Did William Nordhaus exclude 87% of economic industries from his climate change analysis? Economist Steve Keen published in article in 2020 titled, The appallingly bad neoclassical economics of climate change. In it, he alleges that climate economists such as William Nordhaus and Richard Tol have underestimated the impact of climate change by "at least an order of magnitude." Keen primarily responds to two approaches in the economics literature for estimating damages from climate change. The first is the so-called 'Enumeration' approach to estimating damages from climate change. He quotes Tol who said,

In this approach, estimates of the “physical effects” of climate change are obtained one by one from natural science papers, which in turn may be based on some combination of climate models, impact models, and laboratory experiments. The physical impacts must then each be given a price and added up. For agricultural products, an example of a traded good or service, agronomy papers are used to predict the effect of climate on crop yield, and then market prices or economic models are used to value the change in output.

Keen then quotes (Nordhaus 1991) and an FAQ from the IPCC indicating that both Nordhaus and the IPCC explicitly exclude 87% of economic industries from their approach towards estimating damages, since these industries are composed of activities performed indoors. As Keen notes, just because an activity is performed indoors does not mean that it is not affected by climate change. He quotes Nordhaus saying the following,

Table 5 shows a sectoral breakdown of United States national income, where the economy is subdivided by the sectoral sensitivity to greenhouse warming. The most sensitive sectors are likely to be those, such as agriculture and forestry, in which output depends in a significant way upon climatic variables. At the other extreme are activities, such as cardiovascular surgery or microprocessor fabrication in ‘clean rooms’, which are undertaken in carefully controlled environments that will not be directly affected by climate change. Our estimate is that approximately 3% of United States national output is produced in highly sensitive sectors, another 10% in moderately sensitive sectors, and about 87% in sectors that are negligibly affected by climate change.

The IPCC FAQ is quoted as saying,

Economic activities such as agriculture, forestry, fisheries, and mining are exposed to the weather and thus vulnerable to climate change. Other economic activities, such as manufacturing and services, largely take place in controlled environments and are not really exposed to climate change.

Such an obvious blunder on the part of both Nordhaus and the IPCC would be shocking. However, I'm a bit skeptical that Keen fully understands the literature here. In a later section, Keen seemingly admits that he dove into the climate econonomics literature in order to find a flaw, rather than dispassionately reviewing it in order to uncover truth. He says, "While locating the fundamental flaw in the ‘enumeration’ approach took some additional research, the flaw in the statistical approach was obvious in the first reference I read on it." Given my skepticism of his scientific neutrality, I wanted to understand how Nordhaus, Tol and other climate economists would reply to his critique.

Did William Nordhaus exclude 87% of economic industries from his climate change analysis?

No, Nordhaus did not do that but Nordhaus did exclude $\approx 87\%$ economic activity as measured by national income and Keen either confuses economic activity with industry or does not use common terminology in economics (or given that he is known for sensationalism perhaps because $87\%$ of all sectors would be even bigger portion of the economy).
As the table below from Nordhaus (1991) shows, Nordhaus split economic activity into 12 sectors, 11 of which can be considered industries (the word industry is in economics commonly applied to private sectors not to public sector, see also usage in texts such as Mankiw Principles of Economics).
Nordhaus did tabulate effects only on sectors which belonged to category of potentially severe impact and potentially moderate impact which is 7 out of 12 listed sectors so approximately $0.58\%$ sectors were included and approximately $42\%$ sectors were not included in further tabulation.
However, those $42\%$ of not included sectors did generate $87\%$ of economic activity.
In addition, the assertion that Nordhaus excluded these $42\%$ of sectors on account of them 'being indoors' is simply wrong here.
Nordhaus references indoor activity in that section only in one paragraph where he says the following:

The most sensitive sectors are likely to be those, such as agriculture and
forestry, in which output depends in a significant way upon climatic variables.
At the other extreme are activities, such as cardiovascular surgery or
microprocessor fabrication in 'clean rooms', which are undertaken in carefully
controlled environments that will not be directly affected by climate change.

Hence he uses 'clean rooms'  as an example of an extreme case of place not directly affected by climate change. I am an economist not a biologist/climatologist, so I can be wrong, but I think stating that clean rooms - hermetically sealed and sterilized rooms where every single precaution is made that not even a speck of dust will make it in - will not be directly affected by climate change is not completely unreasonable.
Additionally, note the above was never used in the paper as a primary reason for exclusion of the $42\%$ of the sectors (many of the $52\%$ included sectors such as electricity production are also done mostly in indoor environments e.g. nuclear/coal/gas/oil plants).
The true reason why the sectors were excluded is that Nordhaus was simply not able to find evidence for significant negative effects of climate change on those industries:

But for the bulk of the
economy - manufacturing, mining, utilities, finance, trade, and most service
industries - it is difficult to find major direct impacts of the projected climate
changes over the next 50 to 75 years.

Note, Nordhaus does look at various reports like EPA (I988). U.S. Environmental Protection Agency, The Potential Effects of Global Climate Change on the United
States draft to the congress to identify which industries are at risk. One could here levy a criticism that 'absence of evidence is not evidence of absence',  but this is hardly some sort of willful ignorance or a blunder on the side of Nordhaus.


What are the advantages using Otree to progam economic experiments? I need to decide between learning Ztree and Otree. The added cost of complicating the server setup severely should have some benefit. Apart from Ztree being ugly, are there any other advantages of Otree to program experiments?

One irreducible advantage of oTree over z-Tree is that you can easily reuse/imply the entire trajectories of results per entity (session/group/player) when doing repeated games. With z-Tree, it is rather complicated (not to say impossible?) to define dynamically temporal- or individual-based autocorrelated/interdependent/recursive games.
Another notable advantage is that you can also easily run (fully automated) functional tests with oTree, be them headless or not, with random numbers following the distribution(s) or mechanism(s) you like, up to the generation of the final results, which is at best complicated if I remember well with z-Tree. Moreover, keeping in mind that it is increasingly mandatory to provide (in appendix) all the screenshots of the steps of an experiment, you can easily do so with oTree, using libraries such as Selenium as soon as something changes.I remember me doing all that manually, with many zLeafs opened, which was cumbersome to say the least. Having 3 screens was not enough...
Finally, learning oTree will strongly initiate your familiarity with a notorious web-framework, Django (not the CMS), which means that learning oTree means learning it as well. Entirely class-based and PEP8-compliant, it also means that via oTree, you will learn coding in Python the "right way". Moreover, if you wanna tweak your frontend, you will also start learning things such as CSS, HTML and even JavaScript. oTree's dividends are numerous.All that with a concrete goal in mind, which (to me) is the best manner to acquire new skills.
That being explained, there is now a new version of z-Tree which I do not know, i.e. z-Tree unleashed which may be far better than its ancestor.


What is the opportunity cost if I choose the option of highest value I had a confusion about opportunity cost.

Suppose I have three alternatives which can be completed in same time, and can provide me the value of \$30, \$10 or \$40. If I choose the one with \$10 then, the opportunity cost will be \$40 as it is the best alternative that could have been done in the same time.

What if I choose the work with \$40. What will be its opportunity cost in this case as there is no better alternative here.

The opportunity cost of any decision is the value of the next-best alternative forgone.

So in this case, if you choose \$40, your opportunity cost is \$30, because that's the value of the next-best alternative forgone.

(Implicit assumption: These are your only three alternatives and you must choose one of these three.)


mathematical marxian models Ive been doing some superficial reading on Feldman-Mahalanobis Model and have been wondering what other equations and "brand name" models that have been prodouced by marxist economists?

What other mathematical Marxian models are out there?

Paul Samuelson wrote a series of papers formalizing Marxian economics, e.g.
https://www.sciencedirect.com/science/article/pii/B9780123567505500177


Is indefinite exponential economic growth possible? Economic growth is frequently expressed as a % of GDP.  Proponents of a steady-state economy postulate that perpetual exponential economic growth is impossible and that any system relying on it is inherently unstable. Is this true?

Is indefinite exponential economic growth possible?

(See also: Are there fundamental reasons why (exponential) economic growth is highly desirable?)

As Steven Landsburg pointed out, indefinite anything isn't possible given the eventual heat death of the universe.

Von Neumann posited machines capable of indefinite growth in population (and theoretically production and consumption), so in theory, it would be possible for a time.

Sagan's Response however argued if such things existed that we would have already observed them, and that they would destroy whole galaxies en masse.

That's probably the closest answer you can get to a question like this.


Difference between Giffen and inferior goods. Why aren't all inferior goods Giffen goods? What is the difference between an inferior good and a Giffen good?
Are the two following definitions for an inferior good equivalent?

Def 1: An inferior good is a good for which the demand decreases after a decrease of its price.

Def 2: An inferior good is a good for which the income effect leads to a decrease of demand after a relative decrease of its price.

A Giffen good (1) is when after a decrease in price of good (1) the demand for (1) decreases but the demand of some other good (2) increases.

Or is Def 1 just the definition of a Giffen good, which is a special type of inferior good ?

For a Giffen good the Income Effect is strong enough to outweigh the Substitution Effect... But isn't it also the case for all inferior goods? The change in their demand is going to be negative (we consume less after the decrese in price) and that change is equal to the sum of SE and IE, so we also get that Income Effect is strong enough to outweigh the Substitution Effect.

Def 1 is wrong. An inferior good is a good for which the demand decreases after a decrease in the agent's income. (not a decrease in the price of the good).

Definition 2 is trying to define the same concept, "an inferior good" so it is also wrong. Instead, both definitions would be appropriate if they were describing a Giffen good.

Giffen goods are indeed a special case of an inferior good. These are goods for which the law of demand does not apply. Since their demand has a positive relationship with price. The reason why this positive relationship is possible is that when prices change, they create income and substitution effect, and when goods are inferior, these effects go in the opposite direction. Thus, if the income effect dominates the substitution effect, you get the (seemingly counter-intuitive) behavior that demand increases when prices go up, and decreases when prices go down (a Giffen good). Note that to be a Giffen good you need to be inferior, but the reverse is not necessarily true, for example, if the substitution effect dominates the income effect.


Homogeneous Utilities: Anything other than CES? Does there exist any homogeneous utility function, i.e., $u(\lambda \mathbf{x}) = \lambda u(\mathbf{x})$, that is not a special case of the CES (or nested CES) family of utility functions or its limits (i.e., including Leontief and Cobb-Douglas)? I have been thinking a lot about this but cannot think of any example. If so, does homogeneity imply constant elasticity of substitution?

There is an infinity of such functions. You can for instance construct a linear homogeneous function $u$ from any utility function $U$ by using a linear homogeneous function $h: \mathbb{R}^J \rightarrow \mathbb{R} $ as follow:
$$ u(x) = h(x)U(x/h(x)). $$
Example:
$U(x)= \alpha x_1^2 + \beta x_1x_2 + \gamma x_2^5 $,
$h(x)=x_1+x_2$
yields
$$u(x) = (x_1+x_2) \left(  \alpha(\frac{x_1}{x_1+x_2})^2 + \beta\frac{x_1x_2}{(x_1+x_2)^2} + \gamma(\frac{x_2}{x_1+x_2})^5 \right). $$
If you also want to ensure that $u$ is a utility function (increasing in $x$) it is a bit more tricky, however.


Approximation to a second order partial derivative Let's say we want to approximate a second order differential:

$$ \partial_{aa} V(a)$$

Now, say we have a grid of values for $A$, where the step size is $\Delta$, $i$ is the index on this A grid, and we define the short hand notation

$$V_i \equiv V(A_i)$$

Achdou et al (page 13, equation 35) then compute the approximation to the second order differential on the grid at a point $i$ as

$$\partial_{AA} v_i =  \frac{v_{i+1} - 2v_i + v_{i-1}}{\Delta^2}$$

They don't explain why, and I can't follow here. Could someone provide me with the rough ansatz for why this is the case?

I think that this question would be more suitable for math.SE, since it is purely mathematical and not specific to economics.

In your notation, the first derivative would be given by
$$
\partial_A v_i = \frac{v_{i+1} - v_i}{\Delta}.
$$
In order to compute the second derivative, we need to apply this formula with $\partial_A v_i$ itself in place of $v$. This gives
$$
\partial_{AA} v_i = \frac{ \frac{v_{i+2} - v_{i+1}}{\Delta} - \frac{v_{i+1} - v_i}{\Delta}}{\Delta}.
$$
A bit of algebra reveals that this is the same as
$$
\partial_{AA} v_i = \frac{v_{i+2} - 2v_{i+1} + v_i}{\Delta^2}.
$$
Now if your grid is fine enough, you can lower the index $i$ on the right-hand side without changing the result significantly. This gives the equation that you came across. It is generally preferred due to the "symmetry" that it displays between $i-1$ and $i+1$.

For more on this topic, see e.g. Wikipedia at finite difference.


Is the labor force participation rate related more to labour supply or to labour demand? Or both? I have read several Labor Economics books, but I am stuck. Is the labor force participation rate related more to the labour supply or to the labour demand?

If the labor force participation rate decreases in a year (with respect to the previous year), should we worry that employers are not hiring (i.e., labor demand)? Or, is it due to workers that are not currently interested in work (i.e., homemakers)?

Furthermore, from this and this online documents it seems that the LFPR (labor force participation rate) is strongly dependent on labor demand from employers (i.e., positive/negative business cycles) rather than from labor supply. For instance, the LFPR dropped strongly during the Great Recession as employers were not willing to hire workers. Any opinion on these problems?

Traditionally, labor force participation is best thought of as a supply-side phenomena. That is, a certain subset of our nation's population is jobless but is jobless by choice. For example, consider a stay-at-home parent. So, we have agents making a decision to not supply labor.

However, there is a hint of the demand side at play here. For example, a very weak demand side might discourage workers, thereby compelling the marginal participant to exit the labor force.

So, to be explicit: LFP is a supply-side concept. However, there is a mechanism through which demand side forces can factor into LFP.


Why is economic profit for a monopoly per unit demand minus ATC? I understand that the P-ATC gives the per unit economic profit

Why is this the case? Why wouldn't we take price minus the marginal cost?

Price minus marginal cost gives you marginal profit.

\begin{align*}
\text{Total profit}&=\text{Revenue}-\text{Total cost}\\
\frac{\text{Total profit}}{\text{Quantity}}&=\frac{\text{Revenue}-\text{Total cost}}{{\text{Quantity}}}\\
&=\frac{\text{Revenue}}{{\text{Quantity}}} - \frac{\text{Total cost}}{{\text{Quantity}}}\\\\
\text{Per-unit profit}&=\text{Price}-\text{Average total cost}
\end{align*}


Why must the lag length of the autoregressive term in an ARDL model be determined separately? I am estimating an autoregressive distributed lag model, and I've read that I must determine the lag length of my autoregressive term separately from the lag length of the other regressors in the model.

For instance, given the model $y_t=\delta_0 + \alpha_1y_{t-1} + \alpha_2y_{t-2} + \alpha_3y_{t-3} + \gamma_1z_{t-1} + \gamma_2z_{t-2} + u_t$
Wooldridge 18-5 (p.590 in 6th edition) writes, " As a practical matter, how do we decide on which lags of $y$ and $z$ to include? First, we start by estimating an autoregressive model for $y$...Once an autoregressive model for $y$ has been chosen, we can test for lags of $z$."

But, there's no explanation on why we must test for lags of $y$ and then lags of $z$.  Can you provide me with an explanation of why we do this?

Many thanks-
Maurus

The reason for this is that in a most dynamic models, i. e. models where any lags of dependent variables are used, you always have to include enough lags to get rid of autocorrelation.

This is because in dynamic model autocorrelation no longer can be adjusted just by correcting standard errors, because dynamic models in presence of autocorrelation are not just inefficient but also biased. Hence in a dynamic model it is not enough to just use let’s say Newey-West or other autocorrelation consistent errors, you need to get the dynamic specification (the lags) right so there is no autocorrelation in the model.

More formally you can see the reason why this is necessary by considering simple ARDL(1,1) mode:

$$y_t = \beta_1 + \beta_2y_{t-1} + \beta_3 x_{t-1} + \epsilon_t$$

now imagine that in this specification you still got autocorrelation of first order, which would imply that:

$$\epsilon_t= \gamma \epsilon_{t-1} + u_t$$

Now substitute the autocorrelated error back to original equation and you will get:

$$y_t = \beta_1 + \beta_2y_{t-1} + \beta_3 x_{t-1} + \gamma \epsilon_{t-1} + u_t$$

Now if if we lag the entire first equation by one  period we get:

$$y_{t-1} = \beta_1 + \beta_2y_{t-2} + \beta_3 x_{t-2} + \epsilon_{t-1} $$

Hence you can see that in presence of autocorrleation the lagged error term $\epsilon_{t-1}$ causes not just present error $\epsilon_t$ but also lagged dependent variable $y_{t-1}$. Hence this violates the Gauss-Markov assumptions of OLS, to be specific assumption that error term and regressors should not be correlated and in this case estimator is not salvageable as in standard cases where autocorrelation just causes error to be correlated with past errors.


Why can't someone have their money restored if someone they paid in advance goes bankrupt? A friend paid a travel company €3000 for a holiday in Australia. However the travel company went bankrupt so now her €3000 is lost. She has been informed by the liquidator that the company had such debt that she will not be getting any of her money back.

To me she is a victim of a system that requires the customer to pay in advance - we get that you have to pay for a service, but if she paid in arrears like you would, say, a plumber, then she wouldn't have this problem. It seems unfair that some industries require advance payments and so present this problem even though the capitalist system is supposedly based on the idea you pay money for goods and services. She's down money but didn't receive the service.

So my (somewhat hypothetical) question is, why is it not feasible to just "restore" her money, if we assume it's undisputed that she has the right to it? The obvious answer is that since the company is bankrupt, there's nowhere for that money to come from, but why does it have to come from somewhere, isn't money just a representation of your current worth which you have to pay to receive goods and services? And since money is no longer tied to gold or anything - it would seem it's technically possible for the bank to simply add that money back to her balance.

That’s how bankruptcy law works. Every entity that is owed money - including firms that sold products to the bankrupt, workers - wait inline to get paid back according bankruptcy laws.

The state cannot make everyone whole, as that would just encourage fraud. E.g., create a shell company, get rich friends to lend it money, take the money out and declare bankruptcy. Your friends get their money back from the government, and you get their hands on the proceeds as well. You make it worth their while by passing them money back under the table.

The state could bail out certain groups, but that’s an entirely a judgement call by the legislative body.


When a relevant variable unnamed on either axis changes, can the curve NOT shift? Source: pp 40-41, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw

  her demand curve would shift to the left (to curve D3).
  In economics, it is important to distinguish between movements along a curve
  and shifts of a curve. ...
  ♦ There is a simple way to tell when it is necessary to shift a curve: When a relevant
  variable that is not named on either axis changes, the curve shifts. ♦ ...

Is the last sentence of the excerpt (within $\mathbb{R^2}$) above really true ? (I surrounded it with lozenges ♦)

What if ceterus paribus, a fixed number $> 1$, of relevant variables changes? Can they counteract each other, or even equilibrize all the changes so that the curve stands unmoved?

The scenario of counteracting changes in relevant variables is theoretically possible, but unlikely in practice.

As an example that might come fairly close, consider the supply curve of oil in a small developing country which imports all its oil and subsidizes its sale, the subsidy being at a flat rate per unit.  The country will be a price taker for oil, so its domestic supply curve will be horizontal at the world price less the subsidy.  Now suppose the world price falls, and the government takes the opportunity to reduce the subsidy by a similar amount.  Either change on its own would shift the supply curve (the former down, the latter up), but together they could largely counteract each other.  Even in this case, however, it is implausible that the supply curve would never shift at all.  For one thing, the world price of oil is constantly changing, and it would be impracticable to adjust the subsidy so frequently.  For another, there would inevitably be time-lags in adjustments.

So although you correctly identify a theoretical exception to the statement by Mankiw, it is unlikely to be of much practical significance.


What determines the outcome of a price war, and why isn't that outcome reached instantaneously? Mary is making a hefty profit manufacturing and selling widgets. Jim has some money laying around and he is trying to figure out if he shouldn't start manufacturing some widgets too.

In this example assume that the marginal cost to produce a widget is zero (time, money, etc are all close enough to zero to be indistinguishable), but the cost of making a widget manufacturing plant is quite high.

Also assume that the market for widgets is fairly centralized. There are two bins with widgets in them and consumers can purchase their widgets from either bin. They cannot be resold, however. There are laws against that for whatever reason, and they are ruthlessly enforced. You may only sell widgets that you manufacture yourself.

What does Jim decide? And if he decides to enter the widget business, what is the eventual price per widget and how long does it take to reach that price?

More (but not too) formally:
Our situation can be modeled by a few different games,

Case 1: assume the rules of the price war are

players take turns setting a new, lower price from $\mathbb R$ or passing. When both players pass, the prices are locked and the market is allowed to run for t.

in this case the optimal strategy to set your price to whatever price your opponent sets, if the prices are already equal, then pass.

Case 2:

The market is allowed to run for time t after each player sets a (lesser or equal) price, they take turns setting prices, and they choose prices from $\Bbb Z$.

Here the optimal strategy is in fact no different. if the price starts at $p1$, and the sharing strategy is $S$, then there exists some $t$ such that for all $t'$>$t$ $$EV(S,S,t)=\frac{t*p}{2} >tk(p-1) > EV(S,S',t)$$ for all $p,k,S'$

Case 3:

Both players set prices in each timestep t without the information of the price their opponent sets in that time step, and the market is run.

In this case, no pure strategy is guaranteed to exist, because the information is not perfect. I don't know what the Nash equilibrium would end up being, nor do I know that there is any reason to suspect that it converges (as t -> 0) to the same price as the previous two cases, despite the game seeming to do so.

So I guess the question becomes, why does competition even exist at all in this market, and since all of these scenarios seem to converge towards the reality. ?

Answer to question

If we take your assumptions literally, Jim will decide not to enter the widget business. For suppose he did incur the cost of entry and that Mary is selling at price $p_m$. Jim can only sell to consumers if his price $p_j\leq p_m$. The best price for Jim is $p_m-\epsilon$ (where $\epsilon$ is some very small, positive amount). But this would leave Mary with no sales, so she will have an incentive to reduce her price to $p_j-\epsilon$. This is the price war you describe and it will result in both parties reducing their price to marginal cost (i.e. zero—this is known as Bertrand competition).

Since there are no frictions in your model, this will all happen very quickly. Moreover, because a price of zero implies zero profits, Jim has no incentive to incur the cost of entry in the first place, so he will instead opt to stay out of the market.

Effect of relaxing assumptions

This is obviously quite a stylised result owing to the stark assumptions of the model. But it makes a good foundation for thinking about some more realistic settings. For example:

If Mary's factory has a marginal cost $c_m$ and Jim has a patent on a new technology that implies cost $c_j<c_m$ then Jim can profitably set a price below the lowest price Mary is willing to set. Good news: if someone invents a more efficient technology they can enter the market and displace the older, less efficient technology.
Suppose that the factory has a monthly fixed maintenance cost. If Mary is a small, credit-constrained independent firm and Jim is a huge conglomerate with large reserves of cash then Jim can enter and practice predatory pricing. If he is patient enough, he can enter the market, set $p_j=0$, and wait for Mary to run out of money. She will then leave the market and Jim becomes a profitable monopoly. This kind of anti-competitive behaviour is illegal in most jurisdictions (yes, there are laws against setting too low a price!).
Suppose that Mary's widgets are blue and Jim's are pink. Consumers have an idiosyncratic preference for either blue or pink widgets. Then both firms can set a positive price and sell to the consumers who like their colour more. The more are products differentiated the more there is scope for both firms to exist profitably in the industry and Jim may find it worthwhile to enter. This is why firms talk so much about differentiation and unique selling points. There are various ways to model this in economics. Here is an example.

More formally, $p_j=p_m=0$ is the unique pure strategy Nash equilibrium

To be slightly more formal, let us check that $p_m=p_j=0$ is indeed a (Nash) equilibrium of the subgame in which both firms simultaneously set $p\in\mathbb{R}$ with the lowest priced firm capturing the entire (finite) demand. Neither party can profit by deviating to $p<0$ as this yields negative profits. A deviation to any $p>0$ results in the rival firm having a lower price so demand (and profits) are zero---again not profitable. Thus, there is no profitable deviation and $p_i=p_m=0$ is an equilibrium.

Is there another pure strategy equilibrium with some $p>0$? The answer is no. Consider the three possibilities and observe how a profitable deviation can be constructed for each:

$p_m>p_j$. In this case $j$ could increase his price to some $p_j'\in(p_j,p_m)$ without losing any demand. Such a $p_j'$ exists by the connectedness of the real line.
$p_m=p_j=p$. In this case at least one of the firms must be capturing less than all of the consumers. But by deviating to $p'=p-\epsilon$ it can capture all consumers. If we write $D$ for demand before the deviation and $D'>D$ for demand afterwards then the change in profits is $D'(p-\epsilon)-Dp$. By the connectedness of the real line, there exists an $\epsilon$ sufficiently small that this is positive.
$p_m<p_j$. This case is symmetric to $p_m>p_j$.

Thus, the only pure strategy equilibrium of this one-shot game is $p_j=p_m=0$.

If we repeat the game then we can sustain other (collusive) equilibria

What if we repeat the game infinitely many times? Suppose the two firms have an implicit understanding (an explicit agreement would be illegal) that they will both set $p=p^*$ for some $p^*>0$. Moreover, it is understood that if one firm deviates from this behaviour today, then both firms will revert to playing the static equilibrium ($p_j=p_m=0$) forever. Firms discount the future at rate $\delta$. For simplicity, suppose that demand is constant: $D(p)=1$ (not crucial).

If a firm plays in accordance with this understanding (and expects its rival to do likewise) then its profit is

$$\sum_{t=0}^\infty\frac{1}{2}\delta^t p^*=\frac{p^*}{2(1-\delta)}.$$

If a firm cheats and sets $p^*-\epsilon$ ($\epsilon$ small) then it captures the whole demand today, but expects to be punished forever thereafter, so profit is $p^*$. Thus, both firms wish to comply with their implicit understanding if
$$\frac{p^*}{2(1-\delta)}>p^*\iff \delta>\frac{1}{2}$$
(i.e. if they are sufficiently patient).

So, if we repeat the game and firms are very patient, we can sustain any price in equilibrium. But note that these equilibria require not only that firms are patient, but also that a) there is no threat of future entry that could destabilise the equilibrium; b) firms are able to coordinate on a $p^*$ without ever making an (illegal) explicit agreement; and c) firms are able to constantly and accurately monitor rivals behaviour in order to detect cheating.


What gives value to precious, luxury things? What makes things precious? For example Gold, Silver, Diamonds, etc. Once they were unknown to human being. And now they are the more pricey things.

I guess its just the way these things were marketed into the world, Demand and Supply control makes them pricey, there are a lot of other metals too in the world, which are more expensive then gold, but they are not that much poplar or in demand.

Similarly if we look into brands, Gucci, Armani, Micheal Kors, etc. Once they were unknown, no one knew about them. But now, they are the most expensive brands in the market.

I know in brands companies cash their name, that's why they are too pricey. But my concern is what exactly makes things precious and pricey.

Only a surplus on demands will make things "valuable".   So it is about the price level accepted by those who demands it, and the demands can be from essential needs to status show-off.

If you read Netherland history, you will notice there is a period of non-existence luxury goods, due to the shape of religious culture that forbids show off.

OTH, there is also terrible rent seeking like monopoly that make essential goods pricey.


How do we find a competitive equilibrium price when 2 firms operate with different marginal costs? Consider a homogenous goods market with the demand
function Q = 30 − P, where Q and P denote quantity and price respectively.
There are two firms playing a price game in the following manner: firm 1
quotes a price and then firm 2 chooses a price. When they charge the same
price they share the market equally and otherwise the market demand goes
to the firm charging lower price. Firm 1 has a capacity constraint at the
output level 5 units such that upto five units the marginal cost of production
is Rs 3 per unit of output, however beyond 5 units it cannot produce any
output. Firm 2 does not have any capacity constraint, it can produce any
amount with the marginal cost Rs 6. What would be the equilibrium price
in the market?

I'm confused, I think we have to put demand=supply firstly, then if the equilibrium price is 6 (the answer), firm 1 will produce 5 goods but firm 2 has no incentive to produce anything. Will the price not be a little greater than 6?

Ignoring the previously mentioned possibility that the problem may be written incorrectly: The concept of "residual demand" will be helpful here.

As with any sequential game, it's good to work backwards. From the perspective of firm 2, firm 1 has quoted some price $p_1$, and firm 2 has an opportunity to respond.

Firm 2 knows that if it sets $p_2<p_1$, it will take the whole market, and will face the whole demand curve $Q = 30-p_2$. It can choose the profit-maximizing $p_2$, as long as $p_2<p_1$.

Similarly, if it sets $p_2=p_1$, it will share the market.

Finally, if it sets $p_2 > p_1$, it will face the residual demand left over after firm 1 makes its sales. Assuming that $p_1\leq 25$, firm 1 will sell its five units and then not be able to sell any more. The demand left over for firm 2 to sell to after firm 1 makes its sales will then be $q_2=30-p_2-5$, or $q_2=25-p_2$. Firm 2 can choose its profit-maximizing price given this demand curve.

In each of these cases above, Firm 2 is selecting the profit-maximizing quantity by first setting up profit:

$$\Pi_2 = p_2q_2 - 6q_2 = (p_2-6)q_2$$

Plug in the appropriate $q_2$ using the demand functions determined above, then take the derivative with respect to $p_2$ and set the derivative equal to 0, solve for $p_2$. In each case you're constrained by what $p_2$ can be. If you're working with the $Q=30-p_2$ demand function (which you only get if $p_2<p_1$), then if profit-maximizing price is $p_2>p_1$, then you'll actually want to set $p_2$ just a hair underneath $p_1$ (you can just call it $p_2=p_1$ for simplicity).

Now you've got the optimal price for Firm 2 to pick given that it's decided to go for $p_2<p_1$, $p_2=p_1$, or $p_2>p_1$. To figure out which of those three it wants, you'll want to calculate Firm 2's profit under each of those scenarios. These profits should be dependent on $p_1$, since $p_1$ determines the boundaries of what $p_2$ can be while still satisfying $p_2<p_1$, $p_2=p_1$, or $p_2>p_1$.

So now Firm 1 knows how Firm 2 will respond to whatever $p_1$ is. And so Firm 1 will set $p_1$ in a way that maximizes Firm 1's profit. This step is a bit easier, since Firm 1 makes nothing under $p_2<p_1$, so it's unlikely that Firm 1 will want to set $p_1$ so that Firm 2 picks $p_2<p_1$.

To make this a bit more clear, let's do an example for Firm 2 making their decision.

Let's say Firm 1 had picked $p_1=10$.

If Firm 2 sets $p_2<p_1$, it is dealing with the demand function $$q_2 = 30-p_2$$. So we set up profit

$$\Pi_2 = (p_2-6)(30-p_2)$$
$$\frac{\partial \Pi_2}{\partial p_2} = 36 - 2p_2 = 0 \Rightarrow p_2=18$$

which is above $p_1=10$. Since profit is strictly increasing in $p_2$ for $p_2<10$, firm 2 will maximize its profit while holding $p_2<p_1$ by setting $p_2=10$ (a hair below 10, really, but let's just say 10). The profit from doing so is

$$\Pi_2 = (10-6)(30-10) = 80$$

And, of course, Firm 1 sells nothing and makes a profit of 0. Now if Firm 2 sets $p_2>p_1$, it's dealing with the demand function

$$q_2 = 25 - p_2$$

and wants to maximize profit

$$\Pi_2 = (p_2-6)(25-p_2)$$
$$\frac{\partial \Pi_2}{\partial p_2} = 31 - 2p_2 = 0 \Rightarrow p_2=15.5$$

which is above $p_1=10$, satisfying $p_2>p_1$. This price leads to the profit

$$\Pi_2 = (15.5-6)(25-15.5)=90.25$$

In this case, Firm 1 sells their five units and makes a profit of $(10-3)5=35$.

You will also want to check the $p_2=p_1$ case once it's clear what that actually leads to.

And so, if Firm 1 sets $p_1=10$, Firm 2 will want to set $p_2=15.5$, since a profit of $90.25$ is better than a profit of $80$.

This example is worked out with a specific $p_1$, and this particular $p_1$ makes Firm 2 not undercut Firm 1. The next step is to find out what the highest price that Firm 1 can set without getting Firm 2 to prefer undercutting. This will be $p_1$ in equilibrium, and Firm 2's best response to that $p_1$ will be $p_2$ in equilibrium.


Investment and probability Being a mathematician, I am familiar with probability calculations, but I need to ask a question related to investments and probability, and how this is handled seen from an economics view.

Given a project which involves two tasks, task 1 and task 2. Developing task 1 cost X dollars, while developing task 2 takes Y dollars if developed at the same time as task 1 and Z dollars if developed at a later stage.

The probability that we actually need Task 2 i 70%

What is the the "probable cost" of implementing Task 1 and 2 at the same time. vs. Task 1 and then Task 2 with a certain probability.
And what is this called in economics? Probable cost? Estimated cost??

The simplest thing is to calculate the expected cost in the two scenarios:

Scenario 1: Develop both at the same time
$$E[cost]= X + Y$$

Scenario 2: Wait and see if task two is needed
$$E[cost]= X + 0.7Z$$

However I would not say expected value calculation represents the "economics view". The general view rather depends on your preferences, often expressed as utility functions. It could be your utility is a function of expected value, but it could also not be the case. What is your risk tolerance? Some people seem risk-seeking. Are you ok with a a small possibility of ruin (bankruptcy)? Preferences can also depend on reference points and more...

For other simplistic approaches you could start with value at risk.


setting of Lagrangian function Consider a simple consumer's problem:
Max $u(X)$ s.t.  $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$
$w$ is initial endowment.
We can set the Lagrangian function to solve this problem.
$L=u(X)+\lambda (  \sum_i^l p_i w_i +\sum_i^l p_i x_i)$
But some may set the Lagrangian function as differenct way.
$L=u(X)+\lambda (  \sum_i^l p_i x_i -\sum_i^l p_i w_i)$
The result of calculation is same except the sign of Lagrangian multiplier $\lambda$
For the consumer's problem, which way is correct?
Edit after comment: There is a typo at the first equation in this
question. $\sum_i^l p_i w_i +\sum_i^l p_i x_i$ should be changed as
$\sum_i^l p_i w_i -\sum_i^l p_i x_i$

It's a matter of choice how one writes the Lagrangian in the context of Lagrange/KKT. Depending on how it's written, the gradients of the objective and constraint functions are either parallel or anti-parallel at a (suitable) optimum, and the Lagrange multiplier is neither negative or positive. At the end of the day, it is the same (subset of) optima that are covered by the Lagrangian FOC.
However, there is a standard choice---sometimes called the standard form in optimization literature---that ensures the Lagrange multipliers $\lambda$'s are non-negative. (See, for example, Convex Optimization by Boyd and Vandenberghe.)
In the standard form, the Lagranian is always written in such a way that it improves the objective function.
This convention is not always pointed out/followed in economics texts.
When it is followed in economic contexts, then the Lagrange multipliers admit the usual interpretation as marginal values of the corresponding constraints---a kind of "indirect marginal utility". E.g. the standard-form Lagrange multiplier is the derivative of the indirect utility function with respect to wealth---which must be non-negative.
For example, consider the maximization problem
$$
\max_{g(x) \geq 0} u(x)
$$
where $u : \mathbb{R}^n \rightarrow \mathbb{R}$ and $g : \mathbb{R}^n \rightarrow \mathbb{R}^p$. E.g. in a consumer's problem, $u$ is utility function, $p = 1$, and $g(x) = p^T(w-x)$.
Then the standard form Lagrangian is
$$
L(x, \lambda) = u(x) + \lambda^T g(x).
$$
It improves the objective function $u$ when $\lambda \geq 0$. This means that, for maximization problems, $L(x, \lambda) \geq u(x)$ for all $x$ and all $\lambda \geq 0$.
Since you're trying to maximize $u$, improving on $u$ means being larger than $u$.

In the standard form, the Lagrange multiplers must be non-negative., i.e. at an (suitable) optimum $x^*$, we must have $D u(x^*) + \lambda^T D g(x^*) = 0$ for some $\lambda$ with non-negative entries. This is easy to see, especially in the case of single constraint $p = 1$. If constraint $g$ is slack, then $\lambda  = 0$. If $g$ binds, then the gradient $D g(x^*)$ must point toward the interior $\{ g > 0 \}$. On the other hand, $D u(x^*)$ cannot point toward the interior---otherwise there would be an optimum in the interior. So $D u(x^*)$ and $D g(x^*)$ are anti-parallel and $\lambda > 0$.

If the maximization problem is formulated as
$$
\max_{g(x) \leq 0} u(x)
$$
then the standard form Lagrangian is $L(x, \lambda) = u(x) - \lambda^T g(x)$. Again, $L$ is written in such a way that, when $\lambda$ is non-negative, $L$ improves $u$.
Similar for a minimization problem.

"Suitability" of an optimum means that $D g(x^*)$ must be full-rank. This type of condition is called constraint qualification. The general KKT theorem says that the Lagrangian FOC is a necessary condition for local optima where constraint qualification holds. When the objective function is concave or quasi-concave (convex or quasi-conconvex, for minimization), then constraint qualification is not needed and Lagrangian FOC is sufficient for global optima.


Unemployment and Labor Force MacroEconomics Suppose that the total population on the 1st of January 2001 was 40 million people, of which 30 million were above age 16. Suppose that among the latter, 12 million people were not working or actively looking for work, and 2 million were unemployed. 800,000 people are expected to turn 16 during 2001 and half of them are expected to keep studying and not to seek a job, while the other half is expected to seek a job and be ready to start working immediately. How many people have to find a job during 2001 for the unemployment rate to be 8 %.

A short note: please try to show your math work in the original post, and maybe learn some basic Math TeX commands. It makes people much more willing to help with homework questions.

So first, we have our unemployment for the year 2000. Let $U_t, u_t, L_t$ denote the unemployment rate, number of unemployed, and number of laborers at time $t$, respectively.

$$U_t = \frac {u_t}{u_t + L_t}$$
$$U_{2000} = \frac {2,000,000}{2,000,000 + 16,000,000} = \frac{1}{9} \approx 0.111$$

So in the year 2001, 800,000 people turn 16, but half do not enter the labor force. So the question is how many people have to find a job for unemployment to fall below 8%.

$$U_{2001} = \frac{2,000,000 + (400,000 - n)}{2,000,000 + (400,000 - n) + 16,000,000 + n} = 0.08$$

where $n$ is the number of those who found jobs, and $400,000 - n$ is the net change in unemployment numbers. Solve for $n$.

$$\frac{2,400,000 - n}{18,400,000} = 0.08$$
$$\implies 1,472,000 = 2,400,000 - n \implies 928,000 = n$$

So what does this tell us? Even if all 400,000 new 16 year olds found jobs, that would not drive unemployment down to 8%. You need quite a few more people to find jobs.


Autocorrelation test for AR(p) (Breusch-Godfrey LM test) I have a question regarding the test for the error term.
As you know, in AR(p) model

The error term must be i.i.d., so after the regression I want to see if there is no serial correlation in the error term.
The text book says Breusch-Godfrey's LM test is designed to test autoregression in the error term model such as

I wonder if we can use BG's LM test to test serial correlation of the error term in AR(p) model. Is it okay to counstruct the auxiliary regression as usual? By the way, I already know Ljung-Box's Q test.
Thanks, in advance.

Yes you can use  Breusch–Godfrey  (BG) test for autocorrelation also in AR(p) models and dynamic models in general (see Verbeek Guide to Modern Econometrics where BG is applied to dynamic models in some examples - one of such examples is on page 142 in the 4th ed). As a matter of fact BG test is, generally speaking, the preferred test for autocorrelation in AR models and dynamic models  (see Maddala Introduction to Econometrics).
Furthermore, actually Ljung-Box's Q test should not be used for testing AR processes or other dynamic models with lagged dependent variable. The reason for that is that this test is in the models with lagged dependent variable biased towards the null hypothesis. This is because in such models the Ljung-Box's Q test no longer has asymptotically $\chi^2 $ distribution (see the Maddala textbook above for further discussion of this issue).


Why FDI stock in the same year is different in the same reports? From World Investment Report 2003, in 2000, the US has 12.4% GDP as FDI inward stock and 13.2% GDP as FDI outward stock. These numbers are equivalent to 1274720 million USD, and 1356960 million USD. With GDP US in 2003 as 10.28 billion USD.

From World Investment Report 2007, in 2000, the shares above are 12.8 and 13.4, respectively.

From World Investment Report 2017, however, in 2000, the US has 2783235 million USD as FDI inward stocks, and 2694014 million USD as FDI outward stock. If we measure in percentage as above, we would have 27% and 26% equivalently.

Why are the numbers are so different? Do I forget something like the price levels or so?

Many thanks for your help.

In note 1 of the World Investment Report 2017 (page 39), it is mentioned that

  FDI data may differ from one WIR issue to another as data are
  continually revised, updated and corrected by the responsible
  authorities, such as central banks and statistical offices, that
  provide FDI data to UNCTAD.

The last figures are the most accurate ones.


The notation cl co (A) I stumbled upon this notation, (cl co(A+C), while reading "Set Optimization—A Rather Short Introduction" by Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff and Carola Schrage.

Is it right for me to assume that cl co(A) means the convex closure of the set A?

I'm pretty sure that $\mathrm{cl}$ is the set closure operation. I'm less sure about $\mathrm{co}$, but I've seen it being used for the convex hull operation in other contexts.

As a reference, in Fundamentals of Convex Analysis, the same notations are used to refer to the closed convex hull of a set:


Has Stack Overflow been economically beneficial to programmer productivity? Has Stack Overflow an overall positive impact on programmers productivity?

On one side, it takes time and effort to ask and answer questions -- thus decreasing productivity.

On the other side, more and more programmer questions are answered and easily found through Google. This allows many programmers to find solutions to their problems quicker than ever before, thus increasing productivity.

Has Stack Overflow been economically beneficial to programmer productivity? If so, can this be quantified in terms of money?

Note: I've asked this related question on Skeptics.SE, but it turns out there is no peer-reviewed research on the matter.

Exogenous Variation

As Andre says, you are after causality, not just correlation. And the best way to find that is exogenous variation - he mentions experiments. Unless you can randomly force some people to use SO - and forbid it to a control group, there is little you can do with true experiments.

There may be salvation, however. You are after the "treatment of the treated" effect - those who are informed about SO, and decide to use it. That is hard to get, because the latter step will correlate with ability. Some people are just better/more efficient/smarter and hence can benefit from SO more/less.

In order to remove this self selection, you could instead look at something like the "average treatment effect". We inform a subsample about SO. Some will use it, some will not. What is the average increase of productivity in this sample?

IV Estimation

Now, you will either randomly inform individuals about SO (something that StackExchange as a company might actually do), or you/they might already have data on "being informed about SO", lets call that inform. As long as inform does not correlate with relevant characteristics (such as skill), you can use that as a proxy. Unfortunately, one might argue that it actually does: Perhaps smarter/better programmers are also better at creating networks / using google, and hence are more likely to find SO and become inform.

Of course, if you also have data for ability, you can control for that, so this particular bias would be removed.

Then, all you have to do is regress some measure of productivity/success/career advancement onto inform, and perhaps ability.


Is there an accepted metric for "the value of the dollar"? Is there a reasonably well-established scheme for estimating the value of the dollar in international trade, somehow relating it to the "average" of other currencies or to some currency-independent metric?  Something that could be used when researching the relationship between fiscal policies and international trade.

(I would consider using the Euro, but I'd like numbers going back to the 70s, at least, and obviously the Euro doesn't work for that.  In addition, it is perhaps a bit destabilized by, eg, the Greek economic crisis.)

It sounds like you want the Trade-Weighted Dollar Index, which is available from 1973 on for major currencies. It's the value of the dollar against a basket of other currencies, weighted by their share of US trade.


Lower ROI because of zero interest rate I've heard an interview with Bill Gross in which he claims that one of the problems with lower interest rate is that it leads to a lower return on investment (ROI) in the real economy.

How can that be?

Cheap money makes it worthwhile doing stupid things.


Regression with weights I have a question regarding weighing observations by importance.

Suppose I am running the following regression:$$log(y_{it}/y_{it-1})=\alpha+\sum_{i=1}^{N}\gamma_{i}Country_{i}+u_{i}$$ where basically my LHS is GDP growth of country $i$ at time $t$ that I regress on a full set of country dummies (-1).
I want to weigh the observations by the GDP of the country.
If I am doing this on stata, what weights are appropriate? Frequency weights or probability weights? My intuition is that I have to basically run a weighted least squares regression of the form: $$ \hat{\beta}=(x'Wx)^{-1}(x'Wy)$$

where $W$ is a diagonal matrix with the weights (GDP) on the diagonal. I am at a loss regarding how to implement this on Stata. Thanks!

If you check Stata's help file on regress you should understand how to do it. Particularly pp. 16-7 have specific examples of how to apply weights.

I will edit in order to be more detailed.

gen lnyl1y=ln(y)-l1.ln(y)
xi: reg lnyl1y i.country [w=y]

Notice that if the weighted regression is done by dividing all values for observation $i$ by $\sqrt{w_i}$, then
$$\tilde{x} =
\begin{pmatrix}
\frac{x_{1,1}}{\sqrt{w_1}} & \cdots & \frac{x_{1,k}}{\sqrt{w_1}}\\
\vdots &\ddots &\vdots\\
\frac{x_{n,1}}{\sqrt{w_n}} & \cdots & \frac{x_{n,k}}{\sqrt{w_n}}
\end{pmatrix}
$$
Call the diagonal matrix composed of $\sqrt{w_i}$ in the $(i,i)$ element $\sqrt{w}$. Then, the WLS estimator is given by
$$
\hat{\beta}_{WLS}=\left(\tilde{x}'\tilde{x}\right)^{-1}\tilde{x}'\tilde{y}=
\left[(\sqrt{w}x)'\sqrt{w}x\right]^{-1}(\sqrt{w}x)'(\sqrt{w}y)=(x'Wx)^{-1}(xWy)
$$
as $\sqrt{w}'\sqrt{w}=W$ and $\tilde{x}=\sqrt{w}x$ which is exactly the same as multiplying each row of $X$ and $y$ by the sqrt of your weight (analytical weight, here $\sqrt{w}$).

The thing here is to understand how your matrix relates to applying the weights and understanding the connection between your formula and the one which is in the help file. For other ways to calculate weighted regressions in relation with Stata, see e.g. this document


Why is sugar cane still harvested manually in Brazil? I read that up to 70% of Brazil's sugar cane in the Sao Paulo area is still harvested manually. Some workers are paid $1.50 per tonne. Why is it still being cut manually? Some possibilities:

(1) farm owners lack the capital to buy mechanical harvesters

(2) human labor is cheaper

(3) regulatory or legal obstacles are blocking the use of harvesters

Although it seems that it necessarily boils down to the idea that labor is very, very cheap for some reason or another, it is often the case in these situations that workers unions, somewhat perversely, will block the firm´s replacement of labor with capital, in order to keep the workers employed.


difference between demand for money and credit As far as I understand in both cases we are seeking money that we don't have and would like to borrow.

That being said what I find even more misleading is the fact that as a nominal interest rate increase credit increases.

If anyone could clarify this that would be absolutely amazing.

I'm not sure what's the nature of the question but:

Credit isn't necessarily linked to money. It's linked to the fact that people desire to consume today rather than tomorrow. You can have credit, borrowing and an interest rate in a cashless economy.
Demand for money could be abstracted as an extra term in the utility function and/or a cash-in-advance restriction, meaning that the individual seeks to hold money because it makes them happy and allows him to pay for real goods. This, in turn, leads to demand for money.


Year Fixed Effects in a Dynamic OLS Regression with Cointegrated Variables I am estimating a dynamic OLS model since I have variables that are non-stationary, but cointegrated. In addition, the data is a standard time-series (i.e. one observation per one time period) so technically not panel. I know that one should control for year fixed effects when you have panel data. My question is should I still control for year fixed effects even though my data technically isn't panel. Also, does it even make sense to control for year fixed effects when the independent and main dependent variable are non-stationary, but cointegrated?

No in pure time series we generally don't use fixed effects. If you have data on lets say monthly frequency you could include dummies for months in general, e.g. having February, March, April ... dummies but you would include them as a general dummies where all Februaries are part of the February dummy rather having a separate dummy for each time period.
You can also include dummies for time periods where you think that the period was special in some way. For example, you could include a special dummy for all crisis years. Similarly, you could include some dummies for structural breaks and so on.
Furthermore, dynamic OLS model is more of an umbrella term. For example, the above would hold for ARDL/Pesaran error correction model. There might be dynamic models that have further issues, it is always best to survey literature on the particular model you are actually using then just generalizing.
PS:

I know that one should control for year fixed effects when you have panel data.

This is actually not generally valid statement. Yes more often than not you want to control for year  fixed effects in panel data but not always.


How can the VNM completeness axiom be derived from the transitivity and continuity axioms? My teacher mentioned off-hand that the first Von Neumann Morgenstern axiom of completeness is not actually necessary to prove the Expected Utility Property, and that the axiom of completeness is in fact implied by the axioms of continuity and transitivity. I don't see why this is.

I think I just found a counterexample to your teacher's claim.

Suppose completeness is violated completely (sorry for the pun): $L\not\succsim L'$ and $L'\not\succsim L$ for any $L,L'\in\mathcal L$, i.e. no two lotteries are comparable.

Then I claim that transitivity and continuity are both satisfied vacuously.

Transitivity: $L\succsim L'$ and $L'\succsim L''$ imply $L\succsim L''$ for any $L,L',L''\in\mathcal L$. But since no two lotteries are comparable by $\succsim$, the antecedent is always false, and so transitivity is always satisfied.

Continuity: for any $L,L',L''\in\mathcal L$, the sets $\{\alpha\in[0,1]:\alpha L+(1-\alpha)L'\succsim L''\}$ and $\{\alpha\in[0,1]:L''\succsim\alpha L+(1-\alpha)L'\}$ are both closed. Again, since no two lotteries are comparable, both sets are empty and therefore closed. Hence, continuity is satisfied in a vacuous sense.


How do I determine the demand curve of a product through a survey? I'm a student and I want to determine the demand for a product (and consequently determine what will happen if the price of the product increases). I don't have access to any businesses that can provide the data for me (price and demand at that price?

How do I estimate the demand curve or function by surveying people? What I currently have in mind is to ask people what amount of the product they'll demand at a given price and extrapolate this data for a city or a region.

Additionally, what'd be a good sample size for such a survey?

If it is a small project, you may benefit from reading this

At most, you can estimate price elasticity of demand using your survey data to make predictions/forecasts. For this you need to estimate a demand function as @EconJohn suggested, but you need to include additional variables such as income, price of a substitute or complement etc. to hold them constant (i.e. ceteris paribus). Note that the demand function does not allow you to draw the downward sloping demand curve (like the one in textbooks). It is very hard, in practice, to draw actual demand curves using survey data. Even when you have the most complete data, without including the suply function in your estimation, you face so called identification problem. In short, econometric estimation of demand curve does not allow you to draw a downward sloping curve of the type you see in textbooks.

The link above has a good explanation about this point, and it can be used as a guide for your project.


Slope vs elasticity of demand function - Is it the same thing? Do the slope of a linear demand function and the elastisicy of demand coincide when we use specific preferences for pricing. As a paradigm, if we consider the case of CARA normal preferences, by solving the proble of the represenative conumer, we know that the demand function is a linear one. In this case the slope and the elasticity of the demand function coincide, but is this the case in general?

No, slope of a demand function is $\frac{\partial q(p)}{\partial p}$ elasticity of demand is $\frac{\partial q(p)}{\partial p} \frac{p}{q(p)}$. So they cannot be same except in some special cases.


Confusion on Strategy Sets in a simultaneous move game I have a confusion on how to define strategy sets;

I understand that in dynamic games strategy sets are defined as $\textbf{sets of functions}$ - see for example my previous post, link below:

Define and characterize equilibria of the following game

However I cannot square this with what would be a strategy set in a simultaneous move game; as an illustration consider the following simple game:

There are two players competing for a prize of $1$. Each player $i$ choses effort level $x_i \in \mathbb{R}$. When the efforts levels are $(x_1,x_2)$, the probability that player 1 wins the prize is given by $p(x_1,x_2)$ and the probability that player 2 wins the prize is given by $1-p(x_1,x_2)$. Player $i$'s cost of exerting effort $x_i$ is equal to $x_i$.

Players choose their effort levels simultaneously.

Now looking standard textbooks I suspect that the strategy sets of both players would be defined as:
$S_1 = \mathbb{R}$, $S_2 = \mathbb{R}$
Since both players will choose an effort level $x\in \mathbb{R}$.

However given that a $\textbf{strategy}$ for player $i$ is defined as a complete contingent plan, it would be natural for me to define a strategy as a function:
$s_i: \mathbb{R} \to \mathbb{R}$,

that is for each possible effort level $x_j \in \mathbb{R}$ that player $j$ chooses, player $i$ will choose an effort level $x_i \in \mathbb{R}$.
But then since $S_i$ is the $\textbf{set of}$ strategies it would appropriate to define the strategy set of this game for player i as:

$S_i = \{s_i : \mathbb{R} \to \mathbb{R}\}$ , that is the set of all possible functions $s_i$.

But then this definition would conflict with the original one I gave at the beginning.

So which of the two (if any) would be the correct "strategy set" of this simultaneous move game?
Thank You!

There is no conflict. The strategy is a function which maps decision points to actions, that is it tells the player in each decision point (to be precise, in each information set) what she should do. In a dynamic game there are usually more decisions points, so this function nature of the strategy is more obvious.

In the simultaneous game you describe there is only one decision point (information set), players have to make a choice at the start of the game. A strategy maps an action to this one decision point (information set), so basically the strategy space is just the action space.

EDIT:
 You write " it would be natural for me to define a strategy as a function:
$$s_i: \mathbb{R} \to \mathbb{R}."$$
There is some murkiness here, because what is the first $\mathbb{R}$ supposed to be? The strategy set of the other player? But that is not observed by player $i$, so it cannot factor into her decision. As a function the strategy would look actually look like
$$s_i: \left\{I_i\right\} \to \mathbb{R},$$
where $I_i$ is the single information set of player $i$, so $\left\{I_i\right\}$ is a singleton set. There is a bijective mapping between the vector space of these strategies and the space $\mathbb{R}$ so it does no harm to define $S_i$ as equal to $\mathbb{R}$.


Have there been instances where economists have advocated a trade embargo? Due to Comparative Advantage, it is generally accepted among economists that free-trade is the best policy for a country and increases the standard of living for the citizens of that country and the country with which it trades.

I have heard, however, that there are certain instances where it would be more beneficial (on the whole for that particular country) for the country to restrict trade with other nations.  Have there been instances like this where economists have advocated a trade embargo with another nation?

Free trade is, on the whole, one of the few otherwise controversial policy topics on which economists have near-perfect consensus. Historically, this consensus has long been strong in the English tradition (Hume, Smith, Ricardo, Mill), albeit less strong elsewhere. Famously, 1028 American economists signed an unsuccessful petition in 1930 begging Herbert Hoover not to approve the Smoot-Hawley tariff. If the IGM Economic Experts panel is any guide, consensus remains firm today.

That said, off the top of my head, various cases in which some modern economists have departed from advising free trade include:

Import substitution and related protectionist development philosophies in the early postwar era. These were never (as far as I know) advocated by many economists in the neoclassical tradition, but they certainly had support among influential other figures, notably Raúl Prebisch; enough that they were put into action in many parts of the developing world, especially Latin America. The mainstream verdict on import substitution is that it was a costly failure, although there are notable heterodox dissenters like Ha-Joon Chang. Dani Rodrik also has a slightly less heterodox record of free trade skepticism.
Cases of market power. Here, economists do not necessarily advocate departing from free trade in practice; but they do recognize that (in principle) it can be individually optimal for countries with either some monopoly or monopsony power to try to manipulate the terms of trade in their favor via trade restrictions (potentially either import or export tariffs). Many large, developed countries probably have some market power of this kind, and specialized commodity suppliers do as well. This observation is the basis of some economic theories of trade agreements, which are modeled as devices for countries to coordinate on a Pareto optimal free trade regime and overcome their individual desire to manipulate the terms of trade.
Aggregate demand management when monetary tools are limited (due to the zero lower bound). Paul Krugman has discussed how protectionist clauses in stimulus plans could in principle be globally optimal, by allowing countries to retain more of the benefits of their own stimulus and thereby encouraging them to do more. Another case is where countries facing the zero lower bound could impose tariffs on unconstrained countries, in an attempt to redirect expenditure to demand-constrained economies. That said, Krugman still (mostly) favors free trade and doubts that the benefits of such a policy would overcome the cost to the global free trade regime.
Distributional consequences. This is a common refrain in popular critiques of free trade: many pundits argue that even if trade is beneficial in some aggregate sense, its adverse distributional impact (e.g. hurting the already-suffering manufacturing workforce) negates the overall benefit. Indeed, in models where gains from trade arise from differing factor endowments, the whole point is that some factor (the domestically scarce but internationally not-so-scarce one) will lose; this is the idea behind the Stolper-Samuelson theorem.

Traditionally, most economists have argued that it is better to have free trade and address any distributional or insurance objectives through the overall tax-and-transfer system. Whether not this conclusion holds in a formal model, however, depends on exactly what instruments are available to the government; it's conceivable that trade barriers would be an optimal second or third-best policy in some cases. The left-wing heterodox economist Dean Baker has strenuously argued along these lines (though he has certainly not offered a formal model). More in the mainstream, an early version of Autor, Dorn, Hanson (AER 2013) made a suggestive stab in this direction with a back-of-the-envelope calculation showing that the deadweight loss from transfers induced by Chinese trade was a substantial fraction of the theoretical gains from trade - though this calculation was rough and evidently removed from the published version. Notably, Autor was one of the few IGM panelists with an "uncertain" reply about the benefits from trade.


How are workers harmed, from firms' payroll tax revenue aimed at reducing workers' tax? Source: p 130, Question 6.5, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw
= (page unknown), Question 6.5, Principles of Microeconomics, 4 Ed, 2008, by N Gregory Mankiw

  p 125, Case Study: a payroll tax [...] is a tax on the wages that firms pay their
  workers. ... When a payroll tax is enacted, the wage received by workers falls, and the
  wage paid by firms rises.

  $5.$ A senator wants to raise tax revenue and make workers
  better off. A staff member proposes raising the
  payroll tax paid by firms and using part of the extra
  revenue to reduce the payroll tax paid by workers.
  Would this accomplish the senator’s goal? Explain.

  Given Answer: Reducing the payroll tax paid by firms and using part of the extra revenue to reduce the payroll
  tax paid by workers would not make workers better off, because the division of the burden of a
  tax depends on the elasticity of supply and demand and not on who must pay the tax. Because
  the tax wedge would be larger, it is likely that both firms and workers, who share the burden of
  any tax, would be worse off.

I reddened the transfer of tax revenue from firms, and yellowed workers' lost wages. Why are workers worsened, by the transfer of tax revenue from firms (the red as above) intended to repay workers' lost wages (yellow)? The workers receive extra money, right? Please advise if I erred, but I think Question 5 can be graphed as above (I modified Figure 8, p 126).

As the answer says, it depends on the elasticity. The following is a short counter example.

A Short Example

Suppose that the firm is perfectly elastically demanding labor based on

$$L^D (w) = 5 - w$$

And workers supplies labor:

$$L^S (w) = w$$

In equilibrium, we have $L = 2.5, w = 2.5$. Total income = $2.5^2$ Now install a tax:

$$L^D (w) = 5 - (1+t)*w$$

Then, we have that $w = \frac{5}{1+1+t}$ and $L = \frac{5}{1+1+t}$

Hence total labor income is: $wL = \left(\frac{5}{2+t}\right)^2$. We give the payroll taxes to the workers, who then receive $(1+t)*wL$. Let $t=1$.

Total worker income is then $2*\left(\frac{5}{3}\right)^2$ = 50/9. Previously, they got 50/8. They new total income, including payroll taxes, is smaller than before.

So what is the math indicating?

The reason behind this is ignored externalities.

The payroll tax increases the labor cost, making the firms want to hire less
The workers do not take into account that they are going to get the payroll taxes out later on, so they do not compensate for the higher labor costs by accepting smaller wages (see last paragraph)
Given higher costs, we produce less. There is a deadweight loss (DWL) caused by disturbing taxation (production changed), and the only question is: how this DWL is shared. This depends on the elasticities of supply and demand w.r.t. wages. In this scenario, some of the DWL was taken over by the workers. They got partly compensated through taxes, but not by enough

Negative externality

Why doesn't the worker take into account that the more they work, the higher the taxes they receive are going to be?

Basically, because each worker is very very small compared to the total labor force. Think about it this way: Compensating for the increase in labor cost means that the worker has to work for a smaller wage. This will hurt him a lot. The benefits of this $t\cdot wL$ are going to be shared with all other fellow workers. That means that his own share of his "sacrifice" is going to be small. His private gain is too small from doing this. On the other hand, he would receive the positive externality from all the other workers working more. To wit, he would receive some of their tax share - but since the other workers' incentives aren't set correctly either, they don't work "as much" either.


Portfolio choice problem of a CARA investor with n risky assets Ok, I am working on a problem that consists of the following:

I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all of the underlying random variables are multivariate normal.

Problem:

define $\phi$ as the amount invested in each of $n$ risky assets, such that the budget constraint is:

$\Sigma_{i=1}^{n}\phi_i=w_0$ for some initial wealth, $w_0$

Show that the optimal portfolio is:

$\phi=\frac{1}{\alpha}\Sigma^{-1}\mu+[\frac{\alpha w_0-1'\Sigma^{-1}\mu}{\alpha 1'\Sigma^{-1}1}]\Sigma^{-1}1$

where each of the 1's is an $n$-dimensional column vector of 1's.

Work/Attempt

Ok, these are the things I know:

I am dealing with CARA utility, which gives me a utility function of the form:

$u(w)=-e^{-\alpha w}$ where $w$ is my random end-of-period wealth which I believe to be distributed as

$w$~$N(\mu,\sigma^2)$ with $\mu=\phi'\mu$ (a vector of expected returns scaled by the amount invested in each), and $\sigma^2=\phi'\Sigma\phi$ where $\Sigma$ is the covariance matrix of the $n$ risky assets.

So, to find the expected utility of this function, I use the fact that the expectation of an exponential of normals is the exponential of the mean plus half the variance, to arrive at:

$E(u(w))=-e^{-\alpha\phi'\mu+\frac{\alpha^2\phi'\Sigma\phi}{2}}$

Factoring out a negative alpha, and equating the remaining part of the exponential as the certainty equivalent of a random wealth (I might not be explaining that well, but I am almost certain this is the correct path), I can maximize utility by maximizing the utility of the certainty equivalent, which is done by maximizing the certainty equivalent itself.

All that to say, I need:

$\frac{\partial}{\partial\phi}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$

From there I can't seem to get anything even remotely close to the result I am supposed to show. I have

$1'\mu+\alpha\Sigma\phi=0$

which seems to mirror the first term in the result, but I am lost as to where the rest comes from.

Any help would be appreciated. I'm not sure if my mistake is in the multi-dimensional partial derivative, or if it is in obtaining the function that needs to be maximized. The book I am using has a similar problem for a single risky asset which I can work through just fine, but the exclusion of a risk-free asset (which would seem to simplify the wealth constraint) makes it more confusing to me.

The problem is equivalent to maximizing

$$
\max_\phi \left( \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi \right)
$$

subject to

$$
\mathbf{1}' \phi = w_0.
$$

(boldface 1 is column vector of ones, ' stands for transposition).

The lagrangian is

$$
L(\phi, \lambda) = \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi - \lambda \left( \mathbf{1}' \phi - w_0 \right),
$$

and its jacobian wrt. $\phi$ is

$$
\mathrm{D}_\phi L(\phi, \lambda) = \mu' - \alpha \phi ' \Sigma - \lambda \mathbf{1}'.
$$

First order conditions require that the jacobian is equal to zero (in each element), plus the original budget constraint, which together (after transposing the jacobian) yields a system of linear equations in $(\phi,\lambda)$:

$$
\begin{split}
\Sigma \phi + \mathbf{1} \lambda &= \frac{1}{\alpha}\mu \\
\mathbf{1}' \phi &= w_0
\end{split}
$$

We can solve for $\phi$ as a function of multiplier $\lambda$:

$$
\phi = \frac{1}{\alpha} \Sigma^{-1} \mu - \lambda \Sigma^{-1} \mathbf{1}
$$

Plugging this into the budget constraint and some algebra yields an expression for $\lambda$:

$$
\lambda = \frac{\frac{1}{\alpha}\mathbf{1}' \Sigma^{-1} \mu - w_0}{\mathbf{1}' \Sigma^{-1} \mathbf{1}},
$$

and substituting this into the expression for $\phi$ should yield the result as given in the question.


Would saving money outside of banking system be able to reduce monetary base? First of all, here is definition of monetary base taken from Investopedia

  A monetary base is the total amount of a currency that is either in general circulation in the hands of the public or in the commercial bank deposits held in the central bank's reserves.

Thanks to this definition we can see why the Fed is able to reduce monetary base by selling bonds. Money that the Fed gets from selling bonds fall to neither category (the Fed isn't the public, money are out circulation and although they can be deposited in Fed's vaults, they belong to Fed and not to any commercial bank), thus from the point of view of this definition these money just disappear.

It makes me wonder if just taking money from circulation (and consequently - from banking system. Because otherwise money will still "work") will decrease monetary base. Examples:

1.A person named Alex for some reasons keeps all their savings under their mattress

2.Corporation "MacDuck Industries" for some reason keeps its savings in its own building called "money bin". It doesn't loan them, it just sits on its money.

3.The government of Maxtopia has proficit budget and for some reason keeps its excess tax money in its own "money bin". Again, like MacDuck Industries it doesn't do anything interesting with the money, it just sits on piles of cash.

If I understand everything right (including the definition), in all of these cases money are taken from circulation, thus reducing monetary base in all cases. Am I right?

Options 1-2 do not reduce the monetary base. “Money in circulation” are all notes/coins that were issued to the private sector and not returned.

There is no way of measuring how many transactions notes/coins support (or whether they have been destroyed). So reducing “circulation” has no effect on published statistics.

Option #3 would reduce the monetary base if the money is returned to the central bank.


If I employ someone to destroy a building, what is the net GDP impact? Can this one-off cases be extended? For instance, if I employ someone to destroy 50 buildings and rebuild them, does that contribute an increase in GDP?

It certainly does affect DGP, because what you destroy is part of the capital base $K_t$, while the building activity is part of gross investment. The capital base is not part of the DGP, while investment $I_t$ is.

A related question is, what is the net effect on the capital base? That will depend on the current value of the buildings being destroyed, compared to the value of the new buildings.

The more interesting question is : how are we to classify the value of demolishing services alone? Labor and capital will be expended to bring down the old structures, so certainly production factors will get rewarded.

From an accounting point of view, these constitutes also part of the cost of the new buildings, and this has a wise economic rationale: the returns from the commercial exploitation of the new buildings, must be evaluated against the total cost incurred so as to be able to exploit them, and this includes the cost of demolishing pre-existing structures.

Do the economics discipline treats such demolishing services the same way? yes, they are also part of gross investment.

Assume now that we are only demolishing an old building, without building a new one. It seems a little weird to count the cost of demolition as "gross investment" that will increase GDP and tend to positively offset the reduction of the capital base due to the elimination of the old buildings. Investment to what? - we have not erected new structures.

Well, what increases here by the demolition is the value of the land. Why? Because prior to demolition, the land was committed to a specific building, and so the land's uses were restricted. Now that the land is land only, it can be used for many more purposes than before. This has economic value.

Usually, when a piece of land has buildings, we see it as a whole, and we don't make the above distinction. But the distinction is there and justifies treating the cost of the demolition alone as investment.


How does the EBQ formular come out? I figured out how the economic order quantity (EOQ) formula comes out. I know it is quite similar with economic batch quantity (EBQ), sometimes termed economic production quantity (EPQ). I just don't understand why they are different. What makes the difference in holding cost part?

In a way EBQ (also known as EPQ) EPQ is an extension of the EOQ model, the difference being

In EPQ the company will produce its own parts
In EOQ parts are produced by another company

Note that if you produce stuff you need to store it somewhere, which comes at a cost. This is explicitly included in the EPQ model via the holding cost variable


Accounting for machines in the labor theory of value I have read that the Labour Theory of Value holds that the value of a good or service is determined by the total amount of labour involved in its production. To account for this properly, we must count not only the labour directly involved in finishing a good, but also the labour needed to produce the tools used in that process.

I am curious: how does the LTV suggest the labour involved in producing a machine should be divided across the goods that machine is used to make. In particular,

If the output of a machine is arbitrarily reduced, what happens to the value of the goods being produced? If we simply divide the labour put into the machine across the units produced then it seems like their value should increase, but then it seems like the whole notion of value becomes quite arbitrary.
Suppose I built a machine that, in any given hour, produces one unit of penicillin and one unit of randomly mixed chemicals (from the same raw materials). Does this mean that penicillin and mixed chemicals should have equal value?

"I have read that the Labour Theory of Value holds that the value of a good or service is determined by the total amount of labour involved in its production."
That would be Adam Smith's version of the LTV, which according to him only holds in pre-capitalist societies.
Karl Marx's version is thus:
the value of a good or service is determined by the total amount of socially-necessary labor involved in its production.
Socially necessary labor is basically the average amount of labor a society (industry, trading network or whatever) requires to produce a good.
It takes far less labor time on average to produce randomly mixed chemicals than to produce penicillin from chemical precursors.


Why does instrument exogeneity imply conditional mean zero? In the following slide
ECON4150 - Introductory Econometrics Lecture 16: Instrumental variables, Monique de Haan

it says that "instrument exogeneity implies $E[u_i \mid Z_i]=0$" where instrument exogeneity is defined on slide 13 as $cov(Z_i, u_i) =0$. Here $Z_i$ is the instrument and $u_i$ is the error term in the structural model with a single endogenous variable.

Can someone prove to me why this is true? I thought the implication only held the other way, i.e., $E[u_i \mid Z_i] = 0$ implies $cov(Z_i, u_i)=0$ but the slides suggests that $cov(Z_i, u_i)=0$ implies $E[u_i \mid Z_i] = 0$, which isn't true in general?

No, you're right, $Cov[Z_i,u_i]=0$ does not imply $E[u_i|Z_i]=0$ in general. If the author defined "instrument exogeneity" as $Cov(Z_i,u_i)=0$ previously, he/she is being careless here.

"Instrument exogeneity" is often defined as $E[u_i|Z_i]=0$ too, in which case all you need to do would be to understand "implies" as "means". But since exogeneity is defined as uncorrelatedness (zero covariance), this issue has to be addressed. the author'd better fix it in a later version. How about "Instrument exogeneity usually means ..." instead of "Instrument exogeneity implies ..."?


Potential solutions of the $n$-player Prisoner's Dilemma Suppose we want to model a real-world phenomenon as an $n$-player Prisoner's Dilemma.

To be more specific, suppose we use RM Dawes' paper "The commons dilemma game: An n-person mixed-motive game with a dominating strategy for defection, 1973" which generally formalize the Prisoner's Dilemma for $n$ players along with a tragedy of the commons (Hardin's tragedy 1968).

Although modelling might seem "easy," how one can properly analyze in Game theory? How can we justify the validity of the game with the real world?

Also, even if we model the real-world phenomenon as a Prisoner's Dilemma, what would be some of the potential solutions to prevent the tragedy?

To be more exact, what are some of the standard methodologies to prevent universal defection?

I'd appreciate any help.

All the insights to your question about the n-player game carry over from the 2-player game.

The validity in the real world is justified to the extent and in the same way as for the 2-player game, with all its caveats. An n-player game, however is even more realistic, because it is more general and many economic issues involve more than 2 players. Consider for example global geopolitical issues that require coordination, such as climate change or nuclear bombs.

The potential solutions are the same as for the 2-player game as well. Notably, in repeated games over time, reputational effects and retaliation can matter. If those who play the non-cooperative strategy (which is the "bad" action that leads to the prisoner's dilemma outcome that is worse for everyone) are punished over time, then the problem you describe can be overcome. Denesp in his comment to this answer gives some examples.

The reason a static prisoner's dilemma always has a non-pareto-optimal equilibrium (i.e. why it poses a societal problem) is because it is played once or is a one-shot game without the possibility to punish deviators later on. If players interact frequently with each other, (the threat of) retaliation can help overcome the dilemma. There are a number of such strategies. One of the more famous ones to look up would be "tit for tat".


Is it correct to say opportunity cost is cost of giving up quantity of what you are selling? In this video (https://m.youtube.com/watch?v=KEYYN-a_x6E), the instructor talks about the marginal opportunity cost and supply curve. (Begins talking about it at after 1:50).

He states that the opportunity cost of giving up a quantity of apples (what she is selling) is the opportunity cost.

Is this correct? I thought the opportunity cost is the cost of giving up alternative things. But he plots the points of "giving up an apple' on the marginal opportunity cost curve, even though she is selling apples, so wouldn't that not be the opportunity cost.

It seems the instructor is referring to the opp. cost of giving up the profit from selling x amount of apples. Which would be an opportunity cost. Opportunity cost isn't strictly alternative based, for example: If I'm selling 10 apples today, the opportunity cost of be only selling 9 apples tomorrow is whatever profit I stand to lose from not selling the 10th apple tomorrow.


How are historic economic statistics calculated? Sources like Our World in Data can have statistics that date back hundreds of years. Specifically, the Our World in Data source has a graph of GDP Per Capita in England dating all the way back 1270. What kind of resources do these statistics draw upon? Finally, is these any metric to determine how accurate they may be?

The site has actually multiple estimates for historical UK/England GDP (as the Maddison data include data for UK as well). But based on your question I will assume you are referring to this chart:

As the chart itself states the sources for the data are Campbell, Klein, Overton & van Leeuwen(2015) and Bank of England (2020).
Summarizing a methodology and all sources of project that requires whole book to describe is beyond scope of what can be done reasonably within format of this site which holds for all sources as well.
In short, they used output based approach - that is they try to estimate GDP by looking at various historic sources that measure different kinds of output and extrapolate from that. They use very large amount of different sources so I will just give you some examples of historic sources not all (which you can view in their book I referenced above):
Sources on Agricultural Production

Medieval Accounts Database assembled by Campbell (2000; 2007)
Overton’s Early Modern Probate Inventories Database (Overton, Whittle,
Dean and Hann, 2004)
Modern Farm Accounts Database constructed by Turner,
Beckett and Afton (2001)

Sources on Industrial Production:

Hatcher (1973). English Tin Production and Trade before 1550, Oxford: Clarendon
Press.

Mitchell (1988). British Historical Statistics, Cambridge: Cambridge University
Press.

King (2005).  “The Production and Consumption of Bar Iron in Early Modern England
and Wales”, Economic History Review, 58, 1-33.

Sources on Services

Deane and Cole (1967). British Economic Growth, 1688-1959: Trends and
Structure, 2nd edition, Cambridge: Cambridge University Press

Carus-Wilson and
Coleman (1963) England’s Export Trade, 1275-1547,
Oxford: Clarendon.

Fisher (1940) “Commercial Trends and Policy in Sixteenth-Century England”,
Economic History Review, 10, 95-117

ibid (1950). “London’s Export Trade in the Early Seventeenth Century”,
Economic History Review, New Series 3, 151-161.

Davis (1954). “English Foreign Trade, 1660-1700”, Economic History Review, 7,
150-166.

ibid (1962). The Rise of the English Shipping Industry in the Seventeenth and
Eighteenth Centuries, London: Macmillan.

All of the above sources are just examples of course (in fact even calling them a tip of an iceberg would be understating how many more sources are used for these estimates), the lists are not exclusive. For full list consult their book. Furthermore, naturally some of the above sources are not primary but secondary themselves so you will have to dig deeper if you want to know all the primary sources.

Finally, is these any metric to determine how accurate they may be?

There is no straightforward way of measuring the degree of accuracy of this kind of historic data. One way would be to compare it to other historic estimates which use different methods (for example income approach for historic reconstructions is also used) and then see how well they match together. However, a problem to begin with is that even the historic sources themselves might not be accurate as standards of record-keeping/bookkeeping were compared by present day standards low.
Furthermore, as the authors of the data themselves note these are rough estimates and frankly just being able to provide that for such a long span of time is incredible intellectual achievement.

PS: I managed to found an un-gated 2011 paper of the aforementioned authors which serves as the basis for the book. You should still consult the book for full list of sources and description of methodology (as they might have changed something) but if you want to get quick summary of what they done the paper will probably serve you well. In addition in that 2011 paper they also make comparisons of their approach with some of the aforementioned income based approaches.


Math based proof for currency changes (microeconomics) This isn't part of my homework but I am genuinely interested in the mathematical proof behind this question (this is my line of work currently). I tried to work through it but after 2 hours gave up. It is beyond the scope of my intermediate economics class but I wanted to see what a proof would look like for it.

Suppose that the Japanese yen rises against the U.S. dollar—that is, it will take more dollars to
buy a given amount of Japanese yen. Explain why this increase simultaneously increases the
real price of Japanese cars for U.S. consumers and lowers the real price of U.S. automobiles for
Japanese consumers.

So a US consumer picks their non-car bundle of goods, $x$, with prices $p$ denominated in dollars. The price of an American car is $a$ dollars, the price of a Japanese car is $y$ yen, and the price of a dollar in yen is the exchange rate, $r$ (how many yen does a dollar buy?). If the consumer buys an American car, $z = 1$, yielding utility $u_a$ at a price of $a$, and if the consumer buys a Japanese car, $z=0$, yielding utility $u_j$ at a price of $j$, $z \in [0,1]$.

Then the consumer is solving
$$
\max_{x\in \mathbb{R}_N,z\in [0,1]} u(x) + u_a z + u_j (1-z)
$$
subject to $p'x + z a + (1-z) r j \le I$ where $I$ is the consumer's wealth.

The Lagrangian is
\begin{eqnarray*}
\mathcal{L} &=& u(x) + u_a z + u_j (1-z) - \lambda (p'x + za + (1-z)rj - I) \\
&=& [u(x)-\lambda (p'x-I)] + z(u_a-\lambda a - u_j + \lambda rj) -\lambda rj + +u_j
\end{eqnarray*}
The part in $[...]$ is the standard consumer problem: pick a basket of goods $x$ with income $I$ and prices $p$. The $z$ must be in $[0,1]$, with 1 meaning "buy American" and 0 meaning "buy Japanese". So if $u_a - \lambda a > u_j - \lambda rj$, the consumer buys American, and otherwise Japanese.

Now, if we increase $r$, what happens to the consumer's maximized utility?  The envelope theorem tells us that if we want to see how welfare varies with a parameter, we partially differentiate with respect to that, then evaluate the Lagrangian at the optimum. Let's do that:
$$
\dfrac{\partial \mathcal{L}}{\partial r} = (1-z) \lambda j
$$
So if the consumer is buying American, $z^*=1$, and this equals zero: the dollar becoming strong doesn't affect his welfare, because he is buying the American car anyway.  But if $z^*=0$, he is buying Japanese, and a stronger dollar makes it easier to pay the $j$ price denominated in yen.

If you think about the Japanese consumer, everything is the same, except $r$ is going down and not up: he is worse off if he is buying an American car, but the same if buying a Japanese car.

Now, how is $r$ determined in equilibrium?  The above problem gives you a demand curve for yen by American consumers: the ones who want to buy Japanese cars, i.e., they have a high $u_j$ relative to $u_a$.  A similar problem gives demand for American cars by Japanese consumers, i.e. those Japanese households for whom $u_a$ is large relative to $u_j$. As the price $r$ varies, people will opt in or out of the market for the cars made by the other country. Equilibrium in the currency market occurs where the flow of dollars from Americans to Japanese and the flow of yen from Japanese to Americans is equal, just like your regular Supply-And-Demand diagram. This means that if a dollar buys fewer yen in equilibrium, it must also mean that yen buy more dollars in equilibrium, and vice versa.


Why are "transfer payments" not part of GDP? It is well-known that transfer payments are not counted in GDP (e.g. wikipedia 1 and wikipedia 2).

For instance, exercise 2.c in Chapter 2 of Jones book's Macroeconomics ask to calculate how much GDP changes if:

  During a recession, the government raises unemployment benefits by $100 million.

The official solution says:

  No impact. This is a government transfer payment, not a government purchase of a good or service.

But, what if that transfer is spend in goods? Would that not increase C? What if just a portion of the transfer is used to buy goods?

In fact, an article from Business Insider states the following:

  While transfer payments are not included in GDP, they are largely put in the hands of those who spend most of the money immediately. Therefore, transfer payments show up in GDP as increased personal consumption.

I am puzzled. Why transfer payments do not increase GDP? I would have thought that it depends on whether that money is spent. Also, is this completely independent on where that money comes from? What if it comes from borrowing abroad, or nationally, or from taxes?

Transfer payments aren't included in GDP to prevent double-counting.

The reason the author's question is troubling you is because the answer externalizes everything that happens after the payment has been made (which is when the money from that payment DOES get factored into GDP, see below). I assume it does this to reinforce that transfer payments aren't counted in GDP, but you are right to question what's going on here.

That \$100 million is transferred from the government to consumers, who will presumably spend a good portion (say, \$98 million) of those funds on goods and services, at which point those funds would be added to GDP. If we had included the transfer funds in GDP, the change reflected would be \$198 million instead of the more accurate \$98 million.

You can also have an example on a much smaller scale:

You give your child \$5 to buy a burger; none of that is added to GDP.
Your child spends \$4.75 on the burger; \$4.75 is added to GDP.


Would insurance plan be necessary if we had instant access to credit? Assuming one could instantly have access to an loan to cover his recent loss/accident at an affordable interest, would it make sense to pay premium for an insurance plan?

I was thinking that it would make more sense to save the premium money and use it to pay part of a loan to cover the losses if it happens. That way one would save the trouble of paying insurance and never using it and make better use of that money.

But I believe in economics. It wouldn't be such a big thing if it didn't make sense. So, how can one evaluate when it makes sense to pay for insurance or just count on a loan?

As Dave Harris' points out in his comment, I assume that your question deals with events that do not compromise the individual's ability to work, which would prevent her from taking on debt.

Let's take the example of a fire which happens with probability $\pi \in [0,1]$. The damage of a fire equals $D$ dollars, i.e. it costs $D$ dollars to repair the house. If the individual buys full insurance at a price $p$ she ends up with a final wealth equal to $w-p$ in all contingencies since the insurance company covers the damages.

Suppose now that she does not buy any insurance but contracts a loan to cover the damages. In order to repair the house she has to borrow $D$ dollars. This loan costs her $D'$ dollars over her lifetime, taking into account the interest and her time preferences (discounting). She then ends up with final wealth equal to

$w$ with probability $1-\pi$
$w-D'$ with probability $\pi$

Thus, the two distributions that she faces are different under these two solutions. If she buys insurance, she completely eliminates the risk of having to pay for the damages. By contrast, if she contracts a loan ex post she still has a large downside risk of $w-D'$ in case of a fire. Under the standard conditions of insurance theory (e.g. risk aversion, actuarially fair insurance prices) it is clear that buying insurance would be strictly preferable.