Does levying a new tax on production increase GDP? Consider a society with no taxes on (e.g) commercial land ownership.
One day the government impose a small tax on land owned by the foresting industries. I would assume that prices on goods produced on these now taxed lands would increase slightly.
Let's ignore dynamic effects from shifting demand (perhaps this tax is levied on goods with very low price elasticity, like toilet paper).
As we know there are three main ways to calculate GDP that all ought to closely match each other:

Production approach
Income approach
Expenditure approach

My thinking is that levying a tax should not raise GDP, as no extra value creation occurs
With the production and income approach its clear to me that the tax will not increase GDP, perhaps falsely. My thinking is that the profit of the producer would not increase, since the price increase on the goods would just offset the cost of the tax.
With the expenditure approach however it would seem like this new tax would in fact increase GDP at a first glance, since the consumption (based on price) would increase. If the government spending also increases with the new tax, we get even more of an increase.
But after thinking more, I came to the conclusion that spending on other inferior goods must decrease with the same amount. For a household that does no saving, spending 5% more on toilet paper would mean that they decreased their spending on other things with the same amount. For households that do save, they either lessen their savings or spend less on other stuff. And savings is just future consumption, meaning that if they save less, they have less consumption in the future.
Is my reasoning sound? Does taxes like this (with these simplifying assumptions) in fact not increase GDP?

tl;dr: Your reasoning that taxes should not increase GDP by themselves as they do not constitute any new production is on right track but might not hold always. The two main reasons for that are:

If we talk about GDP as the actual statistical measure not just as a theoretical measure we run into practical problems when we try to measure it. The most relevant one for this question is the fact that GDP cannot capture all production as statistical offices have no way how to record non-market transactions but at the same time government can still tax even non-market activity and thus increasing GDP in this way by 'revealing' previously hidden production.

Taxing production of some people and transferring the proceeds to other can in short-run increase output and thus also GDP during recession. This is because consumer spending affects GDP not just directly but also has a multiplier effect and during recession transferring money from people with low marginal propensity to consume to people with high one through taxes actually can increase GDP as spending of those people has higher multiplier attached to it.

Full Answer:
In a closed economy, the GDP/output $(Y)$ under expenditure approach will be given by
$$Y= C + I + G$$
where $C$ is consumption, $I$ investment and $G$ gov. spending. You are completely correct that taxes affect consumption. In fact this is made clear in any undergraduate textbook (such as Blanchard et al Macroeconomics: an European perspective). Following, Blanchard et al. we can explicitly make consumption to be a function of output and taxes and in fact we can write:
$$Y = C(Y-T) + I + G$$
where, $T$ stands for taxes. So GDP is function of taxes but at the same time it is function of output itself so in order to analyze the effect of any part of the equation on output we first have to solve the model for output. In order to do so explicitly and directly we have to make some assumption on the consumption function. A standard textbook one would be just simple linear consumption $C= c_0 +c_1(Y-T)$, where $c_0$ is an autonomous spending - spending that does not depend on income/output/GDP (these are macroeconomically all the same so I will be using them interchangeably when referring to $Y$) and $c_1$ is the marginal propensity to consume people have (for example if people consume $75\%$ of their disposable income and save the rest $c_1=0.75$).
By substituting the consumption function for $C$ one might solve for $Y$ to get:
$$Y = \frac{1}{1-c_1}\left( c_0 + I + G - c_1 T\right)$$
Now, the result above shows, that if all else is held constant if you increase taxes the output would decrease by amount $\frac{c_1}{1-c_1} T$.
Of course, assumption that everything else stays constant is not reasonable here. If we make a reasonable assumption that government will use those taxes to fund its spending which we can do by assuming balanced budget  $G=T$ the taxes will have no effect on output as their effect cancels each other out. To see this substitute $G=T$ into the expression, that gives you:
$$Y=  \frac{1}{1-c_1}\left( c_0 + I + T- c_1 T\right) \implies Y=  \frac{1}{1-c_1}\left( c_0 + I \right) +T $$
In this case the increase in taxes would have no extra effect on output/GDP. The output would increase by amount of $T$ but since $T$ is subtracted from the output/income in the first place the GDP would remain at the same level as it was before changes in taxes. (Note, the same would occur if we would not hold $I$ constant and allow for public saving since $I=S+T-G$ - we would arrive at the same conclusion but in more convoluted way).
This being said there are some caveats here. GDP is an imperfect measure so you might observe that the above wont always hold if you would just look at GDP statistics. The analysis above assumes GDP can capture all output. However, that is clearly not correct. If you have home garden and produce radishes for your own consumption,  clearly such consumption wont be counted on national accounts. However, now if government is able to somehow levy a tax on output of your radish garden, in one way or another, GDP statistics might actually increase because the portion of your output that was previously 'concealed' from national accounts statistics would suddenly become recorded. However, this rather reflects the flaws of GDP as a measure and practical data collection issues than any economic effect.
In fact because we cannot perfectly observe all output and its components you will routinely find discrepancies between trying to calculate GDP using income, production or consumption approach even though they are by construction all equivalent to each other.
Moreover, even more important caveat is that the analysis above assumes that all people have the same marginal propensity to consume $c_1$. This most likely does not hold in practice and taxing people with low marginal propensity to consume and transferring it to consumers with high marginal propensity to consume would actually be able to increase GDP. However, this occurs only in recessions as in normal times saving would just increase $I$ but during recession periods that does not generally hold.

There are some parts in your question that are also not correct that I would like to just briefly address:

My thinking is that the profit of the producer would not increase, since the price increase on the goods would just offset the cost of the tax.

This is not correct because it assumes that any tax is fully passed on to consumers - that is not accurate in most situations. However, when you calculate GDP using income approach you actually account for taxes in the calculation.
Next

For households that do save, they either lessen their savings or spend less on other stuff. And savings is just future consumption, meaning that if they save less, they have less consumption in the future.

This is correct but note future consumption is recorded in future GDP calculation not in the present one. However, this would be reflected on present period GDP through changes in $I$ as dis-saving is negative investment since investment must be equal to private + public savings.


Are debt/GDP ratios calculated with real or nominal GDP as the denominator? As the title suggests, I would like to know whether the numbers are generally calculated with real or nominal GDP. Besides that I would also like to know whether it matters and how significant of a difference this could make.

Usually, the debt/gdp ratio has no unit. For instance, if the debt is equal to 60% of the gdp, the debt/gdp ratio is 0.6. If you use real gdp as denominator and nominal debt as numerator, you end up with a number which is less clear to interpret. It is more relevant to have either both variables in nominal terms or both in real terms. In these two cases, the result is the same.

If you denote D and GDP respectively nominal debt and nominal GDP. As long as you take the same price index P, the ratio in nominal variables, D/GDP, and the ratio in real variables, (D/P)/(GDP/P) are mathematically identical.


Data for producer prices for products I am working with some data that express the environmental byproducts for the manufacturing of specific broad product categories (e.g. milk, soap, leather). The statistics are given per dollar of producer price, e.g. 1$ of soap products results into 0.1kg of CO2 emissions.

I was wondering, where can I find the average producer price for the relevant product categories, so that I can roughly calculate for example how much CO2 emissions are  emitted in average for 100g of soap products?

If you are looking at US producer prices then the Bureau of Labor Statistics is where you would go to find the data.  Producer price indexes are here: https://www.bls.gov/ppi/

If you are looking at other countries producer prices then I would recommend  the OECD:  https://data.oecd.org/price/producer-price-indices-ppi.htm

Or this site as well:  https://www.economy.com/indicators/producer-price-index-ppi


Why can an increase in inflation rate, increase interest rates? A Random Walk Down Wall Street (2015 11 ed). pp. 224 Bottom - 225 Top.

    Changes in the rate of inflation will similarly tend to have a systematic influence on the returns from common stocks. This is so for at least two reasons. First, an increase in the rate of inflation tends to increase interest rates and thus tends to lower the prices of some equities, as just discussed. Second, the increase in inflation may squeeze profit margins for certain groups of companies—public utilities, for example, which often find that rate increases lag behind increases in costs. On the other hand, inflation may benefit the prices of common stocks in the natural resource industries. Thus, again there are important systematic relationships between stock returns and economic variables that may not be captured adequately by a simple beta measure of risk.

Please expound the sentence bolded overhead?

In a free currency exchanges market, the central bank defined goal, in the general case, is to maintain price stability and inflation rate restrained, that is to keep inflation positive but at a low rate,
In order to encourage economic activity

Inflation can be manipulated by the amount of money - more money in the system, the lower its value, and higher prices respectively, and vice-versa.

The amount of money is manipulated by the central using interest rates: higher interest rates encourages deposits in banks and discourages taking loans, both actions leading to less money in the system, and vice-versa


Neoclassical model with proportional taxes In a certain economy, time is discrete with periods $t=0,1,2,...$. The economy is populated by many households and identical firms. The utility of a household is:

$\displaystyle\sum^{\infty}_{t=0}\beta^t\Big(\log c_t - \gamma L_t^{\frac{1}{\gamma}}\Big)$

where $c_t$ is consumption and $L_t$ is labor in hours worked. There are two assets: capital, $K_t$, and a one-period risk-free bond $b_t$.

Capital is rented to the firms at a rate $v_t$. The law of motion for capital is $K_{t+1}=(1-\delta)K_t+I_T$. $I_t$ is gross investment and $I_t-\delta K_t$ is net investment.

The bond, is in zero net supply and yields $r_t$ per period.

Firms are owned by households and produce output $Y_t^p=K_t^{\alpha}L_t^{1-\alpha}$ where $0<\alpha<1$. Wages are $w_t$.

The government has a sequence of public spending $\{G_t\}_{t=0}^{\infty}$ that will be paid with a combination of taxed bonds at a rate $\tau_t^b$, taxed net investment at a rate $\tau_t^I$, taxed labor income at a rate $\tau_t^w$, and taxed consumption at a rate $\tau_t^c$. The government cannot borrow or save, $Y_t=Y_t^p$.

I need to set up the firm's problem, the household's problem, and the social planner's problem but have no idea how to incorporate the tax structure.

For example, here is what I have for the household's problem, but I'm pretty sure it's not right:

$\begin{aligned}
\max_{c_t, L_t, K_{t+1}} \quad & \displaystyle\sum_{t=0}^{\infty} \beta^t (c_t, L_t)\\
\textrm{s.t.} \quad & c_t + I_t + \tau_t = Y_t = K_t^{\alpha} L_t^{1-\alpha} + G_t\\
\end{aligned}$

I'd appreciate any hints for setting up these problems.

In your household problem you have a typo, they maximize their utility so I think you meant $\sum_{t=0}^\infty\beta^tu(c_t, L_t)$. Now their income comes from the profits of the firm $\pi$ their wages $w_tL_t$ and what they borrow-lend (not directly the production $Y$ or government expenditures $G$) and their expenditures are consumption, paying their debts (or receiving money from what they lend yesterday) and what they pay in taxes. Obviously they only pay consumption taxes, bond taxes, and labor income taxes, since consumers don't directly invest. We usually model that investment decisions are made by firms, so investment taxes will enter the firm's problem.

In contrast, the social planner chooses everything so all taxes will be relevant for his problem.


Difference between shocks, innovations and disturbances In macroeconomics and macroeconomic models, shocks, innovations, and disturbances are very prominent and often mentioned in the literature.

In general, is there a difference between macroeconomic shocks, disturbances, and innovations, or can these terms be used interchangeably?

In Greene "Econometric Analysis" the disturbance of linear regression,
$$\epsilon_t = y_t - \mathbf{x_t}\boldsymbol{\beta}$$

is estimated with the residual,
$$e_t = y_t - \mathbf{x_t b}$$

In the text, disturbances $\epsilon_t$ are also called innovations. Qualitative events in an economy that impact a process or model's variables or error term are shocks.


How to enhance one's comprehension of the descriptive meaning of formal economic statements? I found that, though most of the time the math technical parts of economics is not a problem for me, I am somehow troubled with extracting the descriptive meaning of formally stated economic statements. I seem to not have the intuition developed enough to read the economic meaning out of formal conditions. This is manifested in macroeconomics. For example, in the class of OLG models, the budget constraints may take many different forms according as scenario (with money, social security,...). Many times, after I saw how simple the economic meaning a sentence conveys either by lecturer in a class or author in a book, I asked myself how I did not see this.
On the other hand, microeconomics is much more clearer to me.

So my question is how to develop a proper intuition to extract the descriptive meaning of any given economic formal relation.

I faced a similar issue coming from mathematics where context plays very little role in framing theorems. I was a top maths student. But Econ grad school was difficult for this. The solution I found (and successfully so since I publish well) is to read as much “soft Econ” as possible: policy discussions, business reports, newspaper analysis. Then I went back to formalized economics, and it worked. The conclusion I reached is that economics theorems are very context dependent, and typically cannot be grasped without cultural/policymaking/business context.


Solving for optimal consumption bundle Consider a consumer who can consume either A or B, with the quantities being denoted by $a$ and $b$ respectively. If the utility function of the consumer is given by $$-[(10-a)^2+(10-b)^2]$$(suppose prices of both goods are equal to $1$), then solve for optimal consumption of the consumer when his income is $40$.

My approach: I have the problem: $$max(-[(10-a)^2+(10-b)^2])$$ $$s.t.\ a+b \le 40,\ a\ge 0,\  b\ge 0.$$ Looking at the objective function, we see that it's maximum value is $0$ when $a=b=10$.

Am I right here?

Yes, you are correct. That solution implies a utility of $0$, while any other solution necessarily will give you negative utility.

It is an odd problem for its violation of local nonsatiation: It is indeed optimal for the household to throw away the rest of his income.

Update

Let's add the either a or b and see what happens:

$$max(-[(10-a)^2+(10-b)^2])$$ $$s.t.\ a+b \le 40,\ a\ge 0,\  b\ge 0,\ ab=0.$$

The optimal solution set now contains $\{(10, 0), (0, 10)\}$. The preferences between this one are still globally satiated at $(10,10)$, but as the point is not feasible, we set one of the coordinates to that value and keep the other one at $0$.


Subsidies Incidence on Producers and Consumers When calculating the incidence on producers and consumers using a demand and supply curve is dead weight loss a part of the incidence? I know that for example the producer's incidence is a rectangle with a height of the old supply curve price at the new equilibrium quantity minus the new equilibrium price. What I'm not sure about is the width of the rectangle. I know it should start at the y-axis, but I'm unsure of where to stop. The original equilibrium quantity? Or the new equilibrium quantity? If I were to extend the rectangle to the new equilibrium quantity then dead weight loss is included. Is this correct?

According to standard textbooks on public finance (e.g., Tresch, Public Finance, 3rd ed., p. 272) the tax incidence includes the deadweight loss. This is because we are interested in the burden of the tax on consumers and producers and the deadweight loss can be a large part of this. (However, see https://en.wikipedia.org/wiki/Tax_incidence which does not include the deadweight loss.)

There are some difficulties, however. First, the deadweight loss is only a meaningful concept in partial equilibrium settings. The incidence as you described it cannot be used in general equilibrium settings as supply and demand curves shift with the introduction of the tax. Thus, the more common approach in general equilibrium is to determine the welfare loss of different individuals.


Is the Federal Reserve issuing money in a fair way? Basic logic says that, once a central bank is issuing new money for increasing the money supply, the money should be distributed fairly between all the citizens, and that means giving them all an amount equal in proportion to their income (say a 5% of their annual income for achieving a 5% inflation). So they should give the money to the government, and then the  government either distributes the money or it builds public infrastructure that everyone benefits from. Most likely, this is the way the Eastern European countries experienced inflation in the 1990's.

Conversely, when the Fed decides to remove a certain amount of money from the economy, the Government should tax everyone in a fair way (the rich should pay more than the poor) and then give the money to the Fed to burn it. Or to burn it directly, without even giving it to the Fed.

Instead, the Federal Reserve has some sophisticated and arcane ways to move the money, beyond the comprehension of the general public. The Fed and the banks are involved into actually owning those newly issued money, and that looks really strange.

This article says:

  When the Fed wants to expand the money supply, it buys a security -- let's call it Asset A -- from a bank. Then it electronically transfers money to that bank.

Therefore, if the Federal Reserve decides, it can issue new money and buy and own half of the assets in the USA for free whenever they want. The newly issued money get to the banks, and depending on the decision of the banks to use the money or not, they can create inflation - or not. They can give those money in astronomical bonuses to their top executives and those people can send the money to Switzerland for example - therefore no inflation.

Is there something important that I'm missing?

Normative questions require a standard of fairness in order to then be analyzed and answered with some degree of "objectiveness" so that the answer is not just a declaration of philosophical-ideological positions.

The OP starts by writing

  "Basic logic says that, once a central bank is issuing new money for increasing the money supply, the money should be distributed equally
  between all the citizens."

Why "basic logic" says that? Whose "basic logic"? I note that throughout human history and especially in the western civilization in the last 400 years, "fairness" is almost exclusively interpreted as "proportionality", not flat equality (i.e. as "per capita tax" is considered not in accord with current societal values, so are "per capita transfers").

Anyway, under the given fairness standard, certainly "new money is not fairly distributed".

Regarding how the creation of new money works, and how it affects the domestic or global economy (or not), which is an additional question, I believe there are many threads here that provide a lot of information on the matter. Selecting the tag "inflation" or "money supply" will bring up most of them.


Who holds the Reverse Repo and Repo respectively in a Repurchase Agreement? According to investopedia.com in this definition

  A reverse repurchase agreement, or "reverse repo", is the purchase of
  securities with the agreement to sell them at a higher price at a
  specific future date. For the party selling the security (and agreeing
  to repurchase it in the future) it is a repurchase agreement (RP) or
  repo; for the party on the other end of the transaction (buying the
  security and agreeing to sell in the future at a specific date and at a higher price) it is a reverse
  repurchase agreement (RRP) or reverse repo.

However while watching this Khan Academy video from 6:29 to 7:30 (although watching from 0:00 to 6:29 may also be helpful for contextual purposes), it seems that the instructor is switching the definition around where it would be:

  For the party selling the security (and agreeing
  to repurchase it in the future) it is a reverse repurchase agreement (RRP) or
  reverse repo; for the party on the other end of the transaction (buying the
  security and agreeing to sell in the future at a specific date and at a higher price) it is a
  repurchase agreement (RP) or repo.

Thank you everyone for your time and efforts. I really appreciate it!

A repurchase agreement (or repo) is the sale of securities for cash, but with a commitment to repurchase them later at a specified price at a future date.

This is practically a collateralized loan, and I find it conceptually helpful to think of it that way.

If you are taking out the loan, it's a repo for you. You are agreeing to repurchase the security (usually U.S. Treasuries) that you posted as collateral at a later date.

If you are giving out the loan, it's a reverse repo.

Note that this terminology is in terms of the dealer's viewpoint. If the dealer borrow money, it's a repo. If the dealer lends money, it's a reverse repo.

Edit #1 -----------------

Let me respond to the first comment. Although I agree with you that it is confusing, I'm not sure there is an inconsistency per se, in what the lecturer is saying.

Other people can jump in, but I believe what's going on in the video is the following.

Suppose I'm taking out a loan from you that I will pay back in a year. I am "promising" you that I will pay back the principal plus interest. As a proof of this promise, you have a debt contract that enters into your balance sheet as an asset (for me it's a liability). You are the holder of this debt contract.

Similarly, say I enter into a repo transaction with you, posting as collateral a watch (just like in the example given in the video). The "promise" here is that I am going to pay back cash. This promise, along with the watch, enters into your balance sheet as an asset. You are the holder of the repo contract.

But unlike a simple debt, there's another side to this "promise": if I give you your money, you have to give me my watch back! The reverse repo contract essentially represents this side of the promise, and hence I am the holder of the reverse repo contract.


Does a perfectly price discriminating natural monopoly cause externalities or any type of market failure? In perfect price discrimination, all mutually beneficial trades take place, the natural monopoly captures 100% of the surplus, and pareto efficiency is achieved.

However, I've read various sources that mention that, despite this, economists still advocate various taxes, regulations, or nationalization to "fix" the monopoly problem even in theoretical cases of perfect discrimination. Why would these things be advocated if pareto efficiency is achieved?

The only thing I can think of is that, while the lack of competition doesn't remove the incentive for a monopoly to innovate, it hugely slows down the rate of innovation, which would essentially rob society of potential improvements.

Is there anything to this logic? What inefficiency would exist if not pareto inefficiency?

The question seems to be in between theoretical model and reality. In one type of theoretical model one may argue for the monopolist, in another against him. In reality it is worthwhile to mention that no monopoly can achieve first degree price discrimination. It is barely theoretically possible and not technically feasible. Other forms of price discrimination do not necessarily yield Pareto-optimal outcomes. Real world economists usually argue against monopolies in general, not against theoretical first degree price discriminating monopolies.

Even if there were such a monopoly there is an answer to "Why would these things be advocated if pareto efficiency is achieved?"

There may exist a multitude of Pareto-efficient allocations. A rather usual consideration for any social planner is some sort of fairness (or envy-freeness, or inequity-aversion, etc.)

In a setting corresponding to a monopoly with first price discrimination as you say all the surplus is captured by the monopolist. This is such a distribution of surplus/goods/utility that it will probably not meet any of the usual fairness criteria.

So while the monopolist might create an efficient allocation of goods, the allocation might not, and usually will not, meet secondary criteria deemed desirable by society.


Bankruptcy Vs default If a company files for bankruptcy, how come its stock still trades ?
E.g. On June 17, Wirecard was valued at more than $14 billion. Eight days later, it filed for the German equivalent of bankruptcy. Its stock is still trading. Why is it not suspended as is the case for Finablr ( fraud probe ). What is the usual practice ?
Can we say that Wirecard defaulted ? i.e if they filed for bankruptcy earlier.

Declaring bankruptcy is a step taken by companies to get protection from creditors. (They cannot seize collateral unilaterally, etc.) The company can still operate, within the legal framework, and with the obvious limitation that nobody wants to be owed money by the firm. Although equity often ends up worthless after all creditors have their claims settled, this is not always the case. As such, equity shares may not be worthless.
The question of the value of the equity is distinct from being listed on an exchange. This question cannot be easily answered since each exchange has its own requirements to be listed.
For example, the NYSE criteria are here: link to standards.
Note the phrasing: “The NYSE reserves the right...” to delist under various criteria. As such, there are no hard rules on what must happen.
For each company of interest, one would need to go to the exchange(s) where the shares are traded to see the rules.


First welfare theorem and convexity Are convex preferences needed for the first welfare theorem? It would see so. For example, we could have a situation in which B's indifference curve is non-convex, such that, at the equilibrium X, it crosses A's indifference curve. Therefore, a movement to the left (if B's consumption is measured from the bottom left) will make A better off, and B no worse-off. Therefore, X is an equilibrium which is not pareto efficient.

Are convex preferences needed for the first welfare theorem?

No, convexity of preferences is imposed for other reasons. A general sufficient condition is local non-satiation, which says the agent can be made better off by an arbitrary small perturbation of his consumption bundle.
This can hold without the preference being convex.

  It would see so. For example...

As already pointed out by @Shomak, the example of an allocation you suggest is not an equilibrium. It also has nothing to do with convexity. Crossing of indifference curves at an allocation can occur for convex or non-convex preferences. Therefore it does not speak to the role of convexity in the theorem one way or another.

Assuming preferences are represented by differentiable utility functions (as per usual), standard marginalist reasoning tells you the allocation you describe is not an equilibrium.

Regardless of whether utility function is quasi-concave (i.e. whether the underlying preference is convex), the first order condition is always a necessary condition for optimality. Therefore agents must have the same marginal rate of substitution between commodities in equilibrium. When indifference curves cross, they don't have the same MRS.


Kimball (1995) Specification of Final Good Production Kimball (1995) defines production of the final good ($Y$) with intermediate goods $y_l$ in his equation (1) as

$$ 1 = \int_0^1 G\left(\frac{y_l}{Y}\right) dl $$

with $G(1) = 1$, $G'(x) > 0$ and $G''(x) < 0$.

I have never seen such a construct before. What does the $1$ on the left-hand-side mean, is that a normalization or some cost construct? An introduction into this type of production function would be greatly appreciated.

Some input:
Kimball assumes a constant-returns-to-scale (CRS) production of the final good $Y$ from the intermediate goods (and no other inputs are involved in this function). Turn to discrete space for a moment, and this means that we would have something like

$$Y = F(y_1,...,y_l,...,y_m)$$

Since this is a CRS function we have

$$1 = F\left(\frac {y_1}{Y},...,\frac {y_l}{Y},...\frac {y_m}{Y}\right)$$

But also, from Euler's theorem for homogeneous functions we have

$$Y = \sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot y_i \implies 1 = \sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot\frac{ y_i}{Y}$$

Combining and manipulating the index into $[0,1]$-continuity we get something like

$$1 = F\left(\frac {y_1}{Y},...,\frac {y_l}{Y},...\frac {y_m}{Y}\right) =\sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot\frac{ y_i}{Y}\rightarrow \int_0^1\left[\frac {\partial F}{\partial y_l}\cdot\frac{ y_l}{Y}\right]{\rm d}l$$

In a sense, $G(y_l/Y)$ is the elasticity of final output with respect the the $l$-th intermediate good. Given the assumptions on $G()$, it rules out a Cobb-Douglas CRS production function, where the elasticities not only sum to unity but they are constant, and it looks, say, to a C.E.S. production function with constant returns to scale, where the elasticities are variable but always sum up to unity.


TFP in R via estprod package I want to calculate the TFP by using the estprod function (I use R 4.0.2). As far as I understood the only way to calculate the TFP is manual following this logic .
omega_1 = (data$Y - data$Labor*coefs$statistic[1] - data$Capital*coefs$statistic[2] - data$Materials*coefs$statistic[3]) - this is basically the approach other author on StackExchange suggested to use in order to calculate the TFP. However, after running the regressions (LP, OP, Wooldridge) I get only 2 coefficients in the output, for capital and fo labor only, while for materials the coefficient is missing.
Would really much appreciate any comments and support, the dataset is available here:  https://drive.google.com/file/d/1aedWYABus1fQjKWxkOmYOmxv-qSja7hF/view?usp=sharing
The code is the following so far:
remove(list=ls())

library(plm)
library(dplyr)
library(ggplot2)
library(prodest)
library(estprod)
library(broom)

# Set the working directory
setwd("C:/Users/vadya/Desktop/baka")

# Downloading the survey data
Data <- read.csv("LV.csv", header=TRUE, sep=",")
str(Data)

Data$ID<-as.numeric(as.factor(Data$ID))

summary(Data)

# Creating a panel data frame
PData <- pdata.frame(Data, index = c("ID","Year"))

pdim(PData)
pvar(PData)

DataA <- Data %>%
  filter(NACE == 'A') %>%
  filter(VA > 0, L > 0, K > 0, M > 0) %>%
  select(ID, Year, L, VA, K, M) %>%
  summarise(ID = ID,
            Year = Year,
            l = log(L),
            va = log(VA),
            k = log(K),
            m = log(M))

####################################################################################################################################
mod1LP = estprod::levinsohn_petrin(data = DataA, va ~ l | k | m, id = "ID", time = "Year", bootstrap = TRUE, gross = FALSE)
summary(mod1LP)

mod1OP = estprod::olley_pakes(data = DataA, va ~ l | k | m, id = "ID", time = "Year", bootstrap = TRUE, gross = FALSE)
summary(mod1OP)

mod1W = estprod::wooldridge(data = DataA, va ~ l | k | m, id = "ID", time = "Year", bootstrap = TRUE, gross = FALSE)
summary(mod1W)

In order to get coefficient for the materials in the Levinsohn-Petrin model you need to set the gross option true.
 mod1LP = estprod::levinsohn_petrin(data = DataA, va ~ l | k | m, id = "ID", time = "Year", bootstrap = TRUE, gross = TRUE)
     summary(mod1LP)

    Call
estprod::levinsohn_petrin(data = DataA, formula = va ~ l | k |
    m, gross = TRUE, id = "ID", time = "Year", bootstrap = TRUE)

Coefficients
   Estimate Std. Error z value  Pr(>|z|)
l 0.1033754  0.0067384  15.341 < 2.2e-16 ***
k 0.5859219  0.0048667 120.394 < 2.2e-16 ***
m 0.4808831  0.0143580  33.492 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Bootstraped standard errors.

The other two functions unfortunately don't collect the coefficients for proxy variables.
An alternative is to find some different proxy and use $m$ as a control (I do not necessarily endorse such option from scientific perspective but if you are hell bent on having coefficient for $m$ its an option). In your dataset you don't have any additional variables beside $l$, $k$ and $m$ (aside from COGS that is always N/A) so I will just create a new variable to show it would work:
 DataA$m2<-(DataA$m)^2

mod1OP = estprod::olley_pakes(data = DataA, va ~  l |  k | m2 |  m, id = "ID", time = "Year", bootstrap = TRUE, gross = TRUE)
 summary(mod1OP)

    Call
estprod::olley_pakes(data = DataA, formula = va ~ l | k | m2 |
    m, id = "ID", time = "Year", bootstrap = TRUE,
    gross = TRUE)

Coefficients
    Estimate Std. Error z value  Pr(>|z|)
l  0.1057613  0.0040007 26.4360 < 2.2e-16 ***
k  0.4643984  0.0308697 15.0438 < 2.2e-16 ***
m -0.0458770  0.0048684 -9.4234 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Bootstraped standard errors.

   mod1W = estprod::wooldridge(data = DataA, va ~ l | k | m2 | m, id = "ID", time = "Year", bootstrap = FALSE, gross = TRUE)
     summary(mod1W)

Call
estprod::wooldridge(data = DataA, formula = va ~ l | k | m2 |
    m, gross = TRUE, id = "ID", time = "Year", bootstrap = FALSE)

Coefficients
   Estimate Std. Error t value  Pr(>|t|)
l 0.1210159  0.0075386  16.053 < 2.2e-16 ***
k 0.4670722  0.0135809  34.392 < 2.2e-16 ***
m 0.2595440  0.0158134  16.413 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Again such course of action does not seem to make sense to me but it gives you the desired output.


Papers on the rate of returns for government-run pension plans I'm looking for papers studying the implied rate of return for various government pension programs around the world. Both historical and expected returns. Or any study that looks at the taxes for the pension scheme, and what participants receive.

I have searched for papers on Google Scholar and found a few (on the state pension systems in USA, Germany, and Sweden), but now the searches give little yield.

Any help would be appreciated. Thanks in advance!

Here are the papers I've already found, in chronological order.

USA

Boskin, Michael J., et al. "Social security: A financial appraisal across and within generations." (1986).
Boskin, Michael J., and Douglas J. Puffert. "The financial impact of social security by cohort under alternative financing assumptions." (1987).
Duggan, James E., Robert Gillingham, and John S. Greenlees. "Returns paid to early social security cohorts." Contemporary Economic Policy 11.4 (1993): 1-13.
Leimer, Dean R. Cohort-specific measures of lifetime net social security transfers. No. 59. US Dept. of Health and Human Services, Social Security Administration, Office of Research and Statistics, 1994.
Beach, William W., and Gareth G. Davis. Social Security's rate of return. Heritage Foundation, 1998.
Leimer, Dean. "Cohort-specific measures of lifetime social security taxes and benefits." (2007).

GERMANY

Schnabel, Reinhold. "Rates of return of the German pay-as-you-go pension system." FinanzArchiv/Public Finance Analysis (1998): 374-399.
Wilke, Christina Benita. "Rates of Return of the German PAYG System-How they can be measured and how they will develop." (2005).
Lüthen, Holger. "Rates of return and early retirement disincentives: evidence from a German pension reform." German Economic Review 17.2 (2016): 206-233.

OTHER

Auerbach, Alan J., and Ronald Lee. "Welfare and generational equity in sustainable unfunded pension systems." Journal of public economics 95.1-2 (2011): 16-27.


Have Raj Chetty's Opportunity Atlas stats changed or been recalculated since early 2019? During a talk at Google, Raj Chetty said "Fruitvale, in Oakland ... here you see that for black men growing up in low income families in Fruitvale, the average income in adulthood is 5,600 dollars" (see https://youtu.be/X2YzxpWKKz4?t=844).

However, at https://www.opportunityatlas.org/, I appear to see 15,000 dollars for the same statistic (note the selections for parent income, child race and child gender at the top right of the screenshot):

Have the numbers changed?  Has the methodology been adjusted?  What explains the difference?

I noticed a similar discrepancy related to incarceration rates:

During the Google presentation, Raj Chetty says "43% of the black men who grew up in this particular neighborhood [Fruitvale] are incarcerated ... on the date of the 2010 census", but Opportunity Atlas (viewed in December 2019) gives 17% for the same incarceration rate statistic.

I dont think it had been recalculated becouse his plans for the American drean is still on sight,to me he has dispel the myth for of social mobility than anyone else
and that of fruitvale in Oakland the income is highly disappointing for the black people which as such other might think is kind of a racist thing forgive me but is something to look at


A reference for most used utility functions in macroeconomic problems of intertemporal optimization I'm looking for a reference with the most used utility functions in macroeconomic problems of intertemporal optimization. The reference should preferably include a list of properties of those functions, and of why they are usually used.

Any help would be appreciated

I think that this article might be helpful:

“Exotic Preferences for Macroeconomists”

http://www.nber.org/chapters/c6672.pdf

They give a thorough explanation of many preference functionals and show their usefulness with macroeconomic applications.

Im not a specialist in macroeconomics, so I dont know the literature that well, but I think that the most used preferences nowadays are the Epstein-Zin preferences. The article I cite discusses thoroughly their properties.

A nice thing this paper has is that it contains detailed comments  from Peter Lars Hansen and Ivan Werning among others at the back.

Hope it helps!


Comparing rates of return for different assets using $V=Pe^{rT}$ I have been playing with $V=Pe^{rT}$ and have been thinking about how to apply it to situations involving different assets growing at different rates.
Suppose I had data for the historic sale prices of a row of houses in a street, with two data points per house. My assumption here is going to be that each house's price grows according to $V=Pe^{rT}$, with $r$ constant.

House no.
Year A
Sold for
Year B
Sold for

1
2000
\$100,000
2015
\$500,000

2
2005
\$60,000
2019
\$111,000

3
1999
\$30,000
2018
$80,000

4
2002
\$125,000
2005
\$126,000

Using $V=Pe^{rT}$, I could calculate each house's instantaneous rate of interest $r=\frac{\ln{V}-\ln{P}}{T}$, and its APR $\hat{r}=e^{r}-1$, where $T=B-A$ is the number of years elapsed.

House no.
Year A
Sold for
Year B
Sold for
$r$
$\hat{r}$

1
2000
\$100,000
2015
\$500,000
10.73%
11.33%

2
2005
\$60,000
2019
\$111,000
4.39%
4.49%

3
1999
\$30,000
2018
$80,000
5.16%
5.30%

4
2002
\$125,000
2005
\$126,000
0.27%
0.27%

My question is, how should I meaningfully combine the growth rates of the different houses? Say I want a figure for the street's overall growth rate in order to evaluate whether a particular house has been growing more quickly or more slowly than this rate. I have various ideas:

Take the average of the $r_{i}$, which is 5.138%
Use $-1+e^{5.138\%}=5.272\%$
Take the average of the $\hat{r}_{i}$, which is 5.346%
Work out the houses' prices in two base years (say, 2000 and 2021), using $P=Ve^{-rT}$, and work out what my $r$ or $\hat{r}$ would have been if I'd bought the entire street in 2000 and kept the houses until 2021.

House no.
$r$
2000
2021

1
10.73%
\$100,000
\$951,827

2
4.39%
\$48,165
\$121,197

3
5.16%
\$31,589
\$93,400

4
0.27%
\$124,338
\$131,470

Total
-
\$304,092
\$1,297,894

Using the totals above as my new $P$ and $V$, I get that $r=6.91\%$ and $\hat{r}=7.15\%$.
Is this a legitimate method? If so, what is it called? I am concerned that it might not be legitimate, because changing the two base years gives different results:

House no.
$r$
2005
2008

1
10.73%
\$170,998
\$235,930

2
4.39%
\$60,000
\$68,455

3
5.16%
\$40,892
\$47,741

4
0.27%
\$126,000
\$127,008

Total
-
\$397,890
\$479,134

Here, $r=6.19\%$ and $\hat{r}=6.39\%$, which seems strange, since each individual house's growth rate was constant. Does this mean that the "What if I'd bought the whole street in year $X$ and waited until year $Y$" method is wrong, or does it still have an interpretation?
What is the best way of defining the "overall" rate of growth for the whole street of houses?
Thanks

First of all stop using discrete interest in your intermediate steps to keep your work simple. (As a final step you can convert continuous to discrete if you want to.)
The reason you get different results when you change the start and end dates is because the longer the time period the more the result (growth rate of the totals) will approach the biggest growth rate. By implication the dates do not matter; only the length of the period, so 1901 to 1911 gives you the same result as 2001 to 2011. I didn't test this but it seems like it would be true.
You said you want the overall growth rate.
You can take the average (the mean).
You can take the trimmed mean, which is the average after discarding the highest and lowest rate because may be they are anomalies. Since you don't have a lot of data, you might not want to do the trimmed mean.
You can take the price weighted mean of all data or the trimmed data. I would use initial prices, to be specific. Since you might want to buy the whole street (big spender) the price weighted mean might be more relevant to you. If you're only interested in buying one randomly selected house, the mean or trimmed mean is better.
You can take the median if you think higher and lower rates are less useful for your purpose but you would probably want more than 4 houses in your set if you take a median. If you're only interested in buying one house and you have a median-sized budget then the median rate is more interesting for you.


How can I draw an edgeworth box with 3 different goods? If there are 3 types of goods(x,y,z) and 2 consumers (a,b) how can I draw the Edgeworth box? It must be 3 dimensional, right?

Is there a software or online tool that I can use to draw a 3D Edgeworth box?

I assume what you're asking based on your comments is: "How can I visualize indifference curves for 3 goods?" I can think of three options:

1) Use a tool like Matlab, or its open-source equivalent, Octave, to plot 3 dimensional indifference curves. Here is a tutorial on how to do that.

2) Make a series of 2-dimensional indifference curves for two of the goods, and fix a different level of the third good for each one, that is, plot a series of cross-sections of the 3 dimensional box.

3) Using paper and pencil, draw a general $n$ dimensional Edgeworth box, then, set $n = 3$.


LM curve: How interest rate is changing without OMOs by the central bank? The following is the gist from chapters - 3, 4 & 5 dealing with IS-LM model from Macroeconomics, Blanchard & Johnson, 6th edition.

Equilibrium in the financial market implies an increase in income leads to an increase in the interest rate, therefore the LM curve is upward sloping. The LM curve is derived at a given level of (real) money supply. It is further assumed that price level is fixed in the short run. further, it is assumed that  central bank changes money supply through open market operations (OMOs). When the central bank buys bonds, bond prices goes up & interest rate goes down. Conversely, when the central bank sells bonds, bond prices goes down and interest rate goes up. Hence interest rate is directly dependent on the OMOs.

What I could not understand is why the interest rate is changing because of change in income, since there is no change in money supply and hence no OMO is going on?

Also, why would someone will buy the bond from central bank when the bond prices are going down?

Edit: (related to fixed money supply, first question) Since there is no interest rate on check-able deposits and bank money apart from reserves are used to purchase bonds, not for lending, people to people borrowing is an option. However, it does not seem to be a plausible due to the following reasons. The Money demand shift to right for an increase in income and intersects the Money supply curve at a higher point leading to increase in interest rate. Increase in income will suppress the people to people borrowing, leading to opposite of the required result, lower interest.

(related to variable money supply, second question) consumers who have already invested may hold on their bonds but may not purchase new bonds with the extra income. So the interest rate should not change. Moreover, if consumers buy and sell bonds among themselves without Central Bank Intervention and make major changes in the interest rate, they may not cooperate. e.g. CB decides to decrease money supply but consumers may not give up thier bonds as their income may have increase recently.

Hence interest rate is directly dependent on the OMOs.

While open market operations have a large influence on interest rates, there are other ways in which central banks influence interest rates. For example, they may increase or decrease the interest on excess reserves. Moreover, the IS-LM model does not necessarily assume the existence of a central bank. Interest rates have been set by the market far before central banks existed to influence them.

So how do interest rates rise with a fixed real money supply? Increased transactions demand leads to a increase in demand for money. If real money supply stays constant, then interest rates should rise. The following is an example.

Imagine an economy with only one good, tomatoes, and two people, Buyer and Producer. Producer produces all tomatoes. Buyer owns all 10 credits and is disposed to consume everything possible. Imagine the Producer has produced 10 tomatoes, and is willing to sell at 1 credit per tomato. Thus real GDP is 10 credits. If the Buyer purchases all 10 tomatoes from Producer, Buyer will own 10 tomatoes and Producer will own 10 credits.

The next year Buyer has magically regained all 10 credits. Maybe Buyer stole them from Producer; it doesn't matter. Producer produces 20 tomatoes. Hence, real GDP is now 20 credits. Intuitively tomato prices should go down as there is more supply and a fixed demand (from Buyer) but that would imply that real money supply has gone down. So to better illustrate the LM curve, let us assume that Producer keeps prices at 1 credit per tomato. Buyer now purchases 10 tomatoes, and has no credits left. But Buyer is determined to consume everything possible, so Buyer strikes a deal with Producer to borrow the 10 credits that now Producer owns at 10% interest. Buyer then purchases 10 more tomatoes.

In short, real GDP doubled, real money supply stayed the same, and interest rates went up from 0% (no one borrowing) to 10%.

  Also, why would someone will buy the bond from central bank when the bond prices are going down?

Central banks will sometimes sell bonds that they have purchased previously to decrease the money supply. The announcement that a central bank will attempt to do so may attract front-running speculators. In other words, basic market dynamics will apply: if no one wants to purchase bonds, then prices will decline until someone believes prices have declined enough that they will eventually start rising again even under pressure from the central bank.


Understanding the HP Filter The HP Filter has two objectives, with the importance of each objective denoted by the user given value of lambda:

Objective 1: minimize the $\tau_t$ in the term in the square brackets such that we minimize the changes in the estimated growth rate over time.

Objective 2: We want to bring the $\tau_t$ to be as close as possible to $y_t$ to minimize the first sum in the equation.

What I am failing to understand is why these acts of minimization will help us discover what the cyclical component of GDP is? Is there a fundamental gap in my knowledge?

The function $\tau$ is the 'line of best fit', the growth trend. The 'residual' $y-\tau$ is the output gap. So, $\tau$ detrends $y$ over time, to give the detrended business cycle $y-\tau$. $\quad y-\tau$ fluctuates around 0—it is positive when the output gap is positive and the economy is in a boom, and negative when the economy is in recession.

The first term makes the line of best fit follow the points in a more curvy way than just a straight line; the second term makes the line of best fit straighter (the higher $\lambda$ is).


How to determine correct fixed effects in linear regression model using unbalanced paneldata I am currently working on a research project regarding the introduction of mandatory earnings guidance regulation in multiple countries within a time span of 10 years (staggered adoption). As I can see that my results depend on the FE I use, I tried to figure out how to determine which FE I have to use in this setting. Literature doesn't give a clear answer as different authors use different FE in similar models (firm, year, country, industry or a combination of those four). That's why I wanted to ask the community: Is there any rule-of-thumb or a precise answer on which FE I do have to include in my multi-year multi-country model (using linear model with an unbalanced dataset) in the form of:

  Market liquidity measure = b0 + Analyst_following * x1 + US_Listing * x2 + IFRS_User * x3 + MarketCapitalization * x4 + ShareVolatility * x6 + e

I have detailed firm level data for all variables. I would suggest industry-year FE, but don't have any arguments why I should use them (just intuition). Is there any clear guidance on which FE to use or any recommendation for a book adressing this topic in an inuitive way? Most books deal with social research questions (using e.g. race as FE as it does not change) but it is hard for me to transfer this logic to a business regulation setting...
What I want to measure is, whether the staggered adoption of the regulation has a significant impact on market/stock liquidity. Dependant/independant variables are derived from well-established papers in my field.

Any help appreciated!

It seems that you have firm-level panel data over multiple industries in multiple countries.

Year fixed effects control for trends. Do not omit them. For others, a general rule is: ___ FE's control for time-invariant heterogeneity across different ____, but do not control for heterogeneity within the same ____. You replace ____ with "firm", "industry", "country", "industry-country", etc.

Firm FE's successfully control for all firm-level time-invariant effects that are possibly correlated with the RHS variables. So you compare different years within the same firm. In contrast, industry FE's control for time-invariant effects (that are possibly correlated with the RHS variables) across different industries, but do not control for firm-level heterogeneity (fixed effects) within the same industry. And so on.

I personally think a model with firm FE (and year effects) is easiest to defend using. If you can explain why firms within the same industry are homogenous, you can use industry FE, etc.


How to set up the payoffs properly for a division of labor game I'm admittedly a novice when it comes to game theory (currently a few lectures into Yale's intro course lectures), so hopefully people will indulge me what may be a dumb question.
I was trying to understand how Nash Equilibrium affects division of labor (which was one of the examples used in the class). In particular, I was trying to represent a simple marital dispute: who cleans the house. I'm assuming that both partners would prefer equally that the house be clean, and that they have similar cleanliness standards. Furthermore, I'm assuming that if they both clean, they contribute to the work equally (or, at least, in a way that both of them are equally satisfied with). John Gottman's books and personal experience both assure me that these are rather naive assumption, but bear with me.
The two Nash Equilibria are a) where they share the work equally or b) where neither of them clean (the "bad" equilibrium). This is because neither of them wants to be the one that gets stuck doing all of the work. My reasoning behind that is that neither one would like to be the one who has to do all of the cleaning; however, they would both prefer it if the other person did all of the cleaning.
I tried to formulate that as follows:

Husband Cleans
Husband Doesn't Clean

Wife Cleans
1,1
2,0

Wife Doesn't Clean
0,2
0, 0

On the plus side, them sharing the cleaning is effectively a compromise - they both share the cost equally and they both get the benefit equally. On the other hand, the following facts bother me:

Does the way that I've structured the payoffs imply that there's no payoff at all for the spouse that does all the work? Yes, they have the added "cost" of doing all the work themselves, but this would say that the cost of the added work would entirely offset the benefit of having a clean house - it's just as "bad" for them as if the work hadn't gotten done at all (which seems strange to me, because some spouses might prefer to do all of the work themselves than to have it not done at all).
Similarly, there's no "penalty" for having the house not cleaned - they just don't get a payoff. I'm a little bit confused if that's ok, though; if neither people get what they want, should the payoff be negative because they're unhappy about that fact, or should it just be 0 because they didn't get what they wanted (and therefore got no payoff)?
To my previous point: if I adjust the payoffs, I would like to make sure that I do so in such a way that both of them cleaning or neither of them clean are still equilibrium solutions.

Are my concerns well-founded here, or did I do the table correctly after all?

If you want "both don't clean" to be a Nash equilibrium (NE), then your payoff matrix doesn't work. With those payoffs, cleaning is a strictly dominant strategy and "both clean" is the only NE.
A more systematic approach would be to assume that a clean house has utility $b>0$ for both partners, but cleaning alone generates a disutility of $c>0$. Cleaning is likely to have increasing marginal disutility, such that sharing the cleaning work costs only, say, $c/4$. Then the payoff matrix is

Husband Cleans
Husband Doesn't Clean

Wife Cleans
$b-c/4$,$b-c/4$
$b-c$,$b$

Wife Doesn't Clean
$b$,$b-c$
0, 0

If $b>c$, then "only wife cleans" and "only husband cleans" are two strict NE (and there's also a symmetric mixed one).
If $b<c$, then "wife doesn't clean" and "husband doesn't clean" are strictly dominant strategies and "no one cleans" is the unique NE. If $b<c$ but $4b>c$, then "both clean" is Pareto better than "no one cleans" (and uniquely maximizes total payoffs), which implies that your game is a Prisoner's Dilemma.
With this approach, of course, "both clean" is never a NE, since saving the costs of cleaning while enjoying the benefits of a clean house is always preferred to participating in the cleaning. But this is what you seem to assume ("they would both prefer it if the other person did all of the cleaning"). This assumption contradicts your wish for having a NE where both clean.
However, you could go on to argue that the game they actually play is the infinitely repeated version of this 2x2 game. This opens up the possibility that "both always clean" is a NE-outcome of the repeated game, e.g. based on Trigger strategies, even for the Prisoner's Dilemma case (with small enough discount rate).


Unbiased but inconsistent estimator Assume  a random sample X1, ..., Xn with a normal distribution with mean μ and variance σ2.  How do we know the following estimator is unbiased, but inconsistent?

When an estimator is consistent, the sampling distribution of the estimator converges to the true parameter value being estimated as the sample size increases.


Why do wages not equalize across space? In standard economic theory wages are simply prices on the labor market determined in equilibrium by the supply and demand of labor. Looking across space within countries it is however standard to find a relative large variation in local wage levels. At the same time labor is often very mobile again within countries.
For the United States of America the U.S. Census reveals that in 1990 four percent of white males age 25 to 34 moved across state lines, with the cumulative effect that almost a third of this group no longer resided in the state in which they were born.
When individuals move the supply of labor changes and one could expect wages to equalize based on the assumption that the migration itself was motivated by the one state offering higher wages than another. Seen in the light of this it can seem a little puzzling that spatial wage differentials nonetheless continue to exist. What are the main explanations offered by economic theory?

There are several explanations for this in the literature (the order does not necessarily reflect importance of each explanation).

Different Price Levels: Non-trivial portion of the wage differential is due to different price levels/cost of living. Once you adjust for different price levels and compare real wages gap narrows. This is obvious but since you do not mention in your question whether you are talking about real or nominal wages I thought it is useful to mention that.
Regional Institutional Differences: For example Blackaby and Murphy (1991) argue that inter regional real wage differentials are at least partially due to labour market institution heterogeneity (unionisation and the bargaining system). This applies especially to unions/federations like US where there can be large heterogeneity in regional laws & regulations.
Regional Heterogeneity/compensating differentials: Even within one country the geography is usually heterogenous which creates opportunity for compensating differentials  (Farber and Newman, 1989). If you are offered two jobs that are same in all aspects but one job requires you to relocate somewhere to Alaskan wilderness and the other job requires you to relocate to some pleasant state with nice beaches you will likely be willing to accept lower salary from the job in nicer place because you are compensated by the fact that you are living in a nice place. Some regional differences could be also explained by agglomeration economics (e.g. urban vs rural areas have different economic implications), see for example Combes & Gobillon (2015).
Worker Heterogeneity: The wages would only equalize among workers that are homogenous across all work relevant characteristics. However, people are different from each other and often people who are similar tend to cluster together. For example, would it be surprising to find that average wage in silicon valley is higher than average wage in some farming community? This would be just comparison of apples to oranges. Hence looking at average wages can be deceptive and research that takes into account worker heterogeneity find that the wage differentials shrink substantially (see Gerking and Weirick, 1983; Dickie and Gerking, 1987;  Farber and Newman, 1989 and Combes, Duranton and Gobillon, 2008).
Regional Differences in Human Capital: Workers are not just compensated for their labor but also get return for their human capital. Human capital endowments might differ across regions (perhaps due to regional differences in quality of education) leading to wage differentials (see Duranton and Monastiriotis, 2002; Motellón, López‐Bazo & El‐Attar, 2011).

I think the above summarizes all the major factors but there might be some further ones as well for that you can see the sources above and ones cited therein.


How can capitalism be improved? Many people want to try socialism because of valid criticisms of capitalism. So, are there any good ideas on improving the problems of capitalism before we go down the path of socialism?

19th Century American Economist Henry George argued that the real conflict wasn't between Labor and Capital, but rather between the owners of Land and everybody else.  Neoclassical reformulation of the factors of production essentially merged Capital with Land, erasing the distinction and setting the stage for the current Socialism/Communism vs. Capitalism conflict we've had ever since.

Georgists today support a return to the earlier model, which points to a middle way between the two extremes.  They argue that by taxing economic rent (the logical extension of payments made for the use of Land) governments can make infrastructure projects self-financing, eliminate the root causes of financial speculation (and the boom/bust cycles they trigger) and reduce inequality — all while maintaining (or increasing) the freedom of markets.


How can I obtain Leontief and Cobb-Douglas production function from CES function? In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function,
$$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{1}{\rho}}$$

(where the elasticity of substitution is $\sigma = \frac 1{1+\rho},\rho > -1$), has as its limits both the Leontief production function and the Cobb-Douglas one. Specifically,

$$\lim_{\rho\to \infty}Q= \gamma \min \left \{K , L\right\}$$

and

$$\lim_{\rho\to 0}Q= \gamma K^aL^{1-a}$$

But they never provide the mathematical proof for these results.

Can somebody please provide these proofs?

Moreover, the above CES function incorporates constant-returns-to-scale (homogeneity of degree one), due to the outside exponent being $-1/\rho$. If it was, say $-k/\rho$, then the degree of homogeneity would be $k$.

How are the limiting results affected if $k\neq 1$?

The proofs I will present are based on techniques relevant to the fact that the CES production function has the form of a generalized weighted mean.
This was used in the original paper where the CES function was introduced, Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The Review of Economics and Statistics, 225-250.
The authors there referred their readers to the book Hardy, G. H., Littlewood, J. E., & Pólya, G. (1952). Inequalities , chapter $2 $.

We consider the general case
$$Q_k=\gamma[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{k}{\rho}},\;\; k>0$$

$$\Rightarrow \gamma^{-1}Q_k = \frac 1{[a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{k}{\rho}}}$$

1) Limit when $\rho \rightarrow \infty$
Since we are interested in the limit when $\rho\rightarrow \infty$ we can ignore the interval for which $\rho \leq0$, and treat $\rho$ as strictly positive.

Without loss of generality, assume $K\geq L \Rightarrow (1/K^{\rho})\leq (1/L^{\rho})$. We also have $K, L >0$. Then we verify that the following inequality holds:

$$(1-a)^{k/\rho}(1/L^{k})\leq  \gamma Q_k^{-1} \leq (1/L^{k}) $$

$$\implies (1-a)^{k/\rho}(1/L^{k})\leq  [a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{k}{\rho}} \leq (1/L^{k}) \tag{1}$$

by raising throughout to the $\rho/k$ power to get

$$(1-a)(1/L^{\rho}) \leq a (1/K^{\rho}) +(1-a) (1/L^{\rho})  \leq (1/L^{\rho}) \tag {2}$$
which indeed holds, obviously, given the assumptions. Then go back to the first element of $(1)$ and

$$\lim_{\rho\rightarrow \infty} (1-a)^{k/\rho}(1/L^{k}) =(1/L^{k})$$

which sandwiches the middle term in $(1)$ to $(1/L^{k})$ , so

$$\lim_{\rho\rightarrow \infty}Q_k = \frac {\gamma }{1/L^k} = \gamma L^k = {\gamma }\big[\min\{K,L\}\big]^{k} \tag{3}$$

So for $k=1$ we obtain the basic Leontief production function.

2) Limit when $\rho \rightarrow 0$
Write the function using exponential as

$$\gamma^{-1}Q_k=\exp\left\{-\frac k{\rho}\cdot \ln\big[a (K^{\rho})^{-1} +(1-a) (L^{\rho})^{-1}\big]\right\} \tag {4}$$

Consider the first-order Maclaurin expansion (Taylor expansion centered at zero) of the term inside the logarithm, with respect to $\rho$:

$$a (K^{\rho})^{-1} +(1-a) (L^{\rho})^{-1} \\= a (K^{0})^{-1} +(1-a) (L^{0})^{-1} -a (K^{0})^{-2}K^{0}\rho\ln K- (1-a) (L^{0})^{-2}L^{0}\rho\ln L + O(\rho^2) \\$$

$$=1 - \rho a\ln K - \rho(1-a)\ln L+ O(\rho^2) = 1 +\rho \big[\ln K^{-a}L^{-(1-a)}\big]+ O(\rho^2)$$

Insert this back into $(4)$ and get rid of the outer exponential,

$$\gamma^{-1}Q_k = \left(1 +\rho \big[\ln K^{-a}L^{-(1-a)}\big]+ O(\rho^{2})\right)^{-k/\rho}$$

In case it is opaque, define $r\equiv 1/\rho$ and re-write

$$\gamma^{-1}Q_k = \left(1 +\frac{\big[\ln K^{-a}L^{-(1-a)}\big]}{r}+ O(r^{-2})\right)^{-kr}$$

Now it does look like an expression whose limit at infinity will give us something exponential:

$$\lim_{\rho\rightarrow 0}\gamma^{-1}Q_k = \lim_{r\rightarrow \infty}\gamma^{-1}Q_k = \left(\exp\left\{ \ln K^{-a}L^{-(1-a)}\right\} \right)^{-k}$$

$$\Rightarrow \lim_{\rho\rightarrow 0}Q_k =\gamma\left(K^{a}L^{1-a}\right)^k$$

The degree of homogeneity $k$ of the function is preserved, and if $k=1$ we obtain the Cobb-Douglas function.

It was this last result that made Arrow and Co to call $a$ the "distribution" parameter of the CES function.


Effect on revenue of changing price of unitary demand elastic product In my introductory microeconomics class, a problem set had the following question:

  If the price elasticity of demand for Snickers bars is 1, what will a 1% increase in its price do to the total amount spent on it (not quantity purchased, but total amount spent) each week?

The answer in the problem set solution released later was as follows:

  It will leave it unchanged because the percentage change in quantity demanded will exactly offset the percentage change in price.

My problem with this is that isn't total revenue (and therefore total expenditure) supposed to be maximum when the demand is unitary elastic? If so, shouldn't changing the price when demand is already unitary, decrease revenue/expenditure?

I've also tried numerical examples. If the price of a bar is \$2 and 100 units are being purchased, the expenditure is \$200. If the price increases to \$2.02, the quantity should fall to 99. Now the expenditure would be \$2.02 * 99 = \$199.98, which is lower than \$200.

This is an excellent question (and congratulations for spotting an apparent inconsistency in your problem set!) To answer it, we need to distinguish between two slightly different concepts: the arc elasticity and the point elasticity.

The arc elasticity of demand can defined as the percentage change in quantity demanded divided by the percentage change in price. (Other definitions are available.) For instance (to borrow your example), if initially $P = 2$ and $Q = 100$, and later $P = 2.02$ and $Q = 99$, then the arc elasticity of demand is:

$$\frac{\%\Delta Q}{\%\Delta P} = \frac{\frac{100-99}{100}}{\frac{2-2.02}{2}} = -1$$

Notice that to calculate the arc elasticity of demand we need two price/quantity pairs.

In contrast, the point elasticity is defined slightly differently: it is the limit of the arc price elasticity as the size of the change in price goes to zero. In other words, it is essentially the same as the arc elasticity, but for a infinitesimally small price change. As a result, the 'old' price/quantity and 'new' price/quantity are essentially the same, so to compute the point elasticity we need only one price/quantity pair.

A bit more formally, consider a small price change $dP$ that leads to a small change in demand $dQ$. The point elasticity is given by

$$\frac{\%\Delta Q}{\%\Delta P} = \frac{\frac{dQ}{Q}}{\frac{dP}{P}} = \frac{dQ}{dP}\frac{P}{Q} $$

These preliminaries out of the way, we are now in position to answer your question.

Under standard assumptions (downward sloping and differentiable demand), revenue is maximised when the point elasticity equals 1. (You may wish to verify this!)
However, at the revenue maximising point, the arc elasticity need not be one. Conversely, if the arc elasticity is one, revenue need not be maximised (as your example indicates).
Insofar as the point elasticity is a good approximation for the arc elasticity (i.e. when considering small prices changes), the arc elasticity should be close to 1 at the revenue maximising price/quantity.


What is the usefulness of approximating an optimal decision rule that close enough to steadystate in RBC model? I'm reading understanding real business cycle by Plosser.

Here's my crude understanding: For a RBC model, the FOCs of lagrangian together with transversality condition usually forms a nonlinear difference equations system. The standard way of solving this system is by linearization in the neighbourhood of steadystate,  solve this problem as a linear difference equations system.

What insight can we get from this solution, except for gaining  some knowledge of whether the steadystate is locally stable?

Side note: This is one way of solving it - the alternative would be formulating a Bellman equation and iterating on that.

If you assume that the real economy is on or sufficiently close to the steady state, you can also infer about responses to shocks. That is, you can look at the impulse response functions to a change in whatever interests you, and see how the model economy changes. Arguing that we're close enough to that steady state will allow you to see how the economy responds to certain shocks.

Also, in general, you can simulate the economy with (for example TFP) shocks, and look whether the simulated economy looks similar to the real economy. Using this comparison you could judge the model.

This needs arguing that we are close to the steady state - or that convergence happens really fast. Generally, the growth literature around Solow has provided arguments for this.

But your argument is very present in most extensions of the basic RBC model: especially when it is important how close we are to the steady state - when models are more nonlinear. There have been many papers showing that this is the case for standard Neo Keynesian extensions of the RBC model.


Present value of a payment Suppose I've just won 1'000'000 dollars in a game show. At the end of the program they tell me that they will pay me the prize as following: they will deposit in my bank 50'000 dollars every year for twenty years with a constant annual interest rate of 6% (this to prevent me from wasting too quickly the one million dollars I've just won). The present value of my prize is given by:$$V_t=z\frac{1-\left [ \frac{1}{(1+i)^n} \right ]}{1-\left [ \frac{1}{(1+i)} \right ]}$$

with $z=50'000$, $i=0.06$ and $n=20$ I get  that: $$V_t\approx 50'000 \left (\frac{0.688}{0.566}  \right )\approx 608'000$$

My book says that is a really great prize, but I'm not a millionaire at this point. I would have been a millionaire if they'd paid me the one million dollars right at the end of the program. But I can't understand why it looks like I'm less rich. Suppose I won't spend money they'll give me, in twenty years I'll have: $$1'000'000\cdot(1+0.06)=1'060'000$$
Which is obviously greater than 1'000'000. Could you please explain me what's the reasoning that there is behind the present value of 608'000 dollars? Thanks for any help you can provide.


rental cost of capital I am estimating time series for rental price of capital using Hall-Jorgenson method: $$r=\frac{P_k}{P}(i-inf+\delta)$$
here, $r$ is the rental cost of capital, $P_k$ is the price of capital goods (price index for capital goods), $P$ is a deflator, $i$ is nominal interest rate, $inf$ is an inflation rate and $\delta$ is depreciation rate of physical capital stock.

Because inflation rate is high in some periods, the rental cost of capital is negative for these periods; this does not make sense, any little help is welcome!

In theory there is nothing that is preventing the real interest rate from becoming negative. Making the real interest rate negative by keeping the nominal rate at zero and creating inflation has been the goal of the ECB in the recent past. Once you hit the ZLB there is no other way to further loosen monetary policy. With regard to your work it is probably important to carefully select the inflation rate you're using, but you dont describe this in detail. Depending on your question you might want to consider forward looking measures (i.e.Blue Chip or Greenbook forecasts if you're talking about the US).


Is there a reference that plots different countries' debt-to-GDP ratios over (long) times? This old blog post has an interesting plot that demonstrates that the U.K.'s current debt-to-GDP ratio is significantly below its very-long-term (180-year) average.

Is there a plot anywhere that compares the historical debt-to-GDP ratio over long times between countries? Ideally, I would like a single plot that (a) has multiple time series for many different countries (using comparable methodology), to make it easy to compare across countries, and (b) goes back before World War II, in order to show extremely long-term trends. However, a bunch of different separately plotted long-term time series would also be interesting. But a quick search doesn't turn up much that goes back before around 1970.

More broadly, was it common for debt-to-GDP ratios to exceed 1 year before and during the Bretton Woods system? Or was the U.K. unusual in this regard?

Or is it just the case that we don't have any reliable numbers for debt-to-GDP ratios from before the rise of modern econometrics?

Data for the United States 1789 - 2019.
The IMF has the Historical Public Debt Database. It has data for almost every country with varying starting dates depending on country.

From my brief analysis France, Spain, Belgium, and Italy all appear to have exceeded 100% debt to GDP ratios before 1944.


Integration by parts with CDF I am told that the following equality follows from integration by parts:
$$\int_{R-k}^{1}(\theta-R)dG(\theta)-G(R-k)k=\int_{R-k}^{1}(1-G(\theta))d\theta-k$$ Where $R>k>0$ and $G$ is the CDF of $\theta$ which is distributed on $[0,1]$. Can someone explain how integration by parts has been used here? Thank you.

Hint: Simply apply integration by parts to the integral on the LHS. Simplify and you should arrive at the following expression:
\begin{equation}
(1-R)-\int_{R-k}^1G(\theta)\mathrm d\theta.
\end{equation}
Add and subtract $k$ to obtain:
\begin{equation}
(1-R+k-k)-\int_{R-k}^1G(\theta)\mathrm d\theta = (1-(R-k))-k-\int_{R-k}^1G(\theta)\mathrm d\theta.
\end{equation}
Observe that $(1-(R-k))=\int_{R-k}^1(1)\mathrm d\theta$. Substitute that in and collect terms, you'll get the RHS expression.


Open/modern question in labor economics? What are some of the modern lines of inquiry in labor economics and what are some good sources of them (especially, succinct ones)?

If you want a recent state of the art in Labor Economics, you can have a look at the last Handbook of Labor Economics (2011, edited by Card and Ashenfelter). Though it is getting old, it provides an overview of specific approaches or questions in Labor. It is especially relevant if you want to define a research project for a PhD.

To be more specific, here is a (non-exhaustive) list of trendy research questions (you can use these keywords to find research articles and reviews):

Polarization of the labor market. There is evidence of a decline in demand for middle-skilled workers (less employement and reduce in wages). The two main explications are globalization and skill-biased techonological change. Researchers try to weight the role of each explanation from data. In particular, a recent approach interests in the type of tasks in each occupation. This is related to the chapter by Autor and Acemoglu in the Handbook. Jobs with routine tasks are likely to be affected by technology as they can be more easily automated. Looking at the type of tasks, whether it uses cognitive or non-cognitive skills, brings new insigths in other interesting questions (returns to education, gender/racial wage gaps, local labor markets). You can look at some work by David Autor or David Dorn for instance.
Productivity puzzle. In many advanced countries, productivity has not found its before-crisis trend, and we do not know why. Research focused in country-specific situations (the UK productivity puzzle, the French productivity puzzle, etc) to find the reasons of this productivity slowdown in the national labor markets. It is also a puzzle that so many countries are affected despite their differences in the functionning of their economy.
Labor market, household decisions and marriage market. A rich literature studied the labor supply choices within the household as a joint decision. A new research area relates labor market outcomes to the formation of households. The novelty is to consider that individuals are not matched in a pure randomly way on the marriage market, and decisions to marry may account for individual abilities and skills. In particular, it has been shown that high-skilled female workers marry high-skilled male workers more often than in the past, and reciprocally. This has consequences in terms of inequalities and labor supply decisions. There are recent works conducted by Pierre-André Chiappori among others.


Two asset Markowitz Portfolio Optimization and Capital-Market Line construction for a Given Risk Free Rate I practice with some excercises about the Markowitz theory.
If we have a portfolio with two stocks A and B, with given return $r_A$ and $r_B$, the expected return can be computed as: $r_P= w_A \cdot r_A + (1-w_A) \cdot r_B$, where $w_A$ is the weight invested in stock A.

For a given risk free rate ($r_f$), I now have to compute the return of the one fund of the one fund theorem. What I am exactly supposed to do here? I know about the one fund theorem, but I don't know what to do. Any hints?

The Sharpe ratio tells us the amount of excess return we get for taking on each additional unit of portfolio standard deviation.
$$\frac{\mu_p - r_f}{\sigma_p }$$

We are looking for the combination of the two risky assets with the highest Sharpe ratio ($P^*$). Once we do that, we can take linear combinations of that portfolio and the risk-less asset and form the Capital-Market Line. This is usually solved for numerically rather than analytically but it is possible to do so analytically, particularly in the two asset case.

A portfolio $p$ has an expected return of:
$$\mu_p(W_A) = w_A \cdot \mu_A + (1 - w_A) \cdot \mu_B  $$

and a standard deviation of:
$$\sigma_p(W_A) = \sqrt(w^2_A \cdot \sigma^2_A + (1 - w_A)^2 \cdot \sigma^2_B + 2(1-W_A)W_A \sigma_{AB}) $$
where $\sigma^2_A$ is the variance of asset $A$, $\sigma^2_B$ is the variance of asset $B$, and $\sigma_{AB}$ is their covariance. It therefore has a Sharpe Ratio of:

$$\frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } = \frac{w_A \cdot \mu_A + (1 - w_A) \cdot \mu_B}{+\sqrt(w^2_A \cdot \sigma^2_A + (1 - w_A)^2 \cdot \sigma^2_B + 2(1-W_A)W_A \sigma_{AB})}$$

To maximize this you'll want to solve:
$$ \frac{d}{dW_A} \frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } = 0 $$
$\Rightarrow W^{*}_A$ s.t. $P(W^{*}_A)=P^*$
and check that the second order condition is met:
$$ \frac{d^2}{dW^2_A} \frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } < 0$$

The algebra is a bit hairy but there is nothing tricky from here on out.


Monotone transformation of a game What happens to the Nash equilibria and minimax values of a strategic game, when we take its payoff table and modify all payoffs by raising them to the 3rd power?

My conjecture is that it depends:

If, in the original game, all Nash equilibria and minimax points are attained using pure strategies alone - then a positive monotone transformation should have no effect on them. When everything is pure (deterministic), the utility function is ordinal and it is robust to monotone transformation.
But if, in the original game, some Nash equilibria / minimax points are attained using mixed strategies - then a non-linear transformation might change them substantially.

For example, in the following zero-sum game:

4 0
0 4
3 3

The maxinim value is 3 and a maximin strategy for the row player is selecting the bottom row (the column player can guarantee at most 3 by mixing the two columns with equal probability).

But if we raise to the 3rd power:

64 00
00 64
27 27

Now the minimax value is 32, and a maximin strategy for the row player is mixing the two top rows.

So my question is: In what conditions does a positive monotone transformation on a game's payoff not change its maximin and Nash equilibrium strategies?

I interpret the question as follows:
Suppose we have a finite normal form game (with finitely many players and a finite set of actions for each outcomes). For simplicity, assume a strict ordering of payoffs for each player. That is, the payoff matrix of each player does not contain any identical elements.

We furthermore need to be clear what is meant by a positive monotone transformation of the payoffs. The first option is to transform all payoffs by the same increasing function. The second option is to apply positive monotone transformations to the payoffs of each player separately. I interpret the question as meaning the latter, but the results should also go through for the former.

We want to characterize all Nash equilibria such that in any game resulting from an arbitrary monotone transformation of the payoffs the same strategies are also a Nash equilibrium. Call any Nash equilibrium which fulfills this property "robust to monotone transformations".

With your conjecture, you have pretty much answered the question. The set of Nash equilibria robust to monotone transformations is the set of pure strategy Nash equilibria. Here is a quick sketch of why this is so:

Suppose player i plays a mixed strategy. In this case, the player must be indifferent between all pure strategies in the support of the mixed strategy. This indifference is given by an equation which is a linear combination of payoffs. Since none of these payoffs are equal, we can change any payoff by a small amount to break the indifference. It follows that mixed strategy equilibria are not robust to monotone transformations.

We still need to show that all pure strategy equilibria are robust to monotone transformations. A pure strategy equilibrium is given if all players play their best responses to all other players' strategies. This involves maximizing over a finite set of actions. Given the other players' actions, the actions of player i are in one-to-one correspondence with a set of payoffs. Since the monotone transformation of the payoffs does not change their relative ordering, the set of best responses does not change. Since this holds for every player, the pure strategy Nash equilibrium is robust to monotone transformations.


What is the correct way to calculate a selling price from margin and a cost? There seems to be two formula to calculate a selling price.

The first formula that I came upon would be

Cost * (1+Margin) = Selling price
Example : 10 * (1+0.25) = 12.5

However, a lot of people uses the following :

Cost / (1 - Margin) = Selling price
Example : 10 / (1-0.25) = 13.33...

This gives a very different number.

As far as I know, the margin and selling prices can be anything you want for most products.

Also note that the second formula fails if you have a margin of more dans 100%.

My question is why is the second formula the most popular ? Is it only a gimmick to get a bigger price?

Am I missing something subtle at work here ?

Also, what should I do if the wanted margin is more than 100% ?

Thanks!

The "margin" is a portion of the selling price. It is defined as

$$\text {margin} \equiv \frac {P-C}{P}$$

From the above definition, we see that the margin cannot exceed $100\%$.

If one has the cost and he wants to calculate the price in order to have a specific margin, one must calculate

$$P(\text {margin}^*) = \frac {C}{1-\text {margin}^*}$$

The first formula written in the question is wrong, because it uses "margin", but it would become correct if instead one used the concept of "markup".

The "markup" is defined as the percentage increase of cost in order to determine the selling price:

$$\text {markup} \equiv \frac {P-C}{C}$$

It is easy to obtain that the relation between them is

$$\text {margin} = \frac {\text {markup}}{1+\text {markup}}$$


Trying to understanding Mark Blyth’s statement about Keynes I am reading a book called "Austerity, the history of a dangerous idea" written by Mark Blyth. While talking about Keynes, the author writes:

  "he [Keynes] showed that although any worker can accept a wage cut to
  price himself into employment, if all workers did this, it would in the
  aggregate lower consumption and prices, and thus increase real wage
  (the wage-minus-price effect), leaving the worker who "adjusted"
  poorer and just as unemployed"

I don't understand real wage/price effect. If prices drop with wages, how is the employee becoming poorer? If I make less but it costs less, shouldn't I be able to buy the same amount?

Thank you for your input!

Without context, it is unclear what Blyth meant. However, the usual argument runs as follows. The worker starts out with a certain nominal wage, and nominal debts. If wages and prices fall by some percentage, for current spending, the two changes cancel out - goods will cost the same percentage of weekly wages as before. However, the drop in prices does not reduce debt payments, and so those are larger relative to wages.


Allowing the reserve requirements drop to zero On 15 March 2020, the Federal Reserves announced that it would cut its benchmark rate to near zero and that it would also reduce the reserve requirements ratio to zero percent. This is part of its efforts to bolster the economy amidst this downturn. The reserve requirements ratio was put in place to prevent bank runs, which would be damaging to financial institutions as well as households. Why would the Fed deem that it is now safe to put the reserve requirements ratio to zero? Is it no longer afraid that bank runs may occur?

I believe that the Fed is more focused on saving the US economy from collapsing because of the effects of the coronavirus situation.
When the Fed wants to loosen monetary policy and increase liquidity, it lowers the reserve requirements ratio which will make lending cheaper.

Lowering the reserve requirement increases the amount of money that banks have available to lend. Since the supply of money is higher, banks can charge less to lend it.
Thereby more people will access to have loans at a cheaper interest in order to encourage spending.
https://www.thebalance.com/reserve-requirement-3305883

With the fast-spreading coronavirus posing a dire threat to economic growth the Fed is hoping that this move will help prevent the US economy from collapsing.
https://www.cnbc.com/2020/03/15/federal-reserve-cuts-rates-to-zero-and-launches-massive-700-billion-quantitative-easing-program.html

I am sure that the Fed is concerned about bank runs but it may be the case that they only have few options available to help encourage spending to prevent our economy from collapsing.


Did (or can) quantitative easing cause an "everything bubble"? Wolfstreet claims that

  It’s true that despite QE globally – not just in Japan – there has been relatively little consumer price inflation in the countries whose central banks perpetrated it. But it has caused enormous asset-price inflation. We call it the “Everything Bubble” where practically all asset classes globally have become ludicrously inflated.

Is this correct? I'm having doubts because what is a bubble is usually hard to define. How is this "everything bubble" different from just growth that QE is supposed to stimulate?

It’s an editorial opinion, and it’s hard to give it a technical meaning.

The distinction being made in the text is between growth in the “real economy” - which would be measured by GDP and/or consumer price inflation - and growth in asset values. For example, there could be no direct job creation because of QE, just equity prices going up.

However, this is an opinion/claim that is hard to evaluate. One argument is that QE would first lower term interest rates (by reducing the stock of debt outstanding and/or signalling that the policy rate would be lower for longer), and this should raise other asset values by decreasing discount rates. (Please note that how QE affected interest rates was the subject of some theoretical controversy; I believe that other questions here discuss this. I am summarising one standard interpretation.)

The idea is that there is a link between rising asset prices and economic activity (known as the wealth effect). Since we do not know how many jobs would have been created in the absence of QE, we cannot be sure how large this effect is.


Prove that Choice Coherence Implies IIA Prove that Choice Coherence implies Independence of Irrelevant Alternatives (IIA).
From the definition of choice coherence, we have this:
A choice function c satisfies choice coherence if, for every pair x and y from X and
A and B from A, if x, y are in A and B, x in c(A), and y not in c(A), then y is not in c(B).
From the definition of IIA, we have this:
If A is preferred to B out of the choice set {A,B}, introducing a third option X, expanding the choice set to {A,B,X}, must not make B preferable to A.
The converse is not true (IIA does not imply choice coherence) since we can construct a counterexample, as per here:
C({x,y,z}) = {x}
C({x,y}) = {x}
C({x}) = {x}
C({y}) = {y}
C({x,y}) = {x,y}
Since IIA is the weaker assumption of the two, how can we show that choice coherence implies IIA? Thanks for your help.

I think your notation makes this more confusing that is necessary since $A$ and $B$ represent sets of alternatives in one setting and alternatives in the other setting.
So write IIA as:
If $x\in C\big(\{x,y\}\big)$ and $y\notin C\big(\{x,y\}\big)$, then $y\notin C\big(x,y,z\big)$.
To prove this from choice coherence, simply choose $A$ and $B$ appropriately.


What does economics say about "price gouging" during an emergency? Many states have laws against "price gouging" during an emergency.

Are these laws counterproductive?

Take the current situation with toilet paper and water. When someone sees rolls of toilet paper on the shelf, they buy them all. Not because they necessarily need it, but out of fear the next person will buy them all and there won't be any left when they do need it. This creates a positively reinforcing feedback loop of negative cause-and-effect a/k/a/ a "vicious cycle."

But if prices spiked in response to meet demand, the feedback loop cycle would slow because two good things would happen that aren't happening now.

Supply would increase. Increased prices signal suppliers (whose marginal costs remain constant) to manufacture and ship more product in order to maximize profit.
Hoarding would mitigate. Hoarders would have less incentive to hoard supplies of water, toilet paper, etc. unnecessarily if they were paying significantly higher prices than normal. Then there would be adequate supply left for those who need it and aren't simply overstocking out of an abundance of caution. Moreover, would-be hoarders would feel like the next guy also has less incentive to hoard and thus quell their fear of being left with no supply when needed.

Is this a correct economic analysis or am I missing something that justifies "anti-price-gouging" laws?

Actually economics does not even officially use term price gouging. Your analysis is right, actually economists dislike anti-price gauging legislation for this reason.

For example, when the IGM panel of top policy economists were asked about one piece of price gouging legislation in the US, vast majority disagreed with it:

However, an important caveat to keep in mind is that due to menu costs some firm would not be able to change prices quickly enough even in absence of anti-gouging regulation - meaning such regulation is not the sole source of shortages, there would be some even without them although less severe (and of course it goes without saying this is not any justification for them).

However, beside that yes, you are actually missing the main factor that justifies the anti-price-gouging legislation. Which is moral philosophy. There is argument to be made that such practices as gouging are immoral (you can see some ethical arguments against gouging in this paper for example). While I personally dont identify with such moral views, its not our role as economists to pass moral judgements. If people pass anti-gouging legislature well aware of all the negative economic consequences but they still do it despite them then there is no objection an economist can make against them as we are scientists not moral philosophers and such questions are beyond our discipline.


No competitive equilibrium for pooling contracts In class we dealt with insurance economics and, specifically, adverse selection due to information asymmetry. As one possible solution we considered pooling contracts, i.e. the same contract for both high- and low-risk households for some average price. We showed and I understood why the high-risk households now would buy more insurance than the low-risk ones and thus everybody would act as if they were low-risk whether this is true or not. What I am not sure about is why there wouldn't be an equilibrium in a competitive market.

It's clear that the low-risk individuals have reason to go for a lesser premium and, if some other insurer would offer this, then they would go there.

Therefore, my questions are:

1) So why won't there be an equilibrium? I would think that if a new insurer appears with a better deal than the pooling contract, then there would again be the problem with adverse selection. Hence, the new insurer wouldn't appear and offer a better deal in the first place?

2) I also read that this is pareto-inefficient in that it disadvantages the low-risk households (obvious) but does not benefit the high-risk households. The latter point I do not understand since the high-risk households will have a higher demand since the pooling contract situation makes the insurance relatively cheap. Isn't that a benefit?

As one of the comments points out, there are many models of equilibria in insurance markets. But it sounds to me like you are referring to the Rothschild-Stiglitz (1976) paradigm. I will provide a basic summary of the key takeaways here, but there is a more complete explanation in this set of slides and in those from any second-year graduate course in Public Economics.

In this paradigm, mixed strategy equilibria are ruled out. So when we say that "no equilibrium exists", we are actually just saying that no pure strategy equilibrium exists.
There is never a pooling equilibrium. The basic reason for this is that a firm could profitably come in and offer a cheaper package with less insurance and poach away the low risk types. This leaves the existing firm with only the high risk types and it makes a loss.
There may or may not exist a separating equilibrium. If there does not, it is because - starting from a separating equilibrium - there exist pareto-improving cross-subsidies (from high to low risk types). This means that a firm can profitably come in and offer a package that attracts both types (i.e., a pooling contract).
EDIT: To answer your second question, the separating equilibrium - if one exists - is indeed not pareto efficient but it does benefit both high and low risk types. The pareto efficient separating contract would offer full insurance to both type, but the Rothschild-Stiglitz candidate cannot. To see why, suppose that it did. Then both receive full insurance but the low-risk types get it for less. Clearly the high risk types will pretend to be low types and the incumbent firm will make a loss.

In summary, this particular model offers a theory of pure strategy Nash Equilibria in insurance markets with adverse selection and multiple risk types. An equilibrium may or may not exist. If it does, then it is separating.

Additional information

There are competing paradigms that solve this problem with assumptions about the timing of when offers are made and accepted (e.g., Miyazaki-Wilson-Spence). Another classic model to compare is Akerlof-style adverse selection. Neither alternative suffers from the non-existence problem.