Help with this microeconomics exercise The question:

A price-taking farmer produces a crop with labor L as the only input.
His production function is:$$F(L) = 10L^{1/2} − 2L$$

He has 4 units of labor
in his family and he cannot hire labor from the wage labor market. He
does not face any cost of employing family labor.

a. Find out his equilibrium level of output.

b. Suppose that the government imposes an income tax at the rate of
10 per cent. How does this affect his equilibrium output?

c. Suppose an alternative production technology given by:
$$F(L) = 11 L^{1/2} − L −15$$ is available. Will the farmer adopt this
alternative technology?

My approach:

a. As the cost is 0 and the farmer is price taker, he will try to maximize his production, by FOC differentiating F(L) we L=6.25 where function is maximum. But he can employ L=4, hence his equilibrium level of output will be 4

b. Profit when he is taxed $$R(P,L)=PF(L)-0.1PF(L)=.9PF(L)$$
again he will maximize his production and by FOC L=6.25 but he will employ only L=4 because it's his maximum limit for L ,so equilibrium out will not change even after tax.

c. he will not adopt new technology because if we put L=4 in new production function then F(L)=3 which is less than production from his earlier technology.

Is my approach correct for all three parts?

Given the production function
\begin{eqnarray*} F(L) = 10 \sqrt{L} - 2L \end{eqnarray*}
marginal product is
\begin{eqnarray*} F'(L) = \frac{5}{ \sqrt{L}} - 2 \end{eqnarray*}
Since there is no cost for the family for using its own labor and it cannot hire from outside, family will maximize its production subject to the constraint that it has 4 units of labor.
\begin{eqnarray*} \max_{L\geq 0} && 10 \sqrt{L} - 2L \\ \text{s.t.} && L \leq 4 \end{eqnarray*}
Given that $F'(L)$ is falling in $L$ and is positive at $L = 4$ $(F'(4) = 0.5)$, family will use all 4 units in optimum. The corresponding output is $10\sqrt{4} - 2(4) = 12$.

With $10\%$ income tax, the objective of the firm will be to maximize total output net of tax which is equal to $0.9 F(L)$. Solution to this problem will stay the same as before because we are just multiplying the objective by a positive constant.

\begin{eqnarray*} \max_{L\geq 0} && 0.9(10 \sqrt{L} - 2L) \\ \text{s.t.} && L \leq 4 \end{eqnarray*}

Therefore, the equilibrium output is still $10\sqrt{4} - 2(4) = 12$ out of which 1.2 is paid as taxes and 10.8 is consumed.

With new technology the problem is
\begin{eqnarray*} \max_{L\geq 0} && 11 \sqrt{L} - L - 15\\ \text{s.t.} && L \leq 4 \end{eqnarray*}

The above problem again give $L = 4$ as the solution and the corresponding equilibrium output is $3$ which is way less than what the family had earlier. Therefore, it will not use this technology.


How to work out Price with only the Derivative of Profit Function? Best response functions are obtained by differentiating a profit function and solving for q.

In equilibrium, the firms produce 30. But, there's no inverse demand function, so how would equilibrium price be determined?

This question seems to be unanswerable. Even assuming a linear inverse demand function $p(Q) = a - bQ$ and a linear cost function $C_i(Q_i) = cQ_i$ the best response function will take the form
$$
q_i = \frac{a-c}{2b} - \frac{1}{2}q_j.
$$
From the information given, you cannot know if say $c=10$ and $a=55$ or $c = 0$ and $a = 45$, so it is impossible to infer the inverse demand function and the price. (And this is assuming linear functions everywhere.)

Also, notice that the exact market structure (Cournot or Stackelberg) is only given in question c). Yet this would also affect market price. Hence, the question is unfortunately just of bad quality and cannot be answered.


Straub and Werning, 2014, on zero capital taxation This is a test balloon to see whether these kinds of questions are appreciated/welcome on Econ.SE:

I had a hard time grasping Straub and Werning (2014) (no paywall). I understand the general environment, but could someone explain intuitively what exactly the mistake in the original Chamley-Judd calculations was, related to the optimality of zero-capital taxation?

You can download Judd's 1985 paper, here. and Chamley's 1986 paper, here.

The non-mathematical answer I believe is well described in Straub & Werning's last paragraph:

  "In quantitative evaluations it may well be the case that one finds a
  zero long-run tax on capital, e.g. for the model in Judd (1985) one
  may set $\sigma  < 1$ ($\sigma$ is the reciprocal of the intertemporal elasticity of substitution), and in Chamley (1986) the bounds may not bind forever,
  depending on parameters. In this paper we stay away from making any
  such claim, one way or another. We confined attention to the original
  theoretical results, widely perceived as delivering zero long-run
  taxation as an ironclad conclusion, independent of parameter values.
  Based on our analysis, we find little basis for such an
  interpretation."

Just above their Proposition 2, there is also another illuminating passage: S&W write:

  "(...)This shows that the solution cannot converge to the zero-tax
  steady state. Indeed, it actually proves the solution cannot converge
  to any interior steady state, since, we argued, the only possible
  interior steady state is the zero tax steady state."

In other words, it appears that Judd and  Chamley did not fully solved their models, but provided results conditional on the parameters being such that the steady state will be an interior one. S&W argue (I have not checked mathematical correctness), that, depending on the parameter values, the optimal solution may lead to a corner steady state in some aspect (see below for an example) -and in a corner steady state, the capital tax-rate will be positive.  This needs checking because Judd explicitly considers different values for $\sigma$ (so a plain mistake may be involved after all).

Now if you ask me, this mostly indicates that the tools used may not be well suited after all (or we have somehow "misused" them) to solve the specific theoretical problem, and so they are unreliable to affect policy one direction or the other (which, by the way, reminds me of my question...)

Because, exactly of what value is (for either theory or policy) to, say, empirically determine that the elasticity of intertemporal substitution is lower than unity, and then declare "the optimal taxation solution for a government that cares only about the workers entails positive tax on capital and zero consumption of workers" at the steady state?  (see Proposition 3, p. 11 of S& W). Who is going to contemplate seriously such a proposition for purposes of real-world policy?

ADDENDUM

$\sigma >1 \Rightarrow$ the intertemporal elasticity of substitution ($1/\sigma)$ is lower than unity. $\gamma =0\Rightarrow$ the planner puts $0$ weight on capitalists' utility. $c_t$ is the workers' consumption, $C_t$ is capitalists' consumption, and $g$ is governments own consumption (i.e. it is not the part of government budget that goes to the workers as transfers).

Again, I have not checked the mathematics here. Apart from what kind of economic content and relevance one could conceivably provide for workers consumption going to zero, another issue here is that, if the limiting tax rate is going to unity as government own consumption goes to zero, then what happens to the Tax revenues (which are not given to the workers as transfers because then their consumption could not go to zero?)


Regarding the assumption of Classical Linear Regression Model For the given model $y = x\beta + \varepsilon$, the four assumptions would be

The model is linear in $x$ and $\beta$, also additive in $\varepsilon$
The conditional mean of the error terms are zero (i.e. $E[\varepsilon|x] = 0)$
$Var[\varepsilon|x] = \sigma^2I_n$
Exogeneity of $x$ (i.e. either $x$ is fixed or independent of $\varepsilon$)

My questions is that in the case of the following situations which assumptions have been violated.

Case 1. $E[x'\varepsilon] \neq 0$ [I am not sure whether this implies the second or fourth assumption has been violated]

Case 2. $Cov(x_i,\varepsilon_i) \neq 0$ [I suppose the third one has been violated but could it imply the violation of other assumptions?]

Any help will be greatly appreciated.

I  assume that x is a scalar.( but argument is similar if it isn't).

They both violate condition 4) because they both say that x, the regressor, is not independent of $\epsilon$. Case 1 a theoretical statement about the regressor and the noise term. Case 2 is an empirical statement about some of the  actual regressor data and the noise term.   If you want a nice intuitive explanation of econometrics ( not a standard textbook exposition ), I highly recommend Peter Kennedy's text. I think it's called "econometrics"but I'm pretty sure he only wrote one book so you can google for him on amazon.


Relationship between short-run marginal cost(SRMC) and long-run marginal cost(LRMC) While reading Intermediate Microeconomics from Hal Varian, I fell short in understanding the relationship between SRMC and LRMC. I can see how in SR, when fixed factor is chosen at LR optimizing condition, then

                         c(y) = cs (y,k(y))

where, k is the fixed factor.

But at pg. 394 of 8th edition, in Appendix to Ch. 21 (Cost Curves), author writes,
"...the long-run marginal cost will consist of two pieces:
how costs change holding plant size fixed plus how costs change when plant size
adjusts. But if the plant size is chosen optimally, this last term has to be zero!". This is what I am unable to understand, how the 'second' term is zero!

Please, help me understand both intuitively and through calculus.

First, fix an output level $y^*$. Now you solve the (long run) cost-minimization problem conditional to this output level. We obtain the cost function $c(y)$ and the conditional demand function $k(y)$.

Suppose we are at a cost-minimizing size of plant, $k(y^*)=:k^*$, for this output level. If costs decreased when you increase plant size, then it would pay to increase plant size. This contradicts our choice of plant size. Likewise, if cost increases when you increase plant size, then it would pay to decrease plant size. Again, this contradicts our choice of plant size. So, $$\left(\frac{\partial c(y)}{\partial k}\right)_{y=y^*,k=k^*}=0.$$

The calculus treatment is already given in the mentioned page. Let me know if there is still something that bugs you.


Is there an alternative to currency to facilitate exchange of goods and services? Say I have collected a bunch of seashells. Tom wants to have my seashells. Therefore the seashells have value and I have just created wealth. And so Tom buys my seashells for \$10. But even though I have just created value, the amount of money hasn't increased. Instead, I have just caused inflation, since theres more stuff in relation to money. To compensate, the fed prints \$10 (or less) so the inflation rate remains stable and loans it to a bank.
Whats weird about this is, that even though I did all the work, tom became 10$ poorer yet both me and the bank became \$10 richer.
It almost seams money can only be used to exchange a fixed supply of things back and forth and can't actually account for creation of new wealth without "hacks" like the federal reserve.
So heres the question: Is there a system of exchange that would be similarly easy to money, but that accounts for the creation of wealth in a natural and , if possible, fair way?

First, there are many misconceptions and mistakes in your question.

You state:

Instead, I have just caused inflation, since theres more stuff in relation to money.

No, inflation is inversely related to output. If the amount of money is fixed and you create more valuable products that (ceteris paribus) creates deflation not inflation (see Blanchard et al. Macroeconomics: An European Perspective).

You state:

Whats weird about this is, that even though I did all the work, tom became 10\$ poorer yet both me and the bank became \$10 richer.

No this does not make any sense from economic perspective. Tom purchased those seashells from you for \$10 which clearly shows that Tom values seashells more than \$10 and you sold those seashells for \$10 which clearly shows that you value the money more than seashells. Hence after trade you both have higher value than before and you are both richer.
In addition, if Fed loans bank \$10 the bank did not became richer by \$10. Now bank has \$10 liability so their net wealth is zero (since \$10 in cash assets - \$10 in liability =0). So asserting that banks becomes richer just by borrowing from Fed is absurd. Banks earn money on later lending out the funds on intermediation margins and those intermediation margins are their payment for providing financial intermediation services (e.g. screening clients, having accessible branches, etc.).

You state

money can only be used to exchange a fixed supply of things back and forth and can't actually account for creation of new wealth without "hacks" like the federal reserve

No this again makes little sense. Money existed long before Fed. If you have fixed supply of money and amount of products and services in economy increases then prices across economy adjust (decrease via deflation).  That means that even with fixed money supply output can increase indefinitely. As a matter of fact even when Fed prints money to keep prices constant, in real terms value of output is still allowed to increase as in economics what matters is the value of goods and services (e.g. real output) what matters not how price change.
Now to your question:

Is there a system of exchange that would be similarly easy to money, but that accounts for the creation of wealth in a natural and , if possible, fair way?

No, alternative to exchange with money is barter, but barter is much less convenient and inefficient due to the problem of double coincidence of wants. With barter you always have to search for a person that both sells what you want and want what you sell.
At the fundamental level the only difference between barter and exchange with money is that the problem of double coincidence of wants is eliminated. Instead of searching for someone who sells what you want as well you just sell your seashells and then buy what you want from some another person. In the end you are still exchanging just goods and services for other goods and services just with extra step that makes everything more convenient (but clearly less obvious to people without economic training).
In addition I can't see how money is less 'natural' (whatever that is supposed to mean) than barter. Going with dictionary definition of natural as something not man made neither barter or money is natural.
Lastly, what is fair or not fair is not for economists to say (although I have hard time seeing any unfairness with concept of money per se) and requires moral judgements - for that you need to visit philosophy stack.


Marginal tax rate question In relation to an ordinary employee, the Prime Minister of Australia pays?
a) the same marginal tax rate
b) a higher marginal tax ate
c) a lower marginal tax rate

My answer for this was a), as it was explained to me that 'marginal tax rates' refers to the tables that are used to calculate tax, but others are debating whether it is b), as the Prime Minister typically falls under a higher tax bracket than others. Would someone care to explain?

The Prime Minister of Australia seems to earn around AUD $500,000 according to Wikipedia

This would give him a marginal tax rate of 45% plus supplements

The average for full-time employees seems to be close to AUD $80,000 which would make typical marginal rates either 32.5% or 37%

So the Prime Minister of Australia pays a higher marginal tax ate


Free Cash Flow and stock prices There is this concept of free cash flow.
Now, I have heard is the amount of money left after you reduce from the net income all other expenses that relate to labor and investment and so on.

That is the money that you can use freely, and it is a fortune you have gained.  If so, then why if a company has extra X in cashflow, its stock price doesn't grow as $\frac{X}{shares}$? Have you gained ideas too?
and if you haven't?

In fact, the difference in price is closer to 20 or 30 times.

It is however used in the calculation of intrinsic price.
So, I'd like to know how this magic is made.

Do you simply assume price/FCF is constant? Is it better than assuming PE is constant?

How can you roughly estimate how a company is worth given its FCF (and DCF)? and why is this factor of 20-30?

You can't really estimate stock prices this way because it's forward looking (i.e. it considers the future, not so much the past). Example:

Suppose company A lost \$100 million this year, but next year it's going to earn \$100 million. How much is company A worth? If you naively used the P/FCF ratio (price-to-free-cash-flow) you would value company A as worthless today. If everyone agreed with you I'd buy 100% of the company's shares and pocket $100 million next year.

However, this is itself unclear because how can anyone guarantee that the company will earn \$100 million next year? What if, for example, there's another COVID-19 pandemic that hurts the company's performance? Then maybe it will only earn \$50 million. On the other hand, perhaps one of the company's products turn out to be a smash hit, and it earns \$200 million instead. Ultimately, how much the company is worth is going to depend on how much money it makes in the future, and the future is unpredictable. That's why we have a market. Bulls will generally believe that the company will make more money in the future than bears, and therefore value the company differently.

There are other differences, which are entirely dependent on market sentiment. As an example, consider AMZN. Amazon is one of the darlings of the market because the stock price keeps going up even though the company barely makes any money (just look at the P/E and P/FCF). This is happening because Amazon has been fantastic at growing its sales & earnings. There's also the perception that Amazon is working on innovative, cutting-edge things (like Alexa). Therefore the market assigns AMZN a high price. If you based your modelling around free cash flow, you would not have bought AMZN, even though that stock has been extremely lucrative for its holders in the past several years. Of course, this doesn't mean your modelling is wrong; if market sentiment on AMZN shifts its stock price will likely crash by a lot.

In the world of modelling stock prices there is no "right" or "wrong" way. Different people will use different methods or metrics. Further, there is no foolproof way to make money; if there were, then everyone would use it and it would cease to make money.


The median as a welfare function In an economy with several agents who have different utility functions, it is common to define a welfare function, defined as an aggregate of the utility functions of the different agents. Then we can ask if and how this welfare function can be maximized.

AFAIK, the most common welfare function is the sum of utilities, also called utilitarian welfare. A more general function is a weighted-sum, in which each agent has a different weight. Such functions are well-studied. For example, it can be proved (e.g. Varian, 1976) that an allocation is Pareto-efficient, if and only if it maximizes a weighted-sum welfare function, for some choice of weights.

I am looking for references in which social welfare is measured using the median, or a similar statistic (e.g. a certain percentile), instead of a weighted sum. Particularly, I am looking for references about existence of an allocation which maximizes the medial social welfare in various settings.

John Rawls, in his book, A Theory of Justice, discusses the Maxi Min Principle, where he essentially says that societies can be compared based on how well they maximize the welfare of those with the lowest welfare (subject to caveats about certain basic human rights).  If the typical welfare function is a weighted average of individual welfare $U_i$ like: $$\sum a_i \cdot u_i$$ over all $i$, then this welfare function is $$\min u_i$$.
Since the minimum is the 0th percentile, I think this would just barely qualify as an answer to your question. I suspect that's not exactly what you had in mind, but it might be helpful.

This metric has appeared in a number of economics papers including Amartya Sen 's 1977 Econometrica paper,  On Weights and Measures: Informational Constraints in Social Welfare Analysis, The Aggregation of Climate Change Damages: a Welfare Theoretic Approach (Fankhauser, Tol, and Pearce (1997)), and Optimal Redistributive Taxation When Individual Welfare Depends Upon Relative Income (Boskin and Sheshinski (1978))

One more thing:
I've also seen a blog post that compares Swedish and American incomes at all income deciles and shows the level of percentiles at which a poor person would be better off in Sweden:

The Economist's View has a related figure for more countries.


How would you calculate the Price Elasticity of Demand if the Prices are constant, given that there is change in quantity demanded? I have data that I have collected suggests that the prices are more or less constant throughout the month whereas the quantity demanded changes on a day to day basis. Recalling the formula for PED, we can write:

$$PED= {P*\Delta Q \over Q*\Delta P}$$ we can observe that $\Delta P=0$ as I mentioned earlier, giving us $PED=infinity$. However in reality the demand is inelastic. Is there a way to handle this error?

Assumption while measuring the prices: The supply was constant.

These are tuples of the form (Price and Demand)

(275,30)

(275,80)

(275,20)

(275,70)

(282,0)

(282,50)

(282,100)

(282,80)

it does not make sense to use the formula for the elasticity like this.The PED is used for theoretical purposes in static equilibrium analisys. If you want to measure the elasticity empirically, then you must use the correct econometric technique, given your dataset.

One simple way is to run a linear regression (OLS) after a log-transform in your data. The parameter estimated captures de per cent sensibility of you endougeneos variable (demand) to your explicative varible (price).

Don´t forget to check if the parameter statistical significance, you may need more data.


Why do all/majority of loan contracts charge nominal interest rate instead of fixed real interest rate? As far as I understand, all/majority of loan contracts charge nominal interest rate, there are (almost?) no loan contracts that charge fixed real interest rate. Why is it so? Normally I would expect the directly opposite situation, with all (or the vast majority of) loan contracts charging the fixed real interest rate. After all, such loans would be more predictable compared to charging nominal interest rate. Borrowers and lenders would know that neither of them would benefit from inflation/deflation at cost of the other side, so borrowers would be more willing to borrow and lenders would be more willing to lend, real interest rate paid would be the same no matter what happens with the aggregate price level. Of course we will need to know the current inflation rate, but it's not like this information is secret, so it shouldn't be a problem.

There are practical problems with having 'real loan' even though theoretically they would actually be desirable (as paper "The Impact of Inflation on Long-Term Housing Loans" by Tschach, Ingo E.) if they would be practical.

We do not know what the current inflation is.

This statement is simply incorrect:

Of course we will need to know the current inflation rate, but it's not like this information is secret, so it shouldn't be a problem.

When you look at inflation statistics they do not report actual inflation, they report the estimation of what the inflation is. The inflation estimation is based on a basket of goods that is considered representative and most statistical offices try their best to appropriately capture inflation but it is still just the best estimate and it is generally agreed that all inflation measures have some non-trivial flaws. They for example often do not appropriately capture changes in quality of goods due to technology and it is also highly questionable how different items should be weighted. Those two issues are just an examples there are many more, you can have a look at this investopedia article that explores the main issues in measuring inflation (but not all).
This issue could be circumvented by bank and client agreeing that some inflation measure such as CPI would be considered the inflation but that would cause some serious trouble down the road. For example, what if statistical office decides to update its CPI measure in a way to benefit borrowers or vice versa? The way we measure inflation evolves over time and thus using some real loan based on an ever evolving measure instead of actual inflation would not eliminate inflation related risk completely.

Inflation is rarely reported in real time.

Even the best statistical offices usually have about 3-1 month lag in measuring inflation and the most recent measurements of inflation can relatively often be revised later.
This creates another problem as it would make it very difficult to determine each month what the inflation is and hence what actually should be the interest rate paid. All loans would have to be paid with a lag that could even differ per country and due to the possibility of revisions banks would have to send out refunds or send out requests for customers to pay more as the revisions occur.
In future it might be possible to track inflation in real time. There is for example the billion prices project that attempts to do so. However, measuring inflation in real time is still in its infancy and the projects also has its own additional problem of being biased toward goods that are sold by e-commerce and it does not solve the problems in 1.

Even though you are correct in saying:

Borrowers and lenders would know that neither of them would benefit from inflation/deflation at cost of the other side, so borrowers would be more willing to borrow and lenders would be more willing to lend

this is basically what the authors of the paper linked above show in their simulations. You are not correct in stating that this would make the loan more predictable. It eliminates the inflation risk of the loan but as Giskard points out you are paid in nominal terms not real terms so the real value you need to transfer back to the bank is not more predictable per se.


Confused about relation of inflation to a currency's depreciation? I'm a complete newbie to this foreign exchange stuff and was reading about the impact a country's price levels have on exchange rates.

So from what I understand, the general idea is that inflation leads to a decrease in value of a country's currency (depreciation).

increased prices in goods in a country --> the country's exports will be less competitive --> so demand for the exports fall --> demand for the country's currency falls --> and this leads to depreciation of currency

however, it's also said that:

with a depreciated currency -> imports will be more expensive BUT the country's exports will be cheaper and more competitive --> then won't demand for currency rise instead?

If so, doesn't this contradict the first statement? It seriously confuses me. Inflation that leads to depreciation will then lead back to appreciation of the currency again?
I know I'm probably missing something here, but this just makes no sense from the way I picture it.

Your confusion might be mitigated if you thing about these as short- and long-term effects.

That is, the more immediate effect of inflation is that exports fall, demand for currency falls, and so the value of currency falls.

This causes imports to fall, domestic spending to increase, and foreign demand for domestic goods to increase. That is, exports rise. This increase in exports causes an increase in demand for the home currency and this puts upward pressure on the currency.

The short-term effect of inflation is that we have currency depreciation. The long-term effect is less clear cut, since the depreciation should trigger an increase in foreign demand which should counteract the initial effect of inflation.


Is there consensus on why inflation is not rising? I find it kind of odd that we are not experiencing any significant inflation recent years despit the low rates and quantitative easings.

Is there a consensus why this is the case among economists? If not, what are the mos popular explanations?

I am refering to countries in weastern EU and USA.

If your question is about Europe, yes there is. Inflation took time to rise because commercial banks have used injected liquidities to consolidate their balance sheet instead of stimulating demands by creating credits (that did not previously exist when they make loans). This is called the money multiplier. The multiplier was somehow blocked (to ~$1$) because of their precaution behavior.

On March 10, 2016, the ECB has even considered the possibility of giving 200€ per month and per person to stimulate inflation. And this with the main idea of circumventing the commercial-bank system. This unconventional monetary policy is called monetary helicopter.


Housing Supply Elasticity: Proxy for Exogenous House Price Movements Mian and Sufi (2014) say

  We use individual and zip code level data, and exploit cross-sectional variation in house price growth to estimate the impact of rising home values on borrowing and spending. We use the Saiz (2010) housing supply elasticity of a CBSA as an instrument for house price growth, and focus on the heterogeneous effects of house price gains on borrowing and spending.

And, later

  The question we seek to answer is: Are housing wealth shocks also shocks to cash on hand? We answer this question by empirically estimating how changes in housing wealth affect
  household borrowing and spending decisions.

I miss the step on how to get housing wealth shocks as opposed to general equilibrium changes in housing prices. In general, changes in housing wealth are not necessarily exogenous and unanticipated. How does the housing elasticity help to extrapolate exogenous price movements in the housing sector?

The idea is that all else equal, areas with inelastic housing supply (e.g., areas that are surrounded by water, so you can't just build some more houses "over that way") will see greater house price growth, which increases household wealth, and that these differences in supply elasticity are exogenous and uncorrelated with other things that matter. In Saiz (2010), he uses terrain and water data to determine the degree to which the housing supply in different metropolitan areas is constrained by geographic features.

It's a method that really only works (to the extent that it does) when house prices are rising, which would induce housing construction. If house prices are falling, supply elasticity tends to be pretty low— we don't tend to bulldoze homes at a particularly high rate.

With respect to the degree to which house price changes are anticipated, I think the best work on expectations is that done by Case and Shiller over the years, most recently Case, Shiller, and Thompson (2012).


Is there any overlap between evolutionary and behavioral game theory Evolutionary and Behavioral game theory are two different techniques. Both seem to be inspired from biology. So is there any overlap between their theory?

Are the two complementary to each other in some way or instead they overlap with each other? Specific examples in real-world domain would be helpful.

I don't think behavioral game theory is inspired much by biology. Rather it's motivated by the discrepancies between theoretical predictions and the choices observed in lab experiments. It seeks to improve the predictive accuracy of game theoretic models by introducing behavioral assumptions that allow players to behave in ways different from the traditional Bayesian rational benchmark.

Many behavioral game theories such as level-k and cognitive hierarchy are non-equilibrium theories. This is in contrast to evolutionary game theory which studies the stability of equilibria with respect to selected evolutionary dynamics.


Where can I get pizza data I'm studying the pizza market, especially in Utah, and I would like to know if there are any pizza data sources. Can some kind of sales tax data be used to study pizza? Is there a straight up pizza study that collected data? How creative do I have to get to find data? Or how much might I need to pay to have someone collect it?

Any data related to these things will be awesome: pizza input markets, pizza production, pizza demand, pizza labor, pizza firms, anything pizza and economics.

I'm not sure exactly what variables I'm after, I was just going to see which are out there first. I put in a request to Yelp.com to get access to their data sets, but it's only allowed for a few universities right now, not mine. We're on a wait list. Other than that, I'm kind of stumped.

It might not get you very far, but a start would be Google Trends:


What are the general charateristics of matching markets that make then different from regular markets? I would like to know what exactly makes matching markets different from regular markets. If you could illustrate with examples to help me understand better, that would be nice. Thanks in advance.

The deviance from a "normal" market boils down to how that market clears. Matching markets are markets in which prices alone don't clear the market, as thought in mainstream economic theory, and so the market is clear using the help of other institutions.

Take it from Alvin Roth's informative EconTalk interview:

"...labor markets are like these--they are among the markets that I've studied. And in labor markets, as in other matching markets, you can't just choose what you want. You also have to be chosen. So, labor markets have all these other institutions about applications and interviews and offers and acceptances. And sometimes those are used for competitive purposes in various ways. So, for example, right now if you knew someone who was just graduating this month from a prestigious law school and was going to take his first job as a clerk for an appellate judge, there's a good chance he would have gotten that job two years ago, when he was a beginning second-year law student. And he would have gotten that job with an exploding offer--which meant that he had to say yes or no when the offer came, before he could find out what other offers might be coming...And so that's a market that doesn't work the way we normally think a well-functioning market should work, where you can consider lots of alternatives. But it's one of the ways that judges compete with each other. And that's a market where the judges can't compete on price because Congress sets the wage for clerks. But it turns out that other markets work that way, too. When private equity firms are trying to hire young investment bankers, they do the very same thing. When law firms which pay salaries very freely, try to hire new associates, they do the same thing--often with competition through summer associateships while people are in law school. So that new lawyers in general will often, as they go to their jobs after finishing three years of law school, have been committed to those jobs for years."

In some other instances, consider trying to work for Stackexchange, where you can't just choose to work here: you have to be hired. Also this company doesn't actually try to lower it's price in order to clear the market like with commodities, but it will set a wage wide enough to be attractive to many people, in which it gets the opportunity to choose whomever it believes to be the best fit for their company. Or if we want to reverse the situation where you're a well sought after worker (perhaps a sports player, or a consultant) firms will express their strong interest in hiring you, but you can't just be hired because you have the opportunity to pick which of the employers you want to work for. I hope this helps.

Oh, and Consider Roth's book on the subject matter. Cheers.


Has anyone peer-review researched investors buying another, but similarly spelled, ticker? GLDD - Believing Robinhood users are dumb is profitable... : wallstreetbets

Academics need to research the correlation between performing tickers and their closely related (only in name) cousins.

Have any academics done so?

Has any forensic linguist investigated the morphological traits behind such "false friends"?

E.g. in February 2020, many investors misconstrued ZOOM for their desired ZM (Zoom Video Communications). Similarly, I wonder if some investors misconstrued GLDD (Great Lakes Dredge and Dock Company) for their desired GLD, or KDKN (Kodiak Energy) for their desired KODK (Eastman Kodak Company).

Yes, this has been done. Balashov and Nikiforov (2019) studied firms with similar tickers and found 25% had comovement in turnover and estimated 5% of turnover was due to investor confusion


Directed search with privately informed buyers and capacities My question seems to be a basic one and there should be a rather well-known reference in the (IO or labor) search literature.
I will upvote (and comment on) any relevant answer and I will accept an answer pointing to a reference addressing exactly the model below.

In the simplest case, consider the following model:

There are 2 sellers ($j \in \{1,2\}$) and each seller offers a single good.
In stage 1, both sellers set an individual price $p_j$.

There are 2 buyers ($i \in \{1,2\}$) who are privately informed about their valuation $v_i$ which is an iid draw from some distribution $F$.
In stage 2, buyers observe the prices and decide which seller (if any) they want to visit.

Buyers have single-unit demand. Buyer $i$'s payoff is $v_i - p_j$ if they got a good and zero if they got rationed or did not visit any seller. Sellers do not have opportunity costs (and cannot produce additional goods), so their payoff is $p_j$ if they trade.

A guess for a reasonable equilibrium candidate: In equilibrium, both sellers post the same price $p_1=p_2=p$, buyers with $v_i<p$ abstain from buying and high-value buyers ($v_i\geq p$) randomize equally over both sellers.

To evaluate deviations, it matters what happens when. say, $p_1 > p_2$.
Of course this depends on the rationing function, i.e., what happens when two buyers show up at the same seller. Fixing a certain form of rationing (random rationing, efficient rationing, introducing an underlying model in which sellers have additional preferences over customers, etc), one can express the expected utility from visiting each seller in a reduced form and solve. Now I guess my question is: How is this done in the literature (which I find interesting, but don't know much of)? I am happy about references and suggestions and I will provide some myself.

I am aware of classic models such as Kreps & Scheinkman (1983). I am also aware of Burdett, Shi & Wright (2001), but note that their buyers are ex-post symmetric as both of them have value $v_i=v=1$.

I am also aware of the literature on competing auctioneers, e.g., Peters & Severinov (1997) and Virag (2010). The buyers' equilibrium selection strategy in such models is quite cool: If sellers post second-price auctions with reserve prices $r_1>r_2$, buyers of value $v<r_2$ abstain from buying, buyers of value $v \in [r_2,x]$ visit low-price seller 2 with certainty and buyers of value $v>x$ visit each seller with probability $1/2$. One can solve for this cutoff value $x>r_1$. Under a condition on the lower bound of $F$, there is indeed an equilibrium in which both sellers set the same reserve price (which is below the lowest possible buyer value).
Knowing this, one can write down an underlying wasteful costly rationing process that essentially replicates the buyers' selection equilibrium above. However, I find this a little far fetched.

Any other approaches? As I said, this is not my literature - I just find the problem too basic to be neglected.


Solving Stochastic Dynamic Optimization Problems: A difficulty with Lagrange Multipliers In Wickens' Macroeconomics book, in page 552, the author states the following:

«The stochastic problem can be solved using the method of Lagrange multipliers, but there is a problem with this solution.»

However, if I'm not mistaken, it's very usual to see other books using lagrange multipliers to solve this type of problems. Are they wrong? Am I wrong? Is there a way to go around this apparent limitation?

Any help would be appreciated.

Here's a the image:

Following a comment exchange below another answer, the critical detail of a link offered by the OP as an example of by-passing/ignoring Wicken's comment/argument, and one different from Wickens' formulation, is that in the link's eq. $(4)$,

$$\lambda_t = \beta E_t[\lambda_{t+1}(1+r_t)]$$

the multiplier $\lambda_{t+1}$ appears together with the interest rate of period $t$, which is assumed part of the information set at $t$ and so not a random variable in the specific equation. So the authors can proceed in the 2nd line after eq. $(6)$ to write the marginal rate of substitution as they do, without the need to assume uncorrelatedness between the multiplier/marginal utility of consumption, and the interest rate.

This goes back to how they formulate the income resource constraint (page 2 middle), where it is essentially assumed that the household has, at the beginning of period $t$, available assets or debt $(1+r_{t-1})B_t$. The authors explicitly discuss this "timing convention" immediately after eq. $(1)$

  "Note a timing convention -$r_{t-1}$ is the interest you have to pay today on existing debt. $r_t$ is what you will have to pay tomorrow,
  but you choose how much debt to take into tomorrow today. Hence, we
  assume that households observe $r_t$ in time $t$. Hence we can treat
  $r_t$ as known from the perspective of time $t$."


Help with a choice under uncertainty exercise Let $u$ and $v$ be utility functions (not necessarily VNM) representing $\succsim$ on $\mathcal{G}$. Show that $v$ is a positive affine transformation of $u$ if and only if for all gambles $g^1, g^2, g^3$ in $\mathcal{G}$, with no two indifferent, we have:

$\frac{u(g^1)-u(g^2)}{u(g^2)-u(g^3)} =\frac{v(g^1)-v(g^2)}{v(g^2)-v(g^3)}$

I have tried to figure it out, but I am not sure where to start. Anyone who got a hint for me?

Suppose $v=\alpha + \beta u$ for $\beta > 0$ i.e. $v$ is a positive affine transformation of $u$, then
$$ \frac{v(g^1) - v(g^2)}{v(g^2) - v(g^3)} = \frac{\alpha + \beta u(g^1) - \alpha - \beta u(g^2)}{\alpha + \beta u(g^2) - \alpha - \beta u(g^3)}  = \frac{u(g^1) - u(g^2)}{u(g^2) - u(g^3)}$$

Converse is not true because $v=-u$ also satisfy

$$ \frac{v(g^1) - v(g^2)}{v(g^2) - v(g^3)}   = \frac{u(g^1) - u(g^2)}{u(g^2) - u(g^3)}$$

for all $g_1, g_2, g_3\in\mathcal{G}$.

Added Later

By the way, converse is also true. We are given this information that $u$ and $v$ both represent $\succsim$. So, $v=-u$ is not a valid example as they cannot both represent $\succsim$.

Here is the proof of the converse:

Fix any two lotteries $b$ and $w$ such that $u(b) > u(w)$ and consequently, $v(b) > v(w)$. For any lottery $g\in\mathcal{G}$, we know that the following is true

$$ \frac{v(g) - v(w)}{v(w) - v(b)}   = \frac{u(g) - u(w)}{u(w) - u(b)}$$

So, $$v(g) = v(w) + (v(w) - v(b))\frac{u(g) - u(w)}{u(w) - u(b)} $$
or equivalently,

\begin{eqnarray*} v(g) & = & v(w)  - \frac{v(w) - v(b)}{u(w) - u(b)}u(w)+ \frac{v(w) - v(b)}{u(w) - u(b)}u(g) \\ & = &  \left(\frac{u(w)v(b) - v(w)u(b)}{u(w) - u(b)}\right)+ \left(\frac{v(w) - v(b)}{u(w) - u(b)}\right)u(g) \end{eqnarray*}

Therefore, $v = \alpha + \beta u$ where $\displaystyle\alpha = \frac{u(w)v(b) - v(w)u(b)}{u(w) - u(b)}$ and $\displaystyle\beta = \frac{v(w) - v(b)}{u(w) - u(b)} > 0$.


Stiglitz: movement capital vs labor, effect on living standars Stiglitz argues that increasing free movement of labor would be a much more effective way of increasing living standards, compared to free movement of capital (see for instance this video).

Does someone know which theoretical frameworks have been used to derive such conclusions? (I am looking for papers published in peer reviewed journals or theory books).

The Roy model ( a much celebrated model in labor economics) deals with this issue. It can be arguably be directly mapped to this issue.


Situations where revelation principle may not hold Revelation principle is a powerful statement regarding Bayesian Nash equilibrium. However it may not hold always, as where players do not fully know their preferences, or when preference elicitation involves cost.

What are other situations where revelation principle is not applicable? Are there papers which have circumvented these issues with alternate versions of the principle?

You might want to be a little more precise about what you mean by "Revelation Principle" as there are many formulations of the "Revelation Principle" out there, some of which are stronger than others. Each of these formulations makes a different claim and relies on a particular set of assumptions. Of course, the claim will often fail to be true if some of the assumptions are false.

(The following is from notes I got from a microeconomics class.)

Consider for instance the following version of the revelation principle, which is from Repullo (1985), Review of Economic Studies:

  Repullo's Revelation Principle : Let $g$ be a dominant strategy mechanism for the game $\Gamma \equiv ( g, U_1, \dots, U_n)$, where $g$ is some game form. For each equilibrium selection function $s : \Theta \rightarrow S$, there exists an equivalent direct dominant strategy mechanism $h$ to $g$ (where $\Theta$ is the set of types). If in addition the equilibrium selection function $s^* : \Theta \rightarrow S$ is surjective, then the dominant equilibrium outcome under $h$ are a subset of the dominant equilibrium outcome under $g$ for all $\theta \in \Theta$.

The bold part is important. If it is not satisfied, there may still be a non-truthtelling  equilibrium in the equivalent direct mechanism. An example is provided in Repullo (1985), Review of Economic Studies pp 223-229.

$$A \equiv \{a,b,c,d\}$$
$$ \Theta_1 \equiv \{ \theta_1', \theta_1''\}$$
$$ \Theta_2 \equiv \{ \theta_2', \theta_2''\}$$

$$ \begin{array}{c |c c c c}
 & a & b & c & d \\ \hline
u_1(\cdot, \theta_1') & 2 & 4 & 2 & 4\\
u_1(\cdot, \theta_1'') & 1 & 0 & 2 & 4\\
u_2(\cdot, \theta_2') & 2 & 2 & 4 & 4\\
u_2(\cdot, \theta_2'') & 1 & 2 & 0 & 4\\
\end{array} $$

$$S_1 \equiv \{s_1',s_1'',s_1'''\}$$
$$S_2 \equiv \{s_2',s_2'',s_2'''\}$$

The game form is

$$ \begin{array}{c |c c  c}
 & s_2' & s_2'' & s_2''' \\ \hline
s_1' & a & b & b \\
s_1'' & c & d & c \\
s_1''' & c & b & a \\
\end{array} $$

On can check than the following is an equivalent direct mechanism

$$ \begin{array}{c |c c  }
 & \theta_2' & \theta_2''  \\ \hline
\theta_1 & a & b  \\
\theta_1 & c & d \\
\end{array} $$

Yet, when the types are $(\theta_1',\theta_2')$, although telling the truth is a dominant strategy, any other report of preferences is also a dominant strategy. This may be quite bothersome as it means that for some configuration of the types, saying the truth is only one equilibrium among others. As a consequence, we have no real guarantee that "telling the truth"  will be played (it may even be that truth-telling is Pareto dominated by another equilibrium. Using a focal point argument, this may further undermine the relevance of the truth-telling equilibrium).

The above issue is due to the fact that in the original game, some strategies are never played, which is ruled out if $s^*$ is surjective. So Repullo's version of the Revelation Principle (requiering that every dominant strategy equilibrium outcome of the equivalent game be among the equilibrium outcome of the initial game for every possible configuration of the types) only holds if the equilibrium selection function be surjective, and fails otherwise.


How to find the discount rate at which two deals are equally attractive? Dealer A will sell me a car for \$20,000 with \$2,000 down and the remaining $18,000 paid back over 36 months at a rate of 8% compounded monthly.

Dealer B will sell me the same car for \$19,500 with \$4,000 down and the remaining $15,500 paid back over 36 months at a rate of 10% compounded monthly.

(I assume these interest rates are APRs, since having them be monthly rates results in you paying more in interest than the price of the car.)

I'm asked to find the discount rate at which these deals are equally attractive. I assume this means the rate at which if I were to forgo buying a car and deposit amounts equal to the payments instead, I would end up having the same present value for both deals.

But when I try to calculate this I run into problems. I calculated the monthly payments for both deals, but then when I try to equate their present values and solve for the discount rate, I get that there is no positive discount rate for which these two deals are equally attractive. I've checked my math over and over and I'm sure it's correct so I must be thinking about the problem incorrectly. Anyone have any ideas?

Presuming the question is about inter-temporal choices and corresponding discount rate (not central banking).
It is rather easy to prove existence of such discount rate.

At one extreme, when discount rate is 100% (carpe diem mode), the person in question (N) will only look at the immediate value and expense. Dealer A effectively sells the car for \$2,000 (in present terms), while Dealer B sells it for \$4,000. So N will choose A.

At the other extreme, when discount rate is 0% (immortal person without any propensity for investment, perhaps?), N will just sum up all the value and expense streams. So A effectively sells the car for \$20,000 (both in present terms and in nominal terms), while B for \$19,500. So N will choose B.

With discounting function being continuous, it follows that somewhere between 100% and 0% there is a discount rate where A and B have the same present value.

PS: I am not sure if I should have posted this as a comment, but it's too long, and actually proves existence of the answer. I may expand this to include the actual solution if the question is updated to specify whether equal payments or equal principal payments or some other amortization schedule is used.


Overvaluation of a Currency I am interested in knowing more about the valuation of a currency, and how the currency is determined to be 'overvalued' or indeed 'undervalued' in a floating/flexible exchange rate system.

More specifically, in a flexible exchange rate system: what is the currency over/undervalued against, and what role does inflation play in the definition?

Best,

Andrew

Not sure what level of answer you're expecting, but here's an intuition. In equilibrium, countries should have balanced trade. That is, imports equals to exports.

If a country consistently exports more than it imports, the demand for its currency should make the currency appreciate (relative to other currencies). The country's currency would appreciate until its exports are not as competitive, driving down the trade balance and current account.

Likewise, if a country's CA is negative, then the low demand for that country's currency would make the currency depreciate, and it would be able to export more.

By looking at the country's current account, you could "estimate" if the currency is overvalued (should depreciate in the future... if CA is negative) or undervalued (should appreciate in the future... if CA is positive).

This is just the gist of the argument. More considerations need to be put in each country's characteristics such as demography, development status, etc. The IMF (with much criticisms from some of its member countries) issues what's called the External Balance Assessment (EBA) which serves as a "benchmark" of whether it thinks a given currency is over/undervalued. To be sure, some argue that this is somewhat political as the USTR sometimes point to this as a basis for putting some countries on the "watchlist" for being a currency manipulator. Be sure to read the list of caveats and disclaimers of the EBA though.


Why can we see the Current account as variation in NIIP? Let's designate the Net International Investment Position(NIIP:it's the net international wealth of a country), as $B_t$.

Why can we see the current account at time t, as $CA_t=B_{t+1}-B_t$ , when we know that $CA_t=TB_t+i^wB_{t}$?

$TB_t$ is trade balance at time $t$, and $i^w$ is world nominal interest rate.

Intuitively it makes sense, since the variation of a country net international wealth should mimic its international 'consumption/expenditures'. However, I'm looking for a more formal derivation. I think it would probably be related to the Financial and Capital Accounts.

Any help would be appreciated.

In reality the world is much more complicated than those two equations, but the simple theory is that

the current account net flow is the trade balance in that period plus net primary income in that period, and net primary income is the return on investments abroad less the return on foreign investments at home.  The $i^wB_{t}$ term suggests net primary income can be modelled as an implicit rate of return or interest rate on the net investment stock and this implicit interest rate is the same worldwide for all kinds of investment income in both directions (reality is different, in that different types of invest in different places produce different returns)
the change in the International Investment Position net stock over a period should be equal to the net financial flow in that period (reality is again different as there are also revaluations of the value of the components of existing investment positions, for exchange rate changes and other reasons, and these are not reflected in flows)


opportunity cost when inside production possibility frontier I am a bit confused about whether there is a opportunity cost moving from within the PPF to be closer to the PPF and use some of the unemployed resources.
I current thinking is that there is zero opportunity cost of using unemployed resources  to produce more of one good because the alternative is that these resources will not be used and hence no wants or needs will be satisfied. Does my reasoning work? Or is the opportunity cost the used resources? Of units of the second that can no longer be produced?

This would only work if the resources have literally no alternative uses (e.g. think of mana that according to the Judeo-Christian mythology could be only just used for immediate nourishment or it would just quickly spoil and bred worms, hence it had no other uses and could not even be saved or traded).
However, this does not hold in traditional PPF analysis. Usually PPF will look like the picture below (picture taken from Wikipedia). In such case there is always an opportunity cost. If you start at point A and move to point B on the PPF there is an opportunity cost, because instead of moving to B you could move to D. The opportunity cost of choosing allocation B is the allocation D that we forgone.
In order to get rid of opportunity cost you would have to get rid of all other options and have just  one single use to which all resources can be allocated (which would be just a straight line overlapping either $y$-axis or $x$-axis). But such PPFs are not used because resources virtually always have alternative uses. In fact a proper PPF that goes beyond didactic purposes would even have more than just two dimensions, one for each good, as factors (i.e. labor, capital etc.) can be alternatively used in various production processes not just two.


How to determine the price a monopolist will set when the demand curve the firm faces isn't known? I'm reviewing the following question from my microeconomics course notes. I don't understand how to do part (b).

  A firm faces a perfectly elastic demand curve because the product it
  sells has a perfect substitute. The (inverse) demand function is
  $p=2$. Its production function is $f(K,L)=K^{1/4}L^{1/4}$.

  (a) Assuming input prices of $w=r=1$, show that the firm's cost
  function is $C(q)=2q^2$.

  (b) Assuming the firm operates as a monopolist, what price will it
  charge, what quantity will it produce, and what are its profits?

  (c) What is the deadweight loss from monopoly?

  (d) Would the answers to parts (b) and (c) change if the firm could perfectly price discriminate?

Part (a) is easy; I'm stuck on part (b).

If the firm faces a "perfectly elastic demand curve" $p=2$, then the firm isn't in a position to operate as a monopolist, right? A monopolist ought to be able to set a price, not have to accept the market price $p=2$.

If I assume the monopolist sets a price $p$ for quantity supplied $q$, then I can define the profit function

$$\pi(q)=p(q)\cdot q-c(q)=p(q)\cdot q-2q^2$$

Differentiating gives the first-order condition

$$q\cdot p'(q)+p(q)=4q$$

But I can't proceed without knowing the monopolist's demand curve (which, again, I thought couldn't be perfectly inelastic).

If I assume $p(q)=2$, then I can deduce that $q=1/2$. But this can't be right, because then I can't calculate a deadweight loss.

The monopolist is allowed to set the price. However, because his product has a perfect substitute, he can sell 0 products at prices $p>2$. He also has no incentive seting the price $p<2$, because the consumers will buy any number of products he wishes to sell at $p = 2$. This means that $p(q) = 2$. Its graph in the $(q,p)$ coordinate system would be a horizontal line at $p = 2$. Because of this, $p'(q) = 0$. Correcting for this you should be able to easily finish the exercise.


Measuring asymmetry between countries GDP per capita I am looking for an objective way of measuring the "asymmetry" (the overall level of difference) of the GDP per Capita in groups of countries involved in an international agreement. So far, the best measures I found are the variance, the Gini coefficient and the maximum divided by the average. Do you know any better way of measuring this? If not, which one of these measures would be the best in your opinion?

To improve the explanation of asymmetries you could add:

You can look at the standard deviation, and see how the GDP of a certain country is compared to the average +/- standard deviation
You can make subgroups of the countries analysed and compare their averages to the target country. One example of subgroup is the country total population: could be the main explanation of asymmetries for Lichtenstein and Luxemburg to have such high GDPs per capita
Compare the specific elements of the GDP, if the data is available (C vs C + G vs G...)

The best measure depends on what specifically do you want to measure.

If you want to measure the asymmetry per se, then you can focus on alternative indicators and descriptive statistics, as you suggested.


Gresham's Law applied to wampum and gold/silver coins in the 1600s So I was reading an article by Nick Szabo called "Shelling Out"

And in his introduction paragraph he explains how American Indians use bones and shells as a currency called "wampum".

This currency was accepted as legal tender in New England between 1637 and 1661. While the Colonies still accepted coin from the British.

He points out that :

  The beginning of the end of wampum came when the British started
  shipping more coin to the Americas, and Europeans started applying
  their mass-manufacturing techniques.

  By 1661, British authorities had thrown in the towel, and decided it would pay in coin of the realm – which being real gold and silver, and its minting audited and branded by the Crown, had even better monetary qualities than shells. In that year wampum ceased to be legal tender in New England.

Now I'm pretty new to economics but I thought Gresham's Law meant that people will not spend valuable assets such as gold or silver instead they will spend less valuable currency for fear of it depreciating.

So did wampum cease to be legal tender because the government "said so" and if so, why would they make this decision?

A key component of Gresham's Law is that people will spend the less valuable asset such as gold or silver if they legally have the same face value.

Sea shells and beads certainly didn't have the same value or demand within the colonies as British coin. You can't pay for import goods, taxes, or use the company store using beads or shells. Instead the issue that the colonies faced was that the British refused to pay them in coin but rather paid them in goods. If your colony wanted the manufactured British textiles then you would send a shipment of tobacco to receive it. There was an extremely limited supply of British coin in the colonies and so as a result alternatives, such as the wampum which was used by the local Indian tribes had to be adopted.

Eventually the British realized their system wasn't working and began sending over gold and silver coin. These coins were worth more on their face and probably could be considered a less volatile currency and so it was adopted in favor of the wampum which was then phased out of the colonies.


Difference between frictional unemployment and seasonal unemployment Julie spent the entire summer of 2007 working for a tree planting company in Central British Columbia. At the end of July, her contract ended because of weather conditions. She spent the month of August looking for a job but was unsuccessful. In September of 2017 Julie stopped looking for work and instead decided to attend the UBC. As a result, Julie would be considered seasonally unemployed in August but would not be unemployed in September because she is not part of the labour force.

I cannot figure out if this paragraph is true or false. I think it is frictional unemployment in August as she is actively looking for jobs. But it also fits for the seasonal unemployment since she is dismissed because of weather condition. So I dont know how to define this situation.

What "seasonally unemployed" means?

First, to be "unemployed", one must actively be seeking for work, as the OP stated. The adjective "seasonally" here would reflect the fact that Julie is looking for a seasonal job, and couldn't find any. But this information is not provided to us. Irrespective of her past employment, Julie may have been looking for a permanent job, in which case she should not be categorized as "seasonally unemployed".


"risk free interest rate" in Black Scholes In Black Scholes model r is defined as risk free interest rate. Could you please explain what is risk free? Is it same as interest rate on checking account?

Generally speaking, a risk-free rate refers to the yield you get on a government bond (read more here).

On a checking account there's a possibility that the bank would fail. Yes, it's FDIC insured, but up to \$250K.


Implications of differentiable demand function on the utility function properties Suppose you know that the (Marshallian) demand function $x(p,m)$ that satisfies the consumer's problem of utility maximization is such that $\frac{\partial x_i(p,m)}{\partial m}$ is always well-defined for each good $i\in\{1,\dots,L\}$. In other words, the demand function is differentiable with respect to income.

What does this necessarily implies about the properties of the utility function $U(x_1,\dots,X_L)$?

In the (quantity, income) space, not being everywhere differentiable would mean that we would observe gaps, discontinuous jumps, or kinks, in the corresponding graph of the Marshallian individual demand function.

GAPS: if there was a gap, it would mean that for a certain range of income, the demand function does not exist. In other words, it is not defined for a certain range of income (but it still is defined with respect to prices). This could be interpreted that for this range, income does not affect quantity demanded. So "differentiable everywhere" excludes such a phenomenon.

DISCONTINUOUS JUMPS: If there was a discontinuous jump, it would mean that a specific level of income induces a "threshold effect" on quantity demanded. So "differentiable everywhere" necessitates that no such effects exist on quantity demanded.

KINKS: If there was a kink, we would have a "second-order" threshold effect, where quantity demanded would change at a discontinuous rate with respect to income. So "differentiable everywhere" excludes such phenomena.

These should now be "mathematicized" and projected back to the properties of the utility function (if, that is, they are relevant to it).

Note: In many utility maximization problems, income does not enter the utility function, but it is part of the problem's constraints. In others, we postulate a utility function as a bivariate function of one good and income.


Homogenous production function Could one define a production function which is homogenous as having constant elasticity of substitution. Just want clarification .
Thanks

I'm answering if there is any production function which is homogeneous and CES too.

How about a CES production function only.

$y=(x^\rho +y^\rho )^\frac{1}{\rho }$ for $0\neq \rho <1$

This function has Elasticity of Substitution : $\sigma =\frac{1}{(1-\rho )}$ which is a constant.

Considering homogeneity,

$ y=((\lambda x)^\rho +(\lambda y)^\rho )^\frac{1}{\rho }$

$y=(\lambda^\rho (x^\rho +y^\rho ))^\frac{1}{\rho }$

$y=(\lambda ^\rho )^\frac{1}{\rho }(x^{\rho }+y^\rho )^\frac{1}{\rho }$

$y=\lambda (x^\rho +y^\rho )^\frac{1}{\rho }$ which is the initial production function.

Since, the degree of $\lambda $ is 1, so, it's homogeneous of degree $1$.


Walras Law in a production economy with fixed costs Consider a price taking firm with fixed costs $fc \geq 0$:
\begin{align*}
\Pi
&=
\max_{n^D} \left\{ P_c F(n^D) - w\times n^D - fc \right\}
\end{align*}
A representative household owns this firm:
$$\max_{c,n^S} U(c,n^S) \text{ s.t. } P_c c = wn^S + \Pi$$
Equilibrium: prices $(P_c, w)$ & allocations $(n^D, c, n^S)$ s.t. all optimize & markets clear:
1 (Labor market) $n^D = n^S$
2 (Goods market) $c = F(n)$
Rewrite the household's constraint:
\begin{align*}
c
&=\frac{w}{P_c} n^S + \frac{\Pi}{P_c}
\\
&=\frac{w}{P_c} n^S + F(n^D) - \frac{w}{P_c} n^D - \frac{fc}{P_c}
\tag{plug-in $\Pi$}
\\
&=\frac{w}{P_c} \left(n^S - n^D \right) + F(n^D) - \frac{fc}{P_c}
\tag{rearrange}
\\
&= F(n)  - \frac{fc}{P_c} \tag{Labor Market: $n^D = n^S$}
\end{align*}
Observe the household's constraint $c = F(n)  - \frac{fc}{P_c}$
is inconsistent w/ goods clearing $c = F(n)$.
Example:
$F(n)= A \log(n)$
$\Rightarrow w=\frac{P_c A}{n}
\text{ } \&\text{ } n^D(w)= A \frac{P_c}{w}
\text{ } \&\text{ } Y= A \log\left( A \frac{P_c}{w}  \right)
\text{ } \&\text{ } wn = A P_c
$
$\Pi = P_c A \log\left( A \frac{P_c}{w}  \right) - A P_c  -fc  $.
$u(c,n)=c-  \frac{n^{1+\frac{1}{\varepsilon} }}{1+\frac{1}{\varepsilon}}$  s.t.
$P_c c = wn + \Pi$
$U(n)= \frac{w}{P_c} n + \frac{\Pi}{P_c}
-  \frac{n^{1+\frac{1}{\varepsilon} }}{1+\frac{1}{\varepsilon}}
\Rightarrow
\frac{w}{P_c} = n^{\frac{1}{\varepsilon} }
$.
$\Rightarrow n^S(w) = \left(\frac{w}{P_c}\right)^\varepsilon
\text{ } \&\text{ } c=\left(\frac{w}{P_c}\right)^{1+\varepsilon} + \frac{\Pi}{P_c}
$
1 (Labor market) $n^D = n^S$
$\Rightarrow A \frac{P_c}{w} = \left(\frac{w}{P_c}\right)^\varepsilon
\Rightarrow A  = \left(\frac{w}{P_c}\right)^{1+\varepsilon}
\Rightarrow \frac{w}{P_c}  = \left(A \right)^{\frac{1}{1+\varepsilon}}
$
2 (Goods market) $c = F(n)$
$\Rightarrow
\left(\frac{w}{P_c}\right)^{1+\varepsilon} + \frac{\Pi}{P_c}
= A \log\left( A \frac{P_c}{w}  \right)
$
$\Rightarrow
\left(\frac{w}{P_c}\right)^{1+\varepsilon}
+
A \log\left( A \frac{P_c}{w}  \right) - A - \frac{fc}{P_c}
=
A \log\left( A \frac{P_c}{w}  \right)
$
$\Rightarrow
A + \frac{fc}{P_c}
=
\left(\frac{w}{P_c}\right)^{1+\varepsilon}
\Rightarrow \frac{w}{P_c}  = \left(A + \frac{fc}{P_c} \right)^{\frac{1}{1+\varepsilon}}
$
Problem:
Labor market clearing gives:
$\frac{w}{P_c}  = \left(A \right)^{\frac{1}{1+\varepsilon}}  $
Goods market clearing gives:
$\frac{w}{P_c}  = \left(A + \frac{fc}{P_c} \right)^{\frac{1}{1+\varepsilon}} $
They are only identical if $fc=0$.
Question:

is Walras law not supposed to hold here w/ $fc>0$ ?
how do you set up a GE economy w/ production and fixed costs?

Attic:
We can rewrite Goods-market clearing:
$\frac{w}{P_c} n^S + \frac{\Pi}{P_c} = F(n^D)$
$
\Leftrightarrow
\frac{w}{P_c} n^S + F(n^D) - \frac{w}{P_c} n^D - \frac{fc}{P_c} = F(n^D)
$
$
\Leftrightarrow
\frac{w}{P_c} (n^S - n^D) = \frac{fc}{P_c}
$
$
\Leftrightarrow
 \frac{fc}{P_c} = 0
$ if $n^D = n^S$

Partial answer:
for simplicity let $P_c =1$.
The budget constraint: $c= wn + \Pi$
Simplify (plug in $\Pi$): $c= F(n)- fc$
Goods clearing: $c = F(n)$
The household's budget constraint is inconsistent w/ goods market clearing.
The firm pays a fixed cost that doesn't go to anyone. In a "true GE model" all payments have to go to someone in the economy.
One solution is to rewrite the goods market clearing condition:
$c= F(n)- fc$
IE: some of the output good is consumed by the household, and some by the firm...
Alternatively, a common approach in economics is to assume one factor (say capital) is fixed in the short run ($k=\bar{k}$) and rented from households. In this case $fc= r\times\bar{k}$:
\begin{align*}
\Pi
&=
\max_{n^D, k^D} \left\{ P_c F(n^D) - w\times n^D - r\times k^D \right\} \text{ s.t. } k^D = \bar{k} \tag{short-run}
\end{align*}
The household's problem is then:
\begin{align*}
\max_{c,n^S, k^S} U(c,n^S) \text{ s.t. } P_c c = w\times n^S + r\times k^S + \Pi
\end{align*}
GE: prices $(P_c,w,r)$ & allocations $(n^D,k^D,c,n^S,k^S)$ all optimize & markets clear:
1 (Labor) $n^D=n^S$
2 (Goods) $c=F(n^D)$
3 (Capital) $k^D=k^S$
Now the household's constraint is no longer inconsistent w/ Goods market clearing.
\begin{align*}
P_c c
&=
w\times n^S + r\times k^S + \Pi
\\ &=
w\times n^S + r\times k^S + (P_c F(n^D) - w\times n^D - r\times k^D)
\\ &=
P_c F(n^D) \tag{k, n clear}
\\
c &= F(n^D)
\end{align*}
It appears there is no good way of modeling fixed costs in GE w/o having some household in the economy endowed w/ & rent out the factor that is fixed.


Difference between 'surplus' and 'welfare' I see these two terms used interchangeably but I have the feeling they are not exactly the same. My gut feeling tells me this:

1)
Surplus is the additional aggregate utility from the existence of one market, while welfare is the additional aggregate utility from all existent markets

$$\implies  [ W(u_1,u_2,\dots,u_N) \ \mathrm{with} \  u_i=u(c_1,...,c_M) ] \gg [ S(u_1,u_2,...,u_N) \ \mathrm{with} \ u_i = u(c_i) ]$$

Change in surplus is thus an approximation of the aggregate gain in welfare, and is only equivalent under the conditions that: policy changes do mainly affect one market and consumers’ utility is assumed to be quasi-linear in the good at focus.

2) Or a second idea/notion, surplus is only the difference between supply and demand curve (i.e. it entails all aspects of the good that can be monetized), while welfare also includes non-monetizable aspects of a good.

Any thoughts on this?

Surplus and welfare are different concepts, but not for the reasons you state, although there are elements of validity in both (1) and (2).  They may, however, be used interchangeably in certain contexts and where certain conditions are met.

Both “surplus” and “welfare” are terms from ordinary language that in economics are used in more precise senses.  "Surplus" has several senses in economics, and is commonly clarified by the words accompanying it: (in microeconomics) "consumer surplus", "producer surplus" and their sum "total surplus", (in macroeconomics) "budget surplus (of a government)" and (in Marxian economics) "surplus value".  Where it is used alone, the context will usually make clear which of these is meant: the question appears to be concerned with the microeconomic senses.

Consumer surplus and producer surplus are normally defined in terms of supply and demand curves in a single market.  Such curves can in principle be estimated from observations of supplier and consumer behaviour, without reference to utility.  So although consumer surplus can be related to utility via marginal utility theory, utility is not needed in its definition and, as Steven Landsburg points out, surplus should be measured in monetary units, not units of utility.  Welfare, equally, is often related to utility at a theoretical level, but in applied work is often estimated without reference to utility and measured in monetary units.

“Welfare” in economics is used both in macroeconomics, where there have been various attempts to identify an appropriate set of adjustments to GDP to provide a suitable measure of overall welfare in an economy, and in microeconomics, where the focus is on welfare in a single market or a few related markets.   But these are not entirely different senses of the term.  They both retain from ordinary language the essential idea that welfare is good - something policy should aim to increase - and in this respect there is an important difference, even in microeconomics, between surplus and welfare.  “Surplus” is a purely descriptive concept (a term of positive economics), whereas “welfare” is – to borrow a little from philosophy – a thick concept, one that is both descriptive and evaluative.  Let’s consider three situations which can help to clarify our understanding of these concepts.

Firstly, consider a standard supply-demand diagram for a single product, and suppose supply increases (the whole supply curve moves to the right).  Normally, we would say that both surplus and welfare have increased.  Now suppose that the product is one which is harmful to those consuming it.  Would we still say that surplus has increased?  Yes, because “surplus” is a purely descriptive term, so we can quite consistently say that surplus has increased while thinking it might have been better if this had not occurred.  Would we still say that welfare has increased?  Quite possibly not.  Points we might want to consider before drawing a conclusion are whether consumers are aware of the harm and acting rationally, and if not whether their pleasure from consumption outweighs the harm.  This distinction between surplus and welfare can be found in debate on the regulation of smoking (here is an example).

Secondly, consider the effect of a tariff on an imported good.  The tariff raises the domestic price of the good, with resulting changes in the consumer surplus and producer surplus relating to that good. Would we say that the change in welfare is the sum of those changes?  No, because (even if we ignore effects on related markets and on exporting countries) the revenue raised by the tax is also relevant to welfare.  In a simple partial equilibrium analysis, therefore:

Welfare loss  =  Reduction in CS – Gain in PS – Increase in government revenue

This distinction between surplus and welfare can be found in (A) and (B).

Thirdly, consider a change in a market for a good subject to a production externality, say air pollution.  Here we would say that the change in welfare is not just the change in producer and consumer surplus in the particular market, but should also take account of the externality.  However, to draw a distinction between surplus and welfare on this basis would be an oversimplification since, in the economics of environmental valuation, the concept of consumer surplus is routinely applied to non-market goods.  For example (although the details are complex and the results unlikely to be very accurate) a demand curve for air quality (suitably measured) in a region might be estimated using the hedonic pricing method which uses the housing market as a surrogate for a market in air quality.  Once a demand curve has been estimated, it is possible to calculate the change in consumer surplus due to a change in air quality and include this in a measure of change in welfare. Having said that, it is questionable whether methods are available to estimate consumer surpluses for all non-market goods, so to that extent there is validity in (2).

The conditions that would be needed for a change in total surplus in the market for one product to be an accurate measure of the change in welfare resulting from a policy change would therefore be quite stringent, including at least the following:

•  No direct effects of the policy change on other markets;

•  No effects of change in the market on other markets;

•  No harmful effects of the product on those consuming it;

•  No externalities in production or consumption of the product.

Sometimes these conditions will be met, to a reasonable approximation, and then it might be appropriate to use the terms “surplus” and “welfare” interchangeably.

Finally, a brief consideration of the idea that surplus relates to one market while welfare relates to many markets. Because “welfare” is an evaluative term, it makes sense to apply it to situations involving either one or many markets, even if in some cases it is hard to find suitable criteria to determine whether a change is an increase in welfare.  “Consumer surplus”, on the other hand is normally defined for a single market, and the path-dependency problem (see (C)) is a complication in aggregation across markets, except in special cases (this is where, if the connection is made with utility theory, quasi-linear utility is relevant).  To overcome the path-dependency problem, either compensating variation or equivalent variation are sometimes used as alternatives to (ordinary Marshallian) consumer surplus, but then there is the further problem that the theoretical income-compensated demand curves underlying these measures cannot be estimated from observations of consumer behaviour.  To that extent there is validity in (1).

References

A. Koo W W & Kennedy P L (2005) International Trade and Agriculture Blackwell Publishing p 104

B. Mankiw B G & Taylor M P (2006) Economics Thomson Learning pp 172-3 (a European version of the well-known textbook)

C. Johansson P-O (1991) An Introduction to Modern Welfare Economics  Cambridge University Press pp 42-52


Derive a cost production function give prod f, only K In this question, only K is included and L is excluded how would I go about deriving it?

Total cost= Fixed costs + Average costs.

Since the variable input costs r per unit, the variable costs is r times the number of units rQ, hence $VC= rK^\alpha$.

Thus $C(Q)= c_0 + rK^\alpha$

Is the above solution correct?
IS there a more mathematical way of doing it, one that involves grpahs etc.

Here you have to express $K$ in terms of $Q$, since the cost depends on the number of units of capital employed and not on the number of products produced. You have
\begin{equation}
Q = K^a \Leftrightarrow K = Q^{\frac{1}{a}}
\end{equation}
so your cost function will be
\begin{equation}
C(Q) = c_0 + rK(Q) = c_0 + rQ^{\frac{1}{a}}
\end{equation}


Why the treasury is buying back its treasury securities during the coronavirus economic crisis? I have read today that the treasury of my home country (not the US) started to buy back a significant amount of its Treasury securities (bonds, notes ...) from their owners. My question is what happens at a microeconomic scale after the Treasury redeems back its treasury securities ? I've actually read that Central Bank along with the Treasury both agreed and decided to do so before starting printing any new money, I wonder thus why? and why this timing in particular? Why do they have to spend money in order to print new money?

The following link was provided: Link to Moroccan news article, in French

My French is somewhat rusty, and I am completely unfamiliar with the structure of the Moroccan financial system. But I think I understood the basic idea.

As a result of the worries around COVID-19, there was added demand for liquidity in the banking system. The Treasury sits on a buffer of cash (presumably sitting at the central bank). What it is doing is using that buffer to purchase back some short maturity bonds. This injects money into the banking system, which can then be used for liquidity management, or to buy newly-issued long-term Treasury bonds.

The central bank could have done this, but since the Treasury presumably had cash reserves, it did the operation.

From a theoretical point of view, it would not matter whether these bond purchases were done by the central bank or the Treasury - in either case, the private sector is exchanging bonds for cash. However, the central bankers might not agree with that assessment, and there may be institutional rules (that I am unaware of) in Morocco that pushed the Treasury to do this.

Generally speaking, in the developed countries, the central bank would handle such an operation, and the Treasury would not get involved. The closest operation I can think of is in Canada, where the Ministry of Finance lends to the private banks out of its balance at the Bank of Canada.


Educational self sufficiency I heard on UK and US news that higher education is very high and a lot of students are in debt.
In Israel here education is very cheap, because it is heavily subsidized by the government (Israel tries to be self-sufficient as opposed to other western countries because it doesnt have the liberal policies to attract brain drain like the western countries have). It strategy is to produce university candidates by their high school grades and IQ (psychometric) scores (those requirements are mandatory by top universities).

How would such educational policy work for countires like US or UK?
would it be better for US to subsidize their top universities heavily and introduce higher educational requirements (such as IQ scores and better grades)? or would it have disastarous effects on the economy, following such a government policy that Israel has?
I know that education is positively correlated with GDP.

*****UPDATE*****

First, I have to thank you for answering. I live in Israel the tution fees for top univesities (Tel Aviv, Ben Gurion, Jerusalem, Bar Ilan) is no more that \$10,000.. tution fees for harvard for one year only from what I heard is something like \$40,000 a month alone (https://college.harvard.edu/financial-aid/how-aid-works/cost-attendance). My question why wouldnt a government open the market competition for those univesties only to American citizens (that will bring costs down), and also subsidize them, to bring  the costs to around 10-15k, that would be much affordable. In return, Harvard, could make higher requirments for its courses like psychometric tests and very high school grades. If it did that it would gurantee an educational self sufficiency and higher economic independence from immigration.
Wouldnt that be preferable politically (assuming your perception is right wing) instead of being pushed into a tougher rat race with immigrants coming from abroad?

Last point to add, Israel also provides a lot of education for free in the army (Israel has conscription laws). It provides higher degrees for good studens who also performed well on their IQ test.

If Israel provides and believes that there are high returns on their student's educational investment by providing education at low cost (when subsidized) or for free, why other Europen countries dont follow? or is Israel wrong in their approach to education and higher economic efficiency?

Could a country like US and UK pursue educational self-sufficiency or would such a pursuit be a disastor?

According to the OECD: Israel's spending per-student is low relative to other OECD countries. Given that countries like the US have both higher spending per-student and higher enrolment rates, it is not clear what a switch to heavier state subsidisation (both the UK and US already heavily subsidise universities) would achieve. Bear in mind that most of the best US universities privately subsidise talented students from poor backgrounds so that the highest fees are mostly bourne by the wealthy.

It seems that implicit in your question is the idea that Israel rations its heavily subsidised tertiary education by imposing more demanding entrance requirements, but it is not clear to me that this is true. The entrance requirements for elite universities such as Harvard, MIT, Oxford, or Cambridge are among the most demanding in the world. There are 4599 higher education institutions in the US that cater to a very broad spectrum of student types.


Does everyone prosper through development projects? Why don't all investors focus their intentions on development projects in less developed countries? Wouldn't this benefit all parties since the recipient country would be more developed and the investor would make a profit? In other words can it be said that development projects are the type of investments where no one is placed at a disadvantage and everybody prospers? ( p.s. please don't criticize if you find this is a stupid question or I sound redundant, I'm a 17 year old economics student excited to learn as much as I can)

A lot of this comes down to risk. A developing country usually don't have very strong "institution" and are much more prone to various kinds of crisis (think military coup, hyperinflation, terrorist attack, nationalization*). An investor might also not be as familiar with the developing country in question* (and hence the risk).

As a result, investors require higher return (the "premium") to invest in these countries. You can see the risk premium by country here.

This is why developing credibility could be very helpful in terms of drawing foreign investment into a country.

* Added from comments below.


Why is a nations economy is divided into sectors? I understand that modern economists divide a nations economy into sectors such as such as primary, secondary, tertiary, quaternary and quinary.

What is the point of such classification?

That is because in science we try to not just create new knowledge but also classify the knowledge.

In biology biologists don’t just study living things they also classify them as mammals, birds etc based on similar characteristics, in astrophysics physicists don’t just study different astronomical objects but classify different astronomical objects into categories such as planets, asteroids, stars etc. based on similar characteristics, and finally in economics we don’t just study an industry but also categorize industries again based on similar characteristics.

The point of such classification is to have established nomenclature which makes subsequent research more easy. If a economist says “I study primary sector in US” the economist does not need to always explain that he or she studies the sector that produces raw materials/agricultural which saves a lot of time and makes process of science as efficient as possible.


Are there any instances where short run demand/supply is more elastic than long run cases? Conventionally both demand and supply for goods and services are both more elastic in the long run than in the short run.
My Question

Is there ever an instance where demand and/or supply for a given good be more elastic in the short run than in the long run?

That being said if there is never such an instance, does there exist a general proof refuting such an instance?

It is possible.  We normally think demand and supply are more elastic in the long run because consumers/firms have more options in the long run.  For example if gas becomes more expensive consumers can buy more more fuel efficient cars.

So what if consumers or firms are more constrained in the long run?  This could be the case when goods are storable.

Suppose I need to drink coffee every day or I get a headache.  At home I have a canister of coffee that I use every day.  If coffee doubles in price I can hold off buying and hope the price comes down, so I am fairly elastic.  In the longer term though I'll run out of coffee and I'll need to buy more regardless of the price.

Of course in the even longer term maybe I can stop my caffeine addiction so I become more elastic again.

This maybe isn't the best example, but the key thing to think about is how constrained the agent is at each point in time.  When people are more constrained in their options they are less elastic.


Elasticity of substitution between leisure in two periods This is a basic question, but I am new to macro models. The question is from Romer's text.

Assume a household only has one member and has no initial wealth, and the household lives for two periods. We have

$$U=\ln(c_1)+b\ln(1-l_1)+e^{-\rho}[\ln(c_2)+b\ln(1-l_2)],$$
where $c_i=w_il_i$ and $\rho$ is a discount rate in $(0,1)$.

The text says "Because of the logarithmic functional form, the elasticity of substitution between leisure in the two periods is $1$." I tried to verify that the elasticity is $1$ as the following:

Let U_1 denote partial derivative of U with respect to $l_1$. Similar for $U_2$. Then,

$$E_{21}=\frac{d\ln(l_2/l_1)}{d\ln(U_1/U_2}=(\frac{dl_2}{l_2}-\frac{dl_1}{l_1})/(\frac{dU_1}{U_1}-\frac{dU_2}{U_2}),$$
where $U_1=\frac{1}{U_1}-\frac{b}{1-l_1}$ and $dU_1=(-\frac{1}{l_1^2}-\frac{b}{(1-l_1)^2})dl_1$. How chould $E_{21}=1$? In fact, I got something super messy. Therefore, I doubt if my approach is correct or not. Maybe my formula is not correct? Thanks in advance!

I think there are two problems with your calculations. First, as $l_i$ is the time devoted to work, you should care about $\frac{d\ln\frac{1-l_2}{1-l_1}}{d\ln (U_1/U_2)},$ rahter than the $\frac{d\ln\frac{l_2}{l_1}}{d\ln (U_1/U_2)}$ you consider. Second, I don't think your $U_i$ are correct (perhaps it's caused by first). I get

$$\begin{matrix} U_1=\frac{b}{1-l_1} & U_2=\frac{be^{-\rho}}{1-l_1}\end{matrix},$$

that results in the marginal rate of substitution being

$$MRS=\frac{U_1}{U_2}=\frac{1-l_2}{1-l_1}\frac{1}{e^{-\rho}}.$$

If you take the logarithm, you get

$$\ln MRS = \ln \frac{1-l_2}{1-l_1}+\ln\frac{1}{e^{-\rho}},$$

where it is useful to observe that the last term of the RHS is a constant. Now, the total differential is

$$ d\ln MRS = d\ln \frac{1-l_2}{1-l_1}+0,$$

and you get the desired result.


Proving a result In Jones (1999) "Growth: With or Without Scale Effects" Jones (1999) builds his semi-endogenous growth model where output is produced with only one input, labour, in the following "research" function:

$$ Y = A^\alpha L_Y $$

Labour is augmented with technology $A$ which is also researched by
workers, $L_A$, in the following production function:

$$ \dot{A} = \delta L_A A^\phi. $$

We have that $L_A + L_Y = L$ and $0 < \phi < 1$.

He then writes:

  Assuming that the labour force L grows at some constant exogenous rate
  $n$, it is easy to show that there exists a stable balanced growth
  path for the model where

  $$ g_A = \frac{n}{1-\phi}$$

  and

  $$ g_Y = \sigma g_A = \frac{\sigma n}{1-\phi} $$

He says it's "easy to show", but I can't for the life of me show it! We know that $g_A = \dot{A}/A$, and on the balanced growth path $g_A = g_Y = g_L = n$. So I try to plug $g_A = \delta L_A A^{\phi-1} = n$ but that gets me nowhere.

We know that $g_A = \dot{A}/A$ and thus $g_A = \delta L_A A^{\phi-1}$.

Take logs of both sides of $g_A = \delta L_A A^{\phi-1}$ and differentiating with respect to time gives us

$$ \frac{\dot{g_A}}{g_A} = n + (\phi-1)g_A.$$

Multiply both sides of the equation by $g_A$:

$$ \dot{g_A} = n g_A + (\phi-1) g_A^2$$

Given that $\phi<1$, and in the steady-state BGP the growth of the growth rate $\dot{g_A} = 0$, we divide both sides by $g_A$ to get:

$$n+ (\phi -1) g_A = 0 $$

Rearrange to get the desired expression:

$$g_A^* = \frac{n}{1-\phi}$$

The second equation is just the result above multiplied by $\sigma$:

$$g_Y = g_A \sigma =  \frac{\sigma n}{1-\phi}$$