Since the US-China Trade Deficit grew, would the USD be valued more now compared to the RMB? I'm curious to see how broadening or narrowing a trade deficit would affect a currency pair.

The "US-China trade deficit" isn't a real thing. Yes, there are some transactions classified as "exports from US to China" and others classified as "exports from China to the US", and you can subtract the latter from the former and get a negative number, but that doesn't represent anything fundamental about the world economy. Suppose that the US were to stop buying widgets from China and buy them from Taiwan instead, and the country that was buying them from Taiwan were to start buying them from China. Everyone would end up buying/selling the exact same amount of widgets, yet the "US-China trade deficit" would go down despite absolutely nothing fundamental about the world economy changing.
The overall trade deficit of the US does have an effect on the USD, but higher trade deficits make a currency go down in value, not up. Of course, a decrease in currency value tends to decrease trade deficits, which means that this effect tends to be cancelled out.


How to approach rigorous probability theory from an economics background? I am attempting to read around the theory of probability theory from the ground up, coming from a background of economics I have little experience in set/measure theory, whilst I am not new to statistics and econometrics the rigorous treatment of the topics have been difficult to follow. For reference, I have been using 'Probability, Statistics, and Econometrics - Oliver Linton' to read on the subject. What would be the best way to approach this?
I ask this question as this has been recommended in preparation for my MSc in Economics by a lecturer.

If you want a good book with emphasis on rigorous then An Introduction to Probability: Theory and Application, by William Feller is good source.
The book starts completely from the first principles and covers also a lot of applications in statistics. Arguably the book is more suited to graduate as it has a steep curve - the content difficulty increases rapidly, but I think that it can be suitable even at Msc level - especially if you are doing more research oriented Msc.
If you are would be interested in something more focused on econometrics then a primer in econometric theory by John Stachurski is good source although it does not cover probability so broadly as the above source.


Testing Roberts's Law: data request on CEO pay and company characteristics I posted here regarding sources for Roberts's Law.

In this paper it gives a clear qualification of what Roberts's law is. (Page 9, ...Cross-sectional prediction..)

  ..In a given year, the compensation of a CEO is proportional to the size of the firm to the power of 1/3, S(n)1/3, an empirical relationship sometimes called Roberts’ (1956) law.

This gives us the following formula:

$ Pay$~$Size^b$ with $b ≃ 1/3. $

I could not find much info on how to calculate firm size so I dug around and found another paper titled "What Determines Firm Size?
"

I found that there was an equation given and explained on page 10.

  The average firm
  size in each size bin is first calculated by dividing the number of employees by the number of
  firms. The average size for the entire sector is then calculated as the weighted sum of these bin
  averages, using as weights the proportion of the total sectoral employment in that bin. This
  produces a “employee-weighted” average of firm size.

  Employee Weighted Average Number of Employees =$${\sum_{bin=1}^n}(\frac{N^{Emp}_{bin}}{N^{Emp}_{Sector}})(\frac{N^{Emp}_{bin}}{N^{Firm}_{bin}})$$

  Emp
  bin is the total number of employees in a bin, N
  Emp
  Sector is the total number of employees
  in the sector, and NFirms
  bin is the total number of firms in a bin. In contrast to the firm-weighted
  simple average, the employee-weighted average emphasizes the larger firms; note the squaring
  of bin employment in the numerator.

Robert's law is now:

$$Pay={\sum_{bin=1}^n}\left[\left(\frac{N^{Emp}_{bin}}{N^{Emp}_{Sector}})(\frac{N^{Emp}_{bin}}{N^{Firm}_{bin}}\right)\right]^{b}$$
Now I am aware I'm kinda just copying and pasting and my research may be incorrect. But i'd like some data to test Roberts's law.

Where can I find data on CEO pay and company information?

There are plenty of papers studying CEO pay and firm size empirically. Check a long list here. Some examples of data sources:

US: Execucomp, used by Gabaix, Landier, and Sauvagnat (2014).
UK: Hemscott  and Datastream, as used by Ozkan (2009) or Gregg, Jewell, and Tonks (2011). Also, BoardEx and Datastream, as used by Tarkovska (2017). You also have the Bureau van Dijk FAME dataset (used by the Bank of England, e.g. here), but it is proprietary.
Germany: handpicked from financial reports as in Heimes and Seemann (2012), or data obtained from the consulting firm
Kienbaum, as in Fabri and Marin (2015).


Link between elasticity and marginal utility The above is taken from "Economics" 8th Edition by John Sloman, a textbook commonly used at the introductory undergraduate level.

I would like assistance in understanding the bracketed portion - how a decrease in consumption of alternative goods results in a low increase in total utility and thus a small decrease in marginal utility. Thanks.

You may ignore the following picture, "Figure 2". It was uploaded to illustrate something in response to a comment.

We can break this into three parts: (1) price elasticity, (2) substitutes, and (3) marginal utility.

Price elasticity measures price sensitivity (how much a change in price affects quantity consumed). In the example in the book, the product is elastic, which means that a decrease in price increases consumption proportionally more than the decrease in price.

Substitutes are any products that can be substituted for another. This is important because a change in price of a product will not only affect it's consumption, but also the consumption of substitute goods. For example, people who want soda have a choice between Coke and Pepsi. Assume that when priced the same, $10$ people by Coke and $10$ people buy Pepsi. The total consumption is $20$. Now, assume that Coke lowers its price while Pepsi keeps the price the same. Some people who once consumed Pepsi will now by Coke, say $5$ people switch to Coke. The total consumption remains the same, $20$, but its distribution is different. It is the case that a drop in price will encourage consumers to purchase Coke who wouldn't normally do so at the initial price. because a price decrease makes the product available to new consumers the total consumption of soda may increase to $25$.

Marginal Utility is the amount by which utility changes with each unit of consumption. The text doesn't seem to say this, but marginal utility is subject to the law of diminishing returns. This means that at a certain point, consuming more of a good makes a person less happy. For example, if I eat 2 cookies, I'm happy. With each cookie I consume, I grow happier in total, however this is happiness increases at a decreasing rate.

Utility is maximized when marginal utility is zero. After this point one loses...experiences disutility.

Essentially, the text is saying that if the price of an elastic good decreases its consumption will increase. The consumption increase comes from (1) "stealing" consumers from other similar products and (2) new consumers that didn't buy the product at the initial price. Since the total consumption increases as did total utility, the increase was small and thus marginal utility decreased. This isn't saying that utility was taken away, its saying the the rate at which utility increased was slower.


Determining the elasticity of a supply/demand curve visually Hi, I am self-studying microeconomics in preparation for a future course. I am confused about whether or not it is possible to visually determine the elasticity of a demand/supply curve by looking at its slope. Some sources I read tell you that the elasticity can be determined visually based on the steepness of the slope of the demand/supply curve.
However, I encountered other sources that say the opposite--that elasticity cannot be judged by the steepness of the slope. I read that the elasticity of demand changes at different points of a linear demand curve, but I can't find the same assertion repeated for linear supply curves.
Even more confusingly, I read from some sources that any supply curve that is upward sloping from the origin has a constant unitary elasticity regardless of the value of the slope of the supply curve. Doesn't this also contradict the assertion that the slope of the supply/demand curve indicates its relative elasticity, if a curve of ANY slope crossing the origin can be constant unitary elastic?
I am having trouble studying this from different sources. If someone can give an intuitive explanation for what is true (whether or not you can visually determine the elasticity of a demand/supply curve) I would be grateful.

Generally speaking it is not possible to tell if demand/supply is elastic/inelastic just by looking at steepness (although there are some exceptions).
Rigorously, the price elasticity of supply/demand is given by:
$$EL=\frac{\frac{\partial Q}{Q}}{\frac{\partial p}{p}}= \frac{\partial Q}{\partial p}\frac{p}{Q}$$
The slope of demand/supply curve will be given by $\frac{\partial Q}{\partial p}$ so looking at the slope can only tell you whether the first term is high/low but it does not tell you what the second term is $p/Q$. Consequently, generally speaking it is not possible to tell what the elasticity of supply/demand is just from the picture.
This is further complicated by the fact that what matters is the slope of the demand/supply function $\frac{\partial Q}{\partial p}$, but typically what you see on picture is not a demand/supply function but inverse demand/supply function, because by convention we plot price on $y$-axis, so what you see is not $Q(p)$ but $p = f^{-1}(Q)$.
This holds especially for linear supply/demand curves. For example general linear demand is given by:
$$Q = - a p + b$$
(where $a$ is the slope) hence general price  elasticity for demand curve would be:
$$EL= -a \left(\frac{p}{-ap+b}\right)$$
(general formula for linear supply would be almost identical except $a$ would be positive) so as you can see above for linear function the price elasticity with change with price, and consequently knowing slope (steepness) will not help you to know whether the demand is elastic or not. For example, suppose that the slope is $-10$ (which is quite steep), then $\frac{-10p}{-10p+b}$ can still turn out to be less than 1. For example, for $p=0$ the elasticity here would also be zero.
There are some exceptions though. If the whole supply/demand is horizontal flat (as judged when looking at plot of $p=f^{-1}(Q)$ which means that $Q(p)$ is vertical) then we know that slope is infinity. Since, infinity multiplied by any scalar is still infinity we know that in this case the elasticity will be infinite (i.e. such supply/demand will be perfectly elastic) even just by looking at the picture.
Similarly, if the whole supply/demand is vertical (as judged when looking at plot of $p=f^{-1}(Q)$ which means that $Q(p)$ is horizontal) we know that the slope of demand/supply curve will be zero. Since anything multiplied by zero is still zero the whole elasticity will be zero in such case (i.e. such supply/demand will be perfectly inelastic).
Consequently, save for special cases it is not possible to know if a demand/supply is elastic or inelastic just by looking at a steepness of a slope. This is especially true for linear demand and supply where elasticity will range from $0$ to $\infty$ as price change along the demand.

Even more confusingly, I read from some sources that any supply curve that is upward sloping from the origin has a constant unitary elasticity regardless of the value of the slope of the supply curve.

This is true because a linear supply curve that goes through origin is generally given by $Q_s= ap$ which implies that elasticity is:
$$EL_{Q_s} = a \frac{p}{ap}=1$$
This actually holds also for linear demand function passing through origin since general linear demand that passes through origin would be given by $Q_d=-ap $ so the elasticity is:
$$EL_{Q_d} = -a \frac{p}{-ap} = 1$$
So this is another special case where you can exactly tell what a demand or supply elasticity is just by looking at it.


Typical Farming Economics Resource Are there any resources available that provide typical books or accounts for the farming or indeed other industries?

Something that would list, employee numbers, wage bills, expenditure income (sales vs subsidies) and so on in terms understandable to a layman?

I don't have many books to share but there is the website on Farm economy data from the USDA.

Farm Labor
USDA Economic Research Service (ERS)

https://www.ers.usda.gov/topics/farm-economy/farm-labor/

There is also the USDA National Agricultural Statistics Service (feel free to explore their site)

https://www.nass.usda.gov/Data_and_Statistics/index.php

here are some articles

https://arefiles.ucdavis.edu/uploads/filer_public/ad/74/ad7450e7-80ab-4cf7-a147-6b80c2e614a7/chapter_1_the_farm_labor_problem_4-4-17.pdf
The Farm Labor Problem
J. Edward Taylor & Diane Charlton
Agricultural & Resource Economics
University of California, Davis

https://www.bls.gov/opub/mlr/2014/article/agriculture-occupational-employment-and-wages.htm
Agriculture: occupational employment and wages

Labour Market Effects of Large-Scale Agricultural Investment: Conceptual Considerations and Estimated Employment Effects
https://www.sciencedirect.com/science/article/pii/S0305750X17301754

Few books on the subject:

freebook-
The Economics of Food and Agricultural Markets - 2nd Edition
https://open.umn.edu/opentextbooks/textbooks/the-economics-of-food-and-agricultural-markets

Agricultural Economics (3rd Edition) 3rd Edition
by H. Evan Drummond Ph.D. (Author), John W. Goodwin (Author)


Why we need to control for the interation of year and industry fixed effects? Normally, we control for firm and year fixed effects, but in some case I saw people control for the  firm along with yearxindustry or firm and yearxregion fixed effects.
Could you please hint me why they need to do so?

When you control for not just year fixed effects but instead year-region or year-industry it adds flexibility.
The year fixed effects controls in a flexible manner for the time-trend and is more flexible - less restrictive - than for example assuming that the time trend is for example linear $a \cdot t$, second order polynomial $at + bt^2$, exponential $exp(at)$ ect.
Nevertheless, year fixed effects are still restrictive compared to year-region or year-industry because the assume that the time trend is the same for all regions or for all industries instead of allowing different regions to follow different trends or different industries.
Consider for example this plot here

and assume that it show the average wage of women (red) and the average wage of men (black) over time.
We then conclude that the gender wage-gap is closing. However, someone then points out this is not really the case because although female workers on average over time get the same as male workers this is simply because female workers have changed from predominantly working in the low-wage industry to be working in the high-wage industry.
Offcourse you are sharp and you run a regression controlling for industry
$$\log y_{it} = \beta_0 + \alpha \cdot \text{male}_{it} + \lambda \cdot  \text{industry}_{it} + \delta_t + \epsilon_{it}$$
which is a good idea and for the simulated data used to generate the plot above you then get an estimate $\hat \alpha = 3.05$ clearly showing that male workers are given more in wage than female workers.
However, the true $\alpha$ is actually even larger and equal to $4$ so you are still underestimating the degree to which female workers are discriminated on the labor market.
In this example this happens because:
Over time the wage in the high wage sector increases faster than the wage in the low wage sector. So while women over time changed to the high wage sector you are undercontrolling for the sector effect because the early year with a small difference between the sectors bias the sector effect downwards.
In this case what you should have been doing to get an unbiased estimate was to run the regression
$$\log y_{it} = \beta_0 + \alpha \cdot \text{male}_{it}  + \lambda_{st} + \epsilon_{it}$$
where $\lambda_{st}$ is the sector-time fixed effect.
You can simulate this example in R using the following code
library(data.table)
library(lfe)
N1 <- 1000
N2 <- 1000
N <- N1+N2
T <- 10
b0 <- 1

# a1 is the true gender effect
a1 <- 4

gender <- rep(c(rep(0,N1),rep(1,N2)),T)
a2 <- rep(1:T * 2,N)
dt <- data.table(id=rep(1:N,each=T),time=rep(1:T,N),gender=gender)
setkey(dt,time,gender)
industry <- c()
for (t in 1:T)
    {
        temp1 <- sample(c(0,1),N1,replace=TRUE,prob=c(0.5,t/T))
        temp2 <- sample(c(0,1),N1,replace=TRUE,prob=c(0.5,0.5))
        industry <- c(industry,temp1,temp2)
    }
dt$industry <- industry
setkey(dt,id,time)
dt$log_y <- b0 + a1*dt$gender + a2*dt$industry + 0.3*dt$time + rnorm(N*T)
foo <- dt[,.(m_wage=mean(log_y)),by=.(time,gender)]
plot(1:T,foo[gender==0,]$m_wage,ylim=c(0,20),col="red",type="l")
points(1:T,foo[gender==1,]$m_wage,type="l")

dt$i_time <- interaction(dt$time,dt$industry)
# Regression with industry dummy

summary(felm(log_y ~ gender+industry|time,data=dt))
#          Estimate Std. Error t value Pr(>|t|)
# gender    3.05645    0.04164   73.41   <2e-16 ***
# industry 11.13346    0.04215  264.12   <2e-16 ***

# Regression with industry time fixed effects
summary(felm(log_y ~ gender |i_time,data=dt))

#        Estimate Std. Error t value Pr(>|t|)
# gender  4.00873    0.01443   277.7   <2e-16 ***

savePlot("male_female.jpg",type="jpg")


What causes the "imperialism of economics" and how did it start? Wikipedia defines "economics imperialism" as follows:

  Economics imperialism (often economic imperialism) in contemporary economics refers to economic analysis of seemingly non-economic aspects of life, such as crime, law, the family, prejudice, tastes, irrational behavior, politics, sociology, culture, religion, war, science, and research. Related usage of the term predates recent decades.

My question is this. What is it about economics as a discipline that has allowed the it to extend to these "seemingly non-economics aspects of life?" Also, when did economic analysis in these so-called unrelated fields become accepted as part of mainstream economics?

I find the following quotes to be interesting:

  "[Political economy is] a branch of the science of a statesman or legislator [with the objective to provide] a plentiful revenue or subsistence for the people [and] to supply the state or commonwealth with a revenue for the publick services."—Adam Smith, 1776.

  "Economics is a study of man in the ordinary business of life. It enquires how he gets his income and how he uses it. Thus, it is on the one side, the study of wealth and on the other and more important side, a part of the study of man." —Alfred Marshall, 1890.

  "Economics is the science which studies human behavior as a relationship between ends and scarce means which have alternative uses" —Lionel Robbins, 1932.

Together, I think these quotes nicely illustrate the way that economists' perceptions of their own discipline evolved over time. By the 1930s, we have Robbins' startlingly insightful observation that Economics is about choice subject to constraints, which is obviously a very general kind of problem. But even as early as the late 19th century you can already see traces of this kind of thinking in Marshall's words.

This way of thinking about economics is, I think, key to emboldening economists to reach out beyond the confines of market behavior. Indeed, I think one reason why economics has been so imperialistic is that economics is as much a methodology as a body of knowledge. As such, economists have constantly looked for new problems and settings to which they can apply the tools they know.

More recent contributions that have been important are probably:

The work of Gary Becker, whose Nobel citation reads "for having extended the domain of microeconomic analysis to a wide range of human behaviour and interaction, including non-market behavior" Those non-market behaviors include important things such as fertility, marriage, and crime—not traditionally within the domain of economics.
The development of game theory and information economics. These tools equipped people with new ways of modelling social environments. A classical supply/demand type model, or a general equilibrium framework will get you quite far when studying traditional markets, but these tools are much less suitable for studying non-market behaviors that are the preserve of economic imperialism. New modelling frameworks brought about by game theory, the principal-agent model, etc. really opened the way for studying all kinds of new environments.
The Downs model, which was an early application of economic-style (specifically, Hoteling's linear city) modelling to political science—one of the most thoroughly 'colonized' disciplines.


How constant returns to scale leads to zero economic profit? This was mentioned in my textbook Macroeconomics-Gregory Mankiw.

  If the production function has the property of constant returns to scale, as is often thought to be the case, then economic profit must be zero. That is, nothing is left after the factors of production are paid.

It was mentioned that this comes out as a result of Euler's Theorem, but I don't quite understand it, can anyone please clarify this?

The argument is that if there are constant returns to scale then the marginal product of each factor of production is constant as total quantity changes. This makes the production function homogeneous in the first degree and it will look something like
$$Q=\dfrac{\partial Q}{\partial A}A +\dfrac{\partial Q}{\partial B}B +\dfrac{\partial Q}{\partial C}C$$

Then if each factor of production is paid at a rate equal to its marginal product, the complete value of the production will be distributed to the factors of production, since

$$PQ-\left(P\dfrac{\partial Q}{\partial A}A +P\dfrac{\partial Q}{\partial B}B +P\dfrac{\partial Q}{\partial C}C\right)=0$$

so there will be no economic rent or profit

The practical problem with this is that constant returns to scale is a very strong assumption.  The theoretical problem (at least for marginalist theories) is that there is no natural equilibrium for the level of production


Durable/non-durable good industries How to define durable/non-durable good industries? For example, is the industry of  Wholesale and retail trades defined as durable good industry? Thanks in advance.

The US BEA defines durable goods as those lasting three years or more:

  PCE is classified by   type of product as follows. Goods consist
  primarily   of tangible commodities that can be stored or inventoried,
  but they also include certain intangible products, such as software.
  Durable goods are goods that have an average useful life of at least 3
  years. Nondurable goods are goods that have an average useful life of
  less than 3 years. Services are commodities that cannot be stored or
  inventoried and that are usually   consumed at the place and time of
  purchase. ... Durable goods [include]: motor vehicles and parts, furnishings and
  durable household equipment, recreational goods and vehicles, and
  other durable goods.

CHAPTER 5: PERSONAL CONSUMPTION EXPENDITURES (Updated: November 2017)


Log-normality assumption in consumption based asset pricing Consider a very basic discrete time representative consumer maximization problem with CRRA utility. There exist a risky asset with time $t$ price $p_t$ that pays time $t+1$ dividend $d_{t+1}$ , and a riskless asset with price $p_t^f$ that pays a constant payoff 1 at $t+1$. We assume that the dividends are a sequence of random variables that follow a Markov process. Assume further that the consumer has no other income streams (i.e. $y_t = 0 \ \forall t$). At time t consumer invests amount $\pi_t$ in the risky asset and amount $\pi_t^0$ in the riskless asset. Therefore, the maximization problem can be stated as

\begin{align*}
& \underset{\{ c_t, \pi \}_0^\infty}{\text{max}} \ \ E_0 \sum_{t=0}^\infty \ \beta^t \ \frac{c_t^{1-\gamma} -1}{1-\gamma} \\
\\
\
s.t  \ \ \ \ & c_t + \pi_t p_t + \pi_t^0 p_t^0 = (d_t+p_t) \pi_{t-1} + \pi_{t-1}^0  \\
& c_t \geq 0
\end{align*}

Say we want to find the equilibrium riskless rate and expected equity premium. In order to close the model, it is often assumed assumed (see e.g. Claus Munk's book Financial Asset Pricing Theory chapter 8.3) that the log-consumption growth and log-risky gross returns are jointly normally distributed. I.e

\begin{align*}
& ln \ \Big(\dfrac{c_{t+1}}{c_t} \Big) \equiv \bar{g}_{t+1} \sim N(\mu_g, \sigma_g^2) \\
& ln R_{t+1} \equiv \bar{r}_{t+1} \sim N(\mu_r, \sigma_r^2) \ , \\
\end{align*}

where gross returns are defined as
$$R_{t+1} \equiv \frac{p_{t+1} + d_{t+1}}{p_t} \ .$$

What I don't completely understand is where do thelog-normal distribution assumptions "come from". I know that since this is a representative agent economy, consumption of the agent must equal the aggregate dividend in the economy.  But since we assumed that there is no income, $y_t = 0 \ \forall t$, the only exogenous dividend process in the economy is $d_t$ and therefore it should have the same distribution as the consumption growth. However, my impression is that when we say the risky rate has log-normal distribution this actually means the dividend process, since it is the 'random part' in the definition of returns (price $p_{t+1}$ is not exogenous but determined inside the model). To me it seems now that we have made two different assumptions about the same endowment process $d_t$. Where does the assumption for consumption come from or what does it stand for? How would the situation change if the consumer had some income stream $y_t > 0$?

The typical two-period Lagrangian is

$$\Lambda = \beta^t\cdot \Big(\frac{c_t^{1-\gamma} -1}{1-\gamma} + \lambda_t\cdot \big[(d_t+p_t) \pi_{t-1} + \pi_{t-1}^0- c_t - \pi_t p_t - \pi_t^0 p_t^0\big]\Big) \\
+ \beta^{t+1}\cdot \Big(\frac{c_{t+1}^{1-\gamma} -1}{1-\gamma} + \lambda_{t+1}\cdot \big[(d_{t+1}+p_{t+1}) \pi_{t} + \pi_{t}^0- c_{t+1} - \pi_{t+1} p_{t+1} - \pi_{t+1}^0 p_{t+1}^0\big]\Big)$$

The first order conditions with respect to $c_t, \pi_t$ are

$$c_t^{-\gamma} = \lambda_t \implies ... \gamma\ln \frac {c_{t+1}}{c_t} = \ln \frac {\lambda_{t}}{\lambda_{t+1}} \tag{1}$$

$$-\beta^t\lambda_tp_t + \beta^{t+1}\lambda_{t+1}(d_{t+1}+p_{t+1})=0 \implies  \frac {\lambda_{t}}{\lambda_{t+1}} = \beta \frac{p_{t+1} + d_{t+1}}{p_t} \tag{2}$$

and so, using also the definition of the gross return,

$$\ln \frac {\lambda_{t}}{\lambda_{t+1}} = \ln \beta + \ln R_{t+1} \tag{3}$$

Combining $(1)$ and $(3)$ we get

$$\ln \frac {c_{t+1}}{c_t} = \frac 1 {\gamma}\ln \beta + \frac 1 {\gamma}\ln R_{t+1} \tag{4}$$

So we see that at the optimal path, consumption growth is a direct affine function of the log-risk returns. This among other things implies that their correlation coefficient is equal to unity.

The normal distribution is closed under affine transformations (alternatively, under scaling and shifting), so if we assume that log-risky returns are normally distributed, then consumption growth is also normally distributed (with different mean and variance of course).

Note that although in general, the joint normality assumption is an additional one to be made when two normal random variables are not-independent, here, the fact that the one is an affine function of the other guarantees joint normality. By Cramer's condition for bivariate normality, it must be the case that all linear combinations of two normal random variables have a univariate normal distribution. In our case we have (generic notation) the random vavriable $Y$ and the random variable $X = a+bY$. Consider

$$\delta_1X + \delta_2 Y = \delta_1(a+bY) + \delta_2 Y = \delta_1a + (\delta_1b+\delta_2)Y$$

So for any $(\delta_1, \delta_2)$ (except the zero vector which is excluded a priori), $\delta_1X + \delta_2 Y$ follows a normal distribution if $Y$ does. So it is sufficient to assume that log-risk returns follow a normal distribution to obtain joint normality also.


Won’t paying wages to temporarily unemployed due to Coronavirus result in inflation? UK government promises to pay 80% of wages for people unemployed due to coronavirus. I am not economist, so I am probably missing something, but shouldn’t paying wages to unemployed due to coronavirus result in inflation due to decreased products and services created by economy? Is that just a mechanism to redistribute wealth?

Not by itself. The inflation generally depends on the following monetary relation:

$$ MV =PY$$

Where M is money supply, V velocity of money (how much one buck is used in economy), P is the price level (change in which is inflation), and Y is a real output.

Now solving for P so we can focus on inflation gives us:

$$P=\frac{MV}{Y}$$

Hence price level (and consequently) increases when money supply increases, velocity of money increases or output decreases.

Now let’s analyze giving money to those unemployed.

Is it newly created money? If not then money supply won’t increase. If yes then it would increase inflation rate.
Redistribution generally does not affect velocity of money much so we can skip that.
Does this change the output Y? Well not by much those people are already unemployed. If anything it could increase Y as unemployed have probably high marginal propensity to consume so in short run redistribution from a richer person with lower marginal propensity to consume will increase Y. Especially if this spending was not paid out of higher taxes but from debt.

So a priori as long as it’s not financed through printing new money it should not really have effect on inflation. Inflation is ultimately a monetary phenomenon.

Also, yes it’s a redistribution mechanism like most of the things that UK government does. Any policy that uses government funds as a transfer to others, or fund non-public good services (like NHS for example) is a redistributive measure.


Why does the bond market induce efficient allocation of capital? Going from this quote by Bryan Collins of Fidelity on China's economy:

  One of the nice things we see from the development of a domestic bond
  market is that it starts the process of pricing capital better. And
  when you price capital better, you get a much better chance of that
  capital being allocated more efficiently.

I can certainly follow Collins' logic, but I don't understand the economic theory that underpins this logic.

Question

Why is it exactly that a developed bond market leads to more efficient allocation of capital?

This is no different from any other markets... just for money.

On the demand side of the goods, those who want money (those who sell bonds) say how much they're willing to pay in terms of bond yields. The supplier of the goods (bond buyers) come to the bond market and choose the product.

Under a developed bond market, it's guaranteed that those who get the goods (money) are the bond sellers who could pay the highest [risk-adjusted] price, which would infer that they could use that goods most productively. It's also guaranteed that those who sell the goods (bond buyers) are the ones who require the lowest price, i.e. opportunity cost is the lowest.


What are examples of natural monopolies that don't have overwhelming economies of scale? On wikipedia, I read that the currently-used formal definition of a natural monopoly is where “[a]n industry in which multi-firm production is more costly than production by a monopoly”.

I tried looking up things related to monopolies and subadditivity, but couldn't find anything that made it easy to understand.

To further quote wikipedia, "Baumol also noted that for a firm producing a single product, scale economies were a sufficient but not a necessary condition to prove subadditivity."

Note that in the above quote, I'm almost certain the "scale economies" they're talking about are cases where the optimum scale is large enough to serve the entire market.

So my question is, what is another set of conditions that don't involve "scale economies" that is a sufficient condition to prove subadditivity, and thus show that a situation is a natural monopoly?

One answer I found are natural monopolies that form on the basis of significant network effects. While economies of scale decrease the cost per unit as you can produce more units, network effects increase the value per unit as you sell more units. These two things together seem like they cover everything, but I haven't read any sources that mention whether there are other situations than economies of scale and network effects. Btw, I'm including "economies of scope" in my mention of economies of scale, since they're so similar.


Is there a difference between deadweight loss and welfare loss? I've heard these two terms crop up quite frequently, and suspect they may be interchangeable. Deadweight loss - I'm sure I've encountered this in tariff evaluation; for effects of indirect taxes, I've seen textbooks that use deadweight loss. Welfare loss - I've seen textbooks refer to the social costs of profit-maximising monopolies who set prices higher and output lower than that of perfectly competitive firms as welfare loss. Is there really a difference between deadweight loss and welfare loss?

I would say all deadweight loss is welfare loss but not all welfare loss is deadweight loss.

For example an unregulated polluter causing a negative externaly results in a welfare loss compared to the social optimum. There is no 'missing production' and hence no deadweight loss in this case.

Edit: According to Wikipedia, the loss created by externalities is also classified as deadweight loss.


Cost function and short run supply Suppose there is a number of identical firms producing diamonds. Let the total cost function for each firm be given by  $C(q, w)$, where q is the firm’s output level and w is the wage rate of diamond cutters.

I'm supposed to solve for a firm's (short-run) supply curve, given a wage rate $w$, and the industry's (total) supply curve).

I'm very confused with the relationship between the cost function and the supply curve. For part a this question, I get the marginal cost from taking the derivative of the cost function -- is the marginal cost also the supply curve? We know the cost function of the firm, but how can I arrive at the industry's supply curve?

Just like a Production function, in Economics, a Cost function is not an "engineering relation", but a "best response", reflecting minimum cost to produce a given quantity $q$.

Cost-minimizing firms are at first assumed not to determine their scale of operations -what they decide upon is the demand for production inputs,given the inputs' prices.

Still, by duality, the profit-maximizing condition should also hold -these are for-profit private companies, not, say, some public utility without a profit motive. If the firms are price-takers in the diamonds market, this relation is well-known, and provides the answer to a) (remember that there are functions, and then there are inverse functions).

As for b), I see in the question that the suppliers are assumed identical.


How can the abuses of monopoly power lead to market failure? So I know that the abuses of monopoly power can cause market failure, but I don't know why that is. I am guessing that because monopoly's face a lack of competition, they have no incentive to improve their product or service, therefore they are not allocating resources efficiently (market failure), however, in my book, a lack of competition is a seperate cause, so I was wondering if there was anything else to it.

P.S. If you could include a logic chain of reasoning and an example in your answer, that would really help me to understand it. I am also revising for GCSEs so the answer need to be extremely advanced. Thanks

First, let's be clear what we mean by abuse of dominance: this is when a firm is a monopoly or near-monopoly, and attempts to use that position in order to perpetuate or enhance its dominance of the market at the expense of competition.

So we need to know: why would a market with a monopoly perform worse than one with competition. Here are the main typical answers:

A monopoly will increase the price above marginal cost by artificially constraining supply. This means that some units that consumers' value above the cost of production don't get traded, resulting in the usual dead weight loss triangle.
Some people speculate that a monopolist has less incentive to innovate (create new products or means of production) than a competitive firm because there is no competitive pressure to be more efficient or to have a better product than other firms. Others, though, have argued that monopolists have a larger incentive to innovate because they can capture more of the gains from invention. This is the subject of a classic debate between Arrow and Schumpeter. The empirical evidence on whether competition is good or bad for innovation is extremely mixed.
Rent-seeking is a concern. If a monopoly makes profit $\pi_m$ and a competitive firm makes profit $\pi_c<\pi_m$ then a firm would be willing to spend $\pi_m-\pi_c$ to become (or remain a monopoly). This is fine if the resources are spend on something socially constructive. But not if it is spent on wasteful advertising, lobbying, engineering products with poor compatibility, creating needless switching costs, etc.
Another concern is managerial slack / x-inefficiency. If a firm has no competitors then there is very little incentive to exert effort to become more efficient, meaning that the firm may end up spending more resources on production than necessary.

If you want to read more about these things, I would recommend the first couple of chapters of Motta's book on competition policy.


Proving ranking using completness and transivity The book by Jehle and Reny says that one can proof the ranking of consumption bundles using the axiom of completness and transivity.
But the have not proved this in the book and I also have a hard time finding it on the internet.

Could someone be kind enough to at least give me at sketch of the proof?

Kind regards,

The exact quote from J&R is as follows:

  " [Completeness + Transitivity] together imply that the consumer can completely rank any finite number of elements in the consumption set, $X$, from best to worst, possibly with some ties."

Here's a sketch. Take a finite subset of the consumption set, $A \subset X$.

Since $A$ is finite, we can number the elements $A = \{a_1,a_2,\dots,a_n\}$. We want to rank elements in $A$.

Pick $a_1$ and $a_2$. By completeness, either $a_1 \succeq a_2$ or $a_2 \succeq a_1$. Without loss of generality, suppose $a_1 \succeq a_2$.

Now pick $a_3$. By completeness, either $a_1 \succeq a_3$ or $a_3 \succeq a_1$ and either $a_2 \succeq a_3$ or $a_3 \succeq a_2$.

(i) If $a_3 \succeq a_1$, then by transitivity $a_3 \succeq a_2$, so $a_3 \succeq a_1 \succeq a_2$.

(i) If $a_1 \succeq a_3$, then if $a_2 \succeq a_3$, $a_1 \succeq a_2 \succeq a_3$. Otherwise $a_3 \succeq a_2$ and thus $a_1 \succeq a_3 \succeq a_2$

Then go to $a_4$ and repeat.

Since $A$ is finite, the process ends, and you have a ranking of elements in $A$.


Financial economics: Bond Price I know that if the coupon rate on a bond is larger than the yield-to-maturity, then the price must be higher than the par value. Yet I have the bond price equation $P=\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\frac{Par}{(1+y)^T}$, and I wonder how I manipulate this equation to show that what I mentioned above is correct. Grateful for any tips to solve this.

Set the inequality $P>Par$, substitute the bond price equation, and then manipulate the inequality to find your result:

$$P>Par$$
$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\left(\frac{Par}{(1+y)^T}\right)>Par$$
$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)>Par\left(1-\frac{1}{(1+y)^T}\right)$$
$$\frac{c}{y}>Par$$

Let $c=r_cPar$ so that $r_c$ is the coupon rate (this is the fraction of the Par value that the coupon represents. Then the last equality is equivalent to

$$r_c>y$$

Therefore, the first inequality is true if and only if all of the following inequalities are. In particular, the last inequality gives the result you are looking for.


Is the following graph possible? Is the following graph possible?

I've attempted to sketch the TC/TR curves for it but they don't seem to be able to satisfy the following two properties at the same time:

1) Slope of Total Cost (TC) being less than the slope of Total Revenue (TR) until q=b, that is until quantity is at the zero marginal profit level.

2) TC being lesser than TR at first but going on to be greater than it after q=a, that is after quantity is at the zero economic profit level.

Your basic assumptions seem to be
$$
p(q_a) = AVC(q_a)
$$
and as the difference between price and average variable cost is decreasing before $q_a$ and increasing after it you also assume
$$
\frac{d \ \left(p(q) - AVC(q)\right)}{d \ q} < 0.
$$
(More on the necessity of this assumption later.)
 We can rewrite this second assumption to
$$
\frac{d \ p(q)}{d \ q} - \frac{d \ \frac{VC(q)}{q}}{d \ q} = \frac{d \ p(q)}{d \ q} - \frac{MC(q) \cdot q - VC(q)}{q^2} < 0.
$$
Multiplying by $q>0$ yields
$$
\frac{d \ p(q)}{d \ q} \cdot q - MC(q) - AVC(q) < 0.
$$
Adding $p(q) + AVC(q)$ we get
$$
\frac{d \ p(q)}{d \ q} \cdot q + p(q) - MC(q) < p(q) - AVC(q).
$$
The left hand side is now $MR(q) - MC(q)$. It follows from our two initial assumptions that
$$
\forall q > q_a: \ p(q) < AVC(q).
$$
Then for such values of $q$
$$
MR(q) - MC(q) < p(q) - AVC(q) < 0.
$$
Because of this there can be no $q_b > q_a$ such that $MR(q_b) = MC(q_b)$.

On the necessity of the second assumption:
Seems to me that without this assumption the goal function $p(q) \cdot q - C(y)$ would not be concave and hence $MR(q) = MC(q)$ would not be a sufficient condition for optimum.


Theory of Optimal Currency Areas Should a country be more inclined or less inclined to join a currency union (common currency) under the following events:
a) Increase in the size and frequency of AD shocks to the economy.
b) Increase in the size and frequency of money demand shocks to the economy.
c) Increase in the cultural similarities between the country and the country or region it is joining.
d) Decrease in capital mobility restrictions combined with an increase in immigration restrictions.

In case a, I feel as though it depends on whether or not the AD shocks are similar between the joining regions; if they are, then they would be more inclined, but if they're not, then otherwise.
In case b, I think the country would be less inclined since they would have less control over monetary policy.
In case c, I would say more inclined because it would encourage labor movement.
Finally, in case d, I would say less inclined because, although capital would flow freely, a decrease in labor flows would be worse.
What are your thoughts on these four cases?

a) Increase in the size and frequency of AD shocks to the economy.>

More inclined would encourage more demand from other regional countries, which in turn supports the shortage of domestics demand

  b) Increase in the size and frequency of money demand shocks to the economy.>

Less inclined would be better since independent currency regime provides local goverment (ie. the central/state bank) more flexibility in controling supply and demand money in order to deal with money demand shocks

  c) Increase in the cultural similarities between the country and the country or region it is joining.>

More inclined should be a great idea in this case as countries in that region could exchange labor and goods at lower relative cost compared to remain local currency regime

  d) Decrease in capital mobility restrictions combined with an increase in immigration restrictions.>

More inclined to common currency is my answer. Common currency is not the same as establishing an OCA (Optimal Currency Area) so immigration is a less-crucial criteria than capital mobility in considering a decision of commmon currency.


Expected Income Question An urn contains equal number of green and red balls. Suppose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the urn, and the selection process is repeated. Each time a green ball is picked you get 1 Rupee. The first time you pick a red ball, you pay 1 Rupee and the game ends. Your expected income from this game is..

The answer given is 0 but shouldn't it be positive as the as the income rises if you consecutively draw the green ball? Please help me clarify this doubt.

Think of all the possible outcomes. For example, if your first ball is red you loose 1 rupee (this has probability 1/2), if you instead get one green ball and then one red ball you end up with  0 rupees (this has probability 1/4), if instead you get at first 2 green balls and then a red one, you end up with 1 rupee (this has probability 1/8), etc.

The expected value is

$$E(V)=(-1)\frac12+(0)\frac14+(1)\frac18+(2)\frac1{16}\dots=\sum_{n=1}^\infty(n-2)\frac{1}{2^n}=0$$

The last equality comes from two well-known results: $\sum_{n=1}^\infty\frac{n}{2^n}=2$ and $\sum_{n=1}^\infty\frac{1}{2^n}=1$


Finitely repeated Prisoner’s Dilemma with switching cost I'm doing this finitely repeated Prisoner's dilemma with switching costs but I have trouble showing the fact that $\varepsilon$ had to be $1 < \varepsilon < 2$. I do see why and  that it is a subgame-perfect Nash equilibrium, but how do I show it.
My approach:
Consider deviations by player 1, given that player 2 sticks to his strategies in his subgames following histories that end in each of the four outcomes of the game. (By symmetry the below would also account for deviations by player 2 if player 1 sticks to his strategies).
We know that this game is played $T$ times. Let $m$ denote the number of cooperation stages (i.e. number of times the strategy profile $(C,C)$ has been played), let $n$ denote the number of punishment stages (number of times the strategy profile $(D,D)$ has been played), and let $T = m + n +1$. If the players cooperate $T$ times they get $3$ in payoffs. Thus, we can write
$$3T=3(m+n+1)$$
If player 1 deviates at any stage of the game he'll get a payoff of
$$3 \cdot m + (4 - \varepsilon) + 2 \cdot n$$
For the deviation to be profitable for player 1 the deviation payoff has to be greater than or equal to the cooperative payoff. That is,
$$3 \cdot m + (4 - \varepsilon) + 2 \cdot n \geq 3m + 3n + 3$$
We can simplify this
$$n \leq 1 - \varepsilon$$
And this is where I'm stuck. What am I supposed to interpret from this? That when $\varepsilon \in (1,2)$ then the number of punishment stages is negative?
Another question I have is: Assume that the question above got answered. Is that enough? Or do I need to show something more? Because I'm well aware that playing $(C,C)$ $T$ times is a SPNE and playing $(D,D)$ $T$ times is also a SPNE and either way there is no profitable deviations.
So, to summarize: I would like to know if I'm approaching this problem in the right way. I would like to get some help on how to move forward from where I got stuck. And if the above would be enough to show what the problem is asking for.
Many thanks in advance!
[Update]----------------
Below is my updated attempt on a solution with some inspiration from M.J. Osbourne and the feedback I got from @Herr K..

A couple hints.

Regarding the lower bound on $\epsilon$: What happens if deviation occurs at stage $T$? In other words, there is no opportunity for your so-called "punishment stages".
Regarding the upper bound on $\epsilon$: Suppose player 2 deviates at stage $T-1$ but player 1 does not. What must be true about $\epsilon$ in order for player 1 to have an incentive to switch to $D$ at the very last stage?

In general, to show that the given strategy profile is subgame perfect, you need to argue that playing $C$ after $(C,C)$ and $D$ after either $(C,D),(D,C),(D,D)$ are optimal.


Leontief function marginal product of labor/capital Find marginal product of labor of a Leontief production function.  For example,
$$f(L, K) = min\{\frac{L}{a}, \frac{K}{b}\}$$

MY ATTEMPT

Now as I understand it the marginal product of labor, $MP_L$ of this function cannot be found in the "normal" way because this function is not differentiable.  However, by definition, marginal product of labor means the extra output when one puts in an additional unit of labor.  We assume capital stays constant.

Say, we produce originally $q$ units of output.

Thus, if $\frac{L}{a}+\Delta L < \frac{K}{b}$ where we are increasing the quantity of labor, then we are on the vertical portion of the isoquant (Labor on the horizontal axis, Capital on vertical).  Thus, we would increase our output by $q+\Delta L$.

If $\frac{L}{a}+\Delta L>\frac{K}{b}$, we would still be producing $q$ units.

Since you are interested in labour, let's assume for simplicity that the stock of capital is fixed at $\bar{K}$. Then, the optimal choice of capital and labour is given by:

$$\frac{L^*}{a}=\frac{\bar{K}}{b}$$

Therefore, optimal labour is:

$$L^* = \frac{a}{b}\bar{K}$$

The marginal product of labour depends on how actual labour relates to optimal labour:

Case 1: $L = L^*$. In the standard Leontief diagram, with $L$ in the horizontal axis and $K$ in vertical axis, this is any point on the optimal path (which function starts at the origin and has slope $\frac{b}{a}$). In this case, $\dfrac{dQ}{dL}=0$.
Case 2: $L > L^*$. This is when the factors' combination is below the $\frac{b}{a}$ path. In this case, $\dfrac{dQ}{dL}=0$.
Case 3: $L < L^*$. This is when the factors' combination is above the $\frac{b}{a}$ path. The solution here depends on how far $L$ is from $L^*$:

Case 3a: $L^*-L > 1$. This is perhaps the most likely case. Here, the change in labour leaves still with low levels of labour. In this scenario, $MP_L = \dfrac{1}{a}$. This results comes from comparing output before the change in labour: $Q_0=\dfrac{L_0}{a}$ versus after the change: $Q_1=\dfrac{L_0 +\Delta L}{a}$. From here, we conclude that $\dfrac{dQ}{dL}=\dfrac{1}{a}$. Notice that in this case, there is still room for increasing output by increasing labour (i.e. we are still within Case 3).
Case 3b: $L^*-L = 1$. Here, the change in labour leaves us on the optimal path. The change is just as in Case 3a. The differences is that increasing labour further leaves without increases in output. We are then back to case 1.
Case 3c: $L^*-L < 1$. Here, the change in labour is more than what we actually require. Thus, we move from being in Case 3 to being in Case 2. The change is therefore not $\dfrac{1}{a}$, but equal to $\dfrac{1}{a} \times (L^*-L)$. In other words, the change is proportional to the exact amount of labour input we need to be in the optimal path.

A graphical example can be seen below:

For any labour equal or above $L_0$, further increases in $L$ does not change $q_0$. However, when we are above the optimal path (meaning $K>\bar{K}$), the extra marginal worker does add to production. You can see that depending on how far this point is from the optimal, the three scenarios of Case 3 arise.

If interested (in my personal quest in favour of Open Economics), here is the R code of the graph:

plot(c(0,5), c(0,5), type = "n", xlab = "Labour", ylab = "Capital", yaxt='n', xaxt='n', bty='l', mgp=c(1,1,0))
segments(0, 0, 4.7, 4.7,lwd=2)
text(4.9, 4.8, expression(frac(b,a)),cex = 1.3)
points(2, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
segments(2, 2, 2, 5,lwd=2.5)
segments(2, 2, 4.7, 2,lwd=2.5)
text(4.9, 2,  expression(q[0]),cex = 1.3)
text(2.4, 1.75, expression(paste("(",L[0],",",bar(K),")")),cex = 1.3)
points(3.5, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(3.5, 2.3, expression(L>L[0]),cex = 1.3)
points(2, 3.5, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(2.4, 3.5, expression(K>bar(K)),cex = 1.3)


Help needed to understand proof in Myerson and Satterthwaite (1983) The following image is an excerpt taken from Myerson and Satterthwaite's 1983 paper Efficient Mechanisms for Bilateral Trading:

I know what the theorem is all about, all I need is some help with the integrals
 (especially the one at the second equality sign).

I'm assuming by "at [the] second equality sign" you mean the translation of the right-hand side of the first equality sign to the right-hand side of the second equality sign. This can be handled by analyzing each integral separately.

Notice that in the first (double) integral the left-hand side of the inner integral does not vary with $v_2$, so

$$\int_{a_1}^{\min\{v_2,b_1\}} \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] f_1\left( v_1 \right) dv_1 \\\hspace{3cm}= \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] \int_{a_1}^{\min\left\{ v_2, b_1 \right\}} f_1\left( v_1 \right) dv_1.$$

The upper bound of the right-hand integral is just there to cap the CDF, so $\int_{a_1}^{\min\{ v_2, b_1 \}} f_1( v_1 ) dv_1 = F_1( v_2 )$, taking as given that $F_1( x ) = 1$ for all $x \geq b_1$. In terms of the original equation this gives

$$ \int_{a_2}^{b_2} \int_{a_1}^{\min\{v_2,b_1\}} \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] f_1\left( v_1 \right) dv_1 dv_2 \\\hspace{3cm}= \int_{a_2}^{b_2} \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] F_1\left( v_2 \right) dv_2. \hspace{2cm} \text{(Term L)}$$

Notice that the inner integrand in the second (double) integral can be written as

$$v_1 f_1\left( v_1 \right) + F_1\left( v_1 \right) = \frac{d}{dv} \left[ v F_1\left( v \right) \right]_{v = v_1}.$$

Since $a_1 F_1( a_1 ) = 0$ (because $F_1( a_1 ) = 0$) and $x F_1( x ) = x$ for all $x \geq b_1$, this gives

$$\int_{a_2}^{b_2} \int_{a_1}^{\min\left\{ v_2, b_1 \right\}} \left[ v_1 f_1\left( v_1 \right) + F_1\left( v_1 \right) \right] dv_1 f_2\left( v_2 \right) dv_2 \\ \hspace{3cm} = \int_{a_2}^{b_2} \min\left\{ v_2 F_1\left( v_2 \right), b_1 \right\} f_2\left( v_2 \right) dv_2. \hspace{2cm} \text{(Term R)}$$

The second equality is $\text{(Term L)} - \text{(Term R)}$.


Price and quantity in equilibrium I have a market with 1000 suppliers.

The total demand can be described by

$$
Q=-300000 P + 6000000
$$

while the total supply can be described by

$$
Q=200000 \sum(MC^2)-200000
$$

What is the price and quantity in equilibrium?

I have no idea what to do with $\sum(MC^2)$ in the supply.

I guess I should isolate $P$ in the first equation:
$$
P(Q) = 20-\frac{1}{300000} Q
$$
but I don't know what to do now.

Hint: To find equilibrium price and quantity, you want to find when the system is in equilibrium. In equilibrium, quantity demanded should equal quantity supplied.

$$Q_d = Q_s \implies -300,000P + 6,000,000 = 200,000 \sum_{i=1}^n (c_i'(q_i))^2 - 200,000$$

Where $c_i(q_i)$ is the cost function for firm $i$ with respect to $q_i$, firm $i$'s production. I'm assuming when you wrote $MC$ you meant marginal cost.

$$\implies 6,200,000 - 200,000\sum_{i=1}^n (c_i'(q_i))^2 = 300,000P$$

$$\implies P = \frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c_i'(q_i))^2$$

Now that we have an inverse demand function with respect to $q$, we have to solve the firms' maximization problem. Since there's no information you've given to us directly about that, let's assume technology is homogeneous and that firms compete via Cournot competition.

$$\Pi_j = q_j(P(q)) - c(q_j)$$

$$\Pi_j =  q_j(\frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c'(q_i))^2) - c(q_j)$$

Now take the derivative w.r.t $q_j$ and set to zero to find the maximum.

$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \sum_{i=1, i \neq j}^n (c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$

Where that derivative is evaluated with the product and chain rule.

From here, you solve for $q_j$ as an expression of all $q_{i, i \neq j}$. Since the cost functions are homogeneous, it shouldn't be that bad to do. Also remember that the number of suppliers is 1000, so the problem can be further simplified:

$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \cdot 999(c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$

$$\implies \Pi_j' =  (\frac{62}{3} - 666(c'(q_i))^2) + \frac{2}{3} (2c'(q_j) \cdot c''(q_j) \cdot q_j + c'(q_j)^2) - c'(q_j) = 0$$

and then you solve for $q_j$ in terms of $q_{i, i \neq j}$ and do the rest of the Cournot derivation.


Does multicolinearity of variables imply complementary inputs? I've been thinking about how to answer this question How to econometrically identify perfect complements in production? and may think that multicollinearity has something to do with identifying such a process.

If I have a regression such that:

$$y=\beta_0+\beta_1x_1+\beta_2x_2+\mu$$

where $x_1$ and $x_2$ are correlated with each other.

does that imply that some function of the nature $\min\{x_1,x_2\}$ is present in the determination of $y$?

A Leontief production does imply perfect multicollinearity between the inputs, since there is no substitutability between them. Thus, if a firm's production technology features non-substitutability between inputs, one would observe the firm choosing inputs in fixed proportions.

However, (perfect) multicollinearity does not necessarily imply non-substitutability. For instance, the Cobb-Douglas production technology of the form $y=x_1^{0.5}x_2^{0.5}$ is also consistent with a firm choosing inputs in fixed proportions. So the answer to your question is no (from a theoretical perspective).


What is the purpose of monetary expansion during an economic lockdown? Recently, most developed countries central banks substantially lowered their base interest rate. In Australia, this is now at their stated lower bound of 0.25%. We also have QE. This is all in an attempt to stimulate the economy.

At the same time, we are going through a lockdown which shuts down large swathes of the economy.

What is the purpose of monetary expansion in this environment? It is clearly purportedly stimulatory, but with consumers so constrained, can it work in a meaningful way? Is the impact in financial markets sufficient?

What is the purpose of monetary expansion in this environment?

Avoiding (and preventing future) supply and demand shocks

The purpose is to keep as much as possible markets (labor related, of goods and services, etc...) in their pre-crisis configuration -- imperfectly admittedly -- with the intention of avoiding and preventing future post-crisis supply and demand shocks, which empirically translate into (out of control) inflation (positive or negative: deflation).

Doing it well

But when I say "supply" or "demand", I am not only talking about final demands and final supplies. I am talking about supply chains in their entire verticality, i.e. which is related to any intermediate consumption (and related market). And you do not want these intermediary markets to be strongly shocked either, since any shock in those would then propagate to final consumers. As outlined previously, the labor market is a intermediary market like any other from this perspective.

This is where decision makers actually have their role to play, because more than supporting the economy, they have to do it well. Imagine that:

They maintain final demands without adequatly maintaining intermediate markets $\implies$ inflation (which is what happened the most historically), potentially hyper depending on how strong and badly they do it
The net effect of their stimulus boils down to supporting more supplies than demands $\implies$ deflation...

In coordination

Moreover, in a globalized and highly interconnected economy, you may have control on your domestic product, but clearly not on foreign providers whose economic health depends on foreign monetary policies. Hence the need for a coordinated central bank action.

In conclusion and to say it simplistically, what would happen after the crisis if agents use their savings to buy consumer goods to the last company that didn't go bankrupt ? $\implies$ hyperinflation + social tensions.

Targeting a V-shape recovery

  [...] with consumers so constrained, can it work in a meaningful way?

In Germany or in France, partial unemployment -- financed by governments at virtually no (short-term) cost thanks to QE -- is a way to maintain both final demands and intermediary consumptions of productive factors (such as labor force) while favouring a V-shape activity recovery:

As of April 30 2020, 11 million French workers are currently temporarily or partially unemployed. Also, almost 650,000 businesses had applied for Germany's reduced-hours work scheme by April 6th. In the USA, Trump's helicopter money is a manner to protect final demands' purchasing power and limit the number of bankruptcies.

Idem for China, who cut interest rates to be more powerful in supporting the economy, announced on February 5, 2020, 16 preferential policies to sustain the production and operation of SMEs including policies on taxation, finance, social security, subsidies, and rent reduction). China's V-shape recovery follows:

  Is the impact in financial markets sufficient?

Sufficient to what ? As Krugman says:

  I'm a bit disappointed that so many people — including financial reporters! — still assume that the stock market is telling us something about real economic prospects [...]

Not related to Economic Science as such

Then, something you could ask is "why not rely on private capital?" Economic theory stops here. And it all depends on your ideology.


Does the duality of utility maximization and cost minimization hold in practice? I recently learned about the relationship between utilization maximization and cost minimization.

Are there studies on whether this duality holds in real life?

Any information on this topic for a beginner to consumer choice theory would be appreciated. Any tips on key things to consider when starting to learn about the topic would also be helpful.

There is nothing that can "hold in practice".

Utility maximisation and expenditure minimization are theoretical framework that are used to model behavior  by factors that are thought to motivate choices.

This is useful if they approximately or on average correlate with how those factor influence choices. For example, wage will surely be a major factor in the labor supply of an individual.

But utility cannot be measured and arguably does not exist. Both approaches rely on either a utility function or a level of utility, so neither can be practically verified.


Do people really care about higher moments? I have heard that people often handle uncertainty using point estimates not probability distributions. However, I have been unable to find any evidence for this in the heuristics and biases literature. I was wondering if anyone could point me to some relevant papers?

To clarify, there is no question whether some people have some beliefs about the higher moments of distributions. However, it would not shock to me to learn that when (for example) considering by how much the stock market will rise tomorrow, people form a point estimate (perhaps using historic averages) but do really not think about the uncertainty surrounding their estimate.

Perhaps there is some evidence toward your claim, but I would argue that in most situations, people do not use point estimates (although some "smoothing" likely occurs). In particular, there is no doubt that people care about the second movement (variance). The commonly employed, and robustly empirically documented, notion of risk aversion captures exactly this. Given two risky prospects with the same expectation (say different assets) it seems that most people prefer the one with lower variance (i.e., the less risky prospect). Think: which would you chose when given the option between \$50 for sure or a 50/50 toss up between \$0 and \$100? The difference between these two options is entirely in their second moment.

The evidence for this is so vast and varied it is hard to even give an encompassing reference; but since you ask about the stock market, it is worth pointing out that one of the key explanations for why stocks have a higher average return than bonds is that they are more risky. I imagine a google scholar search of 'risk aversion in the stock market" will be fruitful.

It is worth noting that there is also empirical evidence that people care about the third moment (skewness). This working paper by Yusufcan Masatlioglu, A Yesim Orhun, and Collin Raymond is an experiment in which subjects

  reveal a strong preference for positive skew over negative skew; in other words for ruling out more uncertainty about the desired outcome (and tolerating uncertainty about the undesired outcome) compared to ruling out more uncertainty about the undesired outcome (and tolerating uncertainty about the desired outcome).

As for even higher moments, the jury is out, but I would imagine their importance diminishes rather quickly. The complexity of the distributions needed to generate differential higher moments increases and I would think heuristics would start to kick in.


Why will inflation raise sellers' purchasing power? According to my textbook, in the chapter on inflation,

"When a price rises, buyers of that good must pay more, but sellers get more revenue when they sell it. The loss in buyers' purchasing power is matched by the rise in sellers' purchasing power."

I understand the part of falling purchasing power for buyers (since each dollar can now buy less compared to previously), but why so for the seller? For instance, suppose each orange was sold for 1 dollar in the past. The price then rose to 2 dollars (reflecting the rise in general price level in the economy) - but if 2 dollars post-inflation is worth the same as 1 dollar before the price rise, then there really is no change in the sellers' purchasing power right? When the seller uses the 2 dollars to buy other things, it will still fetch him the same as 1 dollar did previously!

I'm not sure if I'm missing something here, and would really appreciate some clarification, thank you!

One price relative to others
The key parts of the statement "When a price rises, buyers of that good must pay more, but sellers get more revenue when they sell it. The loss in buyers' purchasing power is matched by the rise in sellers' purchasing power." are "a price" and "that good".
If the price of oranges increases from \$1 to \$2, then it does not imply that 2 dollars post-inflation is worth the same as 1 dollar before the price rise; we could expect some devaluation, but it would be proportional to the impact of oranges in the whole economy.
Oranges increasing in price means that the purchasing power of people who buy oranges is decreased, but the purchasing power of people who sell oranges is increased, since they can buy more stuff (e.g. apples) from their orange-derived revenue than before.


Why are bond yields so low when Fed is cutting rates? Fed cut its rates for the first time in 10 years in July and it is about to cut the rates at least by 25 basis points during its September meeting.

Usually, when Fed cuts the rates the bond yields should go up as the economy will expand instead of contracting. But why are the yields of the bonds so low?

Perhaps you are confusing bond prices with yields? When bond prices go up, payments relative to the price the bonds are purchased will go down, and thus by definition yields go down.

The Fed mainly controls rates through open market operations, and more recently quantitative easing. In short, when attempting to cut rates, the Fed purchases securities such as US government bonds and in turn injects Federal Reserve Notes into the system. This has the two simultaneous effects: it raises the prices of bonds purchased and thus lowers their yields, and increases the money supply, lowering interest rates.


Principal Components Analysis in Economic analysis I have a data set of annual prices of various energy carriers over several decades. I want to estimate a model of output using these price series - I guess, I will do a co-integration analysis. There appears to be a high positive correlation among the price series.

Would it be appropriate to do a PCA and use the first component as an
explanatory variable assuming it captures most of the variation among
the price series?
How well is the application of PCA accepted in
empirical research in economics?

I know this is a old post, but since I found it and I know the answer I might as well share it.

I would go further than the first component, but it also depends on how much of the variance is actually explained within the component. There is two approaches that I use when determining which components I am using.

The rule of thumb is to use and component with a eigen value greater than one.
Keep an eye on the cumulative variance, for example I have a data set of 150 macro indicators and the first ten can explain 96% of the variance for a particular problem I was attempting to forecast, if I extend the components out to 20 it explains 96.7% of the variance. Is the ten additional factors really relevant?   In this case I would stick to the first 10, but again it all depends on your individual analysis.

To answer your second question, yes..

PCA is process commonly used in Factor Investing, or smart beta.. Google Smart beta and see how many firms are approaching it. As far as strait economics, the answer is still yes. I have seen many working papers published by the Fed that references PCA. It certainly does not hurt to include other methods for reducing high dimensional data. Generalized boosted models comes to mind, it uses a tree based approach. Another would be the minimum redundancy maximum relevance model. Probably the best method would be a combination of models using different machine learning and ensemble modeling.

Either way, I hope this helps even if I am answering the question 3 years after it was posted. Good luck solving the worlds problems with econometrics.

Cheers,
Aaron


What does it mean when the natural rate of unemployment is greater than the actual rate? According to this graph:

I just don't get when natural unemployment is greater than actual unemployment.

Why is this the case?

Natural unemployment is the 'expected equilibrium' rate. In the same way unemployment can go above the natural rate (usually when the economy is in a downturn) it can go below (usually when an economy is doing well).

An economy could report actual unemployment below the natural rate in multiple circumstances, one such being a shortage of labour across the economy. (usually due to an economic upturn). In that case employers may try to reduce the time workers take to switch jobs and they might be temporarily less stringent in their hiring requirements causing some 'structurally unemployed' workers to get picked up. To cope with the labour shortage, firms may also encourage retired or serially unemployed workers who have stopped looking for jobs (who in both cases are excluded from the labour force) to gain employment thus these people may go from not being counted in unemployment statistics to increasing the population of employed workers and therefore influencing the unemployment rate downward.

Another effect that should probably be considered is that real unemployment can be directly measured via surveys and the natural rate is estimated indirectly based on assumptions of what the equilibrium rate of unemployment 'should' be. There is no direct way to validate whether or not a particular methodology is 'the most correct' and when we are presented with only a graph like this we are trusting that the person compiling the graph did their work correctly. If their assumptions were incorrect then the graph is wrong and there isn't much point pulling our hair out to figure out an explanation for the results of the graph. In your case it looks like the graph is coming from a textbook so you're probably expected to believe that the calculation is reasonable and the answer is something lile the above. In real life graphs tend to have hidden assumptions that very well may be wrong (either intentionally or incompetently) so if something doesn't look right it's almost never a bad idea to start asking questions to validate if the result is truly different from your intuition or if somebody made a bad assumption.


Are direct mechanisms always truthful? I am very new to mechanism design so I am a bit lost among the concepts, even the most basic ones.

One thing that is not clear to me is the concept of direct mechanism and its relationship with the revelation principle.

From what I got a direct mechanism is a mechanisms, in which, instead of messages, the agents report their preference/type. Are we assuming this is a truthful report?

I am asking that because the revelation principle is stated as "anything that can be accomplished by any mechanism can actually be accomplished by a direct revelation mechanism that is individual rational and incentive compatible" in this notes but as "a social choice function can be implemented by an arbitrary mechanism (i.e. if that mechanism has an equilibrium outcome that corresponds to the outcome of the social choice function), then the same function can be implemented by an incentive-compatible-direct-mechanism (i.e. in which players truthfully report type) with the same equilibrium outcome (payoffs)." on Wikipedia.

So I don't get if truthfulness is implied by the direct mechanism or if it is an additional requirement.

Incentive compatible here means that you have no incentive to lie. A direct mechanism that is not incentive compatible would entice some agents to report a false type.


Does the marginal product need to be diminishing to have a diminishing average product? We have the rule 'If the marginal is less than the average, then the average declines'.

So if our production function for an input x and output y is concave, does that mean that MP must be diminishing given AP is diminishing?

Suppose the total product is given by $f(x)$. Thus, the average product is $$AP(x)=\frac{f(x)}{x}.$$

Differentiating:
$$AP'(x)=\frac{x f'(x)-f(x)}{x^2},$$

which has the same sign as $x f'(x)-f(x)$. This is definitely negative if $f''(x)<0$ (i.e., if $f$ is concave).

But it can also be negative, even if $f$ is locally convex. Take a look at this figure:

At the point marked with a blue disc: the slope of the red line is $f'(x)$. The slope of the black line is $f(x)/x$. We can see geometrically that $f(x)/x>f'(x)$, or (after rearranging) that $f'(x)x<f(x)$.

Thus, at the point indicated by the blue disc we know that $AP'(x)<0$. But at this point we can also see in the figure that $f''(x)>0,$ implying that the marginal product is increasing.

Since we have a counter-example, we conclude that a decreasing marginal product is not a necessary condition for a decreasing average product.


Bayes Correlated Equilibrium: obedience I would like your help to understand the definition of obedience in an incomplete information game at p.7 of this paper.

Let me summarise the definition provided in the paper.

There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.

There is a finite set of states $\Theta$, with $\theta$ denoting a generic state.

A basic game $G$ consists of

for each player $i$, a finite set of actions $A_i$, where we write $A\equiv A_1\times A_2\times ... \times A_N$, and a utility function $u_i: A\times \Theta \rightarrow \mathbb{R}$.
a full support prior $\psi\in \Delta(\Theta)$.

An information structure $S$ consists of

for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.
a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.

A decision rule of the incomplete information game $(G,S)$ is a mapping
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

One way to mechanically understand the notion of the decision rule is to view $\sigma$ as the strategy of an omniscient mediator who first observes the realization $\theta\in \Theta$  chosen according to $\psi$ and the realization $t\in T$ chosen according to $\pi(\cdot|\theta)$; and then picks a probability distribution from $\Delta(A)$, and privately announces to each player $i$, her lottery to play.

For players to have an incentive to follow the mediator's recommendation in this scenario, it would have to be the case that the recommended lottery was always preferred to any other lottery
conditional on the signal $t_i$ that player $i$ had received.

This is reflected in the following "obedience" condition.

Definition: The decision rule $\sigma$ is obedient for $(G,S)$ if, for each $i=1,...,N$, $t_i\in T_i$, and $a_i\in A_i$, we have
$$
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_i,t_{-i}| \theta) \sigma(a_i, a_{-i}|t_i, t_{-i}, \theta) u_i(a_i, a_{-i},\theta)
$$
$$
\geq
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_i,t_{-i}| \theta) \sigma(a_i, a_{-i}|t_i, t_{-i}, \theta) u_i(\tilde{a}_i, a_{-i},\theta)
$$
$\forall \tilde{a}_i\in A_i$.

My question:

I don't understand fully the LHS (or, equivalently, the RHS) expression
$$\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_i,t_{-i}| \theta) \sigma(a_i, a_{-i}|t_i, t_{-i}, \theta) u_i(a_i, a_{-i},\theta)$$

Saying it in words, this is the sum over $A_{-i}, T_{-i}, \Theta$ of
$$
\text{[Prob that the mediator observes $(\theta, t)$]}\times\\
\text{[Prob that the mediator suggests players to play $a$ under $\sigma$,  given that he observes $(\theta, t)$]}\times\\
\text{[Profit that player $i$ gets from obeying the mediator, given that the other players obey too]}
$$

Let the product just written be denoted by $(\star)$.

Why in the second component of $(\star)$ we DO NOT condition on $a_i$ by taking
$$
\text{[Prob that the mediator suggests the other players to play $a_{-i}$ under $\sigma$, given that he observes $(\theta, t)$ and given that he suggests player $i$ to play $a_i$]}
$$
in place of
$$
\text{[Prob that the mediator suggests players to play $a$ under $\sigma$,  given that he observes $(\theta, t)$]}
$$

To understand the "Obedience" inequality, notice that the player is "integrating out" everything that she is uncertain about, this is why the sum runs over the actions of other players, $a_{-i}$, the types of other players, $t_{-i}$, and the state of the world, $\theta$.

The first term simply gives the probability that the mediator observed $\theta$.

For the second term, note that it is proportional to the conditional probability of the other players being of type $t_{-i}$ when you are of type $t_i$ since $\pi(t_{-i}|t_i,\theta)=\frac{\pi(t_i,t_{-i}|\theta)}{\sum_{t_{-i}}\pi(t_i,t_{-i}|\theta)}$.

Similarly for the third term, you probably expected there to see the conditional probability $\sigma(a_{-i}|a_i,t_i,t_{-i},\theta)$, but this term is proportional to $\sigma(a_i, a_{-i}|t_i,t_{-i},\theta)$.

Technically speaking the agent should multiply his payoff by the probability that others play $a_{-i}$, are of type $t_{-i}$ and the state is $\theta$ conditional on being told to play $a_i$ when being of type $t_i$, so the LHS should read:

$$\sum_{a_{-i}\in A_{-i}, t_{-i}\in T_{-i}, \theta\in\Theta}u_i((a_i,a_{-i}),\theta) \frac{\sigma((a_i,a_{-i})|(t_i,t_{-i}),\theta)\pi((t_i,t_{-i})|\theta)\psi(\theta)}{\sum_{a'_{-i}\in A_{-i}, t'_{-i}\in T_{-i}, \theta'\in\Theta}\sigma((a_i,a'_{-i})|(t_i,t'_{-i}),\theta')\pi((t_i,t'_{-i})|\theta')\psi(\theta')}$$

And the RHS will be identical except for replacing $a_i$ with $\tilde a_i$ in the utility function. Clearly, the denominator is constant over the sum and so can be simplified.

Note: In previous versions on the paper the authors were more explicit about this derivation.


Why $nominal$ interest increases when inflation is expected to increase? Why is it that if lenders expect higher inflation in years to come, they will most likely increase $nominal$ rates of interest opposed to $real$ rates of interest?

Better look at the Fisher rule (Fisher equation).
nominal interest rate (i) = real interest rate (r) + inflation (pi) (i = r + pi)


How often do banks borrow from the Federal Reserve? Though I've seen plenty of articles talk about in what circumstances banks borrow from the Fed, I haven't seen any quantification on that note. On average, what percent of the money that banks lend out is money from the Fed? Alternatively, how much money is borrowed from the Fed every year and how quickly is that money generally paid back?

When banks borrow from the Federal Reserve they can do so through the discount windows:

  The discount window helps to relieve liquidity strains for individual
  depository institutions and for the banking system as a whole by
  providing a reliable backup source of funding. Much of the statutory
  framework that governs lending to depository institutions is contained
  in section 10B of the Federal Reserve Act. The general policies that
  govern discount window lending are set forth in the Federal Reserve's
  Regulation A. As described in more detail below, depository
  institutions have access to three types of discount window
  credit--primary credit, secondary credit, and seasonal credit. All
  discount window loans must be collateralized to the satisfaction of
  the lending Reserve Bank.
  Discount Window Lending

The FRB has published discount window lending activity since 2010. Though this data is published with a two year lag, it may be of use. You can download that data at from their website at the link above.

There is price discrimination at the window:

  In the United States, there are actually several different rates
  charged to institutions borrowing at the Discount Window. In 2006,
  these were: the primary credit rate (the most common), the secondary
  credit rate (for banks that are less financially sound), and the
  seasonal credit rate. The Federal Reserve does not publish information
  regarding institutions' eligibility for primary or secondary credit
  Wikipedia: Discount window

It is somewhat rare to use the window. There are about 8,000 banks in the United States and about four to five hundred banks use the window in a typical quarter (according to that data).

The amounts borrowed are very small. The largest amount borrowed was about 47 million in the latest report. It was borrowed by First National Bank and Trust of Rochelle, a bank with 270 million in assets, so those window loans amount to 18 percent of assets.

But that is not representative of most banks. The second heaviest borrowing bank on the list is Xenith Bank which borrowed less than two percent of assets. More generally, there is stigma for discount window lending. The overall stigma is that since only those frozen out of capital markets use the window, users of the window must be in serious trouble. As shown above, 90 percent or more of banks don't use it in a given quarter.


Why is oil being traded in U.S. dollars? I am trying to understand the concept of money,forex, depreciation of currency,devaluing,...(just the basic stuff). While I was reading about why oil is being traded in U.S. dollars, on the internet, I came upon reasons like at that time the other European currencies were collapsing, the convertibility of the dollar was suspended, some speculate the U.S. used military power to force deal with the Gulf countries to sell oil in U.S. dollars. I could'nt understand these things, why would they force to trade in U.S. dollars instead of just taking the oil from their country by force? Does both the ways give the same result? How is it even beneficial to the U.S.?

why would they force to trade in U.S. dollars instead of just taking the oil from their country by force?

One of the basic tenets of power is that $control$ is superior to $ownership$.  War is expensive, oil must be stored before it is used, and any captured production assets must be defended and maintained at great cost.

Rather than own all of this through waging successful war, it is far cheaper to simply control it.  This is accomplished precisely because the $USD is the world's reserve currency.  It is cheaper for any given sovereign nation to hold one currency in reserve, rather than to hold an adequate amount of all world currencies in reserve.  Much of US global policy in the past has been in support of the US' perceived role as a stable nation supported by rule of law, which builds the necessary trust for most other nations to put their economic "eggs" into the USD basket.

But how does control enter in? As a simple picture, any time international trade is conducted, there are three transactions that take place: the buyer converts home currency into USD; then the USD is exchanged for the goods/services; and finally the seller converts the USD into home or other currencies as required.  Thus there are two points of leverage in this transaction chain where the US can exert control, up to and including blocking transactions altogether.

Have you ever wondered how trade sanctions are actually enforced?  This is how.  Sanctioned nations are simply blocked from using USD to engage in global trade, severely curtailing their ability to transact with other nations.  In the extreme, the US is able to unilaterally exclude any nation not already holding substantial reserves of USD from participating in global trade altogether.  In a global economy, this is often a much more consequential tool than the threat of war.


Is there an economically sound way to decide between buying media or a streaming subscription? How can a consumer (like myself) decide on whether they should pay for a music streaming service based on economic reasoning?

Assumptions:

The budget allows paying for a € 10/Month music subscription with what practically is a near infinite amount of songs.
For 90% of the music, previews or ad-supported versions can be streamed for free. The streams do not offer a high-quality listening experience, but they are good enough to discover and prelisten
The user would, without a subscription, buy 10 albums a year for € 9 each.
The purchased albums can be played infinitely and passed on to family members.
The user listens to music 1 hour a day when listening to purchased albums, 1,5 a day when paying for streaming
80% of the time the user listens to music they listened to before or will listen to again

The numbers are just examples, but key is that streaming services offer near infinite libraries, whereas purchased personal libraries are always limited to a budget.

Partly your question relates to more general questions like "buy versus rent a house", or "buy versus lease a machine". Under neoclassical assumptions of  competition, full information, etc, you can imagine that arbitrage would make these options equivalent for the average of the population (or the representative agent). In practice, heterogeneous individuals might prefer one option than the other.
Having said this, I think the study that is more pertinent for your example is "The economics of subscriptions", by Glazer and Hassin (1982). They evaluate whether it is welfare enhancing for a service provider (in their case, a publisher issuing a monthly magazine), to offer both subscriptions and individual sale of issues, rather than one of them alone. Adapting their terminology and notation to the case of music, define:

$\bar p $: average value to a consumer of an album. Might vary among consumers.
$m$: number of albums available with subscription per year.
$t$: price of each album by itself (assume identical). This might include the transaction cost of buying the albums.
$c$: cost of subscription. (Assume no re-selling is allowed, an important disclaimer!).

Subscription
In this setting, the expected value of a consumer's surplus obtained by purchasing a subscription is
$$ m\bar p -c $$
Individual purchases
Here, we assume that you can choose your favorite albums. Hence, for a given population of albums, you will choose the $n$ that give you the most surplus (your favorite artists). For simplicity, I assume you do you know the universe of albums (To add more complexity, you could instead assume you know a section of it, and hence a subscription could surprise you. I leave this example to you.)
The surplus from buying the best $n$ albums (subset denoted $\Theta$) is
$$  \sum_{i \in \Theta} (\hat p + \epsilon_i - t) $$
where $\epsilon_i$ is the value above $\bar p$ of the $i$-th album you choose to buy. The above is equal to:
$$ n(\hat p - t) + \sum_{i \in \Theta}\epsilon_i $$
Hence, the valuation hinges on this comparison:
$$ m\bar p -c \text{  vs } n(\hat p - t) + \sum_{i \in \Theta}\epsilon_i  $$
Analysis
Since you chose the best albums, $\sum_{i \in \Theta}\epsilon_i  $ is large. However, these albums are also included in the subscription. Hence, define the non-negative function $X$ as the combined value of all the albums in the subscription that you would not buy outside of it. This is:
$$ X = (m - n)\bar p - \sum_{i \in \Theta}\epsilon_i $$
Notice that $\lim_{n \rightarrow m} X = 0$.
The comparison is now:
$$ X \text{ vs } c - nt $$
Thus, you prefer the subscription when:

the number of albums available under subscription is high
the albums you would not buy are very valuable
the cost of subscription is low
the cost of individual albums is high
the value of your best albums is low (if some $\epsilon <0$, then such album is below average)
the effect of $n$ depends on the value distribution (the distribution of $\epsilon$). Clearly, at some point you are buying bad albums for the same price, so you rather buy the subscription.
notice that if you do not know all the albums, then there is a stronger case for a subscription as you might discover some very good albums which value might even be higher than those you would buy otherwise. Here, a modelling of such distribution becomes crucial.
the above does not include a risk-loving valuation from risk (i.e. discovering good music!). Such consumer would then more likely favour subscription.


What's wrong with the "airline marginal cost pricing" argument? This is a very common argument found in introductory classes and texts. Here's Mankiw's version:

Thinking at the margin works for business decisions as well. Consider an airline deciding how much to charge passengers who fly standby. Suppose that flying a 200-seat plane across the United States costs the airline \$100,000. In this case, the average cost of each seat is \$100,000/200, which is \$500. One might be tempted to conclude that the airline should never sell a ticket for less than \$500. But a rational airline can increase its profits by thinking at the margin. Imagine that a plane is about to take off with 10 empty seats and a standby passenger waiting at the gate is willing to pay $300 for a seat. Should the airline sell the ticket? Of course, it should. If the plane has empty seats, the cost of adding one more passenger is tiny. The average cost of flying a passenger is \$500, but the marginal cost is merely the cost of the bag of peanuts and can of soda that the extra passenger will consume. As long as the standby passenger pays more than the marginal cost, selling the ticket is profitable.

Do airlines actually do anything like the above? I think they don't as I have frequently been on planes with mostly empty seats (even before the current pandemic).
Mankiw says the marginal cost of another passenger is "tiny". Let's say it's \$30. Then if a flight has some empty seats 24 hours before scheduled take-off, why doesn't it offer the seats for say \$50 each and fill them all up? Why is it that, in practice and contrary to the above argument, the airline will usually prefer to leave the seats empty?
What, if anything, is flawed or incomplete about the above argument?

One possible answer is that Mankiw's argument takes consumer demand for airline tickets as fixed and given. I would speculate that cheap last-minute tickets are a substitute good for regularly priced tickets, and that if enough people came to prefer them, demand for regularly priced tickets would fall and the airlines would lose revenue. In other words, instead of paying full price, too many customers would deliberately hold out for the heavily discounted tickets.
This is the same reason why so many stores destroy their obsolete/stale inventory instead of selling it cheaply or donating it, and is an example of a perverse market incentive, as it leads to unnecessary waste.


Which parts of health care can be considered market failures? The justification for public health care subsidies often involves declaring the health care sector to have market failures, for example infectious diseases being a public good and the customer having limited knowledge of the products. Are there any other aspects of health care that can be considered market failures, and under what justification?

The most common one I have considered is health insurance as a private market. It becomes a case of the "market for lemons", where in this case the `lemons' are the people seeking insurance. https://personal.utdallas.edu/~nina.baranchuk/Fin7310/papers/Akerlof1970.pdf
The simplified narrative goes something like this:
Assumptions: Say insurance companies sell to everyone who wants insurance at some fair rate, and we cannot discriminate based on pre-existing conditions. Some people are very sick, and cost a lot to insure (lowers profits for the insurer) and others are healthy (profitable for the insurer.)

As sick people, who frequently utilize the insurance buy into the market, the average rates go up.

Healthy individuals, who don't need it at much, leave the market in response to the increasing prices, which only serves to continue increasing the price.

This process of healthy people leaving and sick people joining continues until the market collapses, nearly only sick people are insured.


How time zones affects causality? Considering that markets open in different time zones, in general the opening and closing times are different.
I suppose that a stock market opening later (in the same day) receive information on the other markets opening earlier.

I think I should take into account this if I want to implement i.e. Granger causality analysis and I have daily stock indices prices. What could be a proper method to approach the problem?

I'm skeptical that this will work. I'm concerned that for the sort of shock that is large enough to be interesting and large enough to study that stock prices will move before they open. You could potentially measure this by looking at the changes in stock price from close to open, but I'm not sure what you'd be measuring. Are changes in Tokyo stock prices in any meaningful sense causing the changes in NYSE price? Or is it simply that all stock prices are moving at once, but you can't easily see the NYSE stock prices until the exchange opens?

In support of this concern, consider that there are (thin but functioning) after hours trading markets in many securities. In addition, through the use of American Depositary Receipts, it is also possible to trade US stocks on foreign stock exchanges. As such, for many or even most securities, markets are almost always open for those willing to pay higher transaction costs.

That said, there is some evidence that when stock exchanges are closed that markets are meaningfully affected. See, for example, Stock returns and the weekend effect (French 1980).  Stock Market Structure and Volatility  (Stoll and Whaley (1990)) contribute to the litterature showing the expected returns and return volatility are different when stock exchanges are closed, including at night. While both important papers, they provide dated evidence. More recent evidence from Overnight stock returns and realized volatility (Ahoniemia and Lanneb (2013)) shows that shares, while stock markets are closed, continue to show different return volatility from when stock markets are open through 2007.


When does a high markup mean high productivity? Never? I was reading this article in VoxEU, where the authors (one is a Professor of Economics at Harvard University) make the case that an increasing average mark-up has accounted for much of the late productivity growth, since it leads to an increasing allocative efficiency.

However, isn't this exactly the oposite of what one would want? It seems that they are somehow mixing up the possibility of high profits due to one would usually think of as high productivity, with high profits due to market imperfections... Also, it's also as if the authors would defend a higher 'productivity' but at a lower output, than a lower 'productivity' at a higher output.

Here's a quote from the text which I find baffling:

  «(...)this reallocation of market share from low- to high-mark-up
  firms has considerably improved the allocative efficiency of the US
  economy over the last 20 years. In fact, improvements in allocative
  efficiency account for about 50% of the cumulated growth in aggregate
  productivity during this period. This suggests that ‘true’
  technological productivity growth has been even slower than previously
  imagined.(...)A high average mark-up reduces output and depresses the
  demand for labour and capital, generating low aggregate employment and
  low aggregate investment. It reduces the aggregate labour and capital
  shares, and increases the aggregate profit share.»

allocative efficiency = Using resurces where they add the most value.

Example:
The resources needed to create a branded t-shirt (or any other product with a strong brand) is the same as for the creation of a no-name t-shirt. However, the price/value of the branded product is considerably higher.

So, if you can switch production from cheap no-name products to more expensive branded products then output value increases but resource consumtion is unaffected. This value increase is separate from the resource reduction caused by an increase in productivity.

  However, isn't this exactly the oposite of what one would want?

Depends on who "one" is. For american companies it is good, since it increases their profits. For the resource producers (seamstresses, cotton farmers etc) it is bad since it reduces the demand for their resources.

  It seems that they are somehow mixing up the possibility of high profits due to one would usually think of as high productivity, with high profits due to market imperfections...

That depends on whether you consider branding a market imperfection. When measuring it is very difficult to distinguish quality and branding. If a no-name t-shirt costs \$6 and branded t-shirt costs \$30, how much of the price difference is due to quality and how much is brand name?

  Also, it's also as if the authors would defend a higher 'productivity' but at a lower output, than a lower 'productivity' at a higher output.

From the quoted snippet it looks like they are describing the effects successful branding has had for US companies and for the US economy.