helps: he has the ability to seamlessly inject his -always formal- approach with very down-to-earth comments and examples that are strong in intuition (and he does it better than Savage, I might add). But also, here "formal" does not translate into "complete exposition": he explicitly refrains from formally proving parts of the whole apparatus, mentioning that "this is a two-page proof", and "this is another two-page proof", and "if you want to prove this, good luck". For these parts he falls back on Fishburn's (1970) "Utility Theory for Decision Making" book, chapter 14 "Savage's Expected Utility Theory". And Fishburn is formal alright (more symbols than words in a page).

My impression is that combining these two sources can be beneficial.


Is it necessary to ask permission before including someone in the acknowledgements of a economic research paper? Do I need permission to thank someone, even if I said “all mistakes are our own”?
Do I have to make sure that the person being thanked must read the paper, or, it is fine if he just discussed about the general ideas and the administration stuffs of the target journal?
As a student I don't know how does a specific journal work so I asked a lot of questions. I think I need to ask for the permission before acknowledging, but, if it is unnecessary, then I will be just wasting the time of a super busy professor.
I know that in Math people almost never ask for permissions but in life science people need permissions. I think theoretical economics follows the culture in mathematics?

In short, no it is not necessary. I have never asked anyone for permission and I have never been asked. The people I thank usually have not read my paper (rarely anyone does, to be honest), but we have had chats on visits or after seminars or conferences. If you thank someone make sure they are familiar with the paper. Certainly include someone who presented a formal discussion at a conference.
By all means, never include someone in your thank-you note who you have not talked to. People will see this as a strategic way to influence referee selection. However, if someone made great comments that greatly influenced the direction of the paper, name them. It's free and makes people happy.


WARP implies completeness, transitivity and thus rationalizability. What is wrong with the statement? Let $A$ be a menu and $R$ be a complete and transitive binary relation. Define choice correspondence generated by $R$:

$$c_R(A)=\{x\in A|| xRy \ \forall y\in A\}.$$

Theorem (from Kreps 1988): for any correspondence $c$, the following conditions are equivalent:

$c$ satisfies WARP
There exists a transitive and complete $R$ such that $c=c_R$.

Given $A$ is finite and $c=c_R$, it is obvious that $c$ can be represent by a utility function. Then why do we need SARP/GARP to rationalize a choice correspondence with a utility function? What was wrong with my logics?

In Notes on the Theory of Choice (assuming this is what you refer to by "Kreps (1988)"), Kreps does not appear to mention WARP as such. But he does refer to "Houthakker's axiom" (Houthakker 1950, Economica), which is actually SARP, not WARP.

Your conditions 1 and 2 are equivalent if there are only two goods. If there are more than two goods, WARP in condition 1 must be replaced by SARP to preserve the equivalence. Jehle and Reny (2011) have an intuitive explanation for this:

  [W]ith two goods, the pairwise ranking of bundles implied through revealed preference turns out to have no intransitive cycles. [...] And when this is so, there will be a utility representation generating the choice function.

SARP basically posits conditions that explicitly rule out intransitive cycles in revealed preferences with a general $n$-good environment.

WARP/SARP differ from GARP in that the former presumes the availability of infinitely many choice data to build a choice function with, whereas GARP only requires a finite set of choice data.


When do prices not exist? Often we employ the use of the separating hyperplane theorem to prove existence of price vectors, when discussing infinite economies this proof is substituted for proving existence of linear functionals.
This all being said, I find it difficult to see a case in consumer theory, producer theory and general equilibrium where we cant have a price vector (unless the problem is a planners problem).
To clarify I'm asking in what mathematical environments do prices not exist and what would their economic interpretation be.

There are a bunch of examples for incomplete markets in the finance literature.  The oldest (that I know of) is Hart (1975).  In finance, the problem is that if you have two different assets they have different prices that make the market complete.  But then equilibrium considerations make them have the same price, which forces the market to no longer be complete.  (If the market was complete, then they would have the same price.)


EIB - losses on Greek bonds The EIB is holding large quantities of Greek bonds. if Greece would default on these bonds, how would these (paper) losses be felt by individual tax payers in the other EU countries? Would this be through direct resource transfers out of tax income by the otehr EU governments (to cover these losses)? or would the EIB simply "print" money, and the effect would hence be felt through general price inflation? Many thanks in advance, Wouter

It just arrived to the Swiss Central Banks. Its assets were depreciated of $50 billion in six months.
As fortune summarized:

  Luckily for the Swiss National Bank, it can’t actually become insolvent, since it can simply print francs to cover any losses it suffers. The announcement caused the value of the Swiss Franc to tumble, falling as much as 0.4% against the euro on Friday.

If the ECB would lose money because its assets'value is decreasing, the ECB would print money as well.

By the way, there was an article in voxeu from Paul de Grauwe and Yuemei Ji which title was "Why the ECB should not insist on repayment of its Greek bonds":

  It would be easy for the ECB to solve this problem. The easiest way
  would be to write off the Geek bonds. As we have argued earlier such a
  write-off has no fiscal implications. It means that the circular flow
  of interest from Greece to the ECB and back is discontinued. In
  contrast, the repayment of the bonds by Greece has fiscal implications
  and leads to transfers from Greece to the other member countries. This
  is quite a perverse transfer given the sorry state of the Greek public
  finances.

I am giving you the advise to read the article from de Grauwe and Ji really. I think it would answer most of your questions.


Could a bank simply eliminate a loan from its books? Assume a bank gives a loan to Mr X. for, say 100. The bank's assets (account "loans") increases by 100, and its liability (account "deposits") also go up by 100 (which is the extra 100 added to the current account of that who asked for the loan).

Say Mr. X does not want to pay the loan+interest. The bank is, by law, entitled to them, so it can perfectly use the law to reclaim the money and interest back (maybe possessing some material wealth like a house or other collateral used to secure the loan in the first place). However, can a bank, in theory, simply erase the loan and deposit? Can it simply eliminate the 100 from both assets and liabilities? Would not this way the bank be exactly the same as in the beginning?

PS: sorry I am asking many questions. I am study economics and banking and cannot understand everything very well. It is great to have a place here to improve my knowledge

If Mr. X hasn't used the money, so it's all still in a deposit account at the same bank, then the bank may be able to cancel the loan. Whether they can or not, will depend on the exact contract attached to the loan.

If Mr. X has withdrawn the money as cash, then that transaction represented a drop in the bank's liability to Mr. X of 100, and a drop in the bank's assets of cash-at-hand of 100.

If Mr. X has used the money in a digital transaction, then they can no longer cancel both at the same time, as the asset of the loan is with Mr. X, but its matching liability will be with another party.

In either of the last two cases, the bank can cancel the loan, reducing its assets by 100. The balancing transaction will be a reduction in the bank's equity of 100 - i.e. the bank records a net loss on the loan of 100, and the shareholders make a loss of 100 on the transaction as a result.

(because assets = liabilities + equity)


Weierstrass Thm: Continuous Fn Attaining Extrema on Compact Domain $u:\mathbb{R^L}\rightarrow\mathbb{R}$ This is just semantics, but MWG doesn't use the Weierstrass Theorem in its Math Appendix when using the fact that a continuous function always has a max value on any compact set.

Some books appeal directly to the Weierstrass Theorm.

Is there a right or wrong answer to this? Or is it more just, like I said, semantics?

The Weierstrass Extreme Value Theorem guarantees this. However, it's used so frequently that economists take it for granted or omit it in graduate courses and research papers. MWG is a maturity book. Part of mathematical maturity is being able to fill in the details for a proof.


Why do low-budget films charge the same amount at the box office as super-high budget films? In most occurrences, lower production costs translate into lower sales costs. Obviously there are other factors that influence price, including perceived value (affected by advertising), monopolies, and so on.

However there are many companies that produce films that vary greatly in budget. For instance, the Blair Witch Project came out in 1999 with a budget of \$60,000. The same year, Star Wars Episode 1 came out with a budget of \$115 million. That's almost 2000 times more expensive. Yet the two films sat side by side in the box office charging the same amount of money per viewing.

Why don't the makers of super-low-budget films undercut the big budget films?

Opportunity Cost of the Seats

Once the movie is made the cost of production is sunk and irrelevant to the proper pricing of tickets. Only the marginal costs of serving an additional customer and the opportunity cost of showing a different film would enter into ticket pricing. Since cinemas should be setting the number of screens for each movie so that the opportunity costs of the seats are equated across films, this makes them want to charge the same for all films. In support of this idea, I offer that movies do vary in price by time of day. This is a rational response to the perishability of seats (once the movie starts an empty seat for that show is worthless) and time (of day) varying demand for film-going which varies the opportunity cost of the seats by time of day. The theater can't easily equate the opportunity cost across time the way it can across films.

At the Movies: The Economics of Exhibition Contracts (Filson, Switzer, and Besocke (2004)) provides the following explanation:

  Practitioners provide several explanations for inflexible ticket
  prices. Exhibitors want to avoid menu costs and eliminate consumer
  uncertainty about what the movie will cost. Exhibitors do not increase
  prices of hits because they are engaged in repeat business with local
  consumers, and the potential loss of goodwill from increased prices
  outweighs the potential gain. Charging different prices for different
  movies at multiplexes necessitates employing monitors to ensure that
  consumers see the movies they pay for. Even offering mid-week
  discounts may lead to more time shifting than new demand. Not all
  analysts or practitioners agree that inflexible prices are optimal
  (see Orbach and Einav 2001), although it seems unlikely that such an
  easy-to-exploit profit opportunity would persist. Some practioners
  have experimented with non-uniform prices in  the U.S. in the recent
  past but inflexible prices remain the norm.


Real Business Cycle Theory Which academic paper does this diagram come from? Need to know for citation purposes. Apparently its King and Rebelo (2005) but doesn't seem to be the case.

It's from King and Rebelo (2000) "Resuscitating Real Business Cycles". Here is a link to the NBER working paper version. The graph is in Figure 8 on p.94.


Inflationary recession vs Deflationary recession I heard the terms deflationary recession & inflationary recession in a podcast and would like to understand what it means, what causes them and past examples.

Deflationary recession refers to a recession in which prices are falling. An example was the Great Depression. Below you have a graph of the yearly rate of change in CPI. A negative number means a fall in prices, i.e. deflation. Shaded areas mean recession period according to NBER.

Some argue that deflation was actually a factor contributing towards the great fall in GDP, because the Fed did not act soon enough, keeping the money supply low. The Gold Standard is partly to blame for this inaction of the Fed.

Inflationary recession is when a recession is accompanied with increasing prices. It is commonly known in Macroeconomics as Stagflation. That was the case on the 1973-1975 recession in the US. The graph below shows the yearly rate of change in CPI (again, shaded area represents a recession):

The latter happened mainly because of the 1973's oil crisis where the OAPEC (oil cartel) stopped selling oil to countries supporting Israel (including the US), rising its price.


Effect of a permanent increase in government expenditure in an open economy? In Krugman & Obstfeld's International Economics, 8e, Chapter 16, an involved argument is given showing that a permanent fiscal expansion will not affect output even in the short run. It seems to me that this is a saddlepoint stability argument in disguise. Am I right in thinking this? Can someone please help me with a differential equation formulation?

The intuition for this result is pretty straightforward, and I think one can think about it in terms of saddlepoint stability in a phase diagram, although you don't need any serious technical apparatus - it's all conceptual.

Krugman and Obstfeld posit a model in which government expenditure does not affect the "full employment" level of output $Y^f$ (to which the economy must tend in the long run - note that proving this requires you to look at a phase diagram and rule out the explosive paths that fail to converge to $Y^f$). The money supply $M^s$ is also assumed to be constant. Together, these assumptions imply that the steady state price level $P$ doesn't change.

Now Krugman and Obstfeld, critically, also assume that you're already at the steady state price level $P$ when the permanent increase in government expenditure occurs. And even once the increase hits and the phase diagram changes somewhat, this $P$ is still the steady-state price level (i.e. the price at the unique fixed point of the implied system of differential equations).

In their world, $P$ is the only state variable. (Prices are sticky, but quantities and exchange rates are not; they're jump variables.) Since you're already at the fixed point's value of $P$, the other variables will immediately jump to their fixed point levels and stay there. (This is always true for models where the fixed point is a saddle and equilibrium is unique.) And since the fixed point for output is unaffected by the expenditure shock, this means we stay at the same output even in the short run.

Conceptually, the point here is that we only get nontrivial dynamics from a permanent shock if that shock changes the fixed point values of state variables. This happens, for instance, in the neoclassical growth model following a permanent TFP shock, because an increase in TFP raises the steady state level of capital above the current level of capital. But it doesn't happen here, since under Krugman & Obstfeld's particular set of assumptions there is only one state variable and it's unaffected by permanent changes in government expenditure.


Determining the capital return from the investment in dynamic firms and their growth According to Tech Tour Growth 50, an organization that identifies the. young firms valued between one hundred millions and one billion dollars and which benefits ("chiffre d'affaires" increase by 50% a year) there 284 of such firms in Europe and 833 in United States where the injected funds are five times higher. If 5*284>833 does it means that the return on investing in Europe is greater than in United States ?

I am not sure about the title I have to my question

Information provided by "l'Europe, le continent au plus fort potentiel de licornes" from Les Echos, a French newspaper about Economy

I put the article through translate and it appears that this is the (translated) section you are talking about:

"To achieve this selection of 50 nuggets, the Tech Tour did not scan only the continent, but the whole world. The overall result for Europe is very encouraging since there are 284 companies (including 38 French) , compared to 121 in 2015. They are 181 in China, and 833 in the United States, a country where, nevertheless, injected are five times higher than those irrigating European nuggets."

The criteria for a "nugget" seems to be a company valued between $100 mm and $1 B with annual growth > 50%.

The fact that the U.S. has more of these companies is not surprising. The US economy is equal to that in size of the EU but the tech sector in the US is much more robust. In the US there are 100,000 software and IT companies with less than 500 employees which would put them in this category. US tech makes up 25% of the global marketplace. The fact that there are more US companies that fall into this specific criteria is not surprising.

Whether or not this makes the US a better investment is a different question. The number of investable firms has nothing to do with return on investment. If all the European companies grew in value at an average of 10% and the US companies grew at an average of 5% then the European companies would be a better investment in terms of expected value. It may be that the European companies are much riskier and more likely to default which would offset that number. The number of companies that falls into this arbitrary and narrow criteria has nothing to do about the benefit of investing in either in Europe or the US.

Investment decisions are based on a multitude of factors. European regulatory risk, political stability, etc. may all be factors to consider when investing in a European company but it's difficult if not impossible to make a blanket statement about the quality of a country's investment opportunities.


How does protectionism harm the producers and workers in export sectors in the country implementing the protectionist policies? According to Protectionism

  (Protectionism) also reduce trade and adversely affect consumers in
  general (by raising the cost of imported goods), and harm the
  producers and workers in export sectors, both in the country
  implementing protectionist policies and in the countries protected
  against

How does protectionism harm the producers and workers in export sectors in the country implementing the protectionist policies?

Several channels I could think of:

As you said, protectionist policies tend to raise the cost of goods that could previously be imported. If these goods are intermediaries for the export sector, the firms in those sectors would be affected. This is the most direct channel I could think of.
You could also argue that the other countries will respond to the protectionist policy by imposing their own protectionist policies, which would eventually hurt the exporters. Think of US agricultural product exporters who got hurt as a result of the US' first imposing tariffs on Chinese exports through China's retaliatory tariffs.
In the longer run, protectionist policies keep firms without comparative advantage in business. Of course, there's an argument of "infant industry" who needs protection... but aside from that, without such policies, the resources employed by these "sunset industries" could be diverted and used more productively elsewhere.


Summary data on venture capital Is there a trade group, government, or other publication providing data on aggregate venture capital investment?

Specifically, I'm interested in how overall venture capital funding has changed over time, how it is divided by sector, and how it is divided by funding stage.

I'm specifically interested in data for the US.

The best resource I've found so far is Pitchbook's quarterly summary. It includes some degree of historical data, historical data by funding stage, and historical data by sector.


Is evolutionary game theory a useful modelling tool? Has evolutionary game theory ever been used to successfully describe an ecosystem or a reasonably closed subsystem of the economy? A frequently used example is about side-blotched lizards but I don't know of any others. And has it ever been proven that in this case the main factor is really relative fitness of strategies and not some other factor, such as the dynamics of gene recombination?

You're looking at it from a particularly biological point of view. I know nothing of side-blotched lizards, but I can say that evolutionary game theory is often used for equilibrium selection. In games with more than one equilibrium (whether it be Nash or otherwise), evolutionary game theory can allow us to select a particular equilibrium that is a better prediction than others in that a population is, in some sense, more likely to reach it and stay there than it is to reach and stay at other equilibria. Have a look at Population Games and Evolutionary Dynamics by William H. Sandholm if you want to see more of how economists use evolutionary game theory. What your question gets at is really more about creating a model with evolution to describe an ecosystem rather than evolutionary game theory, though the two are not unrelated.


Two Funds Separation & CAPM I've read that, concerning the CAPM, in equilibrium all portfolio weights are strictly positive. Why is that? You can also go short in the risk free asset right? And then you're on the right of the Tangency Portfolio.

Furthermore, I read that by definition, the MVP can contain short positions. How does this relate to my first question?

Hope you can help out!

An equlibrium is a property of an entire market, not of any given investor's portfolio. The CAPM is an equilibrium theory.

In equilibrium, all (nontoxic!) assets must have non-negative prices. Hence, the weights of all assets in the market portfolio must be non-negative as well. The market portfolio weights will be strictly positive only if the assets have positive value, i.e., if their prices are strictly positive.

Within an equilibrium, it may be possible that some individual investors will find it optimal to be short some assets. But that does not imply that it's possible for market portfolio weights to be negative.


Hayek influenced by? I am going to write an essay about Hayek's views on economic and political (individual) liberty. I am trying to find books to develop a positive discussion on the topic, but I do not know where to search. I started with J.S. Mill's "on Liberty" and Hobbes' "Leviathan", but I cannot find where Locke discuss subjects such as individualism. Also, I was thinking about reading anarchist idea on the topic as an opposing view. Is there any good reading I've missed so far?

Hayek's The Constitution of Liberty is in my opinion the place to start. It has a lot of references to John Stuart Mill (which whom he strongly disagrees, btw), and puts forward his argument on freedom. There is also a very recent (but rather long) analysis of Hayek's book (available online here), which dissects his influences.


Why did the Greek GDP grow so fast between 2002 and 2004? Here is the history of the Greek GDP. In 2004 year it was 239.7\$ billion, but in 2002 it was 157.7\$ billion. It grew by approximately 50%. How did it grow so fast?

You are using USD exchange rate GDPs, which are very volatile. The main effect is probably the fact that the Euro began to circulate in 2002, replacing the cheap domestic currency.

Here are some more general reasons why the peripheral countries did well before the recession:

There was abundant oil supply and the developed countries were splurging on FDI and debt in the periphery
Greece had just joined the Euro and was getting tons of transfer payments
The Euro was still fairly high and hadn't fallen yet.
Greece may have fudged its numbers a bit to get into the Euro.

Eur/usd since 2000:


"The citizens know their own utility functions better than the politicians" In informal discussions about public policy, I often hear the claim that the government should not intervene in a certain way, because the citizens know what is good for them better than the politicians. As an example, a discussion about a public housing program can proceed like this:

  Supporter: The government should help the poor by giving them apartments for a low rent.

  Opposer: It is better to help the poor by just giving them cash. For a fixed government expense, giving cash is better than giving apartments, since the poor can use the cash  either for renting apartments or for buying other things that are more important for them.

  Supporter: But if you give them cash, they might waste all the money on alcohol or gambling and will remain homeless.

  Opposer: A citizen knows what's best for him/her better than the politicians. If a citizen decides that the best way to use the money is buying alcohol or gambling, then this provides the highest increase in utility, and thus it is the best way to help him/her.

There are many things to say about this claim, but here I do not ask whether this claim is correct or not - my question is: does such claim appear also in main-stream economic publications? For example, is such claim mentioned in a paper published in a respectable economics journal (in the context of public economics)? If not, is it mentioned in a blog-post by a respectable economist?

This idea in the title of your question sometimes falls under the name of consumer sovereignty.†
Macmillan Dictionary of Modern Economics (edited by David W. Pearce, 1992):

consumer sovereignty. The idea that the CONSUMER is the best judge of his or her own welfare. This assumption underlies the theory of consumer behaviour and through it the bulk of economic analysis including the most widely accepted optimum in WELFARE ECONOMICS, the PARETO OPTIMUM. The notion is, however, challenged in that governments do provide MERIT GOODS, while ADVERTISING is held, especially by J.K. GALBRAITH, to distort consumers' preferences.

William Vickrey (Microstatics, 1964, p. 217):

consumers are to be considered the court of final recourse as to the relative satisfactions that they derive from different situations, and that they are capable of predicting accurately the satisfaction which they will derive from the consumption of various contemplated combinations of goods.

Abba Lerner ("The Economics and Politics of Consumer Sovereignty", 1972):

The basic idea of consumer sovereignty is really very simple: arrange for everybody to have what he prefers whenever this does not involve any extra sacrifice for anybody else. … One of the deepest scars of my early youth was etched when my teacher told me, “You do not want that,” after I had told her that I did. I would not have been so upset if she had said that I could not have it, whatever it was, or that it was very wicked of me to want it. What rankled was the denial of my personality—a kind of rape of my integrity. I confess I still find a similar rising of my hackles when I hear people's preferences dismissed as not genuine, because influenced or even created by advertising, and somebody else telling them what they “really want.”
… as an economist I must be concerned with the mechanisms for getting people what they want, no matter how these wants were acquired.

Joel Waldfogel ("Does Consumer Irrationality Trump Consumer Sovereignty?", 2005):

According to economic theory, individual utility—and social welfare—are maximized when individuals make their own consumption choices. This justifies the doctrine of consumer sovereignty that underlies standard lessons of economics: for example, that lump-sum grants are more efficient than price-changing subsidies or government grants in kind. An obvious corollary is that consumption choices made by others tend to generate less satisfaction than one’s own choices

Lester Thurow ("Cash Versus In-Kind Transfers", 1974):

the received wisdom of economics seems to support transfers in cash (the 'New Federalism') rather than in kind. Governments may modify the market distribution of purchasing power, but then should stand aside and allow consumer sovereignty plus competitive markets to work. Any further intervention lowers consumer utility below what it otherwise could be. In a federal government, presumably an ideal system of cash transfers would consist of unrestricted cash transfers from the federal government to individuals and to state and local governments, who would in turn make unrestricted cash transfers to individuals. The result would be an optimum distribution of income within each area with each individual free to make his own expenditure decisions.

†I avoid the term consumer sovereignty because it is a rather muddled and vague term that has at least two distinct interpretations other than the one given above. Rather than use the term consumer sovereignty  at all, to express the ideas discussed here, I would simply say that:

"Each individual is the best judge of her own welfare"; and
"In general, cash transfers are better than in-kind transfers".


How do saving and investment work in macroeconomics? It is stated that saving equals investment. However, suppose the interest rate rises and people save more. However, businesses will borrow less so investment decreases. Hence investment is less than saving now. What happened?

Kenny LJ and FooBar already did a great job of clarifying the broader conceptual issue here: in equilibrium, prices don't just increase or decrease without a cause (a shift in supply or demand), and any deviation from the equilibrium price will result in a surplus or deficit.

But since you're talking about interest rates, I assume that you're interested in what happens when a central bank changes the interest rate. This is legitimately very confusing, because at first glance it doesn't seem like the central bank is affecting either the supply of savings or the demand for investment - and hence, by changing the interest rate, it seems like the central bank will break the equality of savings and investment.

This isn't true. But it took me years to wrap my head around this, and I suspect that the vast majority of economists would get it wrong if asked to explain it (as evinced, perhaps, that someone as smart as FooBar misfired a little on this point). Indeed, John Taylor - an extremely well-known macroeconomist - recently fell victim to a fallacy here, as Miles Kimball pointed out (see point 7 here; as further proof of the difficulty of this topic, I'm not sure that Kimball got it quite right either!).

To clarify, let's split into two cases.

Case 1: flexible prices.

Note that what really matters for savings and investment is the real interest rate, which is the nominal interest rate minus expected inflation, while the central bank sets the nominal interest rate. Suppose that the central bank increases interest rates, and at first suppose that inflation expectations remain constant. Then as you point out, desired savings will increase while desired investment falls.

This means that there will be a glut in the economy: there will be more goods supplied than demanded. How is this resolved? In a world where prices are flexible, the price will fall until inflation expectations rise exactly enough to offset the central bank's increase in the nominal interest rate, leaving the real interest rate ultimately unchanged (so that desired savings and investment balance again). For instance, if we have a certain expectation of where prices will be next year, then when prices fall today, that fall creates more expected inflation over the next year.

This is the neoclassical, flexible price case, and it resolves your paradox in the simplest possible way - by saying that expected inflation will adjust such that real interest rates, and hence the desire to save and invest, are unchanged. But as you can probably guess, this is fairly unrealistic: prices don't always adjust that quickly, and it's counterfactual to say that higher nominal interest rates result in an equal rise in expected inflation.

Case 2: sticky prices.

Let's suppose that prices are "sticky", meaning that they don't change in response to events very quickly. This is the kind of assumption made in "New Keynesian" models (and, to some extent, "Old Keynesian" and monetarist models before them).

Indeed, for our purposes, let's suppose that prices are so sticky that expected inflation is a constant, so that a change in the nominal interest rate translates exactly one-for-one to a change in the real interest rate. (This is not a bad assumption empirically, according to Nakamura and Steinsson!) This means that we can't use the dodge from before: when the central bank pushes up the nominal interest rate, the real interest rate that matters for the savings/investment market rises too, and it seems like we should get some kind of disequilibrium.

Indeed, if not all sides of the market react right away, something like this might happen. Suppose that in response to an interest rate hike, consumers quickly increase their savings, cutting back consumption, but producers are slow to respond and keep producing the same amount of goods. (Let's ignore the response of desired investment to interest rates, which complicates the story but doesn't change the basic lessons.)

Then there will be a surplus of unsold goods, which will accumulate as inventories. This inventory accumulation is a form of investment (e.g. line 14 in the US national accounts here), so $S=I$ will continue to hold - but it will hold in a weird way, where consumers' saving is matched by producers' unintended inventory accumulation.

Alternatively, suppose that producers react right away. Then when consumers try to save more and consume less, producers will produce less, and there will be less income earned by workers and capitalists. How will consumers deal with this fall in income? Partly, they'll save less - because it makes less sense to save when your earnings are low relative to their usual level. Partly, though, they'll consume even less - which means that producers make less and incomes fall again, and so on and so on.

This is just the Keynesian multiplier, and the process ends when income $Y$ has fallen by $\Delta Y = \Delta S/MPS$, where $\Delta S$ is the initial increase in desired savings and $MPS$ is the marginal propensity to save out of income. At the point, the decrease $MPS\times \Delta Y$ in desired savings from the $\Delta Y$ fall in income precisely offsets the initial increase $\Delta S$ in desired savings arising from the higher interest rate. (If there is an initial fall $\Delta I$ in investment in response to the higher interest rate too, then this becomes $\Delta Y = (\Delta S - \Delta I)/MPS$; and it becomes more complicated still if investment changes in response to income, etc..)

In short, $S=I$ continues to hold because desired savings depends on income - which you can't see from the simple, partial equilibrium diagram showing S and I relative to the interest rate. Consumption (and therefore income) will eventually fall enough to cause a decline in savings that restores equilibrium.


Testing for weak instrument I have 3 endogenous variables in my model and I have identified 3 (or max 4) instruments. I wanted to know appropriate test for identifying whether the instruments are weak or not given that I have heteroskedaticity (cluster option is being used) in my data. I read that one should employ the Kleibergen-Paap F statistic in this case and compare with Stock and Yogo Critical values but in the case of 3 endogenous variables the table in their paper starts with a min of 5 instruments. I am confused as to which test to employ in this case. Any suggestions would be useful in this regard.

Staiger and Stock (1997) formalized the definition of “weak
  instruments” and most researchers seem to have concluded (incorrectly)
  from that work (or hearsay) that if the F-statistic on the excluded
  instruments in the first stage is greater than 10, one need worry no
  further about weak instruments.Stock and Yogo (2005) go into more
  detail and provide useful rules of thumb regarding the weakness of
  instruments based on a statistic due to Cragg and Donald (1993).
  Stock, Wright, and Yogo (2002) provide a summary of this work.

Weak instruments:  An overview and new techniques

Austin Nichols (2006)

While an  F-statistic cutoff of 10 may be a crude rule of thumb and not sufficient to conclude that a set of instruments are not weak, Stock, Wright, and Yogo (2002) have a table about weak instrument cutoff values that goes down to 1 instrument.

GMM, Weak Instruments, and Weak Identification


Research on the existence of utility functions? I've been having difficulty finding some research on indifference curves/utility functions and methods in constructing them in an applied manner.

Does anyone have any papers or pdf books to recommend?

The most famous one is probably Afriat's 1967 paper titled "The construction of a utility function from demand data".

You can also find the basic idea in Hal Varian's Intermediate Microeconomics: Revealed Preferences Chapter or you can read the generalized version of the theorem in the paper.


Monotonicity and continuity implies that all bundles are weakly preferred to 0 Suppose a consumer has a preference ordering $\succsim$ on $X$ that is complete. Show that if preferences are continuous and monotone, then $x\succsim0$ for any $x\in\mathbb{R}_{+}^{N}$, where $0$ is the $0$ vector in $R_{+}^{N}.$
My approach so far is the following.
Case 1. $x$ has more of all commodities, in which case the result follows from monotonicity.
Case 2. $x$ has more of only one commodity, in which case monotonicity is not sufficient. We want to show that $x$ is in the upper contour set of $0$, i.e. that $x\in\{\phi\in X:\phi\succsim0\}$. If we can show that there exists some sequence of bundles in this set with bundle $x$ as its limit, then the result would follow from continuity. But I am not sure how to do this.
Case 3. $x:=0$. Trivial.

Without loss of generality, suppose $\mathbf x=(x_1,0)$ where $x_1>0$. Consider the following sequence
\begin{equation}
\mathbf y^n=\left(x_1\Bigl(1-\frac1n\Bigr),\,\frac1n\right).
\end{equation}
Clearly, $\mathbf y^n\gg\mathbf 0$ for all $n\in\mathbb N$, and thus $\mathbf y^n\succsim\mathbf 0$ by monotonicity. It is also the case that $\lim_{n\to\infty}\mathbf y^n= \mathbf x$. Hence by continuity, we have $\mathbf x\succsim\mathbf 0$.


How does an increase in the minimum reserve requirement affect the monetary base? Monetary base is defined as currency+currency held by banks+deposits of banks at central bank. We assume that the current reserve is below the minimum reserve requirement. Which component of monetary base is affected?
Please note the distinction between money supply and monetary base.

In a fractional reserve banking system, the maximum portion of the money supply represented by bank-issued loans relates to the reserve requirement as $M_{loans} = \frac{deposits}{r} - deposits$ where $r$ is the reserve requirement expressed as a decimal.

For example, a bank holding \$1000 in deposit liabilities facing a reserve requirement of 1% is free to issue $\$1000/.01 - \$1000 = \$99,000$ in new money in the form of loans.  Doubling the reserve requirement to 2% would reduce this number to $\$49,000$.

Reserve requirements in the US depend on the total amount held by a given bank and range from 0% to 10%, with larger banks facing the larger requirement.  https://www.federalreserve.gov/monetarypolicy/reservereq.htm.  Some countries, like Canada, do not have a reserve requirement.  Banks determine their reserves according to their internal risk models, including assumptions on inflation risk.

Of course, banks don't have to lend money out until they hit the reserve requirement, and as noted in comments above, the money supply in the US is largely decoupled from banking operations thanks to quantitative easing measures.


I don't understand the MV = PY I'm not a native speaker so please excuse my English.

When they draw the result that the central bank's monetary supply determine the inflation rate, didn't they suppose many variables sticky?

With money supply matching up with the demand, they start their train of logic as follows. MV = PY, here the Y is fixed since it's exogenous, and V is fixed for the convenience of computation, so M controls the P... Okay I get that part, but what I wonder is that, when you fix the V, doesn't that make the k also fixed, which means kY, the money demand function is eventually fixed.

Then, how can they manipulate the M when they assumed M equals to the fixed demand of money?

Money demand is fixed because it depends on the real economy....that's an assumption.
So when M changes only P can change. That's the theory. It's not really a theory. Just an accounting equality, since V is never defined exogenously. It's bad economics. :/


What is Natural Capital? "European movements for degrowth, decroissance, are rooted in the idea
  that the perpetual growth assumption built into modern economics fails
  to take into account the depletion of natural capital. Hence the
  importance of sustainability."
  - The end of progress

What is natural capital?

Natural Capital is an analytic accounting approach: it is a system of putting a monetary value on environmental assets such as ancient woodland. It is also the name of those assets, within that approach. So it's one way of internalising the externalities of environmental change, within macroeconomic indicators.

For more, see in particular the work of Paul Ekins, including "A Framework for the practical application of the concepts of critical natural capital and strong sustainability" (pdf)

The philosophy of decroissance - of deliberately engineering a decline in traditional economic growth indicators - is one attempt to find an optimal and sustainable economic path, taking into consideration both the traditional economic indicators such as year-on-year changes in GDP, and changes to the population's wellbeing, together with changes to the national asset base, which includes changes in Natural Capital.


Why is Japanese Currency tied to gold? I was listening to a Bloomberg market update, and the journalist mentioned that the Japanese currency was down as Gold rose.

I've heard that investors park their money in Yen. Is the reason the Yen is down because investors are moving their money out of Yen and re-investing in gold?

I am using the answer function as I do not have enough rep. to comment.

Take note that gold, or most of the commodities available for trading, are priced in USD. By saying yen is down because of gold is therefore a rather speculative statement in my opinion.

Why I say this is because when the price of gold moves, it should have no effect on the JPY/USD quote. I think the journalist saw that there was a rise in gold prices, as well as appreciating USD/ depreciating JPY which led him to conclude JPY has depreciated in terms of gold prices. But how strongly are the two events correlated should be up to your own research and conclusions.


Why is the Chilean economy so much better than the rest of the countries in South America? Why does Chile have a better economy than all of the other countries in latin America?

It is labeled as a high income country by the IMF,
It is the only Latin American country that is part of the US Visa Waiver Program
It is one of two Latin American countries that is an OECD member.
Unlike some of the other developed countries in South America,Argentina , they don't have chronic inflation problems.

What was is that set Chile apart? I've read a little bit about how "The Chicago Boys" played a big part of the country's success. What did they do differently in Chile? Didn't a lot of the policies that they helped to put in place take place under Pinochet? I read that Chile's chronic inflation problem was also solved under then Pinochet regime. If so, how do you think that affects the legacy of Pinochet?

This is honestly a pretty complicated question - it's very difficult to pinpoint any exact factors that have led to Chile being so successful relative to its Latin American counterparts. I'm not particularly knowledgeable on the political factors, so I'll focus on some economic ones.

As a developing country, Chile has several factors that have really helped with exports.

Copper

Chile is by far the world's largest producer of copper. Copper is essential for electronics - meaning that Chinese factories have been buying massively - and Chile has been able to use its natural resources to boost its own economy. Copper is also far less volatile in price than oil; prices have only begun to go down because of slowing growth in China.

Unique position in Southern hemisphere

Since the seasons are swapped between the northern and southern hemispheres, Chilean farms can produce fruits and vegetables to sell to the Americans in the middle of winter. If you live in the States, go to a local supermarket in the middle of winter and look at the fruits. Some of them (for me, I see grapes) are from Chile.

Chile also exports fish. It has the longest coastline in the world, and has had some success in fish farming; both of these factors help Chile sell a lot of fish.

This is just part of why Chile has been successful (I haven't even mentioned government policy!) but hopefully it'll give you some background on why Chile has been more of a success story than other Latin American nations.

Source: http://atlas.media.mit.edu/en/profile/country/chl/#Exports


How to intuit a necessary optimality conidtion involving growth rate of Lagrange multiplier in Ramsey growth model? I encountered a necessary optimality conidtion involving growth rate of Lagrange multiplier in this note on Ramsey model

$$\frac{\dot \lambda}{\lambda} = \mathcal {v} - (f^\prime(k)-\delta-\xi)$$

in which $\mathcal {v}$    is the time preference rate, $\xi$   is the population growth rate, and $\delta$   is the depreciation rate.

Is there some way to make sense of it in a similar fashion as Euler equation?

The multiplier is equal to marginal utility of consumption. In case of CRRA utility (i.e. $\lambda_t = c_t^{-\gamma}$), its rate of growth is then related to consumption growth:

$$
\frac{\dot \lambda}{\lambda} = \frac{\mathrm{d} \log \lambda}{\mathrm{d} t} = \frac{\mathrm{d} (-\gamma \log c)}{\mathrm{d} t} =  -\gamma \frac{\dot c}{c}
$$

The condition can be then rewritten as

$$
\frac{\dot c}{c} = \frac{1}{\gamma} \left[ (f'(k) - \delta - \xi) - \nu\right],
$$
so that the consumption growth rate is proportional to return on capital (less rate of time preference), which is pretty much the same as Euler equation in discrete time.


Would directing more money to science and research and development be an effective response to recessions? Typical responses to recessions is to follow Keynesian economics and support demand. Would directing money towards science and research and development be a better long term solution to recessions because in the long run it will create new industries (that would drive demand higher) and drive costs lower (aka increase long run supply)?
I know innovation and technological progress are difficult and can take a long time to show benefits (for example the PC was invented in the early 70s but productivity due to computers didn't really jump until the 90s) so would this method be too slow to be effective?

tl;dr:
R&D spending cannot stimulate economy during just by increasing long-run aggregate supply because recessions are fluctuations around the long-run aggregate supply and not necessary affected by the long-run aggregate supply.
A in which spending on R&D in principle could be more effective in fighting recessions than other spending. The effectiveness of spending on increasing output through the multiplier depends on marginal propensity to consume. Hence, a redistribution of resources from people with low marginal propensity to consume to people with high marginal propensity to consume is helpful in increasing the effect government spending has on the economy. If you could argue that people who work in R&D have higher marginal propensity to consume than other people you could argue spending on R&D is more effective. However, I would be extremely surprised if this would be the case, because it is established that marginal propensity to consume decreases with people's incomes and people engaged in R&D tend to be one of the better payed workers (see For example above mentioned Blanchard et al).

Full Answer:
Not in general, even though spending on R&D might be beneficial for long run growth, it is not, in general, any more effective in stimulating economy in recessions or more effective response than some other spending. There are several reasons for this.
First, a recession (especially demand driven one) is, from a perspective of macroeconomics, a short-run phenomenon.  As opposed to microeconomics in macroeconomics long run has slightly more vague definition but scholars generally agree on that it is a very long time period.
For example, Blanchard et. al. in Macroeconomics a modern perspective offer probably most vague definition of long-run:

long run A period of time extending over decades.

More advanced texts such as Romer's Advanced Macroeconomics use the term long-run to define a period in which all relevant variables (wages, prices etc.) adjust to the state of the economy.
Depending on the book/research paper the definition might slightly vary but these definitions imply that in general recession is a short-run phenomenon, not a long-run one and this hold especially true for demand driven recession.
To clarify, recessions is defined as defined as a period of negative growth of economic output (and for statistical/research purposes often as negative growth of output for at least two consecutive quarters - again see the above cited Blanchard et al.).
Under either of the above definitions of long-run recession would generally not qualify as a long run phenomenon. I cant think of any historical example of recession spanning decades, and even if there were such example it would be an exception rather than the rule. Under the second more narrow definition recession could only occur in long-run if the economy's own long-run growth path would itself be negative which is very implausible.
Moreover, under both Real Business Cycle theory and New Keynesian theory of business cycles recessions are considered transitory phenomena. In both of these it does not really matter what is the long-run aggregate supply for the recession - in fact the business cycle occurs as economic output fluctuates along the long-run aggregate supply (See Romer's Advanced Macroeconomcis for more detailed treatment of business cycle theories).
Consequently, the long-run changes in aggregate supply do not affect recessions. Of course, it is desirable to increase the long-run AS as higher output will lead to higher material welfare but there is no reason why increasing  economy's productive capacity should have any effect on reducing the size of fluctuations themselves.
In addition, the idea that spending on R&D even affects rate of economic growth and hence the rate at wich long-run aggregate supply expands is somewhat contentious one. In a standard Solow-Swan growth model a economy will growth at an exogenously given rate no matter what is it's R&D spending as technological progress in such models is completely exogenous. In an endogenous growth theories (e.g. Romer model) spending on R&D actually increases long-run economic growth (and thus also rate at which long-run aggregate supply expands). Unfortunately, an empirical evidence is not yet settled on this issue and Solow model is the more accepted one (again see Romer's aforementioned book).
However, there is a way in which spending on R&D in principle could be more effective in fighting recessions than other spending. The effectiveness of spending on increasing output through the multiplier depends on marginal propensity to consume. Hence, a redistribution of resources from people with low marginal propensity to consume to people with high marginal propensity to consume is helpful in increasing the effect government spending has on the economy. If you could argue that people who work in R&D have higher marginal propensity to consume than other people you could argue spending on R&D is more effective. However, I would be extremely surprised if this would be the case, because it is established that marginal propensity to consume decreases with people's incomes and people engaged in R&D tend to be one of the better payed workers (see For example above mentioned Blanchard et al).
Thus to sum up, generally you cant claim that R&D spending is more effective in stimulating the economy and aggregate demand in the short-run than lets say social or infrastructure spending. It also does not make sense to talk about 'long-run solutions'  to recessions as they are by definition short-run phenomena (where I take the solution to mean response to a recession and not a policies that might try to lets say reduce likelihood of recession occurring in the first place).


Why is deflation bad for economy? As now many people are talking about the inflation of US dollar on the account of the COVID-19 pandemic and the printing of trillions of US dollar bills, the price of decentralized cryptocurrencies like Bitcoin, which are not controlled by any central banks has skyrocketed. Also these cryptocurrencies are deflationary, i.e. they tend to get value increasingly more expensive.
However, many economists state that deflation is worse than inflation. I just can't understand why. Will it not be good for people to have their money valued? Or the prices go down? Why is it bad for economy? Or is it just a story told to make people believe in it?

First of all it is important to make clear what economists are actually saying. Most economists believe that moderate inflation is better than moderate deflation, while high inflation or deflation would be considered undesirable. This is because both inflation and deflation are examples of price instability and in fact if all markets would be perfect, competitive and all prices fully flexible models tend to show that it would be desirable to have neither inflation or deflation over business cycle (you can see some example of such models in Romer Advanced Macroeconomics).
However, unfortunately we do not live in a world where all markets are perfect, competitive and where all prices are flexible. For example, wages might not be flexible because of minimum wage laws or because in many countries contracts are set for long term (for example, in the Netherlands where I live it is very difficult to fire someone with permanent contract or to force them to renegotiate their contract). However, even in countries where labor markets are more flexible it might be difficult to renegotiate wages or contracts quickly.
These rigidities create problem for an economy during recessions where in order for markets to equilibrate you usually need people to accept lower wages. Inflation, solves the issue above because it decreases people's wages in real terms without any need for negotiations. This can help to moderate recession because if real wages are too high during recession it will lead to above equilibrium level unemployment. In turn higher unemployment means people have less income to spend or invest which then just further reinforces the recession.
Also note while I focused mainly on wages the inflation has also other benefits as in the short-run it stimulates the economy (since higher product prices motivate firms to produce more before they realize they are higher due to inflation), it also helps to relax government budget constraint since government as a monopolist producer of money benefits from inflation and has some further benefits (see discussions of this in standard macro textbooks such as Blanchard et al Macroeconomics: A European Perspective or Mankiw Macroeconomics).
Lastly note that economy usually has no problem when it needs to increase aggregate wages during expansion. Wages are much more rigid downwards than upwards and consequently economy usually does not need deflation during expansions when real wages tend to increase.
Hence to sum up, most economists believe that it is better to have moderate inflation (most economists would argue around 2% per year), because it helps to 'grease' the wheels of economy. Inflation helps to moderate business cycle and to the extent that one agrees it is better to have milder recessions it is obvious policy prescription given our current understanding of macroeconomics.


Why's Federal Reserve buying individual corporate bonds, when corporations have been selling them successfully? On Jun 15 2020, the Federal Reserve is expanding its Secondary Market Corporate Credit Facility into corporate credit to now buy individual corporate bonds, on top of the exchange-traded funds it already is purchasing. I know the Fed is supposed to be apolitical.

  The program has the ability to buy up to $750 billion worth of corporate credit. Its March 23 initial announcement is largely considered a watershed moment for the financial markets, reeling from the coronavirus threat spread.

  The Fed says it is going to start buying individual corporate bonds

  Under the latest guidelines, the Fed said it will buy, on the secondary market, individual bonds that have remaining maturities of five years or less. Those purchases will go along with the ETFs the Fed already has been buying, which are balanced toward investment-grade indexes but also include some junk bond funds that track debt which had been investment grade before the crisis but had been downgraded after.

I'll  list just some corporations that succeeded in selling bonds. Total's over US$1 trillion at May 28 2020.

  Exxon Mobil Corp. and Verizon Communications Inc., braved the financial turmoil created by the coronavirus pandemic and sold a combined US$12 billion of bonds in a single day.

  Unlike previous years when companies deployed debt for acquisitions, almost all of the proceeds have been used to boost their cash. Boeing Co. sold US\$25 billion in the largest deal this year, followed by Oracle Corp.’s US\$20 billion. T-Mobile US Inc.’s US\$19-billion bond sale has been the only notably large offering for M&A, which helped finance its takeover of Sprint Corp.

  Companies even increasingly double-dipped to get additional funding, often within weeks of their first offering. Nearly 40 non-financial companies — especially consumer ones like Coca-Cola Co. and McDonald’s Corp. — raised more than US\$100 billion in debt through May 13, up from six that borrowed twice in the same period in 2019, Citigroup strategist Daniel Sorid said in a report.

  While several companies like Discovery Inc. and General Electric Co. have borrowed to refinance outstanding debt, many like Boeing and General Motors Co. have taken on additional obligations in what’s also been a period of rising net debt issuance. That may contribute to an even higher number of borrowers getting cut to junk this year.

  On April 13, 2020, Marriott International, Inc.  entered into a commitment letter providing for a 364-day senior unsecured revolving credit facility in an aggregate principal amount of $1.5 billion (the “364-Day Credit Facility”).

On June 5, Royal Caribbean Cruises Ltd. sold $2 billion of bonds on Thursday, the cruise liner’s second outing in the credit market in less than a month as its seeks to boost liquidity while the coronavirus keeps ships at port.

On June 8, General Electric Co said on Monday it reopened portions of its prior debt offerings for $3 billion in total proceeds to boost its cash reserves amid the coronavirus crisis.

On June 10, Delta Air Lines Inc. returned to the bond market within weeks of its last visit, looking to support its liquidity and repay outstanding debt.

On June 15 2020, American Airlines Group Inc. is in talks to raise new financing to help it weather the collapse in travel demand during the pandemic, according to people with knowledge of the matter.

"...Its March 23 initial announcement is largely considered a watershed
  moment for the financial markets, reeling from the coronavirus threat
  spread..."

As the quoted statement implies,
people are spurred into buying corporate bonds in the current environment precisely because of the Fed's previous announcement that it would purchase such bonds. The market has been---more precisely, believes it has been--- front running the Fed, which achieves intended effect of the policy. As shown by the examples listed, corporates have been able to tap into the bond market for financing. The Fed now has to demonstrate it can/will implement its announced program to signal its credibility, and thereby maintain the effectiveness of its current and future policies.


Positive interaction term with one negative component? Say I have an augmented growth regression, where my Y is GDP growth, and my Xs are the classical MRW variables, + international aid + corruption + the interaction term between the two.
Basically, I'm interested in seeing if lower corruption increases the effectiveness of international aid, which means I'd like to see a positive sign on my interaction term.
However, I also think I should see a negative sign on corruption, as the higher corruption, the lower should growth be.
Now, my questions:

Is this possible as I describe it? Can I see a negative component but a positive interaction term? In which case, how should I interpret it?
Or is there a reason the coefficient on corruption should be positive, and still indicate that higher "institutional quality" has a positive impact on growth by itself and also in interaction with aid?

Ignore the other variables. Let $g$ be growth, $C$ corruption and $A$ aid. You look at

$$g = \alpha C + \beta A + \gamma A\cdot C$$

You ask, "does lower corruption increases the effectiveness of foreign aid?"

For this you have indeed to examine the cross-partial derivative

$$\frac {\partial g}{\partial A\partial C} = \gamma$$

This is "how growth is affected at the margin by Aid, if corruption changes"... So if lower corruption increases the effectiveness of Aid, $\gamma$ should be negative, not positive...

...if your corruption index is mapped as "the higher the value of $C$ the higher the corruption".

If your "corruption index" is mapped in reverse (and so in reality it measures "non-corruption"), the anticipated sign reverses also.


What is this graph saying? This is from: https://eml.berkeley.edu//~dromer/papers/RomerandRomer.pdf

What does it mean that we are looking at a "tax increase of 1 % of GDP". What does a tax increase have to do with the GDP? If the tax is 20 %, and we raise it to 22 %, that's an increase of 2 percentage points, or 2/20 = 10 percent. What does GDP have anything to do with it?

The tax increase is first measured in dollars. Since the dollar amounts are large, and not comparable across countries, they divide by the size of nominal GDP. This is standard practice in discussing macroeconomic policy. This way, we can quickly judge the relative size of a quantity without having to run to an economic database to see statistics on the size of the economy.

In this case, they are looking at a tax increase that is assumed to equal 1% of nominal GDP. That is, if nominal GDP were \$100 billion, that is a tax increase of \$1 billion.


Survival Rates of Firms Jovanovic (1982) cites Du Rietz (1975) as a source for

Survival Rates of smaller firms are smaller than for larger firms

It's his doctoral thesis, and using Swedish data. Is there any recent evidence on this for the United States?

This is unlikely to be a published result in its own right, simply since it is publicly available here  (click on the row Firm Age).


Are there economic studies surrounding the efficacy of gun control? In light of a recent mass shooting in Nova Scotia, the Prime Minister of Canada intends to ban a lot of firearms. I have seen both sides of this debate, notably whether or not gun control is an effective strategy to reducing violent crime. Gun control itself is a supply-reduction policy whereas loosening gun control coupled with more severe penalties for offenders who use guns is a demand-reduction policy. Both sides of this have valid arguments, though, which one is more effective? This is an economics policy question.

I have not seen very much attention given to this topic from economics. Is there a well defined economics literature surrounding gun control and its efficacy (does it work)? Maybe econometric studies examining gun control laws imposed in certain states as natural experiments?

I am not simply asking "Does gun control work" as this would attract opinion-based answers. I am looking for studies that have examined the effects to better inform myself on the subject, on which I have not formed an opinion.

To answer the headline question directly: yes, there are.

While economists are among the central participants, this is really an interdisciplinary policy-oriented question. The Science of Gun Policy is a fairly recent and comprehensive review of the research overall, published by the Rand Corporation. It's over 400 pages long and I haven't read it. But if you glance through it you will get a sense of the many complex angles that this question has been examined from. Relevant outcomes are not limited to violent crime per se.

Table 21.1 on pages 332-333 summarize the evidence for which policy measures may affect which outcomes, according to the current evidence. Regarding violent crime specifically, there is moderate evidence ("two or more studies—at least one of which was not compromised by serious methodological weaknesses... and contradictory evidence was not found in other studies with equivalent or stronger methods.") to show:

waiting periods reduce total homicides
background checks by dealers reduce firearm homicides (evidence for private-seller background checks is less conclusive)
firearm prohibitions for individuals subject to restraining orders decrease intimate partner homicides.

I have not looked through the relevant studies for these particular findings to see which ones are specifically by economists, but in any event, these are probably the most conclusive findings you will find. Someone else might be able to give you a more in-depth look at the economics research per se.


Why does changing the value a currency help/hurt exports if sellers can just inflate/deflate prices to match the change? 1) I make widgets and sell them for $10.

2) The government reduces the value of my currency by 10%.

3) The market value of my widgets is the same as before, so I raise my
prices to $11 and I can still sell them them because the (real) value hasn't changed.

Following this chain of reasoning means that devaluation of a currency wouldn't encourage exports, but that's not true.

An increase in the value of another currency relative to mine may allow you to buy more of MY dollars, but the market value of my goods is still the same.  The only people who would notice a difference when this happens would be those holding currency and not assets.  Are those people the target of currency devaluation/appreciation, or is there something wrong with my reasoning?

What is missing in your reasoning is the effect of devaluation on the seller's profit.

To keep things simple, assume that the inputs needed to produce the widgets are all bought domestically. Then costs will not change following devaluation.  But prices in terms of domestic currency can increase, without as you say increasing the cost to foreign buyers in terms of their currency.  So profits per widget - selling price less costs per widget, both in terms of domestic currency - can increase.

There is another possibility.  A seller might take advantage of devaluation to try to increase total profits by increasing the volume of exports. If so, they could increase the price in terms of domestic currency by less than 10%, or perhaps not at all, with the effect of reducing the price to foreign buyers.  Unless demand is perfectly inelastic, this will increase the volume of export sales and, if domestic sales are unchanged, the total volume of sales.  An increased production volume may in the long run permit economies of scale, reducing costs per widget.

Either of these strategies could yield an increase in total profits.  Which yields the larger increase will depend on both the elasticity of foreign demand and the scope for economies of scale.

Through the profit motive, therefore, devaluation can certainly encourage firms to increase the value and/or volume of their exports.


Housing econometrics: interpretation of a quadratic variable I'm working a location based home pricing model and I have a regression which looks like this:

$$Price=\beta_0+\beta_1x_1+\beta_2x_1^2+\beta_3x_2+\beta_4x_2^2$$

where $x_1$ is longitude and $x_2$ is latitude.

I'm finding statistical significance in longitude, longitude squared and latitude squared but not in regular latitude itself. I know that in economic theory quadratic terms are used to capture the effect of diminishing returns, but in this case I find it odd that the quadratic is significant but the regular variable isn't.

How do I interpret these results?

When both the level and the square affect the dependent variable (and if the coefficient on the square is negative), we have a case where the variable in question initially affects positively the dependent variable, but after a point it affects it negatively.

When only the square is deemed to affect the dependent variable, then the effect is monotonic and accelerating, while its direction is indicated by the sign of the coefficient.

Real world interpretation: can you rationalize why latitude could have a monotonic effect, while longitude has not? Can you match this to the geographical areas you use data on? Is there anything special along latitude as regards the weather, socioeconomic conditions etc? And also, why longitude has an up-and-down effect? What could be the possible reason here?

These are the cases when empirical studies become thrilling: when unexpected econometric results provide new insights into the real world.


Can A Utility Function Take On Negative Values? Can someone provide a rigorous definition of a utility function? I had thought that a utility function only needs to the preserve the order of preferences. Thus a utility function can take on negative values as long as it preserves the order of preferences. Others have told me that a utility function cannot take on negative values. Is this a condition of a rigorous definition of a utility function?

A utility function can certainly be negative. The utility function is nothing more than a way to represent a preference relationship. This is an important conceptual point. In several theorems that typically show up in introductory texts, we show that sets of preferences that satisfy certain regularity conditions can be represented as utility function.

Also, there are different decision theory frameworks that allow the utility function to be transformed. You alluded to something like this in your question. In the traditional framework without uncertainty, the utility function is defined up to a monotonic transformation. Under certain kinds of uncertainty, we get Von Neumann–Morgenstern utility functions that are unique up to affine transformations. You can read more about this elsewhere. For now, the consider the following definition of a utility function. It is taken from Advanced Microeconomic Theory by Jehle and Reny (3rd edition):

A preference relation $\succeq$ is defined as follows:

where the axioms references are these:

Axiom 1: Completeness. For all $x^1$ and $x^2$ in $X$, either $x^1 \succeq x^2$ or $x^2 \succeq x^1$.

Axiom 2: Transitivity. For any three elements $x^1$, $x^2$, and $x^3$ in $X$, if $x^1 \succeq x^2$ and $x^2 \succeq x^3$, then $x^1 \succeq x^3$.


Why do falling oil prices take stocks with it? The price of oil fell again today and is now below $30/barrel for the first time in 12 years. Along with the fall came most of the US stock markets. If I didn't know any other information except for "the price of oil fell today", my gut reaction would be that the oil companies stock would fall, but the rest of the markets would rise (on average) due to the lower operations cost (and/or increased consumer spending) leading to an increase in profits. This should be especially true for the transportation industry, such as airlines, which consume a lot of gasoline. Yet, in the sharpest decline of the oil price in the past month, airline stock prices have dropped as well. Is my gut feeling incorrect, or is it being overpowered by a stronger force than the price of oil?

Update: Not long after I asked this question I did some more research on it and came to the conclusion that the premises for this question were wrong. I had been basing my question on what I had regularly been hearing/reading in the news, which was something similar to "stocks fell again today due to oil taking another dive". I added an answer which shows my conclusions. (I don't like to accept my own answer to my questions, so as of yet I've left it open.)

First of all, the fall in prices does not only concerns oil but most commodities, in particular raw materials such as iron ore, copper, nickel and many other, though it is indeed most noticeable for gas and oil. If you want to see the full extend of this drop in price, you can check out the IMF Primary Commodity Prices and their monthly reports, from which the following chart is extracted:

What are the causes of lower commodity prices?

There is no secret: the demand is falling and the supply increase. The demand is falling mainly because of the slowing growth in China. As China imports less commodities to fuel its industry, a large chunk of the demand on these markets has disapeared, leading to lower prices. But simultaneously, the supply keeps steady or increasing on most of the commodity markets. On the oil market for example, 2015 has been a great year of surge or expected surge: OPEC countries are committed to keep flooding the market, mainly because the Gulf countries can afford lower prices thanks to low production costs, while their competitor might not sustain low prices very long. Furthermore, the ban on Iranian exports is going to be lifted, as well as the ban on US exports. The market anticipate already this formidable future new supply, which drives prices down. The US are quasi independent as far as Oil is concerned, as the Shale gas industry has been rising in this country.

What are the consequences of lower commodity prices?

The first consequence is not really caused by lower commodity prices, but rather by the root of this price drop: China's growth is slowing. Therefore their is more uncertainty than before about this market. As a consequence of this, exporters to China begin to worry as well as firms who have factories in China, as a slowing growth and reduced imports might be the sign of a stabilization and potential lower inner demand. China being one of the world's biggest market, and one with a still relatively low penetration for technologies, it is indeed worrying. Thus, the stock of a lot of companies have been going down all over the world because of the situation in China, reflected in Oil prices.

Another very important factor comes into action: most firms in the commodity sector have huge debt. Indeed, this sector requires big investments to begin the exploitation. Among others, Glencore's $30bn debt has been discussed a lot, but it is only an example among many others. In particular, their are two areas that are under threat: shale gas and renewable. This two sectors require higher investment than others. For shale gas, because the exploitation is costly and technically complex. For the renewable, because the R&D is costly and the production as a whole is costly as it uses high technologies. Lately, the partial default of Abengoa gave us a taste of what might follow. The debt accumulated by this industry is by no mean sustainable with such low commodity prices.


How can you model/simulate the price/value of a good? Let's say that you have a good which price can change quite often because of its demand. Let's assume that you developed an algorithm that changes continuously the supply of the good to make its value as constant as possible and equal to $1. How would you simulate this system?

What I cannot figure out is how to simulate the changes in price of the good based on the demand. For example, let's take these as the initial conditions of the system:

$$initial-supply = 1000 goods$$
$$initial-value = $1/good $$
$$ production-rate-of-the-good=500 goods/second$$

Now let's assume that the demand is 500 goods/second. Does this imply that the price of the good will stay constant?

If the demand becomes 2000 goods/second for some interval, how will the price of the good change if the production rate remains the same?

I am a beginner in these concepts and I hope that this question makes sense

You're question is unclear...

If you're in a competitive market, as a supplier, you can only see the market price at a given time, where it is set essentially by all suppliers and consumers at once

If you're a monopoly, then you can estimate the quantitive demand based on price, simply because you control the entire market, and set the price level at the marginal revenue, where you gain the most

The only ways, other than a monopoly, to estimate demand are:

If demand is completely inelastic, like with fuel: people won't change their driving patterns on marginal changes in price
meaning quantity will remain the same in any case

completely elastic: price will remain the same no matter the quantity

A government set price, quantity or any other specific regulation on the product. Although, with "too much" regulation, in some cases a black market can be created, etc.


Combining Compliment and Substitue Goods Given
1) Perfect substitutes: X and Y
2) Compliments: Z and X
is it possible to generate a utility function that relates them?

X and Z must always be consumed together, and you can substitute Y for X.

I will leave the clarity of the question aside.

1) If what is meant by perfect substitutes/complements do not imply that these products are necessarily relevant for the preferences of an agent, then the answer is positive, with the utility function $U(x,y,z)$ being defined as
$$
U(x,y,z)=0 \text{ for all } x,y, \text{ and }z.
$$
2) If it is implicitly understood that increasing $x$ and $z$ together would yield a higher utility and one for one transfer between $x$ and $y$ consumption yields the same utility for the agent, then the answer is negative. The reasoning can be seen as follows.
$$
\begin{eqnarray}
\mathbf{U(1,2,0)}&=&U(0,2,0) \text{ by perfect complementarity}\\
&<&U(1,2,1) \text{ since $x$ and $z$ consumption together is good}\\
&=&U(0,3,1) \text{ since $x$ and $y$ are perfect substitutes}\\
&=&U(0,3,0) \text{ due to perfect complementarity between $x$ and $z$}\\
&=&\mathbf{U(1,2,0)} \text{ since $x$ and $y$ are perfect substitutes}\\
\end{eqnarray}
$$
leading to a contradiction. A general proof can be written when the assumptions are relaxed, but I believe this is sufficiently clarifying.


Stimulus (Rescue) Package With regard to the recent 2 trillion dollar coronavirus package, is the money just being printed or borrowed? A gentleman from the tax payer union claims the taxpayer will eventually be on the hook.

If it is borrowed and budget deficits become a problem then the Fed will step in anyway.

The USA situation is the same as with the BoJ, which helps fund a Japanese 250% debt-gdp ratio: the Fed's profits are donated to the govt anyway. Purchasing govt debt with printed money essentially cancels the debt as any returns paid on the bonds are donated back to the govt.

The 2t coronavirus package, unless a vaccine is found, will not nearly be enough to stop a depression. Helicopter drops will be needed for at least the next 6 months. However, the US only has about 110% (?) debt-GDP so it can easily afford another 20t stimulus before it even approaches Japanese debt levels.


Generalized KPR: Frisch Elasticity Consider the following version of KPR preferences (with $l$ being leisure):

$$ U(c,l) = \left(\left(c\right)^\gamma l^\omega\right)^{1-\sigma}$$

I'm after the Frisch elasticity:

$$ \frac{\partial(1-l)}{\partial w} \frac{w}{1-l}$$

Note that $\frac{\partial(1-l)}{\partial w} = -\frac{\partial(l)}{\partial w}$.

In general, one can compute the first component as $$  w\frac{\partial l}{\partial w} = \frac{u_l}{u_{ll} - \frac{u^2_{lc}}{u_cc}}$$

Hence, the Frisch elasticity is given by

$$  \eta = \frac{u_l}{\frac{u^2_{lc}}{u_cc} - u_{ll}}\frac{1}{1-l}$$

With the given preferences, I have that

$$ u_l = \omega l^{\omega-1} c^\gamma K\\
u_c = \gamma c^{\gamma-1}l^\omega K\\
u_{cc} = (2\gamma-1)\gamma c^{\gamma-2}l^\omega K\\
u_{ll} = (2\omega-1)\omega l^{\omega-2}c^\gamma K \\
u_{cl} = 2\gamma c^{\gamma-1} \omega l^{\omega-1}K
$$

and then the Frisch elasticity boils down to

$$ \eta = \frac{1}{\frac{4\omega}{2\gamma-1} + 1 - 2\omega}\frac{l}{1-l}$$

It makes sense to some degree: as $\sigma$ is the inter temporal elasticity of substitution, it does not appear here. However, the relative curvatures on $c$ and $l$ appear, which is fine. However, the elasticity is not independent of $l$. As I'm somewhat in the KPR environment (despite adding curvature to $c$), I didn't expect this.

Are my result correct? Does anyone have more insights on this matter?

Regarding the presence of $l$ in the Frisch elasticity, your result is correct. It is my impression that a Frisch elasticity free of $l$ requires very specific functional forms.

For example we know that the following utility function

$$U(c,l) = ln(c) +\alpha \frac {(1-l)^{1+1/v}}{1+1/v} \tag{1}$$
gives a Frisch elasticity independent of $l$ (and then $\eta=v$).

We could compare this with your utility specification and say "ah, so we must have additive separability to obtain a Frisch elasticity independent of $l$"...
No, this is not enough. Note that essentially $(1)$ does not express utility as a positive non-linear function of leisure but as a negative non-linear function of labor, and so the term $1-l$ appears. This is critical to eliminate $l$ from the final expression.

Because, if  we assume additive separability in general, then the cross-partial derivative is zero and we would be left with

$$\eta = \frac{u_l}{- u_{ll}}\frac{1}{1-l} \tag{2}$$

If the additively separable utility function is expressed as a non-linear positive function of $l$, and not as a negative non-linear function of $1-l$, $(2)$ would also be dependent on $l$.

--

Also, with your utility function specification, I do not find that $\sigma$ is out of the final picture. Instead I find

$$ \eta = \frac{\gamma(1-\sigma)-1}{(\omega+\gamma)(1-\sigma)-1}\frac{l}{1-l}$$

If $\omega = 1-\gamma$ this simplifies to

$$ \eta = \frac{1-\gamma(1-\sigma)}{\sigma}\frac{l}{1-l}$$

But these are treacherous algebraic calculations - you should recheck.


What is the general rule for classifying factor that only shift one supply/demand curve or two supply/demand curve? What is the general rule in words/mathematical description for a factor that only shifts one supply/demand curve or two supply/demand curves simultaneously?

It is pretty confusing to me, because no single textbook I found had such a rule.

Treating supply and demand as functions that relate the price and quantity for a good, a variable that appears as a parameter in both functions will shift both curves. In macro and micro theories, the details of the model will determine what these are.

When we consider supply and demand with econometrics, the fact that prices and quantities affect each other through two processes (are jointly determined)  results in endogeneity problems. Wooldridge's Introductory Econometrics has an excellent chapter on simultaneous equations models.


How helpful is Hotelling's Rule in determining the price of oil? After posting this question, I realized that there was quite a bit of controversy surrounding the use of Hotelling's Rule to estimate the price of oil.  Participants on this site seemed to be evenly split on the issue.

From an Economics standpoint, How helpful is Hotelling's Rule in determining the price of oil?

From my analysis, it doesn't make sense that the price of oil should depend on Hotelling's rule instead of the traditional supply and demand models.

For example, when fracking was "new," its use was limited, making the impact on the oil industry negligible.  Because the amount of oil produced was small, supply (for the most part) remained constant; businesses would also be incentivized to raise prices to market levels, increasing their profit margins.
However, as fracking became more widely used, the oil entering the market increased, affecting the price of oil (and, originally, causing concern for OPEC).
Years from now, when the Middle East is out of oil, companies will shift to drilling less accessible and more expensive oil, increasing the prices of oil for the consumer.  Eventually, the last barrel of the oil will either cost an enormous amount of money or will be worthless because of the rise in substitutes (solar, hydro, and wind power; electric cars).

Thoughts or other analysis?

It would be easy enough to subject the question to simple quantitative analysis - get the time-series of historical oil prices and away we go.

Or, we can take a quick qualitative approach and get the same answer:

The Rule just sets out what the most economically efficient path would be. That's nothing at all to do with how things happen in reality: it's just how things behave in textbooks for beginners. For example, it relies on an assumption of perfect foreknowledge; whereas, in the real world, we have previously-unanticipated innovation that radically changes both industry's cost structures, and the amount of known reserves - as your fracking example illustrates.