Extensive form games (pure strategies) Why do we include actions even on nodes which will not be reached if the strategy is followed, while defining the pure strategies of an extensive form game?

Think about a player (female) that moves first and is considering the profitability of deviating. Let's say that according to the strategy she is supposed to play action $x$ and then the second player (male) is supposed to play action $a$. It will not be correct for the first player to think that if she instead chooses $y$, the second player will necessarily still play $a$. Remember this is a sequential game, so after seeing that she chooses $y$ instead of $x$ player 2 can also deviate to some other action,  say $b$. Therefore, by specifying the actions that each player will take in each node (even in the ones that are not supposed to be reached according to the strategy) we completely specify the strategy of each player, and can correctly check that no player has profitable deviations.

In fact, in many games, the only reason why the player that moves first chooses an action $x$ is because of the response of the other player after observing $x$, and the knowledge that choosing something else will also affect the decisions of other players, etc.

In a practical example, if you are playing chess and consider moving your queen to get one step closer to eat an important piece of the other player, you must anticipate they will react to this movement and protect their piece, or attack something more valuable of yours, etc. This is why when thinking about the strategy of the other player you must consider how different they will react depending on which piece you move next and to where.

BTW, chess is really hard to fully specify as a sequential game precisely because the number of possible nodes is massive, and properly specifying a complete strategy for each player has proven extremely hard. However, hopefully it gave you some insight of why actions and strategies are not the same things in an extensive game.


Confusion over homoskedacity assumption In my introductory econometrics course, we have been discussing the GM-assumptions and homoskedasticity. Unfortunately, I have some confusions and interlinked questions, so I am wondering if somebody could help me with my understanding please?

model: $y_i = \beta_0 + \beta_1 x_i + u_i  $

What is the difference between $E[u] = 0$ and $E[u|x] = 0$? Does the first imply the latter?

I understand that the latter marks that the conditional mean, ie that the expected value of $u$ is $\sigma^2$ irrespective of $x$, but isn't this implied by the unconditional mean? Is one of them a stronger assumption?

Do the above imply $cov(u,x) = 0$ and how? What is the difference between $cov(y,x)$ and $cov(u_i,x_i)$?

We have stated that the homoskedasticity assumption implies zero covariance between $u$ and $x$, but how does it come? And why is it labelled a "weaker assumption" (at least by Woolridge)? What really confuses me is that we often use the notation $cov(u_i,x_i)$ instead of $cov(u,x)$ - is there any difference? If the first one relates to a specific $i$, does it really have a meaning at all?

Apologies if these questions are rather trivial. And thank you very much for your help!

Intuitive explanation might help.

(i) $E[u]=0$ vs $E[u|x]=0$: Imagine partitioning the population by the value of $x$ so that each slice of population has the same value of $x$ in it. You can then get the mean of $u$ for each slice. "$E[u|x]=0$", which is a shortcut notation of "$E[u|x=c]=0$ for (almost) all $c$", means that the average of $u$ in every partition is zero. You can also get the mean of $u$ for the whole population. "$E[u]=0$" means that the average of $u$ for the whole population is zero.

For example, let $x$ be years of education. $E[u|x]=0$ means that the average of $u$ for those with 9 years of education is zero, the average of $u$ for those with 10 years of education is zero, etc., while $E[u]=0$ means that the average of $u$ for the whole population is zero.

You see that "$E[u|x]=0$" implies some sort of "$u$ being not related with $x$", and it's called mean independence. On the other hand, $E[u]=0$ does not say anything about the relationship between $u$ and $x$. $E[u]=0$ makes sense without $x$ ever being involved, while $E[u|x]=0$ makes sense only in relation to $x$.

(ii) $var(u)=\sigma^2$ vs $var(u|x)=\sigma^2$: Imagine partitioning the population by the value of $x$. You can get the variance of $u$ for every slice. '$var(u|x)=\sigma^2$' means the variance of $u$ for every partition is $\sigma^2$. Here, the key point is that $\sigma^2$ is a constant and is not involved with $x$. It means that all the slices have the same variance. $var(u|x)=\sigma^2$ is very informative. Next, you can also get the variance of $u$ for the whole population, and $var(u)=\sigma^2$ means that that variance (of $u$ for the whole population) is denoted $\sigma^2$, where $\sigma^2$ is just a notation.

$var(u|x)$ can depend on $x$ (heteroskedasticity), but it is totally nonsense to question whether $var(u)$ depends on $x$ because $var(u)$ has nothing to do with $x$ from the beginning (unless you mean $var(u|x)$ by $var(u)$).

(iii) $cov(x,u)=0$: This just means that $cov(x,u)=0$, where $cov(a,b)$ is defined as $E[(a-Ea)(b-Eb)]$. In your case, $cov(x,u)=E[xu]$ because $E[u]=0$. If you need intuition about the meaning of $cov(x,u)=0$, imagine you have $(x,u)$ values plotted on the XY plane for the whole population, with $x$ on the horizontal axis and $u$ on the vertical axis. You draw a nice straight line (where being straight is important). $cov(x,u)=0$ means that the straight line is horizontal. It says something about $u$ and $x$ being not related, and when it happens, we say that "$x$ and $u$ are uncorrelated."

$E[u|x]=0$ implies $cov(x,u)=0$ but not vice versa, which is the reason why $cov(x,u)$ is weaker than $E[u|x]=0$. You can prove it using the law of iterated expectations: $E[xu] = E[xE(u|x)] = E[x\cdot 0]=E[0]=0$. For a counter example for the other way around, suppose that $x\sim N(0,1)$ and $u=x^2-1$. Then $E[u|x]=x^2-1$, which is not zero except $x=\pm 1$ but $cov(x,u) = E[x(x^2-1)] = E[x^3]-E[x]=0-0=0$.

(iv) What is the difference between $cov(y,x)$ and $cov(u,x)$? If $y=\beta_0 + \beta_1 x +u$, $cov(y,x) = cov(\beta_0+\beta_1 x+u,x) = \beta_1 cov(x,x) + cov(u,x)$, where $cov(x,x) = var(x)$ by definition. That's the difference.

ADD:

I've just noticed the OP's confusion about $cov(x_i,u_i)$ and $cov(x,u)$. First, $cov(x,u)$ can be intuitively understood as the population property explained above in (iii). $cov(x,u)$ is about the population, and does not say anything about the sample. The notation $cov(x_i,u_i)$ is on the other hand about the sample, and should in fact mean something like "$cov(x_i,u_i)$, $i=1,\ldots,n$", i.e., $cov(x_1,u_1)$, $cov(x_2,u_2)$, ..., and $cov(x_n,u_n)$. They are all the same if the first person, the second person, ..., the $n$th person are independent random draws from the same population (which is the meaning of $iid$). If they are drawn from different populations, $cov(x_i, u_i)$ may be different for different $i$, but I am pretty sure you are assuming $iid$.

Now, what is $cov(x_1,u_1)$ then? This is the hard part. To understand its meaning, you should understand that $(x_1,u_1)$ is a random vector, whose value can change when you repeat sampling in your thought experiments. (This will keep confusing you until you understand this point.) The "first person" in the sample will keep changing when you repeat drawing the sample over and over again in your mind, and thus the $(x_1,u_1)$ value will keep changing over repeated samples. So, as you repeat sampling indefinitely, the values of $x_1$ and $u_1$ will change and make a (joint) distribution. $cov(x_1,u_1)$ is the covariance of that joint distribution. Likewise, you can understand $cov(x_2,u_2)$ as the covariance of $x_2$ and $u_2$ over the repeated samples. If you keep drawing the first observation independently from the same population as you talk about $cov(x,u)$, then $cov(x_1,u_1) = cov(x,u)$.

So when we talk about the population property, we use the notation $cov(x,u)$. When we talk about the covariance of $x_1$ and $u_1$ (over repeated samples), we use the notation $cov(x_1,u_1)$. The notation $cov(x_i,u_i)$ is a (sloppy) shortcut notation of $cov(x_1,u_1), \ldots, cov(x_n,u_n)$. "$cov(x_i,u_i)$" itself does not make much sense until you say what $i$ is, like in "$cov(x_i,u_i)$ is zero for every $i=1,\ldots,n$", or "$cov(x_i,u_i)$ is nonzero for every $i=1,\ldots,n$ but they are the same for all $i$." The statement "$cov(x_i,u_i)=0$" is usually a lazy (or clumsy) writing of "$cov(x_i,u_i)=0$ for all $i=1,\ldots,n$", which means $cov(x_1,u_1)=0$, $cov(x_2,u_2)=0$, ..., $cov(x_n,u_n)=0$.


In Barro's (2009) Rare Disaster Model in AER: How to derive equation (5) In Barro (2009) http://piketty.pse.ens.fr/files/Barro2009.pdf
My question is reference to equation #5, whereby Barro is deriving the reciprocal of the market value 1/v, and I am trying to derive this equation, and having trouble. In particular, without further specifications of distribution of v.
So far, I have:

But can't seem to arrive at the same result, without further information/assumptions on the distribution of the disasters.
Barro's Result for equation (5)

Any help would be greatly appreciated.

There is probably an error in your formula for $\sum_0^s v_i$.
You can use a trick to directly compute $E_0 e^{(1-\gamma)v_i}$ using
\begin{align}
E_0 e^{(1-\gamma)v_i}& = \Pi_0^s E_0e^{(1-\gamma)v_i}.
\end{align}
Notice $e^{(1-\gamma)v_i}$ is a random variable equal to 1 with probability $1-p$ and $(1-b)^{1-\gamma}$ with probability $p$. Therefore,
\begin{align}
E_0 e^{(1-\gamma)v_i}& = \Pi_0^s (1-p+pE_0(1-b)^{1-\gamma}),\\
& = (1-p+pE_0(1-b)^{1-\gamma})^s.
\end{align}
The term $E(1-b)^{1-\gamma}$ now appears your expression.


Estimating Price Elasticity from sales I'm looking at data for an online shop with clothing items listed at different prices. I'd like to use this information to do some kind of profit maximization (i.e. I'd like to have some understanding of the impact on sales if I raise/lower price of an individual item).

Unfortunately, I don't have information of the variety "Q sales for clothing item X at price P" for a variety of prices; this would be hard to estimate due to the small number of items in each category.

However, I do have a distribution of sales vs. price and the distribution of price of items in the shop. I'm thinking if I compute
$$
\frac{\text{% of all sales at this price}}{\text{% of available items at this price}}
$$, that gives me something analogous to a "quantity demanded". For simplicity, I bucketed the prices and sales into $5 increments.

Having done this and doing your traditional log-linear fit
$
\ln(P) = a\ln(Q)+b
$ I get a pretty good fit and a reasonable number (around $-.9$).

Question 1: Does my methodology make sense? If so, what did I just calculate? I feel like it's an "aggregate" price elasticity of all items in the shop. The clothing items for sale are mostly substitutable with each other, the price elasticity of an individual item must be a lot higher (since there are substitutes available).

Question 2: Is there a way to figure the "average" price elasticity of each item from this (approximating that they all have about the same price elasticity)?

EDIT: Based on the suggestion from @optimal-control and @NickJ, it appears the issue is that I don't really have a good model of what's going on. Here's what I'm imagining:

Suppose you have $n$ unique items, each with a fixed price $p$. Each item has a limited supply available $s$. I observe $q(p, t)$ sales for each unique item, limited by $s$. Let's say $q(p, t)$ is something like a homogeneous poisson process, for simplicity.

The above is starting to look like a markdown optimization problem; I could imagine finding an optimal price for each item based on these factors and my cost structure (storage cost, time value of money, etc.), if I could, say, change prices and estimate the change in $q(p, t)$. Unfortunately, this function is so small, and $s$ is so small (maybe 1 to 5 items) that this may be impossible to estimate. What I'm trying to figure out is some way to estimate the response of $q(p, t)$ with respect to $p$ without modifying prices, off of the assumption that the $n$ items are partial substitutes for one another.

If they were perfect substitutes, I would expect ALL of my lowest price items to sell out, and then my next lowest, etc. going on upward and I could use these results to estimate $q(p, t)$. But clearly this isn't the case, because some more expensive items sell first. The problem is, $q(p, t)$ is dependent on the mix of items and prices available. Any thoughts on how to approach this?

In this kind of cases, I think you can not use a simple OLS estimation because of endogeneity of your variables.

Probably, there would be causality between prices and sales data, this means that sales are influenced by prices but also prices are influenced by sales. So, in this case, in order to do a non-biaised estimation, you should use instrumental variables in order to avoid endogeneity issues between your variables.


How to prove that GDP grows at the same rate as consumption on Lab Equipment Model? I'm studying the Lab Equipment Model (Barro-Sala-i-Martin, Chapter 6). I'm having trouble when trying to prove that every variables grows at the same rate as consumption.

I was able to prove that $\dot{C}/C = g_{c}$, which is constant. Also, I have that

$b \dot{N} = Y - X - C$ where $b$ is a parameter,

$Y/N$ is a constant related to the interest rate and

$X = a^{2}Y$, where $a$ is a parameter from the production function.

From the second and third equations above, it's clear that $Y,X$ and $N$ grow at the same rate. However, it's not clear that this rate should be $g_{c}$ or even constant at all. I'm trying to use the first equation to show it, but I couldn't do it.

The book forgoes this proof. It says that it's quite similar to the AK model since there's no transitional dynamics. Is there a simple way to show this result? I think it should follow from a simple manipulation of the first equation. Any ideas? thanks a lot in advance!

The household side of this model is pretty standard. Denote $K(t)$ be household's assets at time $t$. Then the transversality condition (which is an optimizing condition, not a constraint), is

$$\lim_{t \to \infty} [e^{-\rho t}\lambda(t) K(t)] = 0$$

where $\lambda(t)$ is the current value multiplier on assets in the Hamiltonian. Given the assumed form of the utility function we have the f.o.c.

$$\lambda(t) = \frac {1}{c(t)^{\theta}}$$

So we want

$$\lim_{t \to \infty} [e^{-\rho t}\frac {1}{c(t)^{\theta}} K(t)] = 0$$

and since in the model we have a constant consumption growth rate, $g_c$ we have

$$c(t)^{\theta} = c(0)^{\theta}\cdot e^{g_c\theta t}$$

Ignoring inconsequential constants, we require

$$\lim_{t \to \infty} [\frac {1}{e^{(\rho+g_c\theta) t}} K(t)] = 0$$

Use the relation between household assets  and the number of goods $N$, the relation between income (from labor and assets) to aggregate output, as well as the expression of aggregate output as a function of $N$, to obtain a first-order linear differential equation in $N$, and proceed  as in pages 208-209 of the book.


Computing optimum efforts Consider the following cost function:

$$c(e_1, e_2) = (\beta_1e_1 + \beta_2e_2)^2$$

The value function is:

$$v = v_0 - [l_1(1-e_1) + l_2(1-e_2)]$$

How do I compute the optimum efforts $e_1$ and $e_2$, with $e_{i} \in [0,1]$ ? I seem to be ending up in an inconsistency.

If I look at $v - c = v_0 - [l_1(1-e_1) + l_2(1-e_2)] -  (\beta_1e_1 + \beta_2e_2)^2$, and then I take the partial derivative with respect to $e_1$ and $e_2$ respectively, I obtain:

$$\beta_1e_1^* + \beta_2e_2^* = \dfrac{l_1}{2\beta_1} = \dfrac{l_2}{2\beta_2}$$

What am I missing? Does the inconsistency simply mean that the optimal efforts are corner points?

You are right. The problem here is a corner solution.

Let us define the optimal combination $X^* = \beta_1e_1^* + \beta_2e_2^*$.

The first derivative gives you

$$X^* = \dfrac{l_1}{2\beta_1} $$

whereas the second gives you

$$X^* = \dfrac{l_2}{2\beta_2}$$

They can only be true "by chance". This is, the existence of an interior solution cannot be assured for every possible parameterisation allowed by the parameter space.

Actually, if you take a look at the two functions, this is clear. The cost function generate linear isoquants. You can check this here. This is because the cost function is equivalent to $c = X^2$ and $X$ is a linear combination of both $e_1$ and $e_2$.

Similarly, the value function also gives linear isoquants. This is because the function has the form $e_1 = a + be_2$.

Then, the most likely case is that the isoquants will join in a corner. The interior solution happens just by chance. Notice that when this is the case, any interior solution is optimal! The equilibrium set is then infinite.


Is there any way to sufficiently cover the cost of a UBI? Note: I'm talking about the cost of a UBI, and the cost only. Not the social benefits, or the political barriers, or all that stuff. Just the cost.

If you've been following American politics for a while, then you've probably heard of Andrew Yang. He wants to give everyone over 18 and under 65 $1000 every month, free and clear, and no questions asked.

However, how do we (or in this case, the government) pay for it ($2.4+ trillion)?

On his site, Andrew Yang suggests the following:

500 billion saved from welfare programs (you either chose UBI or current benefits)
100 billion saved from healthcare related needs
800 billion generated from a VAT
600 billion from economic growth

This all adds up to $2 trillion.

500 billion saved from welfare. Okay. Another $800 billion generated from a tax. Okay.

But what about the $100 billion saved from healthcare needs? Is that an actually reason, which studies to back it up?

Also, at what point do we keep some of that money as profit?

So with all of these factors combined - can we fully pay for a UBI?

Since you say that you want to ignore social benefits etc. then answer is trivial. As long as the cost of UBI is not so high that taxes would cross the peak of Laffer curve then answer is always yes, after the peak always no (peak of Laffer curve is a point of maximum possible revenue government can extract from economy through taxation).

On left side of Laffer curve just raise taxes high enough and you can pay for it. On right side the program would be too expensive to pay for it as simply it’s cost would outstrip the maximum resources government can extract from economy. So the candidate can decide to raise VAT taxes higher or to increase federal income tax etc as long as the candidate does not cross that point.

You can also cover any slack by government borrowing - there a maximum sustainable point would be where the growth rate of debt just matches the growth rate of economy. If you consistently borrow more than that you put economy on a path where in the limit debt to GDP goes to infinity- which is not good. If you consistently borrow equal to growth rate or less than that then you are on either stable path or path where Debt to GDP goes to 0 in limit.

I am also not aware of a study that would say that UBI reduces healthcare costs, however my feeling is that the point that candidate was trying to make is that thanks to UBI gov. can reduce some healthcare welfare spending as every person already gets subsidy that can be spend on medical care.

Also there can be no profit. UBI is a transfer of resources from one point in the economy to other. It does not create any profit. You could argue that compared to some other transfer it can create less distortions and less deadweight loss, but I would not call that profit. A profit is a difference between total revenues and total costs of some activity UBI has no revenue side only cost side as it’s a welfare-transfer measure.

However, as always question is if the value of UBI to society outweighs the costs they have to bear for it, but that is ultimately just a value judgement. So thats a problem for moral philosophers to worry about and answer not for economists.


What are some currencies that are not pegged and fixed? Are there any currencies that are not pegged to the dollar or euro and are purely fixed exchange rates?

According to the table from this website, these are countries that peg their currencies to the Euro:

Bosnia and Herzegovina
Bulgaria
Comoros
Denmark
Sao Tome and Principe

Other than these, Bhutan fixes their exchange rate to the Indian rupee, Lesotho and Namibia fix their exchange rates to the South African rand.


Does GDP Income Approach hold? Let me introduce my question with an example. There is a company producing cars. $20$ cars are produced, with a value market of $20000u$.  $\textrm{GDP} = 20\cdot20000u = 400000u$. Assume that the company doesn't purchase any intermediate good, so all the income, $400000u$ goes for the company, and you can compute (income approach) $\textrm{GDP} = \textrm{S} + \textrm{B}$, where $\textrm{S}$ stands for salaries and $\textrm{B}$ for benefit. But we are somehow assuming that the company will get $400000u$ for selling the cars. What would happen if they only sell 15 cars? They will only get $15\cdot20000u = 300000u$, so $\textrm{S} + \textrm{B} \neq \textrm{GDP}$. I have read that this is written down as if the company buys the $5$ remaining cars, but there is no money transfer here, so they can't distribute the $100000u$ worth of cars between $\textrm{S}$ and $\textrm{B}$. Why would the income approach for GDP hold in general?

Using the income approach, GDP is the sum of compensation of employees, gross operating surplus, gross mixed income, and taxes less subsidies on production and imports. Gross operating surplus includes not only corporate profits but also increases in inventory valuation.

In the above example, using the income approach, the 15 sold cars will reflect under corporate profits and the value of the 5 unsold cars will reflect in an increased inventory valuation. Using the expenditure approach, the 15 sold cars will reflect under consumption expenditures and the 5 unsold cars will reflect changes in inventory valuation under investment. In practice, you cannot determine the value of a car until it is sold, so most accounting standards will conservatively use the cost to produce the car for inventory valuation. Since the cost of production in the example is unrealistically zero, the change in inventory valuation would also be zero in both approaches. Note that even if a less conservative accounting standard was used, both approaches would lead to the same result as it would affect the valuation of the 5 unsold cars in the same way.


Is there an "ideal" ratio of tax revenue to GDP? Or, at least, can I assume a higher ratio is better? At the link below, you can see countries have a huge difference in ratios of tax revenue to GDP. Is it fair to assume that a higher ratio is better? Does raising a substantial portion of your GDP in taxes mean that your country is more or less financially stable?

I see that Algeria is #1 worldwide at 63%, while UAE is the lowest at 1.4%. I assume this is because the UAE has a huge GDP from oil (and thus taxes don't play a big role in its GDP) and because Algeria has a tiny GDP (and thus, a fairly-efficient tax system results in a fairly high ratio).

https://en.wikipedia.org/wiki/List_of_countries_by_tax_revenue_to_GDP_ratio

There is no ideal ratio as the “ideal” (I choose to interpret the ideal as optimal/efficient) level of tax to GDP depends on the moral philosophy. Generally Ralwsians prefer more taxation in order to help poor people, through redistribution, than libertarians/classical liberals, who would prefer private solutions such as charity.
Higher is not always better. First, there is a Laffer curve - at a 100% tax rate the collected tax is zero because nobody is going to willingly become a slave by working only so that government takes all of their money. So there is always a maximum tax rate and implied tax to GDP ratio that is the maximum feasible tax any rational government can implement. Where exactly is the peak (gov revenue maximizing) tax rate changes from country to country, although in advanced countries it is usually around 60-70% (see for example this study) of all income meaning tax to GDP ratio would be 0.6-0.7.
Second, actually the peak of Laffer curve is higher than a welfare maximizing top marginal rate even under Ralwsian preferences (which are the most commonly used redistribute preferences) so if you care about welfare of population instead of just the revenue of government you would usually keep the tax to GDP ratio lower.
Country can be financially stable at any tax to GDP ratio. In order to be financially stable you just need to have debt to GDP ratio be on non-explosive path. The explosive path is a one where debt grows faster than the overall economy in long run. As long as the debt grows at the same rate as economy or at smaller rate the country’s public finances are stable.
Having high tax to GDP ratio does not mean your tax system is efficient. Algeria is not really known for its tax efficiency.


Are spectrum auctions a tax? According to Wikipedia,

A spectrum auction is a process whereby a government uses an auction system to sell the rights to transmit signals over specific bands of the electromagnetic spectrum and to assign scarce spectrum resources.

In  2000, for example, the UK government auctioned off five 3G licenses and raised a total revenue of $£22.5$ billion -- or $2.5$ percent of the UK's Gross Domestic Product at the time. In The Undercover Economist, Tim Harford argues that the auction could not have resulted in higher prices for mobile phone users, because

firms will charge the maximum they possibly can, under any circumstance [whether the licenses are cheap or not]. We also know that their ability to do this is limited by their scarcity power... In the UK there was scarcity built into the amount of radio spectrum available:
five licenses were the most the technicians could offer. It does not make any
difference to the customer how much the licenses cost, but it does matter to the
taxpayers, who would like the government to get a lot of money for this valuable
public resource.

Let's say for the sake of simplicity that each license sold for an equal price, $£4.5$ billion (the prices varied, but this doesn't affect my argument). This amount had to come from somewhere. Let's say they borrowed this money. The bills they send to their customer will be affected by three things: infrastructure costs and wages, profits, and the repayment of the loan used to pay for the license. In a free market, the companies would have roughly the same costs and profits. Harford is right that neither of these is influenced by how the licenses were distributed: scarcity is baked into the amount of spectrum available, and only scarcity power determines profit.
But consider the last of these costs. Over time, companies would recoup the licenses' cost by charging their customers. Sure, they could postpone this recoupment, but they will get it someday (and will pass on the interest to the customer). It should "make a difference to the customer how much the licenses cost". If my thinking is right, the windfall from this auction was right out of the taxpayer's purse.
In reality, telecom companies in the UK nearly went bust after the auction. But that needn't have happened. The telecom companies without 3G licenses were the most seriously afflicted. Elsewhere, too, telecom companies have paid exorbitant sums for spectrum licenses and come away just fine.
Tim Harford got his economics degree from Oxford. I, on the other hand, haven't ever read an economics textbook. I'd bet the error in reasoning is on my part rather than Harford's, but I'd like to know where.

Suppose you write some software that you can then freely sell at practically no cost per unit (the wonders of the internet). You want to make as much profit as possible. Since you have almost no per unit cost, maximizing your profit will amount to maximizing the revenue, the price per unit times the number of units sold.
Note what did not enter the calculation: The cost of developing the software. How much money you might eventually get might influence whether you put in some effort to create the software, but it will not influence your pricing decision once you made the software.
The companies bidding for spectrum are essentially in the same situation. Once they have gotten their spectrum rights, how much they paid will not influence how much they charge..
Of course someone has to pay, but these are companies bidding for monopoly rights. Apart from the cost, the license gives them a huge profit and this profit is what the cost is paid from.


How to design / interpret a difference in differences regression I have a set of active scientists and editors. I want to study the effect that becoming an editor has on the different outcomes of a scientist (such as citation count and publication rate).
I decided to do a difference in difference regression. I'm not sure though is all the steps taken are correct.
I match scientists who did not become editors (control group) to scientists who did become editors later in their career (treatment group) such that they share certain characteristics in the beginning of their career. These characteristics are gender, rank of affiliation, first year of publication (to estimate "era" and "experience"), and discipline.
I define the year in which a scientist becomes an editor as year0. Then I look at the effect becoming an editor has on different outcomes over different years post editorship (year1, year2, ... , year10).
Question 1: Is it correct to have your treatment and control groups matched at the beginning of their career? I read many different opinions on the matter online.
I run a regressions as follows, and explain it as follows:

Question 2: What is the difference between gamma and delta, I know both explain the effect of being an editor, but one is the interaction and one isn't?
This is the way the final table looks like:

The "interaction terms" are the values for delta.
Question 3: Would it be correct to state that:

The impact (number of citations) of a scientist increases every year they become an editor
The impact of a scientist increases more than three folds by their 10th year as an editor compared to a comparable scientist that did not become an editor?
Becoming an editor has a slight negative impact on productivity over the years?

The point of control group matching is to ensure similarity between the groups, to isolate the effect of the treatment variable. If are assuming that the development paths of all academics is identical, and at some point, all academics have a chance to become editors, and some do, then you a reasonable to match at career start.
That's not a good assumption. Academics differ widely in quality/competency/publication/citation over the course of their careers. Much better to find matches by identifying academics that were similar those the became editors, who (for whatever reason) didn't, so that your control group is identical to your treatment group.


Mechanisms to incentivise social welfare Suppose there is a set of agents causing loss $l_i$ to themselves and aggregate loss $L$ to the society, both of which can be minimised by their investments $c_i$.

The social planner who is concerned with minimising $L(c_i)$ wants to incentivise players to invest more. Players have utility $u_i=-f(l_i)-c_i$ and they decide on whether to invest more based on utilities.

What are some mechanisms which can incentivise agents in this framework? Can VCG-type mechanisms be adapted to this situation?

(PS: Sorry if the question is vague; please feel free to add assumptions)

You say that losses are reduced by investment, so we can write the (private) loss as a function of own investment and others' investment: $l^i(c^i,\mathbf{c}^{-i})$ with $l^i_1(c_i,,\mathbf{c}^{-i})<0$.

Player $i$'s utility is $U^i(c^i,\mathbf{c}^{-i})=-f(l^i(c^i,\mathbf{c}^{-i}))-c^i$ and he invests until the first-order condition is satisfied:
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-1=0.$$
The social loss is
$$\sum_i\left[-f_i(l^i(c^i,\mathbf{c}^{-i}))-c_i\right]$$
so that the first-order condition for the optimal $c^i$ is
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})\right]-1=0.$$

This now looks like a standard externality model, where the external benefit of $i$'s investment is
$$b^i(\mathbf{c}^{-i})=-\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})>0.$$

If $l^i(\cdot)$ are publicly known then the problem can easily be solved as by imposing upon $i$ a standard Pigouvian subsidy of $b^i(\mathbf{c}^{-i})$ per unit of investment. This means that $i$'s payoff (including the subsidy) when $j\neq i$ choose $\mathbf{c}^{-i}$ is
$$\underbrace{-f(l^i(c^i,\mathbf{c}^{-i}))-c^i}_{U^i(c^i,\mathbf{c}^{-i})}+b^i(\mathbf{c}^{-i})c^i.$$
A rational $i$ will maximise this subsidy-augmented payoff; his first order condition is
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})\underbrace{-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,c^{-j})\right]}_{b^i(\mathbf{c}^{-i})}-1=0,$$
which is exactly the same as the first-order condition used to maximise social welfare. It is therefore a Nash equilibrium for each $i$ to choose the socially efficient level of investment given that everyone else does the same.

If the $l$s are heterogeneous and private information then things get more complicated because we have to get people to reveal their private information. Assume there is a benevolent central planner with the power to monitor agents' investments. One way to do this is then, indeed, to use a VCG mechanism. It would work like this: first, the central planner asks each agent to report their $l^i(\cdot)$. Write $\widetilde{l}^i$ for the reports. He then solves the first-order condition for the maximum of social welfare given the (possibly dishonest) reports. Thus, he maximises
$$\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})=\sum_i\left[-f_i(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c_i\right].$$
Next the planner instructs all players to invest up to the optimal level. In return, he promises to pay to each $i$ a subsidy equal to
$$S^i=\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)|c^i\right]-\max_{\mathbf{c^{-i}}}\left[\widetilde{W}(\widetilde{\mathbf{l}},c^1,\ldots,c^{i-1},0,c^{i+1},\ldots,c^n)-(-f(\widetilde{l}^i(0,\mathbf{c}^{-i})))\right].$$
Woah, what is that ugly thing!? The first set of square brackets is the welfare everyone other than $i$ gets. The second is the welfare everyone other than $i$ would get if $i$ were to drop out of the market all together. Thus, the difference between the two is $i$'s positive externality on everyone else.

Now let's look at $i$'s payoff:

$$U^i=S^i-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i$$

When evaluated at the equilibrium $\mathbf{c}^{-i}$ some terms cancel and we have

$$U^i=\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})|c^i\right]-\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},c^1,\ldots,c^{i-1},0,c^{i+1},\ldots,c^n)-(-f(\widetilde{l}^i(0,\mathbf{c}^{-i})))\right]$$

The second square bracket here does not depend on $c^i$ (remember, it is the surplus if $i$ were to drop out of the market) so $i$'s problem collapses to choosing $c^i$ to maximise $\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})|c^i\right]$. Thus, $i$ has the right incentive to choose $c^i$ to maximise social welfare with respect to the reported $l$s.

What about the incentive to report $l$ honestly? The social planner is going to chose the optimal $c$s to maximise
$$\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})=\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)\right]+(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)$$
(this is just $\widetilde{W}$ with $i$'s utility added on and then subtracted again). $i$ would like to maximise
$$U^i=S^i+(-f(l^i(c^i,\mathbf{c}^{-i}))-c^i)$$
which is
$$U^i=\underbrace{\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)\right]+z}_{S^i}+(-f(l^i(c^i,\mathbf{c}^{-i}))-c^i)$$
where $z$ is the second term in $S^i$ (does not depend on $\widetilde{l}^i$). Thus we see that the social planner's choice will coincide with $i$'s optimum exactly when $l^i=\widetilde{l}^i$ so that $i$ has an incentive to report $l^i$ truthfully.


Does commercial paper usually carry a higher yield than corporate bonds? Why? In Banking: A Very Short Introduction, Goddard and Wilson write:

  [Commercial paper] usually carries a higher yield than corporate bonds.

But aren't corporate bonds longer term than commercial paper and so all else equal should pay higher yields/interest rates?

The wording seems incorrect, the only thing that appears to make sense is “higher spread,” since as you note, bonds are issued with longer maturities. The yield differential would depend on the slope of the curve.

Whether “higher spread” is accurate is yet another question. I did not spend much time looking at corporate bond data, but spreads normally increase as maturity lengthens for firms that are strong credits. A bond that is secured against specific collateral might trade tighter, but most “corporate bonds” are unsecured (as opposed to asset-backed securities). However, commercial paper may have wider spreads for banks, as the bonds may be senior versus commercial paper. (I’m guessing about that.)


Different values for US Public Debt, why? In this OECD link we have the US General government debt. The value is close to 120% of GDP.

However, the wikipedia has the US Federal Gov debt at 80%.

Why is there such a difference? What debt is being considered by the OECD, that is not by the wikipedia graph?

Any help would be appreciated

The OECD number includes intragovernmental debt, which is debt not held by the public, while the Wikipedia number does not. One of the largest of these debts (several trillion dollars) is related to Social Security, which is a debt owed by the federal government to the Social Security Trust Fund. A review of the history of this debt and its implications is available from the Social Security Administration.


The relationship between gross profit and tax I tried to do following question but I am not sure about my solution.  Please tell me your opinions

Since gross profit is strictly concave, I can say that $R’’(y)-C’’(y) <0$
Now I maximize the net profit and
FOC is $R’(y)-C’(y) -t=0$
SOC is hold because of strict concavity in y.
That is $MR(y)=MC(y)+t$ from FOC.
Now show that the implicit relationship btw y and t can be solved for the explicit choice function
$$y=y^*(t)$$
FOC with respect to y combine with above function
$R’(y^*(t))-C’(y^*(t)) -t=0$
Derivative wrt t
$$R’’(y) (dy^*/dt)-C’’(y) (dy^*/dt)-1=0$$
$$\frac{dy^*}{dt}=\frac{1}{R’’-C’’}<0$$
which means that the output will decrease as the tax goes up.
Think again gross profit
$G\pi =R(y^*(t))-C(y^*(t))$
I am not sure about my solution after this point especially.
Suppose the firm is price taker then $R(y)=py$
So
$$G\pi =py^*(t)-C(y^*(t))$$
That is, since the output will decrease as tax rises, revenue $R(y) $ will decrease as well. Accordingly gross profit will decrease.
———————
I am not sure about the last part especially. Thank you.

Most of what you have done was correct. Here are the steps to finish the argument.

Evaluating the profits at the optimum, you get
\begin{equation}
\pi_G=R(y^*(t))-C(y^*(t))
\quad\text{and}\quad
\pi_N=R(y^*(t))-C(y^*(t))-ty^*(t).
\end{equation}
Differentiate the net profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi_N}{\mathrm dt}&=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-t\frac{\mathrm dy^*}{\mathrm dt}-y^*\\&=\underbrace{[R'(y^*)-C'(y^*)-t]}_{=0 \text{ due to FOC}}\frac{\mathrm dy^*}{\mathrm dt}-y^*<0,
\end{align}
which is true assuming $y^*>0$.
Differentiate the gross profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi_G}{\mathrm dt}&=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}=[R'(y^*)-C'(y^*)]\frac{\mathrm dy^*}{\mathrm dt}<0.
\end{align}
This is true because from the FOC, we know that for any positive tax, $R'(y^*)-C'(y^*)>0$, and since you've correctly derived $\frac{\mathrm dy^*}{\mathrm dt}<0$, their product must be negative.


Signalling models and out of equilibrium beliefs Consider the simplest version of the Spence signalling model. There are two types of worker, with either productivity $\theta_H$ or instead productivity $\theta_H < \theta_L$. The proportions of the workers are $p_H$ (share of high types) and $1 - p_H$ respectively. Risk-neutral and competitive firms pay wages that may depend on a worker's education level $e \in \{0, 1 \}$. Finally, getting education ($e = 1$) costs $c_H$ for the high types and $c_L > c_H$ for the low types.
My question is simple:

What are the possible wage schedules that sustain a pooling
equilibrium? In particular, can a pooling equilibrium wage schedule specify a wage of
zero following $e = 1$?

I will briefly elaborate on my understanding of the problem and what is unclear to me. In any pooling equilibrium, workers must be paid their expected productivity. So following $e = 0$, the specified wage must be
$$
w(e = 0) = \mathbb{E}(\theta) = p_H \theta_H + (1 - p_H)\theta_L
$$
Furthermore, to ensure that nobody wants to deviate (by getting education), we only need to ensure that the high types won't deviate. (If they won't want to deviate, then neither will the low types). This requires that
$$
w(e = 1) - c_H \leq w(e = 0) = p_H \theta_H + (1 - p_H)\theta_L
$$
So the wage following $e = 0$ is uniquely determined in the pooling equilibrium (the equation). Moreover, the wage following $e = 1$ must be sufficiently low to ensure that nobody gets education (the inequality). I am wondering if there are any additional constraints that the wage schedule in a pooling equilibrium must satisfy, or whether any wage schedule satisfying these constraints will sustain pooling (as a PBE).
To make the issue a bit more salient, suppose that we propose some wage $w(e = 1) < \theta_L$. For example, suppose you make the wage for those who get education zero. Clearly, nobody will choose education, i.e. the inequality holds. Moreover, wages don't need to be set optimally following $e = 1$ since this happens with zero probability in equilibrium. On the other hand, no possible type could have a productivity of zero, so this wage schedule does not follow from any possible firm beliefs. This suggests that the wage following zero education needs to also satisfy
$$
w(e = 1) \in [\theta_L, \theta_H]
$$
Is this indeed a further constraint (in any pooling equilibrium where both types choose $e = 0$)?

In my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply $E[\theta]$. If, in your example, a worker sends the off-path messags $e=1$, the wage must be $w(e=1)=w_1=b_1 \theta_H + (1-b_1) \theta_L$, where $b_1$ is the belief in the information set of $(e=1)$ that the type is high. Since $b_1 \in [0,1]$, it must be that $w_1 \in [\theta_L,\theta_H]$.
This information set is not reached on equilibrium path such that we are free to determine the out-of-equilibrium belief $b_{1}$. Only $b_{0}=p_H$ is a PBE-consistency requirement.
The set of all no-education pooling PBE is then given by the following:
$e_H=e_L=0$,
$w_{0}=E[\theta]$,
$w_{1}=b_{1}\theta_H + (1-b_{1})\theta_L$,
$ b_{0}=p_H$,
$b_{1}: b_{1}\theta_H + (1-b_{1})\theta_L -c_H\leq E[\theta].$
Note that the beliefs are part of the equilibrium. To get rid of equilibrium multiplicity, have a look at the numerous refinements such as the intuitive criterion that kills unreasonable out-of-equilibrium beliefs (and thereby unreasonable equilibria). The restriction on belief $b_1$ originates from the incentive constraint of the high type. We want to make sure that $\theta_H$ does not want to deviate get eductation, and because the wages are competitive they are implied by the off-path belief which we can set freely.
The idea to set $w_1<\theta_L$ would violate the competitive nature of firms.
This sentence can be interpreted in two ways.

Wages are just exogenously set this way.
There are two firms (having the same beliefs) who both set a wage menu $(w_1,w_2)$ and a la Bertrand the worker always goes to the firm setting the higher wage.
For some off-path beliefs, one firm would deviate from $w_1<\theta_L$ and may attract workers by setting a wage $w_1=b_{1}\theta_H + (1-b_{1})\theta_L - \varepsilon$ and thereby make a profit. Because you did not specify the off-path belief, we have no idea whether this deviation on the wage would be profitable.


Why is JPY considered a "safe currency"? When the economic prospect around the world feels getting worse and/or getting higher uncertainty, usually the value of JPY is getting higher, because investors are said to start to buy JPY to shelter their money in a safer place. In some cases even if the bad economic prospect is not on US or Europe, it seems that JPY still tends to be stronger even compared to USD or EUR (e.g. near the French election this year or Brexit, JPY was getting stronger compared to USD).

But it seems ridiculous to think of JPY as a safe currency, given that Japan's insanely high amount of debt.

So what makes JPY considered so safe to buy in case the economic situation feels getting worse and/or getting higher uncertainty around the world? And is JPY a safe currency to begin with, even compared to other major currency such as USD, EUR, CHF, CHY, CAD, etc...?

Because:

they are a massive exporter, meaning their capital account is negative, and thus they own a lot of assets abroad. Therefore, as this site explains:

When a crisis erupts, the Japanese repatriate that money – they bring it back home. In that process, they’re selling assets in other currencies to buy yen. And at that point it’s a supply/demand issue. The supply of others currencies is rising as the Japanese pull out (those currencies fall in value) and demand for yen rises as the Japanese bring their money back home (the yen rises in value).

they have had and are likely to continue to have among the lowest interest rate in the world (-0.1% now). Therefore, as this site explains:

In what’s known as a carry trade, investors will borrow money in a low-interest rate environment and then invest that money in higher yielding assets from other countries. Japan’s long-standing policy of near-zero interest rates has caused it to become a major source of capital for these types of trades.
When uncertainty subsequently hits global markets, it can cause investors to unwind these trades, which then results in additional demand for the yen.

hoard behaviour + path dependence in financial markets. As the site above explains:

Part of the reason the yen continues to act as a safe-haven currency is simply because everyone acknowledges that it is. Investors around the world have come to embrace the yen as a reliable place to go for safety, and by attempting to take advantage of this, they actually create the intended effect.
If you know that whenever the world panics, the yen is going to rise, then it makes it a no-brainer to buy the yen even if you’re unsure exactly why everyone else is.

Some differ though. An IMF study concedes that:

Safe haven currencies tend to have low interest rates, a strong net foreign asset position, and deep and liquid financial markets. Japan meets all these criteria.

But they find that:

Surprisingly and in contrast to the experience of the Swiss Franc, yen risk-off appreciations appears unrelated to capital inflows (cross-border transactions) [large foreign asset ownership] and do not seem supported by expectations about the relative stance of monetary policies [low interest rate].

They state:

The third possibility is that risk-off appreciations of the yen come about because of portfolio rebalancing transactions that are not fully captured by BoP statistics, such as derivative transactions. For example, when risk perceptions change, exporters or overseas affiliates of Japanese companies may decide to lock in the exchange rate through hedging, or given Japan’s sizeable net foreign asset position, holders of foreign securities may take a long yen position. As such, these derivative transactions may not just reflect short-term speculative behavior, but may just reflect prudent risk management. We find support for this conjecture as risk-off episodes are associated with increases in net non-commercial exposures to the yen in the derivatives market.


Is an EU single currency still a trade advantage in 2019? When the single currency was planned, there was no internet and there all currency exchange was manual.

Today the web can automatically convert prices using multiplication of the numbers in the price field, and robotic cash exchange can count and exchange currencies better and cheaper than a human exchange.

Does the UK suffer an considerable economic handicap from having not adopted the Euro? Is the single currency concept nearly obsolete?

Very good points about the role of modern tools (the web, robotic cash exchange, etc.) are raised. The  classic argument is that trade should expand within the eurozone following (1) increase in price transparency, (2) elimination of currency conversion costs, and (3) elimination of exchange rate fluctuations.

Are the classical arguments challenged by modern tools?

Price comparison across border is surely much easier today with the web than it was in the nineties.
In  its  report  One  Market  One  Money,  the  European  Commission  (1990)  calculates  that  eliminating  costs  of  conversion  of  currencies  and  costs  of  cover  for  exchange  risk  will reduce costs by .25 to .5 of 1 percent of total output in the EU. The report also stresses additional  gains  that  would  come  from  the  elimination  of  “in-house  costs”  associated  with multiple  units  of  account.  These  are  reductions  in  costs  connected  with  record  keeping,  accounting and decision-making. The question: How much lower are the conversion costs thanks to modern tools? Even if they are now lower, they are still positive.
Despite modern tools, exchange rates still fluctuate.

So, the adoption of the single currency may still boost trade.

The role of modern arguments

To these classical arguments, we can add more subtle ones. For instance, in some industries, components cross borders many times before being assembled and sold as a final  good,  even  a  small  reduction  in  transaction  costs  can  increase  trade  flows non-linearly (Flam and Nordstrom, 2006).

In a series of papers, Richard Baldwin adds that adopting a single currency may (1) reduce collusion among competitors and eliminate pricing mark-ups and enhance trade (p. 63-64); (2) increase the entry of small firms by reducing the fixed cost of trade.


Usefulness of the value for the slope of the demand curve I see the elasticity value(s) of demand being used frequently to determine the relation between demand and variables such as price and income. I have however been unable to find any uses for the slope of the demand curve. Are there any?

Also, is there any relationship between the slope and the elasticity?

Price elasticity of demand is related (but not equal) to the inverse of the slope of the demand curve.

This Wikipedia article defines PED as:

  Price elasticity of demand (PED or Ed) is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price, ceteris paribus. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. [emphasis added]

Look at the following (canonical) supply-demand graph.

Let's define $P^*$ the price of market equilibrium, $Q_D$ as the quantity demanded and $E_D$ as price elasticity of demand. Then:

$$ {slope} = \frac{rise}{run} = \frac{\Delta P}{\Delta Q_D} $$

And in mathematical terms, the verbal description of the PED reduces to:

$$ E_D = \frac{\frac{\Delta Q_D}{Q_D} \cdot 100}{\frac{\Delta P}{P^*} \cdot 100} $$

$$ E_D = \frac{P^*}{Q_D} \cdot \frac{\Delta Q_D} {\Delta P} $$

$$ E_D = \frac{P^*}{Q_D} \cdot \frac{1}{slope} $$

Edit

Constructive comments have been incorporated into this answer.

In particular, @Kontorus pointed out:

  The PED is a percentage change, the slope of the demand curve is an absolute change.

And @denesp pointed out that:

  ... linear demand curves have constant slopes but changing elasticity. Cobb-Douglas derived demand (hyperbole) has changing slope but constant elasticity.


House allocation: Data notation correction with Top Trading Cycles (TTC) example I am trying to understand TTC algorithm for house allocation. I have got a small problem. I am trying to understand the following example:

  IE=[i1 , i2 , i3 , i4], IN=[i5], Ho=[h1 , h2 , h3 , h4], and Ho=[h5 , h6 , h7]

The above example is using the Ho variable twice (which is not correct). Somebody please guide me what could be the other variable:

I got the following image:

The link of the paper is:

House Allocation with Existing Tenant

Somebody please guide me.

I think this is a typo in the paper. As far as I see, the houses $\{h_5, h_6, h_7\}$ are not occupied, so it should be that $H_V = \{h_5, h_6, h_7\}$.


In the C.E.S. utility function do the parameters need to add up to unity to obtain the Cobb-Douglas utility function? Consider the C.E.S. utility function

$$U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}} $$

Is it true that we must have $a+b=1$ in order to obtain a Cobb-Douglas utility function as $c\rightarrow 0$?

Yes.

Write

$$U(x,y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}} = \exp\left\{\frac {-1}{c}\ln \left(ax^{-c} + by^{-c}\right)\right\} \tag{1}$$

Now if $a+b=1$ then as $c\rightarrow 0$, expression

$$\frac {\ln \left(ax^{-c} + by^{-c}\right)}{-c} \tag{2}$$

will be an indeterminate form $0/0$ and so we can apply L'Hopital's rule on it to get

$$\frac {1}{-c}\cdot \frac {-ax^{-c}\ln x + by^{-c}\ln y}{ax^{-c} - by^{-c}} \rightarrow \frac {a}{a+b}\ln x + \frac {b}{a+b}\ln y,\;\;\; c\rightarrow 0$$

where we have assumed $a+b=1$.

So (by the uniform continuity of the exponential function)

$$c\rightarrow 0,\;\;\; U(x,y) = \exp\left\{\frac {-1}{c}\ln \left(ax^{-c} + by^{-c} \right) \right\} \rightarrow \exp\left\{a\ln x + b\ln y \right \} = x^ay^b$$.

But if  $a+b\neq 1$, then as $c\rightarrow 0$, the argument of the logarithm in the numerator in eq. $(2)$ would not have been equal to unity, and so the logarithm would not equal zero (which would give us the indeterminate form, and would allow us to use L' Hopital's rule). Instead, eq. $(2)$ would have gone to infinity. So $a+b=1$ is needed in order to arrive at the Cobb-Douglas function.


Are there papers in evolutionary game theory that are influential in economics? The concept of evolutionary game theory has been around for about forty years. The mathematical details are worked out quite well. Yet this discipline is usually a side note in game theory classes and is seldom applied in economics research. Have there been any papers in this field that are considered influential in economics?

To clarify:  There are a lot of papers about the connection of game theory and evolutionary game theory, but only a few that seek to apply evolutionary game theory to explain some economic phenomenon.

I know of this paper by Vega-Redondo and this paper by Schaffer.

Kandori, Michihiro & Mailath, George J & Rob, Rafael, 1993. "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages 29-56, January.

Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.

I disagree with your assertion that EGT is seldom applied or that it has not been  influential.  Both of the papers referenced above appear in RePEc's list of the
top 1% of citations.

Allowing preferences to 'change' can generate just about any equilibrium you want, so the premise behind EGT is a tough sell in many economic models - not to say compelling research hasn't done so.  The bottom line is that it isn't a part of the atmosphere where mere mortals can tread.

EGT has, in fact, been applied to many problems in economics as well as those traditionally parsed by the other social sciences - particularly the evolution of social norms, political representation, voting, and distributive justice.


How can I prove that a utility function does (or does not) satisfy diminishing MRS? I have this CES utility function:
$$U(f, c) = (f^\alpha + c^\alpha)^{1/\alpha},$$
with $\alpha > 0$.
The problem set asks does it "satisfy the principle of diminishing marginal rate of substitution for any value of $\alpha$".
To which the answer is "False, if $\alpha > 1$ the indifference curves are concave, so they don’t satisfy the principle of diminishing MRS"
But short of graphing the indifference curves and hoping that it's visually obvious, I'm not sure how to tell this.
How can one tell from a utility function whether it is or is not non-concave?

If you remember, in a two-dimensional curve, its concavity or convexity (the slope of its slope) is given by the second derivative. For a three-dimensional function, you want to look at the Hessian table (the table of all second derivatives).
If the Hessian is negative definite for all values then the function is strictly concave, and if the Hessian is positive definite for all values then the function is strictly convex. If the Hessian is not negative semidefinite for all values then the function is not concave, and hence of course is not strictly concave. So on, so forth.
This particular function will be a bit more tricky to find second derivatives, I presume.
There are various examples shown in a very nice website here.
There is even a Cobb-Douglas example, which I'm certain you'll find valuable.


Real Exchange Rate vs PPP rate I'm having troubles to understand the difference between the Real Exchange Rate and the PPP rate.

I know the first one is calculated using a basket of goods and services so that the non-tradeable basket costs the same in 2 countries (imagining a 2-country world).  Now, isn't that the rate that would hold if PPP applies?

I think I'm wrong if I say that the real exchange rate is the one that makes PPP become valid.  Any help appreciated.

There are several exchange rate concepts that need to be distinguished. There are bilateral and multilateral (aka "effective") exchange rates, nominal and real exchange rates, and market-price versus PPP (purchasing power parity) exchange rates.

Let's start with the simplest concept: A bilateral nominal exchange rate, e.g., 117 yen/(US dollar), 0.8 pounds/euro, 7.7 (Swedish krona)/(Swiss Franc), etc. These are the exchange rates that are commonly reported in newspapers and in any number of online outlets.

Aside: There are two distinct quoting conventions for exchange rates: the "E" system (with the home currency in the numerator) and the inverse, the "R" system. A nominal exchange of 1.2 Swiss francs per euro is expressed in the "R" system from the point of view of a euro area resident, and in the "E" system from the point of view of a resident of Switzerland. The "R" convention is conceptually easier to work with, since an increase of the numbers of foreign currency units per unit of home currency corresponds naturally to an appreciation of the home currency. Using the "E" system, the opposite is the case. There seems to be little or no standardization with regard to quoting conventions of bilateral exchange rates. However, multilateral exchange rates are always expressed using the "R" convention, such that a larger numerical value corresponds to an appreciation of the currency, and vice versa.

A bilateral real exchange rate is the bilateral nominal exchange rate multiplied by a ratio of price indices of the two currencies. (How the ratio is taken depends on whether the currency pair is quoted in the E or R method.) Naturally, many different price indices can be used; most common, though, is the use of headline CPIs (consumer price indices). Because CPI series are generally monthly (or even lower frequency) time series, real exchange rates too are generally monthly time series. The home currency appreciates in real terms against a foreign currency either if the home currency appreciates in nominal terms or if the home country's inflation rate is lower than that in the foreign currency.
While bilateral exchange rates are straightforward to report, interest often centers on multilateral exchange rates. The home economy has trade and financial ties not just with one other economy but (usually) with many different economies. To form an index of the multilateral exchange value of the home currency, it's necessary to weight the various bilateral exchange rates according to the importance of the foreign economies. Again, many different weighting schemes are possible; a common scheme uses merchandies trade weights based on imports and/or exports between the home economy and the most important foreign economies. Because bilateral exchange rates are ratios of relative prices, an exchange rate index (aka a multilateral or efffective exchange rate) is computed as a Tornqvist index (a geometric rather than arithmetic average).
Up to now, we've mainly dealt with market prices for both nominal and real exchange rates. While that's perfectly OK for many applications, it's not the most helpful way to proceed if one wants to make comparisons of standards of living across countries. Purchasing power parity (PPP) tries to facilitate such comparisons by figuring how much a standard "basket of goods" -- frequently, but not always, the goods in the CPI baskets -- costs in various countries.

A famous case of a bilateral PPP exchange rate is the "Big Mac Index" of The Economist magazine: Since a Big Mac is pretty much the same in all countries where it's sold, one can work out the PPP-based bilateral exchange between the U.S. dollar (say) and all other countries as the exchange rates that would make the cost of purchasing a Big Mac the same. (Obviously, The Economist makes no claim that the Big Mac features prominently in anybody's consumption basket.)

By comparing nominal and PPP-based exchange rates it is possible to make statements such as "currency A is overvalued (or undervalued) against currency B". Of course, different choices of consumption baskets will lead to different PPP exchange rates. Hence, depending on the choice of basket, conflicting answers may arise as to whether a currency is over- or undervalued.

For more about the Big Mac index, check out the Economist Magazine's own website.

For a non-technical discussion of the dollar indices used by the staff of the U.S. Federal Reserve Board see, e.g, the 2005 article Indexes of the Foreign Exchange Value of the Dollar in Federal Reserve Bulletin.


Negative interest rate and land I currently deal with interest rate theories, reading amongst others Paul Samuelson (1958) and Peter Diamond (1965).

In Samuelson the possibility of a naturally occuring negative interest rate is introduced due to consumption smoothing and without some central bank mechanism. Often times I heard (or read) that the interest rate is impossible to decrease below zero as soon as some commodity is introduced that preserves its value over time, like land or some kind of neutral money. What I don't understand is, why? What is the idea behind that and what kind of concept is this neutral money as opposed to "traditional" money?

By neutral, the author probably refers to something that can hardly be influenced by external forces. Money can be affected by market forces and by central banks, and the consequences are variations on inflation and exchange rates. However, if you have gold or land, these will hardly lose their relative value, at least that was the idea in the 50s-60s.

Coming back to the negative interest rates, for a rational investor that would need to decide between investing his money in return for a negative interest rate, or use the money to buy gold or land, which yield a neutral or positive interest rate, he/she would take the latter.


Change in the way unemployment is calculated in the US? In the last 8 years has their been a change in the way unemployment is calculated? Is there a nuance that has changed in the polling info? Many of my right wing comrades claim the Obama administration has changed the calculation in some way to make it seem as though the number is lower than what it should be?

The most likely explanation for discrepancies that political partisans like to pick at is that presidents like to cite different measures of unemployment. The BLS has 6 alternative categories of labor under-utilization. The U-6 measure of unemployment with seasonal adjustment stands at 9.7%. Although the official rate may be under 5% as in the BLS's September 2016 release here, many people may not feel like the economy is much better, and partisans can argue over which measure of employment or labor force participation rate is most useful.

Most politicians are not part of insidious, cynical political schemes, as the media may make them out to be. Just that when incidents happen, we hear about it non-stop, and it reinforces the public's view that political workers are untrustworthy as a whole, allowing for weird specious claims from the left and right. Take your friends' words with a grain of salt, and when you hear any president make a claim about the unemployment rate, you can look up the BLS's numbers yourselves and see how they are measured and which one the president is citing.

It is also worth noting that the BLS regularly updates how they measure unemployment.


What's the difference between equity capital and market capitalization? Is there any difference between the equity capital and the market capitalization of a company (or a bank)?

If yes then what's the difference?

Equity capital = Assets − Liabilities.

Market capitalization = Number of shares outstanding × Current share price.


A general drought in the world, raises the total income that farmers receive? I need to write 6-7 lines explaining why is the following statement true:

A drought around the world raises the total revenue that farmers
receive from the sale of grain, but a drought only in
Kansas reduces the total revenue that Kansas farmers
receive.

I have few answers in my mind, but I have no economy training so it's really difficult for me to express my self using economic terms.

Is anyone kind to provide me a comprehensive answer.

The demand for the world grain is relatively inelastic, while the demand for the Kansas grain is relatively more elastic. As a worldwide drought reduces the supply of the world grain, the world price of grain rises significantly and the quantity of grain falls insignificantly. Therefore, total revenue that world farmers receive rises. On the other hand, as a drought only in Kansas reduces the supply of Kansas grain, the price of grain rises insignificantly and the quantity falls significantly. Therefore, total revenue that Kansas farmers receive falls.


Markets that are linked to options prices I recently went through a serious of lectures on economics and markets, and I was surprised by the lectures on options. The lecturer, said that the future prices of certain commodities like rice (food) could be closely estimated with option prices of the present. Commodities like fuel are also closely linked, but there are problems when there is war and diplomatic issues. But for things like shares, options prices are irrelevant to the prices of the future.

While I understand the second example, I was to surprised with the other two because no reason was given. If anything, shares are controlled by the society, and therefore, should have been easier to control compared to say food, which is susceptible to natural calamities.

So my question is what are the characteristics of markets that could be predicted with options prices of the present?

what are the characteristics of markets that could be predicted with options prices of the present?

There aren't any. Options might be descriptive of the present, today's expectations, and of today's uncertainty; but they cannot be predictive of the future.

Putting aside the intrinsic value of in-the-money options, option prices reflect traders' expectations about the future price of the underlying asset. But expectation does not mean determination or prediction of the future. In fact, traders try to factor volatility in their valuation of options, which further weakens the notion that option prices are predictors. Volatility represents uncertainty.

Therefore, it is false that option prices predict what the future price of the underlying asset will be.

The lecturer is wrong in the distinctions he stated. Had he substantiated his rationale for the distinctions, we would be able to either follow his point or identify where exactly his argument goes astray.

What could happen is that an option or its underlying asset are under- or overvalued, thereby creating arbitrage opportunities while price adjustments take place. However, that has nothing to do with options prices predicting any future prices.

  While I understand the second example, I was to surprised with the
  other two because no reason was given. If anything, shares are
  controlled by the society, and therefore, should have been easier to
  control compared to say food, which is susceptible to natural
  calamities.

Whether an asset is controlled by either nature or society has no bearing on this matter because nature and society are not even disjoint factors, and they are not mutually exclusive factors.

Trade and production of rice are subject not only to nature calamities, but also to acts of man: strikes, increase of technology, changes in work culture, sudden news that rice cures a chronic disease, and so forth.

Conversely, the stock price of an oil company might go down as a result of nature calamities and thus be unrelated to the acts of man.


Effects of Decentralized Currency What effects could the growing use (and abuse) of Decentralized crypto-currency have on the rest of the world economy? For example, one cause for concern is the simplicity of laundering the currency. Is there any comparable event in history that could help forecast what might be in store? Is there even any cause for worry?

I want to get some outside opinions on this thought experiment.

I agree that cryptocurrency can be used to launder illicit gains, but I think this question needs to consider a broader societal context. It is estimated that there is at least $20 trillion hidden within tax havens around the world. Much of this is laundered by the very institutions (banks) now claiming that cryptocurrency is a potential space for criminal activity.

I think a good example of this is China's recent clampdown on Bitcoin within the context of capital outflows. Although it was speculated that Bitcoin trading within China could be used to avoid capital controls, the comparative amount of Bitcoin was a fraction of the overall amount. Bitcoins entire market cap is about USD18 billion, whereas Chinas capital outflows were USD220 billion in 2016.

I suppose my answer is that cryptocurrency will only ever be a cog within a broader illicit machine. Reducing criminal activities like laundering will require systemic and governance changes within our societies, starting tax avoidance and effective regulation on those who are most likely to carry it out - the super rich.


K-step ahead forecast for an AR(2) Given an AR(2) process $y_{t}=\phi_{1}y_{t-1}+\phi_2 y_{t-2}$, I understand that the two-step ahead forecast (that is, $\mathbb{E}[y_{t+2}|y_t]$ is given by $(\phi_{1}^2+\phi_{2})y_{t}+\phi_{1}\phi_{2}y_{t-1}$. However, I am unable to derive a general expression for the k-step ahead forecast (that is, for $\mathbb{E}[y_{t+k}|y_{t}]$. How might I go about doing this? Thank you.

Let's define $C_k$ such that:
$$
\mathbb{E}(y_{t+k}|y_t) = C_k y_t.
$$
Then:
$$
\begin{align*}
\mathbb{E}(y_{t+k+1}|y_t) &= \phi_1 \mathbb{E}(y_{t+k}|y_t) + \phi_2 \mathbb{E}(y_{t+k-1}|y_t),\\
&= \phi_1 C_k y_t + \phi_2 C_{k-1} y_{t},\\
&= (\phi_1 C_k + \phi_2 C_{k-1}) y_t
\end{align*}
$$
This gives us the following recursive second order difference equation:
$$
C_{k+1} = \phi_1 C_k + \phi_2 C_{k-1}.
$$
Or, updated one period:
$$
C_{k+2}- \phi_1 C_{k+1} - \phi_2 C_k = 0.
$$
This is a second order homogeneous difference equation, which is of the form $C_k = \alpha a_1^k + \beta a_2^k$. Computing $a_1$ and $a_2$ can be done by solving the following characteristic function:
$$
(a^2- \phi_1 a - \phi_2) = 0
$$
This gives the following two roots:
$$
a_1 = \frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2}}{2} \text{ and } a_2 = \frac{\phi_1 - \sqrt{\phi_1^2 + 4 \phi_2}}{2}.
$$
So in general we have:
$$
C_k = \alpha a_1^k + \beta a_2^k
$$
To find $\alpha$ and $\beta$, we need two starting values, say $C_0 = 1$ and $C_1 = \phi_1$. In other words, we assume that $\mathbb{E}(y_t|y_t) = y_t$ and $\mathbb{E}(y_{t+1}|y_t) = \phi_1 y_t$.
This gives the following two conditions:
$$
\alpha + \beta = 1\\
\alpha \left(\frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right) + \beta \left(\frac{\phi_1 - \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right) = \phi_1.\\
$$
Solving this system in terms of $\alpha$ and $\beta$ gives:
$$
\beta = \frac{\sqrt{\phi_1^2 + 4 \phi_2} - \phi_1}{2 \sqrt{\phi_1^2 + 4 \phi_2}}\\
\alpha = \frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2} }{2 \sqrt{\phi_1^2 + 4 \phi_2}}
$$
This gives the following closed form expression condition:
$$
\begin{align*}
\mathbb{E}(y_{t+k}|y_t) &= C_k y_t,\\
&=\left[\begin{array}{ll} &\frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2} }{2 \sqrt{\phi_1^2 + 4 \phi_2}}\left(\frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right)^k\\
+ &\frac{\sqrt{\phi_1^2 + 4 \phi_2} - \phi_1}{2 \sqrt{\phi_1^2 + 4 \phi_2}} \left(\frac{\phi_1 - \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right)^k\end{array}\right] y_t
\end{align*}
$$


Question about Euler condition In dynamic macroeconomic model(without production sector), Euler equation is $$U'(C_t)=b(1+r)U'(C_{t+1})$$

I found another equation related with Euler which is called Euler condition.
In New Keynesian model,
$$Y_t-E(Y_{t+1})=-\left(\frac{1}{\sigma}\right)(i_t-E(\pi_{t+1})-\rho)$$

which means Euler condition. In this equation, what is the Economic meaning for Euler condition?

The Euler Equation typically refers to the interior optimal choice between consumption today and tomorrow (or some similar intertemporal choice).  That is, it equalizes the marginal utility of consuming a unit today vs saving that unit in order to consume tomorrow.  This yields the first equation you listed (where $\beta=\frac{1}{1+\rho}$):

$$
u'(C_{t}) = \frac{1+r}{1+\rho}u'(C_{t+1})
$$

I have always referred to the second equation you describe as the "New Keynesian IS Curve" as in Gali's textbook and several NK model papers.  But it does come from  the same intertemporal consumption choice Euler Equation.

Based on the equation you've provided, my guess is the model assumes a few things:

CRRA utility of the form:  $ u(C) = \frac{C^{1-\sigma}}{(1-\sigma)}$
No capital investment such that $ Y_{t} = C_{t} $

Using these assumptions, the Fisher Equation ($ r_{t} = i_{t} - E_{t}[\pi_{t+1}] $), and the Euler Equation you gave above, you would arrive at what you're calling the NK Model's Euler Equation. (Note that your second equation is also probably in log terms)

At the end of the day, the two are the same thing: The first order condition that is making consumption equivalent to saving in terms of utility.  The latter version just tacks on additional features of the NK model to arrive at an equation that's useful for describing aggregate dynamics (along with the Phillips Curve and a monetary rule).


Rule of thumb for confidence level Typically in statistical software, the default confidence level is 95%. The higher the better, I suppose. But this is still not a rule of thumb, right? In Stock and Watson's paper (https://www.aeaweb.org/articles/pdf/doi/10.1257/jep.15.4.101), they use a 66% confidence level. While they did not explain why 66, not 95, I guess it makes some IRFs (which would be insignificant at 95%) more significant. For example, in their original paper, interest rate shock to inflation appears significant at a 66% confidence level, but my estimation at a 95% confidence level shows it is not significantly different from zero. I am not criticizing this paper or anything like that, but it appears the choice of the level of confidence is at the discretion of the researcher, either arbitrarily or purposefully. If this is not the case, what would make one choose a confidence level of 66% and not 95%, or even 99%?

Yes confidence level always depends on choice of a researcher (at least in principle). There is no reason to use $90\%$, $95\%$ or $99\%$ level either  - it is more or less just convention. For example, in physics they use three-sigma or five-sigma significance which corresponds to confidence levels of $99,7\%$, $99.99997\%$ (and even higher ones can be reported).
However, this being said once literature settles at a particular convention you will be fighting an uphill battle. For example, you would have hard time publishing in physics results that are not at least $3\sigma$ or $5\sigma$ significant without very good explanation. In economics, usually people settle for $95\%$ level. Just because you can pick any significance/confidence level that does not mean others will accept it.

They actually do explain why there is $66\%$ confidence interval. According to Stock and Watson:

Also plotted
are $\pm 1$ standard error bands, which yield an approximate 66 percent confidence
interval for each of the impulse responses.

When it comes to VAR impulse responses (and forecasting in general) it is conventional to plot $\pm 1$ standard error bands. This is also often done when people estimate DSGE models so it is more or less conventional practice. This is ultimately arbitrary choice on which people somehow settled.
Also note reporting 1 standard error deviation error bars does not mean the variable was significant at only  $66\%$ confidence level - for example $\hat{\beta}=30$ with $SE(\hat{\beta} )=1$ would have $t$-stat $=30$, would be significant at any conventional significance used in economics (in any reasonably sized sample). That does not prevent you from constructing 1 standard error bars (i.e. $30 \pm 1$).


Uncovered Interest Parity question I am trying to understand the fact that, given that the uncovered interest parity holds, a rise in dollar interest rates will cause the dollar to appreciate. Is this a good intuition?

Uncovered interest parity (UIP) is defined as ${R_F=R_$}+{({E_{F/$}}^e-{E_{F/$}}^c)}/{E_{F/$}}^c$ where $$ $$ $R_F$ is the annual return of foreign currency, $$ $$ ${R_$}$ is the annual rate of return (the interest rate) for holding dollars in a bank, $$ $$ ${E_{F/$}}^e$ is the expected exchange rate (how many foreign currency units could be bought with one dollar) in one year's time, and $$ $$ ${E_{F/$}}^c$ is the current exchange rate (also per dollars). $$ $$

Example. Supposing ${R_$}$ is 4%, ${E_{F/$}}^c$ is 0.67, ${E_{F/$}}^e$ is 0.69, the rate of return for holding foreign currency $R_{F}$ is 7%. But now, a rise in dollar interest rates (${R_$}$ is now 8%) it means I fix the rate of return on dollars at 7% and see how ${E_{F/$}}^e$ changes. Is that the correct intuition? I use this approach and get that ${E_{F/$}}^e$ is 0.66, which means the foreign currency appreciated, but that means the dollars didn't appreciate, which contradicts my original claim...

Your example does not contradict the theory.

Uncovered interest parity (UIP) indicates the degree and direction of movement of exchange rates in the long term (though I see some economist supposes it's mid term). The intuition behind is that the currency of higher interest rate county has the expectation of depreciation in the future, to the percentage as the difference in the interest rates, to make arbitrage not profitable during a period of time.

Come back to your example. Given the foreign interest rate $R_{F}$ is 7% and and the domestic interest rate $R_{$}$ is 4% (less than 7%), the domestic currency (dollar) is expected to appreciate about 3% in the future according to the model, which is consistent with your calculation (from the current exchange rate $E_{F/$}^{c}$ 0.67 to the future exchange rate $E_{F/$}^{e}$ of 0.69, a 3% change). If the domestic interest rate $R_{$}$ is 8%, while the foreign interest rate $R_{F}$ is still 7% (now domestic interested rate is higher), it is expected that the domestic currency will depreciate for 1%, which is indicated by a change from 0.67 to 0.66.

This theory does not account for short term movement of exchange rate. The underlying assumption implied by this theory is that exchange rate is determined by capital flow, which is driven solely by interest rate spread, and it does not account for market impact of rate hike as well.

In addition, a interest rate hike, according to the theory, will change the direction of appreciation or depreciation only in the circumstance that it causes the interest rate spread to change direction, which means that a rate hike is not necessarily associated with appreciation or depreciation, but only the degree of appreciation or depreciation as the interest rate spread may increase or decrease under a rate hike.


question regarding econometric testing of the natural rate hypothesis Hi All: I have the Lucas, 1972b paper "Econometric Testing Of the Natural Rate Hypothesis" but unfortunately not as a pdf. ( it's from a book that contains a lot of Lucas' papers called "studies in business cycle theory").  Assuming that someone reading this question has read that paper and understood it, could you explain how equations 11) and 12) come about. Thank you very much for your help.

ADDENDUM TO ORIGINAL QUESTION:

I am going to write the equations that lead up to the two equations
that I don't follow the derivation of. My hope is that not many people have
the pdf of the file so  the equations will obviously help.

1) $y_t = a(P_{t} - P^{*}_{t})$  (aggregate supply function ).

7) $y_{t} + P_{t} = x_{t}$ ( aggregate demand schedule )

8) $x_{t} =\rho_{1} x_{t-1} + \rho_{2} x_{t-2} + \epsilon_{t}$ ( AR(2) for $x_{t}$ ).

9)  $ P_{t}^{*} = E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t}) + \eta_{t} $
( Definition of agent's expectation of the price at time $t+1$. )

10) $ (1+a) P_{t} - a P^{*} _{t} = x_{t} $
(obtained by straightforward algebraic manipulation of  (1) and (7). )

Meanings of variables:

1) $P_{t}$ is the price level at time t.

2) $P^{*}_{t}$ is the expected price level at time t+1 where the expectation is taken at time t. ( the agent's expectation of price level ). Essentially, $P_{t+1}$ is the next period's price level, the variable of which, $P^{*}_{t}$, is a forecast.

3) $y_{t}$ is the output at time t.

4) $x_{t}$ is the log of the money supply and is a shift parameter. It is part of a policy rule which gives the current value of $x_{t}$ as a function of the state of the system.

After these equations are explained,
the author then says:

"We shall seek linear solutions $P_{t}$ and $P^{*}_{t}$ to equations 9 and 10: "

11) $P_{t} = \pi_{1} x_{t} + \pi_{2} x_{t-1} + \pi_{3} \eta_{t} $

12)  $ P^{*}_{t} = \pi_{4} x_{t} + \pi_{5} x_{t-1} + \pi_{6} \eta_{t} $

Equations 11) and 12) are the equations whose derivation escapes me totally. Thanks a lot for your help.
                                               Mark

                                                     Mark

P.S: For any newbies wrestling with the theory behind RE, this paper is the clearest and simplest that I have found, aside from the derivation of 11) and 12). I highly recommend it. I only found the book recently and have decided that out of all the top RE people, Lucas really stands out as far as the clarity of his writings.

Equations $11$) and $12$) are not derived, they are assumed.

Specifically, with eq. $9$) Lucas defines/assumes that the value-expected for the $t+1$ price level is the conditional expectation based on $x_t, x_{t+1},\eta_t$ (plus $\eta_t$ on its own). This is imposing the Rational Expectations Hypothesis.

He then says "we will seek linear solutions..." which is translated "assume that the functional form of $E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t})$ is linear in its arguments, and then determine (or estimate) the coefficients". Certainly, this is in all likelihood an approximation to the true functional form of this conditional expectation.

So assuming that

$$E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t}) = \pi_{4} x_{t} +\pi_{5} x_{t-1} + \beta\eta_{t}$$

and inserting this into $9$) we get $12$) with the mapping $\pi_{6} = 1+\beta$.

Inserting $9$) in $10$) we get

$$(1+a) P_{t} - a [\pi_{4} x_{t} + \pi_{5} x_{t-1} + \pi_{6} \eta_{t}] = x_{t}$$

$$\implies P_{t} = \frac{1+a\pi_{4}}{1+a}x_t +\frac{a\pi_{5}}{1+a}x_{t-1}+\frac{a\pi_{6}}{1+a}\eta_{t}$$

which is $11$), after compacting and mapping the above coefficients to the $\pi_1, \pi_2, \pi_3$ ones.


In a binary-dependent variable model (logic) will the inclusion of a scale introduce endogeneity? This question is somewhat related to my previous question, see here: https://stats.stackexchange.com/questions/463221/why-when-do-attitudinal-scales-introduce-endogeneity

I wonder why a scale as an explanatory variable will always make a binary-dependent model endogenous?

This is what an editor told me along with rejecting my paper, and as long as I don't change it/acknowledge the possible endogeneity I cannot resubmit the paper.

Based on the clarification in your comment this is not because you use scale variables but because the scale variable that you use is self-reported.

It is well known in literature that self-reporting leads to endogeneity. See for example: Lindeboom, M., & Kerkhofs, M. (2004). Subjective health measures, reporting errors and endogeneity in the relationship between health and work.

The reason why that is when people self-report some outcomes those are often affected by other unobservables that you do not include in your model which is a form of endogeneity. For example, studies that measure happiness show that the self reported happiness can vary depending on which time of the day/week/season you make the measurement.

Moreover, you can have also endogeneity in more narrow sense unemployed people can self-report that they cannot work as a rationalization of their unemployment status and hence you would get reversed causality where the unemployment actually causes the self-reported inability to work even though the person actually would be able to work as measured by objective measurable criteria (in a model where employment is regressed on self-reported ability).

So it’s not that there is any problem with having scale variables. For example, having some 1-10 index but rather the problem is that in your case the scale was made from self reported data.


How the Federal Reserve Manages Money Supply I'm trying to understand how the Federal Reserve manages the money supply via open market operations. According to this post, when Fed buys securities, that increases bank reserves, which allows banks to lend more money, which increases the money supply. That makes sense. But when the issuer pays off the bond, wouldn't that reduce bank reserves, thus reducing lending, thus reducing the money supply? It seems like the overall effect of bond buying would be to increase the money supply in the short term but in the long term there may be a slight reduction in the money supply if the bond is paid back with interest.
Please let me know what I'm misunderstanding here. Thanks!

How the Federal Reserve Manages Money Supply?

Federal reserve manages money supply in various different ways (see Blanchard et al Macroeconomics ch4). The main ones are:

By managing interest rates on a reserves: lower interest rate reserves allow banks to lend more and expand money supply and vice versa.
By open market operations: where purchase Fed purchases debt in exchange for new money (which eventually increases reserves as that money finds it way to banks). Issuing new debt increases money supply and repayment of the debt decreases it.
By changing reserve requirement. Currently they are abolished by Fed, but nothing prevents Fed from bringing them back if they desire so. Here higher reserve requirement contracts money supply and lower one expands it.
Other: Fed can also affect money supply through other means such as banking regulation, although that is too broad of a topic to discuss in detail in one SE post. You can find some explanation of that in McLeay et al (2014) and sources cited therein.

But when the issuer pays off the bond, wouldn't that reduce bank reserves, thus reducing lending, thus reducing the money supply?

When the bond is payed off it does lead to reduction in money supply unless the money are lent again. However, money supply is reduced just to the original point. If Fed injected 100 USD to the economy through OMOs and if through fractional reserve banking system that injection multiplied up to 1000 USD then repaying that debt only destroys the 1000 USD not any additional money (e.g. if M before injection was 1 billion USD after repayment it will again be just 1 billion USD).

It does not reduce money supply further than it was before it was expanded. The interest rate is paid to the bank. In case of commercial banks they keep the money as a profit (after paying for all costs of course) so it will still be circulating in the economy. In case of US government paying interest to Fed, again money supply is not additionally destroyed because Fed is government institution and it sends all profits from interest back to the treasury (e.g. see WSJ article about profits they send back in 2020) so the money used to pay interest stays circulating in the economy.


Transforming a matrix of explanatory variable in regression Given the partitioned regression equation (into $X_1$ and $X_2$), I want to transform $X_1$, say $X^*_1$, such that $X_2$ and $X^*_1$ becomes orthogonal ie. $X_2^T$. $X_1^*$= 0.
A matrix can be transformed as $X_1^*$ = $X_1$.p where P is transformation matrix.

If we premultiply $X_1^*$ = $X_1$.p with $X_2^T$ we will get $X_2^T$.$X_1$.p = $X_2^T$.$X_1^*$= 0. How can I find P?

Part (ii) is easy to solve. Parameters of $X_2^T$ and $X_1^*$ will depend on their separate regression with Y.

$\boldsymbol X^*_1$ is the residual of the projection of $\boldsymbol X_1 $ on $\boldsymbol X_2 $. The 'residual maker' matrix is $\boldsymbol I - \boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T$. Therefore, $$ \boldsymbol X_1^* = \left(\boldsymbol I -\boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T\right)\boldsymbol X_1.$$


The Economical "justification" of mass killings Throughout history people have engaged in wars and mass killings.

I'm sure similar stories exist in all parts of the globe, throughout the world.

So what is the story? If the world revolves around economics, then how does economics profit from mass destruction and killings? If we are to stop this, we must first understand this.

Sorry to ask, but is it that the money made from jobs involved in setting up concentration camps, producing equipment of mass destruction, and all of that, more profitable than what the bastards can earn from the people designated to be killed? Is this why we have all these killings throughout history?

Or are you going to tell me that all the mass killings that happened in history were motivated by something other thanan economical reward optimization problem?

Let us find a solution (or a set of solutions, together) so that this does not happen (again!). So many people spend their time wondering about so many things not realizing their own lives are in danger. TikiI think this is a priority we should be focusing on.

I'm not really sure why this is tagged [social-welfare] or [welfare economics].  It seems to me that more appropriate tags would be [history] and/or [international economics] (since the question involves international relations), and/or [game theory] (since the question involves putatively rational behavior in a multi-agent context).

Having said that, here is my (uninformed) response.  In a case where a much stronger group murders, enslaves and expropriates a much weaker group, there can be a net economic advantage (for the stronger group).  The European conquest and colonization of the Americas and their abduction and enslavement of millions of Africans are probably the most obvious examples.  Of course, genocide, slavery and mass expropriation do not guarantee long-term economic growth, as can be seen by comparing the economic histories of North America and South America.  Furthermore, the influx of easy resources following an easy conquest can create something analogous to a "resource curse" or "Dutch disease".  For example, the massive influx of silver from Spain's American conquests initially made Spain "rich", but eventually it led to inflation.  It also caused the Spanish economy to stagnate relative to the dynamic economies of northern Europe (Corden, 1984).

However, in a military contest between actors who are roughly evenly matched, I believe the historical record shows that it is always a negative-sum game.  Even if there is a clear winner, the wealth which the victor gains from plundering and/or enslaving the vanquished state almost never justifies the amount of blood and treasure that was needed to obtain the victory.  So the upshot is that these military conquests cannot be explained in terms of the economic self-interest of the warring states.

But it might be a mistake to treat states as monolithic rational actors.  Every state has a governing class, and it may be the case that the war was economically beneficial for the governing class, even if it is economically detrimental for the country as a whole.  For example, it may be that the Iraq war was a huge net economic loss for the United States, as argued in the book The Three Trillion Dollar War, by Joseph Stiglitz and Linda Bilmes.   But at the same time, it may be that certain individuals and firms associated with people in the Bush Administration (e.g. Haliburton) reaped a large economic benefit.  So from the perspective of those people, the war was "economically rational".  (Note that I say "it may be....", because I don't have enough information to back up this speculation.)

It is also likely that some wars and mass-murders just have non-economic motives.  For example, Hitler was motivated by race-hatred at least as much as by economics.  Stalin's purges, massacres, and genocides were motivated by a desire to consolidate his political power, at least as much as by economics.  And the Crusades and the Thirty Years War were motivated (at least initially) by religion, not economics.  (Certainly, they were a net economic loss for almost everyone involved.)

Afterthought.  (added later)  Here is another data point.  At the beginning of the twentieth century, the economies of Western Europe  were closely integrated in a big trading block.  Many people thought that a major war between Western European powers was impossible, because it was so obviously against their economic interests.  We saw how that turned out....


Does one include inflation when computing elasticity? I am attempting to compute the price elasticity of a good with the help of a time series, and am unsure whether to correct the prices for inflation. Is there a standard argument for one or the other option? I would also be grateful for a reference.

Elasticities are partial derivatives -- so to estimate them you need to hold all other factors fixed! That would include the prices of other goods.


Using the Dividend Discount Model when no current dividend exists? Say a company is doing an IPO and we expect them to start paying dividends in 5 years?
I understand how we could use the Dividend Discount Model to predict their stock price 4 years from the IPO. But what about the day of the IPO?

How can we use the Dividend Discount Model to value this stock before it pays dividends?

With what you say and using the DDM we know that

$P_{t=4} = \frac{D_{t=5}(1+g)}{r-g}$

Via the above equation, it is assumed that from year $5$, dividends will annually grow at a $g$ rate and that investors' required rate will be $r$, both forever. Since this calculation is nothing but a simplification (future growth and return rates will a fortiori change, etc...), we will not do more complicated.

Taking into account the share of debt in the capital structure of the company, say $X\%$, we can compute the average cost of capital (average over equity and debt), $r'$, needed to discount $P_{t=4}$. One may have an idea of $X$ or make assumptions about it. It follows that $r'$ is such that

$r' = X\% r_D + (1-X\%) r$

where $r_D$ is the interest rate or the cost of debt (naturally $r_S < r$). Thus the company's IPO may be valued by involving that

$P_{t=0} = \frac{P_{t=4}}{(1+r')^4}$

Finally, as you can see there are many assumptions in this valuation, which is the job of experts, with the most fundamental about $D_{t=5}$.