In difference-in-difference approach, would it be possible to add a dummy variable indicating whether a data point belong to treatment group? Since there is the possibility that there are some factors result in whether a data point belong to treatment group, I think we should add a dummy variable indicating  whether a data point belong to treatment group when using difference-in-difference approach.

However, this was not done in research generally, e.g. this paper:
Chhaochharia, Vidhi, and Yaniv Grinstein. "CEO compensation and board
structure." The Journal of Finance 64.1 (2009): 231-261.

Are there reason for not adding the dummy variable? Would colinearity be an issue? (I don't think it is an issue in the above example.)

It depends on the estimated model. Let me use the example provided in Mostly Harmless Econmetrics.

The experiment

Suppose we are interested in the effect of the minimum wage on employment. Card and Krueger (1994) use a dramatic change in the New Jersey state minimum wage. On April 1, 1992, New Jersey raised the state minimum from \$4.25 to \$5.05. Card and Krueger collected data on employment at fast food restaurants in New Jersey in February 1992 and again in November 1992. Fast food restaurants are big minimum-wage employers. They also collected data from the same type of restaurants in eastern Pennsylvania, just across the Delaware river. The minimum wage in Pennsylvania stayed at \$4.25 throughout this period.

Using their data, they computed diferences-in-diferences estimates of the effects of the New Jersey minimum wage increase. That is, they compared the change in employment in New Jersey to the change in employment in Pennsylvania around the time New Jersey raised its minimum.

The Fixed Effects Model

Differences-in-differences (DD) is a version of fixed-effects estimation using aggregate data. To see this, let

$y_{1ist} =$ fast food employment at restaurant i and period t if there is a high state minimum wage.
$y_{0ist} =$ fast food employment at restaurant i and period t if there is a low state minimum wage.

Let $D_{st}$ be a dummy for high-minimum-wage states, where states are index by $s$ and observed in period $t$. In the absence of a minimum wage change, employment is determined by the sum of a time-invariant state effect ($\gamma_s$) and a year effect ($\lambda_t$) that is common across states. Assuming that $E[y_{1ist} - y_{0ist}|s,t)$ is a constant, denoted $\delta$, we have:

$$Y_{ist}= \gamma_s + \lambda_t + \delta D_{st} + \epsilon_{ist} ~~~~~~ (1) $$
(I corrected $\lambda_s$ here.)

The Regression Model

We can use regression to estimate equations like (1). Let $NJ_s$ be a dummy for restaurants in New Jersey and $d_t$ be a time-dummy that switches on for observations obtained in November (i.e., after the minimum wage change). Then

$$Y_{ist}= \alpha + \gamma NJ_s + \lambda d_t + \gamma(NJ_s \times d_t) + \epsilon_{ist} ~~~~~~ (2) $$

is the same as (1) where $NJ_s \times d_t = D_{st}$.

Your Question

Regarding your question, a key difference is that equation (2) includes the dummy variable indicating whether a data point belongs to treatment group ($NJ_s$ the dummy for restaurants in New Jersey). This dummy is not present in model (1) but the two models are equivalent.


Differentiating a conditional expectation: RBC models with uncertainty Is the following true? \begin{equation}\frac{\partial}{\partial X_{t+1}}E_t(f(X_{t+1}))=E_t(\frac{\partial}{\partial X_{t+1}}f(X_{t+1}))\tag{1}\end{equation} where $f$ is some affine function (e.g., $f(x)=a+bx$), $E_t(X_s)=E(X_s|I_t)$ denotes the conditional expectation of $X_s$ given information $I_t$ in time period $t$, and where $X_s$ denotes the value of some variable $X$, say financial asset holdings, in time period $s$, such that $X_s$ is unknown (i.e., stochastic) given information at time periods $0\leq t<s$ but known otherwise.

Edit 3: I think we should view $X_{t+1}$ as a function of information in time period $t$, i.e. $X_{t+1}=X_{t+1}(I_t)$. This hinders $(1)$ from being zero (see Alecos' comment below). If this is the right way to look at the problem (see "Edit 2" below; I do not know how to look at the problem, that's why I'm asking!) then $(1)$ may be written in detail as \begin{equation}\frac{\partial}{\partial X_{t+1}}E(f(X_{t+1}(I_t))|I_t)=E(\frac{\partial}{\partial X_{t+1}}f(X_{t+1}(I_t))|I_t).\tag{1'}\end{equation}

I often notice that $(1)$ (or $(1')$) seems to be used when studying real business cycle models with uncertainty where the Lagrange method is applied (this framwork is outlined in e.g. Gregory C. Chow's Dynamic Economics: Optimization by the Lagrange Method).

For example, we may have a situation where we have to differentiate \begin{equation}-\lambda_t A_{t+1}+E_t(\lambda_{t+1}(1+r_t)A_{t+1})+\Phi\tag{2}\end{equation} w.r.t. financial asset holdings $A_{t+1}$ in time period $t+1$, where $\lambda_t,\lambda_{t+1}\geq 0$, $r_t\in\mathbb{R}$ and $\Phi$ are independent of $A_{t+1}$ (i.e., they are not functions of $A_{t+1}$), and seem to use $(1)$ to arrive at the conclusion that the partial derivative of $(2)$ w.r.t. $A_{t+1}$ is \begin{equation}-\lambda_t+E_t(\lambda_{t+1}(1+r_t)).\tag{3}\end{equation}

I neither understand if $(1)$ is used to justify the implication from $(2)$ to $(3)$, nor if $(1)$ is true, within the framework mentioned in the parenthesis above, but know of other treatments which begin by discussing measure theory and stochastic processes (e.g., Lecture notes for Macroeconomics I: Chapter 6), and then deriving similar, but not exactly analogous, first order conditions to that derived by using fact $(1)$ and the Lagrange method.

Edit 1: I was asked to post a reference. The reference is lecture notes written by my lecturer. It says the following. If we have a representativve household's maximization problem $$\max_{\{C_t,A_{t+1}\}_{t=0}^{\infty}}E_0\sum_{t=0}^{\infty}\beta^tu(C_t)$$ subject to the budget constraint $$C_t+A_{t+1}=Y_t+(1+r)A_t,\quad\forall t\in\mathbb{Z}_{\geq 0},$$ then we want to investigate the first order conditions for the Lagrangian\begin{equation}\mathcal{L}=E_0\left[\sum_{t=0}^{\infty}\beta^tu(C_t)+\sum_{t=0}^{\infty}\lambda_t[Y_t+(1+r)A_t-C_t-A_{t+1}]\right].\tag{4}\end{equation} One first order condition is the partial derivative of $\mathcal{L}$ w.r.t. $A_{t+1}$: $$-\lambda_t+E_t[\lambda_{t+1}(1+r)].$$ (To be exact he writes that the above is a first order condition for the Lagrangian $\mathcal{L}$. I've interpreted that as meaning that he partially differentiates $\mathcal{L}$ w.r.t. $A_{t+1}$.)  My question is then: Why is that true?

Edit 2: I've also used the following reference: Chapter 5. Real Business Cycles. See equation $(5.7)$. How does the author derive that equation? Does he differentiate inside the conditional expectations, as expressed by me in $(1)$?

Edit 4: To be more exact, $I_t$ may capture the value of e.g. output $Y_t$ (compare with the budget constraint above).

As I noted in a comment, to consider non-trivially a derivative, we must have a non-constant function in the first place. So, in generic terms, we are looking at the conditional expectation function $E(X\mid Z)$ and not at the conditional expected value of $X$ given a specific value $Z=  z$.

Then, $E(X\mid Z)=g(Z)$, i.e. it is a function of $Z$ only, not of $X$, so it appears that its derivative with respect to $X$ should be zero.

But: the way the problem is formulated, we treat the $X$, ($A_{t+1}$ in the OP's example), as a decision variable. Namely as a variable for which we can choose the value that it takes. But this conflicts with the fundamental characteristic of a random variable, i.e. that it is not under our control and its actual realization is unknown.  By treating it as a decision/command variable, we effectively neutralize any aspect related to a random variable, the conditional expectation aspect in our case. And it is in that sense that we can consider the derivative "ignoring" the existence of the conditional expectation operator.

Note that, for the problem laid out at the end of OP's question, if income $Y_t$ is known at the time of the decision for current consumption, then $A_{t+1}$ is also completely determined, there is nothing random/unknown/uncertain about it. This then allows, in a Lagrangian formulation, to treat $A_{t+1}$ also as a decision variable and consider derivatives with respect to it.


Calculating the amount paid for a loan in today's dollars So I was trying to figure out the amount paid for a loan in today's dollars using two different methods but they give me different results. I was hoping that someone could explain which method is right (or if neither are) and what mistakes I was making in the other method.

The problem assumes a constant rate of inflation.

Method 1

Calculate the real interest rate using the Fisher equation: $i_{real} = \frac{i_{nominal} - inflation}{1 + inflation}$
Use the real interest rate instead of the nominal rate in the loan payment formula: $payment = \frac{i * A}{1 - (1 + i)^{-n}}$ where i is the interest rate, A is the amount borrowed, and n is the number of payments
Find the total amount paid by multiplying the payment by the number of payments $total = payment * n$

Method 2

Find the payment using the nominal interest rate. Again the payment formula I used is  $payment = \frac{i * A}{1 - (1 + i)^{-n}}$
Convert each year's payment to today's dollars. To convert to today's I did $dollars_{today} = dollars_{future} * (\frac{1}{1 + inflation})^n$. n is the number of inflation periods.
Sum up the results from step 2 to get the total paid.

An example where the methods give different answers

Loan Amount: = $1000
Nominal Interest: 10%
Number of payments: 20
Inflation: 5%

Using Method 1 we have

Real Interest Rate = $\frac{0.10 - 0.05}{1 + 0.05} = 0.048$
Each Payment = $\frac{0.048 * \$1000}{1 - (1 + 0.048)^{-20}} = \$78.63$
Total Paid = $\$78.63 * 20 = \$1572.61$

Using Method 2 we have

Each payment = $\frac{0.10 * \$1000}{1 - (1 + 0.10)^{-20}} = \$117.46$
Here is the spreadsheet with the work and a picture of it
The total paid is as you can see $1463.81

So you can see that the methods differ in amount by $108.80.

So can anyone explain which is right (if either of them is) and why the wrong one is wrong? My only guess so far is that

   the loan payment formula always gives results in nominal dollars and all I did is change the interest rate by using the real interest rate. Not sure if this is really the case because I studied CS when I was in school, not Econ.

Thank you for your help in advance.

The monthly payment formula tries to ensure that you pay a constant amount each month. Hence, in real terms, you are actually paying more up front as $117.46 today is worth more than in 20 years assuming positive inflation. The more you pay up front, the less you have to pay in total. Even if you managed to correctly calculate payments such that they were constant in real terms, you would end up paying more in nominal terms than if payments were constant in nominal terms.


Find a perfect Bayesian equilibrium Each of two sellers, $1$ and $2$ owns one indivisible object that a buyer
would like to buy. The two objects are identical. The buyer´s valuation depends on
the number of objects he gets. The valuation of any of the two objects is $0.7$ while
the valuation of the two objects together is $1$. Seller $i$'s valuation of his object is $0$
($i$ = $1$, $2$).
Consider the following bargaining game. In period $1$, seller $1$ makes a take-it-or-leave-it (TIOLI) offer $s_{1}$  $\geq$ $0$ to the buyer. If the buyer accepts the offer, he gets the object and pays $s_{1}$. If the buyer rejects the offer, then there is no trade. Seller $2$ does not observe what happens in period $1$, In period $2$, seller 2 makes a TIOLI offer $s_{2}$.
The payoff of each seller is equal to the price that he receives from the buyer. The
buyer´s payoff is equal to the difference between the valuation of the objects that he
gets and the prices that he pays.
Find a perfect Bayesian equilibrium of this game.

In the second period, the buyer accepts any offer $s_{2}$ $\leq$ $0.7$ if he has rejected the first offer and any offer $s_{2}$ $\leq$ $0.3$ if he has accepted the first offer.
Given this, there are only two offers that may be optimal for the second seller: either $s_{2}$ $=$ $0.3$
or $s_{2}$ $=$ $0.7$. Let $\mu$ denote the probability that the second seller assigns to the fact that the buyer has rejected the first offer. The optimal offer in the second period is:
\begin{equation}
 s_{2}=\left\{
  \begin{array}{@{}ll@{}}
    0.3 & \text{if}\ \mu < \frac{3}{7} \\
    0.7 & \text{if}\ \mu > \frac{3}{7} \\
 \end{array}\right.
\end{equation}
any randomization between $0.3$ and $0.7$  $\text{if}\ \mu =  \frac{3}{7} $
Suppose that the equilibrium is such that the buyer rejects the first offer. In the second
period, the second seller will offer $s_{2} = 0.7$ and the buyer's payoff is equal to 0. Because of this, the buyer should accept any offer smaller than $0.7$ (the buyer can guarantee a positive payoff by accepting the first offer and rejecting the second one). But then the first buyer
should offer $s_{1} < 0.7$ and get a positive payoff. In other words, we have shown that there is no PBE in which the buyer rejects the first offer.
Let us now see whether we can construct an equilibrium in which the buyer accepts the first offer. In this case $s_{2} = 0.3$. This implies that the buyer will accept the first offer only
if $s_{1} < 0.3$ (in fact, by rejecting the first offer and accepting the second one, the buyer
guarantees a payoff equal to $0.4$). Finally, given this, it is optimal for the first seller to offer $s_{1} = 0.3$
To sum up, we have the following pure-strategy PBE.
The first seller offers $s_{1} = 0.3$ In the first period, the buyer accepts an o¤er if and only if $s_{1} = 0.3$
In the second period, the second seller assigns probability $\mu = 0$ to the fact that the buyer has rejected the first offer.
Thus, the second seller offers $s_{2} = 0.3$
Finally, the buyer's strategy in the second period is described above.


Literature on the effects of third-party certification on industrial dynamics? Is there any comprehensive theoretical studies on how third-party certification adoption may affect industrial performance/dynamics of an industry?

For models of the informational role/effects of third-party certifiers on an industry, you might like to look at

Lizzeri, Alessandro (1999). “Information Revelation and Certification Intermediaries”. RAND Journal of Economics 30.2, pp. 214–231.

Albano, Gian Luigi and Alessandro Lizzeri (2001). “Strategic Certification and Provision of Quality”. International Economic Review 42.1, pp. 267–283.


What economic factors should be considered in attracting suitable staff and determining pay for temporary project work? Getting qualified workforce for temporary projects can be difficult. If found from within the organisation, they may have to abstain from their regular work, and this may create additional complications to their worklife. Given that the right people may be difficult to find, they may have to be paid higher than if the same post would be a year-round job (if the quality of work is the same).

Many factors might need to be taken into account by an employer seeking staff for such projects.  What economic factors are relevant here?

Characteristics of a "project", as per the Project Management Institute, are that it is temporary and unique (ie. non-routine).

For a person employed to work on a project, the temporary nature of the work is likely to be a source of additional cost and risk, as compared with a person in a permanent job. This is because, as the project comes towards its end, the person is likely to have to find another job.  That involves both the transaction costs of finding a job (searching, assessing whether jobs are suitable, negotiation, etc., implying time and effort if not monetary costs) and the risk of not finding a suitable job when needed and therefore experiencing a loss of income. These factors suggest that to recruit to a project role will require somewhat higher pay than to recruit to a permanent job with a similar skill level.

An exception to the above is where a current staff member in a permanent job is seconded to work on a project. Provided they can be sure that at the end of the project they can return to their original job, they will not be subject to the sort of risk described above. On the other hand if a current staff member is to be asked to work on a project while also keeping their regular work going, they may decline unless offered higher pay (for reasons of workload, not risk).

A further issue is how pay for project work might be structured. Suppose A is working on a project which is 90% complete. The employer will want A to stay and complete the project.  But A knows that staying until the end of the project will increase the risk that they will find themselves without a job. So they will be tempted to apply for other jobs before the project is complete. Hence there is a risk for the employer, ie that the departure of a key person will mean that completion of the project becomes problematic.  To address this, an employer might consider a pay structure consisting of a regular monthly salary plus a bonus payable on successful completion of the project.

But that in turn raises the issue of asymmetric information. Because of the non-routine nature of project work, it may be that only those working on the project really know how near (or far) the project is from successful completion. This is especially likely where the output of a project is not very visible or tangible, an ICT system implementation for example. So even if the project is only 90% complete, and A knows that, there is a risk for the employer that A will leave to take another job while also persuading management that the project is complete and receiving the bonus.


Why would a government bail out private banks? If the private banks never learn from history and keep giving credits to it's clients for overpriced acquisitions, and then they enter default why would the government give them free money or buy those banks?

If I buy overpriced apples to sell them in my fruit shop and I fail to sell them, the government will not bail me out and pay my debts.

Just as my company running the fruit shop should go bankrupt, those banks should go bankrupt, other private investors should buy their assets and start doing better banking business, removing that incompetent staff who buy (or give credits for buying) overpriced things (in this case: houses).

On a first thought, that's my conclusion. And also on a second and third thought.

The whole point of default and bankruptcy should be to allow those competent businesses to survive and those incompetent to leave the sector and do something else. And that looks like a fundamental piece of a working capitalist system that encourages the players to be competitive. Therefore the bailout of private banks looks like a bonus for incompetence. Sponsoring incompetents doesn't look like a good option for making the banking sector work well and totally not suitable for restoring confidence of customers in such banks.

Example: If Compaq is selling bad computers and Gateway is selling good computers, then Compaq should go bankrupt instead of getting government subsidies in order to keep being incompetent. Their owners can try to do something else, like for example Barbie toys, maybe they can do it better than making computers. Gateway will buy their assets and they will survive and grow because they are competent.

I would like to know the arguments used by the governments for bailing out the banks, like for example in Spain where they gave 100 billion Euro for the private banks.

Because the banks are not loaning their own money but that of depositors. One may also argue that the depositors should know the risks and if their bank makes bad investments it is partly their fault so they cannot complain should they lose their deposits. But a functioning banking system is essential to the current economic system. The bankruptcy of a big bank would cause panic and banks runs, which because of fractional reserve banking would probably result in the insolvency of other banks, banks that perhaps made good investments but ones that cannot immediately be liquidated. After the closing of the banks considerably less savings would be deposited in financial institutions. This will hinder consumption because credit/debit card traffic would cease and getting investment funds would also be considerably more difficult.

So the governments have considered the damage the bankruptcy of a bank would cause to their economy and decided it is better to bail them out. ("Too big to fail.") Of course since banks have a very large revenue stream they can also afford lobbying for intervention as well, but this does not mean that the intervention is not beneficial to the public. Furthermore, if you lose your deposits you may very well (wrongly) blame the government, so a bail-out is also politically prudent.

If you are interested in what could be done I recommend reading about the Glass-Steagall act.

If you are not satisfied with my answer there are any number of conspiracy theories that offer unconvincing alternative explanations.


Where do public savings go? In Mankinw's Macroeconomics, he mentions that public and private savings form national saving, which ends up in the financial sector, which then people and firms use to purchase investment goods. But what happens in real life? If the US federal government runs a budget surplus, where does it go?

Most governments (outside of typically oil-producing nations that have sovereign wealth funds) are net debtors, so public deficits show up as increases in government debt outstanding, and public surpluses (public saving) show up as decreases in debt outstanding.

In the model used in Mankiw’s book, the assumption is that government spending never constitutes investment (i.e, it doesn’t create productive capital). In that model, it’s also assumed that households hold government debt: government debt just soaks up some of the funds that would otherwise be available as “loanable funds” to private firms to produce “productive capital,” shifting the supply curve to the left.

So, in that model, the quantity axis represents something like “private investment,” and government saving shows up as an outward shift in the supply curve, which results in lower interest rates and more private investment.

There’s another way of thinking about it, which doesn’t require that one assume that government spending never forms capital like roads, bridges, airports, etc., nor that government deficits are paid for directly out of private saving. In that model, government deficits just show up as increases in demand for loanable funds, which increases interest rates. Implicitly, this does still mean that less private sector investment will occur. The opposite is of course true for government surpluses. The interpretation of the quantity axis then changes to something like “amount of financial intermediation.”

Either model of loanable funds is, of course, a radical simplification of how financial intermediation works, and like anything in an intro course, it’s worth remembering that tons of caveats apply.


Why do top professional sport players get usually paid so much? I'd like to find an answer to this question as I can't see how something like this can be happening. I can see three reasons that they would earn that much but any of them if thought better is wrong.

1) They generate all the money they win. I don't deny that they generate lots of money, but news like this one: http://www.euroweeklynews.com/news/axarquia-malaga-east/item/126022-la-liga-clubs-threaten-strike, or this one http://www.forbes.com/sites/mikeozanian/2015/01/22/nba-adding-1-1-billion-of-debt-while-getting-its-best-credit-rating-ever/ make clear that there are going to be a lot of times where their salary is clearly overvalued, and according to only economic terms they should earn maybe something that's a third of their salary.

2) They generate the money on its own. I consider that also mostly false, if they generate that much is because mass media is always talking about them and showing them, if they wished they could take any other thing and with time put it on top being that thing much less expensive for them on the beginning.

3) What the do not earn economically , they earn in entertaining people and making them less aware of their problems, ok, so do a lot of other things, like computer games or books or whatever, and none of them get that much of hype usually, moreover, they are likely to be victims on piracy with a lot more frequency than, let's say, a televised sport event.

Maybe this is applied for other professionals like top actors or singers, but I don't know of such huge debts on their case, and I get the impression that there much more top sport players with huge winnings rather than actors or singers.

I'm not a economist, but I find this question intriguing, hope you can help me to understand why.

Thanks for your time.

tl;dr There is a lot of demand for what they produce.

A proper answer should have some evidence, perhaps I will add that when I have more time.

Even within a category, say, football (soccer) players, you see a lot of discrimination: Top players at top clubs in top leagues get most of the cake. This is anecdotal evidence for the demand side story:

Only the best leagues earn much money through television rights, because only there is most of the demand (how much media coverage do you have for your national third league anyways?)
Only the best teams in these leagues get most of this money, mostly because leagues partly pay teams based on their performance ranking wise
Only the best players get the highest share, which is a combination of the previous two facts, and that these guys are quite unique and have hence an even higher demand:

They make more marketing money (people really want to see them)

Due to their uniqueness, they can bargain better deals with teams and sponsors

Note that the last point is really the only supply-side story. If you ignore super stars, the supply of good players is huge. Most of the twist really comes from some of them being marginally better / lucky, making it into the better leagues, which happen to be the one that mass media demand, and get the high compensation.


Why is food less expensive in India than the United States? India has 4x the population, less land, and far worse technology and infrastructure than the United States. Shouldn't food cost in the US be less expensive? I've never been to India, but it seems like food costs far less there from what I see on TV (10 cents a meal). Why is this?

I think your intuition is right that food is cheap there. The Economist Big Mac price index suggests that India has some of the cheapest big macs in the world. More generally, the index indicates that there is a strong relationship between GDP per capita (a proxy for labor costs) and Big Mac prices:

Labor is a significant part of the costs of making food (25-35 percent in America). Transportation is a significant part of the cost of food grown far away (world bank data indicates that even exports fees alone can be substantial). But when food is grown locally and prepared locally, the total costs are significantly determined by local labor costs. In addtion, Indian restaurants probably also serves much smaller portions than in rich countries, particularly the United States. They may also make use of less expensive ingredients generally, more stables and seasonal vegetables and less dairy and meat.


How is the ARIMA model a valid approach for forecasting Economic Variables? Ive noticed the use of univariate forecasting methods (i.e the ARIMA model) employed in forecasting inflation (see references below).

How is this a valid approach in terms of forecasting economic variables? does this approach not omit the large amount of economic theory which determines how these variables are determined (i.e via equilibrium and interaction of variables other than themselves?)

1.https://scholar.harvard.edu/files/stock/files/forecastinginflation.pdf see pages 6-72.http://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdfhttp://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdf
3.http://www.unagaliciamoderna.com/eawp/coldata/upload/pakistans%20inflaction_arima%20model.pdf

So your question is basically asking why a purely statistical model would be better at forecasting than a model based on theoretical relationships. For example, think about a simple form of the ARIMA model i.e. an AR model where you're just using lags of the process to explain the process itself. Now think about what is contained in the lags of the process... If the true process (i.e. the DGP) is determined by theoretical relationships between other variables then these will automatically be included in the lag. So the ARIMA model is a statistical way of modeling the process to make sure you capture the statistical properties of the underlying relationship but does not necessarily provide an explanation for what is going on. In forecasting, especially when series are volatile and erratic, you often find that simple statistical models such as these do a better job since the underlying relationships may not be very stable or may be more complicated than any theory we can come up with. This is related to a recent article  which you may find interesting: http://www.sciencedirect.com/science/article/pii/S0169207017300997


Unit Elasticity Calculation Question I'm struggling with a math concept dealing with unit elasticity at an undergraduate textbook level. In most problems, we are given a starting price and an ending price as well as corresponding quantities demanded. For example:

When bread was ${$2.00}$ a loaf Bobby wanted ${4}$ loaves, when bread prices raised to ${$2.50}$ Bobby wanted just ${2}$ loaves. What is bobby's price elasticity of demand?

To calculate these we say the percent change in price is ${\% \Delta P = \frac{$2.50-$2.00}{$2.00}=25\%}$. The percent change is measured relative to the starting price. Meanwhile ${\% \Delta Q = \frac{2-4}{4}=-50\%}$. Making elasticity ${-2.0}$.

However, I came across a problem where essentially it asked what is the elasticity for someone who wants to spend exactly ${$10}$ on gasoline. I first tried to think about this intuitively, "If price doubles, she'll be able to get half as much quantity no matter the starting price," which seems like elasticity of ${1}$. However, I set it up algebraically I get the following:
$${Q*P=10 \implies Q = 10/P}$$

So I put in some random values of ${P}$, for example ${P_1=$2.00}$ and ${P_2=$2.50}$ for ${\% \Delta P = \frac{$2.50-$2.00}{$2.00}=25\%}$. Meanwhile, For quantity we have ${Q_1 = \frac{10}{2}=5}$ and ${Q_2 = \frac{10}{2.5}=4}$. So that ${\% \Delta Q = \frac{4-5}{5}=-20\%}$.... Which isn't unit elastic.

So I turned to calculus and did the following:

$${Q(P)=\frac{10}{P} \implies \frac{dQ}{Q}*\frac{P}{dP}=\frac{dQ}{dP}*\frac{P}{\frac{10}{P}}=-\frac{10}{P^2}*\frac{P^2}{10}=-1.}$$

Which is unit elasticity. Clearly the discrepancy here is caused by the fact that the derivative has the "starting quantity" = "ending quantity" since it's the same point. But should we be measuring percent change relative to starting price and quantity? Or relative to starting price, but ending quantity? Or does it depend? And why?

You have to use the midpoint method to resolve this (if I recall correctly).

https://quickonomics.com/how-to-calculate-price-elasticities-using-the-midpoint-formula/

The reason I may not recall correctly is because the second you introduce calculus into the study of economics, you discard these formulae entirely and (try to) forget they ever existed.  You seem to already know calculus, which is probably causing you to be even more confused rather than less, and understandably so.

The primary purpose of these non-calculus formulations of elasticity is to teach you the intuition around what elasticity is as a concept - I definitely have strong opinions about how well it accomplishes that. :)

EDIT: Just wanted to add that, if indeed the problem is as you're saying it (come hell or high water, Billy is going to spend \$10 and only \$10 on gas) then your intuition is right - Billy has unit price elasticity of demand for gas, although in a kludgy sort of way.  This is supported by your calculus-based analysis.


Proving existence of utility function In Advanced Microeconomic theory by Jehle and Reny there is a proof of the theorem which states the existence of utility function.

In order to prove the existence of a utility function $u(\mathbf{x})$ which represents binary relation $\succeq$ if it is complete, transitive, continuous, and strictly monotonic, it is suggested to consider a mapping

$u: \mathbb{R_+^n} \to \mathbb{R}$  such that $u(\mathbf{x})e\sim \mathbf{x}$ is satisfied, where $\mathbf{x}$ is a bundle, $u(\mathbf{x})$ is some number and $\mathbf{e}$ is a bundle which contains one of every good.

So first we need to show that there always exists such number $u(\mathbf{x})$.
To do this consider two sets:

$A \equiv \{t \geq 0 \mid t\mathbf{e} \succeq \mathbf{x}\}$

$B \equiv \{t \geq 0 \mid t\mathbf{e} \preceq \mathbf{x}\}$

if $t^* \in  A \cap B$, then $t^*\mathbf{e} \sim \mathbf{x}$, so we need to show that $A \cap B$ is nonempty.

The continuity of $\succeq$ implies that both A and B are closed in $\mathbb{R_+}$. By strict monotonicity, $t \in A$ implies $t' \in A,$ $\forall$  $ t'\geq t$. So $A=[\underline{t}, \infty)$.
Similarly $B=[0, \overline{t}]$

For any $t \geq 0$, completeness of $\succeq$ implies that either $t\mathbf{e} \succeq \mathbf{x}$ or $t\mathbf{e} \preceq \mathbf{x}$,
i.e., $t \in A \cup B$

$\mathbb{R_+} = A \cup B = [0, \overline{t}] \cup [\underline{t}, \infty)$.

So in order to $A \cap B$ be nonempty it should be that $\underline{t} \leq \overline{t}$.

But is it always like this? I can't see why the last inequality should hold always.

Το take this one out of the Unanswered queue:

The completeness property of the preference relation, implies that for all non-negative $t$ we will be able to form and declare the preference relation. Hence

$$A \cup B = \mathbb{R_+}$$

By monotonicity we have $B=[0, \overline{t}],\;\; A=[\underline{t}, \infty)$.

Ad absurdum, assume that $\overline{t} < \underline{t}$. Then there exists an open interval $(\overline{t},\underline{t})$ that does not belong to the union of $A$ and $B$. But then we arrive at $A \cup B \neq \mathbb{R_+}$ which contradicts the implications of the completeness property.

So we conclude that $\overline{t} \geq \underline{t}$, which implies that the intersection $A \cap B$ is non-empty (even if it may contain a single element, when $\overline{t} = \underline{t}$).


Inada Conditions in Plain English I wondered if someone could potentially explain the Inada conditions in plain English, and the implications.
$$lim_{c_t\searrow0}\frac{\partial U}{\partial Ct}= +\infty$$
$$lim_{c_t\nearrow+\infty}\frac{\partial U}{\partial Ct} = 0, \forall_t$$
$$\Longrightarrow C_t^*\in (0,+\infty),\forall_t$$
I've stated the conditions using:https://faculty.babson.edu/lcdstein/210/sectionall.pdf

I'm not sure what exactly in plain English means, but here is a try:
Think about it as consuming water.
Look at condition 1. If you have no water, you would die, making you extremely unhappy.
If someone came up to you and offered one bottle of water, it would make you infinitely happier than having no water, as you would live! (You would pay any amount of money to get that single bottle!)
The second condition states that eventually giving you extra water gives you less and less extra happiness. That first bottle was life saving, the second helps a lot,... but after the thousandth bottle, having extra water is nice, but having a thousand bottles versus a thousand and one does not increase your happiness too much.
The implication of the assumptions ensure that if I asked a consumer how much water she would like (for some price usually), she would pick a number between 0 and $\infty$.
She wouldnt pick 0, as we argued before, and eventually she cares less and less, so she would not buy $\infty$ bottles.
A famous example is $U(c) = \log(c)$.


How to calculate Shapley Value with uncertainty? So I'm trying to estimate a Shapley value in a game with uncertain payoffs. Specifically, imagine a game where the payoff function as as follows

(A) = 1
(B) = 2
(B,C) = 4

For instance, you can imagine this game representing Person A and Person B deciding whether to go into business by themselves or together.

Calculating the Shapley value for this game is very straightforward by listing all the permutations along with the marginal contribution of each person in the given permutation divided by the number of permutations:

[A, B] = [1, 3]
[B, A] = [2, 2]

[A, B] = [(1+2)/2, (3+2)/2]
[A, B] = [1.5, 2.5]

So Person A should get \$1.5 and Person B should get \$2.5 in the grand coalition.

However, I am struggling to figure out how I would go about computing the Shapley value in the presence of uncertainty. For instance, imagine I change the payoff function to use confidence intervals as follows:

(A) = 1 ± 0.2
(B) = 2 ± 0.6
(B,C) = 4 ± 0.5

This is important as I am trying to compute the Shapley value in an real life setting with experimental data.

What will the Shapley value be in this case WITH uncertainty?

It sounds like you may have a quadratic loss function, but I am guessing a little.  If the players have played enough to be experienced in how the game works, then I would simply use the sample mean reward as my calculation point.

You should read up on the theory behind loss functions.  The players are making estimates, certain types of estimates are the natural minimization of certain loss functions.  Most gambling type behavior results from quadratic loss, though there are particular exceptions.  Equity securities center of location is the mode because there can be no such thing as a mean return.  Equity securities behaviors should result from an all-or-nothing loss function.  Physicians are another example where there is an asymmetric loss.

If a physician fails to diagnose you with a disease, then they lost the revenue from treatment and face malpractice claims.  If they misdiagnose you as having a disease but you do not, then they still collect the medical billings and may not face malpractice claims, especially if they do not reveal it was a misdiagnosis.  A diagnosis is a form of gamble.

Take a look at the losses experienced from failing to take a particular type of behavior.


Does salary contribute to GDP under the expenditure approach? The question really confuses me. On one hand, we can consider the wages as purchasing services from employees. On the other hand, the salary is also part of the cost of the products, thus it isn't "final services".
Consider the following scenarios:

Firefighters' salary

Yes, it contributes to GDP under the expenditure approach

Salary for the employees in a private firm

No, it doesn't

A tycoon pays the salary for a personal guard directly.

Yes
Am I correct to say so? Can anyone explain this to me? Any help is appreciated. Thanks.

No under expenditure approach neither salary or wages are directly counted. An expenditure approach to GDP calculates GDP as follows:
$$GDP=C+I+G+NX$$
Where $C$ is consumer spending on final goods and services at market prices, $I$ is the investment spending, $G$ is government spending and $NX$ are net exports.
All wages and salaries are of course indirectly included in as prices of final goods and services indirectly may include them but under spending approach they are not counted directly. Even if you would hire lets say language tutor from some company where company pays almost everything directly to that tutor it would be the cost of service that would appear on national accounts not the wage of the tutor under spending approach.
If you would want to count wages & salary directly you could use income approach to GDP where the GDP would be calculated by summing all returns to labor (wages & salary), profits, rental income and interest.


IS curve, interest rate and consumption It is possible for the interest rate to affect the expenditure with consumption. An increase in the interest rate could, in principle, lead to increases in savings and, that way, to a decrease in consumption, given the income level. Suppose that consumption is, indeed, reduced by an increase in the interest rate. How will the IS curve be affected?
Considering the following equation for the IS curve:
$$
Y=C(\overbrace{Y−T(Y)}^{+})+I(\overbrace{r}^{-})+G+NX(\overbrace{Y}^{-})
$$
I assume it would be changed to
$$
Y=C(\overbrace{Y−T(Y)}^{+}, \overbrace{r}^{-})+I(\overbrace{r}^{-})+G+NX(\overbrace{Y}^{-})
$$
And thus, what was
$$
\frac{dY}{dr} = \frac{dI(r)}{dr} < 0
$$
now is
$$
\frac{dY}{dr} = \frac{dC(Y-T(Y), r)}{dr} + \frac{dI(r)}{dr} < 0
$$
and, therefore, the curve now is more downward-sloping than before.
Is this right? Kinda feels like I am missing some indirect effect.
Thanks!
EDIT: I was on the right track, but not correct. The trick is to apply the total derivative.
Going from $Y = C(Y-T(Y)) + I(r) + G + NX(Y)$, we have:
$$
\operatorname{dY} = \underbrace{\frac{\partial C(\cdot)}{\partial Y}}_{\equiv C_Y > 0} \cdot \operatorname{dY} + \underbrace{\frac{\partial I(\cdot)}{\partial r}}_{\equiv I_r < 0} \cdot \operatorname{dr} \therefore \operatorname{dY} - C_Y \cdot \operatorname{dY} = I_r \cdot \operatorname{dr} \Leftrightarrow \boxed{\boxed{\frac{\operatorname{dr}}{\operatorname{dY}}\bigg\vert_{IS} = \frac{1-C_Y}{I_r} < 0}}
$$
Now, with the new equation, we have:
$$
\operatorname{dY} = \underbrace{\frac{\partial C(\cdot)}{\partial Y}}_{\equiv C_Y > 0} \cdot \operatorname{dY} + \underbrace{\frac{\partial C(\cdot)}{\partial r}}_{\equiv C_r < 0} \cdot \operatorname{dr} + \underbrace{\frac{\partial I(\cdot)}{\partial r}}_{\equiv I_r < 0} \cdot \operatorname{dr} \therefore \operatorname{dY} - C_Y \cdot \operatorname{dY} = (C_r + I_r) \cdot \operatorname{dr} \Leftrightarrow \boxed{\boxed{\frac{\operatorname{dr}}{\operatorname{dY}}\bigg\vert_{IS'} = \frac{1-C_Y}{(C_r + I_r)} < 0}}
$$
Since $|C_r+I_r| > |I_r|$, the IS curve is now flatter and thus the income responds more to changes in the interest rate than before.


What is preventing the United States from printing USD to buy up prized assets from foreign countries given that USD is the world's reserve currency? USD is accepted all over the world. U.S can easily print USD to buy up the most valuable assets from all over the world. Ethics aside. What is preventing the U.S from doing so? What did I miss out?

To a certain extent it can.  But not too much, else it will create a run on the dollar.  If speculators no longer trusted the dollar, they would sell it for other stores of value (like Euros, gold, etc...).  Foreign dollars would have to come home and be redeemed by real wealth which means less real wealth for the US domestic economy and inflation.  This can feed on itself and cause a self-reinforcing inflationary fire that would be difficult to put out.


Interest rate Effects on labour supply I was analysing, free of specificities, the effects of an increase in the real interest rate on labour supply in a two period setting. The obvious effect on leisure is the intertemporal substitution effect as leisure in the present becomes more expensive than leisure in the future. Nevertheless, all the books I have looked at do not mention any income effect from this. Shouldn't the latter one exist as well? The increase in interest rates, if one is a net lender, allows one to increase leisure in both periods.

Now looking at the typical optimization FOC:

$$\frac{\frac{\partial U}{\partial l}}{\frac{\partial U}{\partial l'}}=\frac{(1+r)w}{w'}$$

(1+r) increases, thus, the ratio needs to increase. Nevertheless, that can happen by increasing both levels of leisure, but by different amounts. Am I reasoning incorrectly?

{Please disregard from mentioning all other intertemporal and cross effects, ex. consumption and leisure as I am aware of them}

Yes, there is an income effect, as the individual needs to work less in order to obtain the same amount in the numerator (relative to before the increase in exchange rate). The assumption is that the substitution effect dominates the income effect here, regardless of whether the individual is a borrower or a lender.


Is there a typo in Sims(2002) paper on solving linear expectations model? Here's the Sims(2002) paper. It's equal to the published version.

The author gives the following example:

And tries to write it in the matrix notation of (1) in the paper.

However, his solution - in (4) - puts $\epsilon (t)$ in the first equation of (2) instead of $\nu (t-1)$, and $\nu (t-1)$ instead of $\epsilon (t)$. Is this correct, and why?

Also, when I try to write the the first equation of (2) the time notation doesn't match. The other two equations in (2) do match.

Any help would be appreciated.

Yes the $z(t)$ column vector, as it appears in $(4)$ is written upside-down in the translation. Also the first equation in $(2)$ is translated lagged once, i.e. the translated system describes the $w(t-1)$ equation.

Finally there is a typo in the $\Gamma_1$ matrix, there is a void 6th column that should be ignored, and $\theta$ should be in the 4th column not in the 5th (because it multiplies $u(t-1)$.


given someone's past investing history, is there a way to calculate his risk aversion? given someone's past investing history, is there a way to calculate his risk aversion?
Say, we know this client's investment history for example his past return, is there a way to calculate his risk aversion and use this parameter to portfolio optimization?

Generally speaking no. You wouldn't be able to distinguish re-balancing for risk aversion reasons from re-balancing motivated by changes in expected returns or the co-variance of returns.

Consider the simple case of a household periodically re-balancing their investments in across both a fixed index fund and an equity index fund. The econometrician sees the investor reduce their equity holdings and increase the fixed income investment. Any of the following could induce such a change:

An increase in risk aversion
Lower expected returns for the equity index
Higher expected returns for fixed income index
Greater positive co-variation of the two funds

And in reality, the situation is much worse because there are many investments, taxes, transaction costs, behavioral issues, and rational-inattention.

I'm sure you could simplify and cook up a theoretical setting where you could do this (for one, just assume that all the other things cannot change), but in general this won't be possible.


Pareto set with Cobb-Douglas and Leontief preferences If $U_A(x_A,y_A)=x_Ay_A$ and $U_B(x_B,y_B)=min(x_B,y_B)$ and the total endowments are (8,4), is the Pareto set given by the line joining the kinks of B (black line shown in the diagram)? Shouldn't the bottom left origin and the flat portion on the horizontal axis also be Pareto efficient?
Suppose we are at the bottom left origin, there is no way to improve one person's utility without hurting the other. By this definition, the origin should be included in the Pareto set. Please point out any flaw in the logic.

The bottom right origin is actually not in the Pareto set.  At that point, $(x_A,y_A)=(8,0)$, so $U_A(x_A,y_A)=0$.  Similarly, $(x_B,y_B)=(0,4)$, so $U_B(x_B,y_B)=0$.

As an example, $B$ could give one unit of $y$ to $A$ and thereby raise $A$'s utility to 8 without hurting $B$'s own utility (which would remain zero).  As a matter of fact, any allocation other than the top left origin is a Pareto improvement over the bottom right origin.


Would the military be more efficiant if they were unionized? I hear about the extremely long work hours in the army. I understand it hardens them for readiness for combat situations. With the technology and increase of intelligence need to operate it would a more normal work schedule be more efficient and less mistakes?

How much more would it cost the military to have union like rules?

European countries offer a variety of different unionisation models, so provide reasonable evidence on the usual consequences of unionisation.  Dutch military personnel are represented by a union, for instance.

The European Association of Military Organisations appears sanguine about the idea, although many European military unions don't have the right to strike (emphasis added):

  The President concluded by emphasizing that EUROMIL strongly believes
  that military trade unionism does not harm military efficiency. On the
  contrary, being represented in trade unions and professional
  associations can enhance the [morale] and loyalty of the troops.

Less information is available about the consequences of unionisation on working hours; in general, it seems that unions appear to understand and accept the reasons for long working hours under certain circumstances.


Verifying the constant elasticity of the demand model Q= aP^b "It is well known" that the elasticity of $Q$ relative to $P$ is the constant $b$ in the equation $Q(P)=aP^b$. We can verify it using the convenient definition of elasticity $= \frac{dQ}{dP} \frac{P}{Q}$, we can verify it in Wikipedia.

Based on this, when I create a spreadsheet with parameters $b=-2$, $a=100$ and P ranging from 1-5 by 1 I expected to recover an elasticity of -2, where elasticity =  % change in Q/ % change in P.  This is not the case. Instead the elasticity  varies substantially as the % change in P varies.

I'm forced to conclude that 'constant elasticity' does not mean a fixed value regardless of the magnitude of the price change, but only invariant with regards to the value of price OR that I have a spreadsheet error.

I've pasted an image of the spreadsheet results below.  Obviously, without seeing the formulas, no one can verify that I don't have an error, but I'm hopeful that someone will give this a spin for themselves and get very different numbers and then I'll know my failure to verify the constant elasticity in this function is a spreadsheet error.

The problem with your calculations is that you have very big changes in price

If you made price increases of $1\%$ in price, you would find the quantity fell by about $2\%$ as shown in the table below leading to an elasticity of about $-2$

If you made the price changes even smaller, the calculated elasticity would get even closer to $2$, and in a sense you are looking for the limit of elasticity as the price change approaches $0$, which will indeed be the constant $b=-2$

 a       b      P       Q=a*P^b pct change Q    pct change P   elasticity

100     -2      1       100.00
100     -2      1.01     98.03      -1.97%          1%          -1.97
100     -2      2        25.00
100     -2      2.02     24.51      -1.97%          1%          -1.97
100     -2      3        11.11
100     -2      3.03     10.89      -1.97%          1%          -1.97
100     -2      4         6.25
100     -2      4.04      6.13      -1.97%          1%          -1.97
100     -2      5         4.00
100     -2      5.05      3.92      -1.97%          1%          -1.97


Why don't consumers subsidize monopolies? Monopolies don't maximize welfare because they set prices above the equilibrium price, leading to dead-weight loss. It is possible for the government to provide a per unit subsidy to a monopoly until the producer's marginal cost equals the consumer's marginal benefit at the monopoly's chosen quantity, which would maximize welfare. This change would increase producer surplus and consumer surplus in this market.

To a monopolist producer, a per unit subsidy is essentially equivalent to shifting the demand curve up by the value of the subsidy. It would be possible to shift the demand curve up until the optimal quantity produced is the higher equilibrium quantity. This requires that the subsidy provider knows the complete demand curve to set the optimal subsidy.

The subsidy must come from somewhere, like a tax on a different market. It seems plausible that there would be situations where even if all tax incidence fell on consumers, consumer surplus would still be greater after the subsidization. If consumers and producers are overall better off after a subsidy, why don't consumers do this without government intervention?

As you pointed out, the problem of inefficient supply by a profit maximizing monopolist can be solved via subsidizing the monopolist to increase his marginal revenue. The subsidy can be paid by the consumers or by a central government.

Consumers unfortunately perceive the problem as a collective action problem: The marginal utility gain from subsidizing the monopoly is much lower than the marginal welfare gain of society, making individuals reluctant to subsidize the monopoly.

Governments, however, can solve the collective action problem by raising a tax and thus financing the subsidy. However, governments usually do not only pursue efficiency but also interpersonal equity. If the government cannot transfer the monopolist's earnings lump sum to the consumers, then this may conflict with the goals of interpersonal equity.

To give a simple example: Suppose rich individual A owns a monopoly on water. There are many consumers of water who are all poor. A sets the profit maximizing price/quantity. If the government maximizes total utility and all consumers have identical concave utility functions, the government would like to impose the efficient subsidy and tax the rich individual A lump sum to finance the subsidy and possibly transfers to the poor. However, if the lump sum tax on A is not available, we are in what is called a second-best solution. In this case, it may be optimal for the government to permit some inefficiency from the monopoly rather than taxing the poor to pay for the subsidy.

As an aside: Most interestingly, if there is an omniscient government which maximizes social welfare and can make lump-sum transfers, then under some regularity conditions the collective action problem disappears without the government imposing the subsidy! Thus, I would have an interest to subsidize the monopoly on my own. This is because the utility possibility frontier is pushed outwards by the subsidy and the government will reimburse me lump sum for my cost.


Panel Data and Forecasting I have balanced panel data for around 130 countries, over three years. I ran a fixed effects regression using 'country' as my panel variable, and adding dummies for 'year'. I want to forecast the value of the dependent variable for the next 20 years, for only one country in particular. Could somebody suggest the best way to do this?

I am using Stata for this purpose.

Probably this question is better placed at stackoverflow. Technically, in order to predict out of the sample, you need the full set of variables in the estimation. So after carrying out the estimation, you could do something like this. I assume your last observed year is 2010.

* expand US data for the last observed data point.
expand 20 if country == "US" & year == 2010
bys country year: replace year = . if _n > 1
bys country (year): replace year = year[_n-1] if year == .
estimates restore panel_estimation
predict outcome_var
* check whether that worked
tw line outcome_var year if country == "US"

This will predict a value based on your panel estimation also for the newly created observations. I am not judging whether it makes sense to extrapolate a time-trend over three years to twenty years ;)


Monte carlo simulation for macroeconomist I would like to know if it is useful to learn Monte Carlo simulations if I want to focus on macroeconometrics ? Or even is it useful outside of academics ?

The main challenge in macroeconometrics as opposed to microeconometrics is a the small sample size, because of short recorded time-series and less than 200 countries providing data. To get around this problem increasingly the more popular approach is Bayesian estimation, which often relies Monte Carlo Markov Chains to construct the posterior distribution of the parameter.
So if you want to understand how the results are obtained, you will need to have a rather deep knowledge of the topic.


Is $(\mathbb{R}^m)^n$ the real coordinate space of dimension $m\cdot n$? Very simple question here: say that there are n individuals and each individual $i\leq n$ has a consumption bundle $x_i\in \mathbb{R}^m$ (i.e. there are $m$ types of goods). Suppose that social welfare is a function $f(x_1, x_2, ..., x_n)$. The domain of $f$ is clearly $(\mathbb{R}^m)^n$. My question whether the set $(\mathbb{R}^m)^n$ is the same as $\mathbb{R}^{m\cdot n}$?
In other words, is $(\mathbb{R}^m)^n$ the real coordinate space of dimension $m\cdot n$?

No. And yes. For any set $X$ we have (by definition) $$X^k=\underbrace{X\times\cdots\times X}_{k\text{-times}}=\{(x_1,x_2,\ldots,x_k)\mid x_i\in X\text{ for }i=1,\ldots,k\}.$$
Now let, for example, $m=2$ and $n=3$. Then
$$(\mathbb{R}^m\big)^n=(\mathbb{R}^m\big)^n$$ $$=\big(\mathbb{R}^2\big)^3=\Big\{\big((x_1,x_2),(x_3,x_4),(x_5,x_6)\big)\mid (x_1,x_2)\in\mathbb{R}^2,(x_3,x_4)\in\mathbb{R}^2, (x_5,x_6)\in\mathbb{R}^2\Big\}$$ $$=\Big\{\big((x_1,x_2),(x_3,x_4),(x_5,x_6)\big)\mid x_i\in\mathbb{R}\text{ for }i=1,\ldots,6\Big\}.$$
This is different from $$\mathbb{R}^{m\cdot n}=\mathbb{R}^{2\cdot 3}=\mathbb{R}^6$$
$$=\Big\{(x_1,x_2,x_3,x_4,x_5,x_6\big)\mid x_i\in\mathbb{R}\text{ for }i=1,\ldots,6\Big\}.$$
So they are not the same, formally $(\mathbb{R}^m\big)^n\neq \mathbb{R}^{m\cdot n}$ However, there is clearly a natural way to associate points in one space with points in the other space and this way to connect them preserves a lot of structure. They are equivalent as vector spaces, topological spaces and whatnot. For this reason, one might often identify these two spaces and unless one does very strange things, little can go wrong.


Who is working right now? (coronoavirus) Is there any research, graphs, or any formal (non-opinion) information on where I can get information on what people are working right now? I want to know, who is still collecting a paycheck.

I am a sales representative that sells knives, and does virtual appointments online (over the phone, video conference, etc.), but I really need to show people that are working. So, I imagine police and hospitals are working right through this coronavirus (COVID-19), but someone who works in the party industry in Las Vegas are probably OUT of work, as their jobs are not essentials, and aid in the spreading of the virus.

So, instead of being "I assume" or "I guess" these people are working, what are the statistics of who is working right now? Where can I find them? When I google "people working right now" or "people working coronavirus", I guess a bunch of unrelated news, saying "the affects of working at home", which is useless.

Data on employment usually comes out a month or two (sometimes a year) after the reference period.   The Bureau of Labor Statistics (https://www.bls.gov/) puts out data on employment (who is working right now)  but to see the effects of our current crisis will take time to show.

Here are some links to the employment data that the BLS produces:

National employment (monthly):  https://www.bls.gov/ces/

State and local employment (monthly): https://www.bls.gov/sae/

State and county quarterly employment:  https://www.bls.gov/cew/

JOLTS data or Unemployment claims data can show you jobs that are hiring, laying off, separations,  and how many are collecting unemployment claims.  This could give you an idea about how many people are out of a job.

Job Openings and Labor Turnover Survey (monthly): https://www.bls.gov/jlt/data.htm

Unemployment Insurance Weekly Claims Data (weekly) : https://oui.doleta.gov/unemploy/claims.asp


What can explain increasing exports after default episodes? I am going through a paper on sovereign default by Mendoza and Yue (2012), in which defaults are explained in terms of how firms that use foreign intermediate inputs lose access to international credit markets once the governement defaults on its debt. This means that after a default production decreases and therefore the model predicts that exports should decrease, but whenI look at periods of default of Argentina, Ukraine and Panama, the data tells exactly the opposite: exports increased in the wake of the sovereign default. What can explain this? What papers can I look for?

at periods of default of Argentina, Ukraine and Panama, the data tells
  exactly the opposite: exports increased in the wake of the sovereign
  default. What can explain this?

Sovereign default may trigger a depreciation of the domestic currency. When that happens, it has a two-fold effect on the domestic economy: On the one hand, foreign immediate inputs (that is, imports) become more expensive. On the other hand, exports become cheaper (that is, more attractive) to foreign markets.

The net effect depends on the particularities of the economy being analyzed. For instance, a depreciation of the domestic currency might prompt firms to resort to domestic substitutes of the foreign intermediate inputs, in which case the drawback of sovereign default is reduced or mitigated.

An exporter is more immune to the detrimental effect of sovereign default if (1) its need for foreign inputs is not too high, (2) it hedged against a depreciation of the domestic currency, and/or (3) it does not compete for or need subsidies from the government because these could be harder to obtain in the context of sovereign default.


Are unit root tests necessary or useful on small samples of time series data? I have a 16 year time series (annual frequency with 16 observations). I will conduct an OLS regression. In this setting do I need a unit root test?

Do you have additional suggestions for things that I should do to pre-analyze my data?

Cochrane (1991) says that unit root tests have low power, so you need lots of data (at best) to distinguish among non-unit root alternatives:

  This paper exploits the fact that any time series with a unit root can
  decomposed into a stationary series and a random walk. Since the
  random walk component can have arbitrarily small variance, tests for
  unit roots or trend stationarity have arbitrarily low power in finite
  samples. Furthermore, there are unit root processes whose likelihood
  functions and autocorrelation functions are arbitrarily close to
  those of any given stationary processes and vice versa, so there are
  stationary and unit root processes for which the result of any
  inference is arbitrarily close in finite samples.

A critique of the application of unit root tests

However, depending on the test and the realistic alternative, this may be fine. David N. DeJong, John C. Nankervis, N.E.Savin, Charles H. Whiteman (1992):

  Monte Carlo methods are used to study the size and
  power of serial-correlation-corrected versions of the Dickey-Fuller
  (1979,1981) unit root tests appropriate when the time series has
  unknown mean. The modifications do not cause serious size distortions
  or power deterioration in the white noise case. While studies in the
  literature have investigated the operating characteristics of these
  tests in the presence of moving average errors, of particular concern
  in this paper is the performance of these procedures in the presence
  of autoregressive errors. The Philips and Perron (1988) and Choi and
  Philips (1991) procedures are found to suffer from serious size
  distortions and have very low power when errors are autoregressively
  correlated. We conclude that even in the most favorable cases, these
  tests perform poorly against trend-stationary alternatives which are
  plausible for annual, quarterly, and monthly macroeconomic time
  series. The augmented Dickey-Fuller procedure, on the other hand, is
  reasonably well-behaved.

The power problems of unit root test in time series with autoregressive errors

My impression is things have improved a bit in the power of more cutting edge estimators. See Haldrup and Jansson (2005)

  A frequent criticism of unit root tests concerns the poor power and
  size properties that many of such tests exhibit. However, the past
  decade or so intensive research has been conducted to alleviate these
  problems and great advances have been made. The present paper provides
  a selective survey of recent contributions to improve upon both size
  and power of unit root tests and in so doing the approach of using
  rigorous statistical optimality criteria in the development of such
  tests is stressed. In addition to presenting tests where improved size
  can be achieved by modifying the standard Dickey-Fuller class of
  tests, the paper presents theory of optimal testing and the
  construction of power envelopes for unit root tests under different
  conditions allowing for serial correlation, deterministic components,
  assumptions regarding the initial condition, non-Gaussian errors, and
  the use of covariates.

Improving Size and Power in the Unit Root Testing

Nevertheless, I can't help thinking that 16 annual observations is not much data to discipline time series estimation, and there will likely be lots of somewhat plausible specifications you will not be able to reject. Instead, you could use economy theory to justify the functional form and the statistical tests you use. For example, in efficient markets (crudely), price levels are highly serially correlated (random walks) but returns are independent. So if you knew your data were in returns, you might just argue rather than test that they are not a unit root.


Behavioral economics and utility function I'm not very familiar with behavioral economics, but as a recommendation of a teacher I'm reading "A Behavioral New Keynesian Model" by Xavier Gabaix. There's a part that I don't understand, here it goes:

And the footnote says:

  In particular I use the objective (not subjective) expectations. Also, I do not include thinking costs in the welfare. One reason is that thinking costs are very hard to measure (revealed preference arguments apply only if attention is exactly optimally set, something which is controversial). In the terminology of Farhi and Gabaix (2015), we are in the “no attention in welfare” case.

What does he mean with "This is as in much of behavioral economics, which views behavioral agents as using heuristics, but experience utility from consumption and leisure like rational agents"? Which author says that behavioral agents has utility like rational agents?

Thanks in advance.

I don't know the paper, but I can make a general point.

In behavioral economics, agents do not arrive at their optimal choice by maximizing a standard expected utility function. They may maximize some function with other characteristics or, alternatively, they use heuristics to make choices. That is, they may end up with a choice that is not optimal in terms of "objective" expected utility. This paper uses this objective expected utility as welfare criterion - with the true probabilities and true outcomes, not the perceived probability weights or perceived outcomes that led to the choice of the agent. The argument behind this approach is that although the agent makes choices according to a perceived world, he experiences utility in the true world. Therefore, instead of using the expected utility perceived when making the choice as a welfare measure, it makes sense to use the "objective" expectation of the objectively experienced utility.

I interpret the footnote without having read the paper: The agent uses subjective probability weights to make choices and incurs 'thinking costs'. The welfare measure ignores these thinking costs.


Can the Fed raise the M1 money supply permanently w/o helicopter money? I just read parts of an introductory economics book. I read about the monetary system in the U.S. and came up with the following question:

Can the Fed raise the M1 money supply permanently without having either to buy further Treasury bills or use helicopter money (i.e. to refrain from recieving the principal payment at maturity) ?

I think about the following model:

There is the Fed $F$ the companies $C$ the households $H$ and the banks $B$. We can partition the system into $\{F\}$ and $\{B,C,H\}$. At time $t=0$ there are $\alpha$ $\$$ in $\{B,C,H\}$ as currency i.e. as physical bank notes. No matter what happens in $\{B,C,H\}$ $\alpha$ cannot increase. Suppose $F$ wants to increase the money supply. As far as I know the Fed primarily uses open market operations (OMOs) to increase money supply. Let's try this. To simplify things let's consider a very simple bond.  Someone in $\{B,C,H\}$ issues a bond for  $\beta \$ $. The terms of the bond will grant the buyer a payment of $\gamma \$ $ at $t=1$ and $ \delta = \gamma - \beta > 0$.

So, as the Fed wants to increase money supply it undergoes an OMO, i.e. prints $\beta \$ $  and buys our bond. The money supply in $\{B,C,H\}$ at $ 0 < t <1 $ is $ \alpha + \beta $. So far so good. We could consider the money multiplication buy loans during $ 0 < t <1 $ but that would not change the situation at $ t > 1$. At $t=1$ the issuer of the bond i.e. someone in $ \{B,C,H\}$ has to pay $F$  $\gamma \$$. So, at $t > 1 $ there is less money ( $ \alpha - \delta$ ) in $\{B,C,H\}$.

So, to me all of that seems controversial. $F$ wants to increase the money in $\{B,C,H\}$ but effectively reduces it. Of course $F$ could disguise the situation by buying another bonds and thus increase the time until maturity but this does not really change the situation. So, this question comes up:

Can the Fed raise the M1 money supply permanently without having either to buy further Treasury bills or use helicopter money (i.e. to refrain from recieving the principal payment at maturity) ?

I am going to just discuss the situation for the monetary base, not M1. It is possible that the monetary base and M1 could move in opposite directions.

The following three instruments are the main components of the monetary base.

Currency in circulation: dollar bills, and coins. Currency holdings are the result of voluntary decisions by the private sector, and the Fed has very limited ability to influence those decisions.
Required reserves of banks held at the Fed. In order to increase this, loan amounts outstanding must increase. One can argue that Fed decisions influence borrowing decisions, but it is not something that they can easily directly control. (We would need to go back to the Volcker Fed to see a historical attempt to do this; its effectiveness has been debated. Sorry. no reference handy.)
Bank settlement balances in excess of required reserves (excess reserves). These balances went from near zero to being quite large as a result of “quantitative easing.”

The Fed cannot drive excess balances negative, as that would imply that some banks are short reserves - which is not allowed by regulators. However, it is free to make excess reserves as positive as they wish, as quantitative easing proved.

They do this via:

buying something;
or lending against something (which includes repurchase agreements, or repos).

That “something” is usually Treasury securities, but there are classes of private debt securities is can buy or lend against (as was done in the Financial Crisis). There is no other choice; the Fed does not have a legal mandate to dump money from helicopters (or whatever). So the answer to your question is that they essentially need to buy enough Treasury bills/bonds to increase the monetary base, unless they want to lend against other private debts.

I am not able to give a legal opinion, but it is unclear that the Fed has the ability to not demand payment for matured bonds/bills. Even if they did so, it would just be an irregular loan to the Treasury. (Since Fed profits are returned to te Treasury anyway, there is no point not taking the contractual Treasury payments.)


Is IS-LM still used in research or we learn it just to understand basic macro? Since i had my "Introduction to advanced macroeconomics" and discovered DSGE i've been wondering if IS-LM is used in research, i know it was used in the 80's or so, but now?

Yes, it is still used a recent example of IS-LM model used in literature is (Bonyah et al. 2019).
Also, it is important to understand those two are not substitutes for each other. IS-LM is a general equilibrium (GE) model. The way how you get dynamic stochastic general equilibrium (DSGE) model is that you take a general equilibrium model and add dynamism and uncertainty there.
As a student you were probably exposed more to the DSGE model based on real business cycle (RBC) model (which is based on Neoclassical GE models) as they are simpler, but New Keynesian DSGE models which are in essence DSGE version of IS-LM exists as well (see discussion of New Keynesian DSGE in Frische & Größl, 2010) and are relatively widely used (in fact I would say that after 2008 they became increasingly popular in the literature).
You will see DSGE models dominating over GE models in the literature, but this is because GE models are mostly used to derive some theoretical insights into how economy works, whereas DSGE can be used for actual practical simulations or even forecasting. You can't make any serious forecasts  with theoretical GE model - no matter whether it is IS-LM GE model or some other one (only predictions about variables and relationship but that should not be confused with forecasting). Consequently, you will see DSGE models dominating because it makes sense to make separate paper about DSGE model calibrated for let's say economy of the Netherlands as in Lafourcade and de Wind (2012) (coincidentally they actually apply IS-LM based DSGE), whereas applying any GE whether IS-LM or some Neoclassical one to a particular country would not be very useful (at least not much beyond what you can learn from the general version already). So for every one theoretical GE model, no matter what variety, you can easily find more than dozen DSGE models.


Partial vs. general equilibrium Microeconomic models are usually classified as partial and general equilibrium models. As a layman, I understand that partial equilibrium focuses attention on a few economic variables to find the equilibrium, while general eq. models capture a larger interaction.

Besides this, what are the key differences between the two kinds of modelling? Are there advantages and downsides to each of them?

Let's put the succinct answer by @TheAlmightyBob into an abstract model:

We want to model the labor market.

Markets' structure assumptions: goods market and labor markets are perfectly competitive. All participants are "too small" economically, and they cannot affect equilibrium price through their quantities demanded/supplied - they are "price takers". Markets "clear" - i.e. prices adjust so that quantity actually supplied equals quantity actually bought.

Agents assumption: There are $n$ identical workers, and $m$ identical firms, that participate in the market. Both populations are fixed.

Other assumptions: a) deterministic environment, b) one perishable good produced, c) model in "real terms" (real wage etc, scaled by the price of the good produced).

The typical firm produces according to the technology
$$Y_j = F_j(K_j,L_j;\mathbf q)  \tag{1}$$

where $\mathbf q$ is a vector of parameters. Perfect competition in the goods market, and a perishable good imply that all output produced is sold.The goal of the firm is maximization of capital returns over the choice of labor.

$$\max_{L_j} \pi_j = F_j(K_j,L_j;\mathbf q) - wL_j$$

We are modelling the labor market, so we are interested in the first-order condition

$$\frac {\partial \pi_j}{\partial L_j} = 0 \tag{2}$$
and the corresponding input demand schedule

$$L_j^* = L_j^*\left(K_j, \mathbf q, w\right) \tag{3}$$

Total Labor demand is $L_d = m\cdot L_j^*$.
The labor market equilibrium assumption implies

$$ L_d = L_s \Rightarrow m\cdot L_j^*\left(K_j, \mathbf q, w\right) = L_s \tag{4}$$

which implicitly expresses the equilibrium wage as a function of technology constants, of per-firm capital, and of labor supplied. In order to fully characterize the labor market, we need to derive also the optimal labor supply.

Each identical worker derives utility from consumption and leisure, subject to a biological limit of available time, $T$, and the budget constraint that consumption equals wage income:

$$\max_{L_i} U(C_i, T-L_i;\mathbf \gamma),\;\; \text{s.t.} \;C_i= wL_i$$

where $\gamma$ is a vector of preference parameters, indicating the relative weight between utility from consumption, and from leisure.
This will give us individual labor supply as

$$L_i^* = L_i^*(T,w, \mathbf \gamma)  \tag{5}$$

and total labor supply is $L_s = n\cdot L_i^*$. Plugging this into $(4)$  we obtain

$$mL_j^*\left(K_j, \mathbf q, w\right) =n L_i^*(T,w, \mathbf \gamma) \tag{6}$$

If we stop here, we have a partial equilibrium model that examines the labor market. We have fully described the market, and the goals and the constraints of the participants in it (firms and workers), related to the specific market. We can perform comparative statics in order to see how the various components of $(6)$ affect the equilibrium wage. Among them, there is the capital-per-firm term, whose effects on wage we can also consider based on $(6)$, by treating it as varying arbitrarily.

In order to turn this model into a general equilibrium model:
a) We need to specify things about capital: who owns it/controls it/makes decisions on it. What is the objective functions of these decision makers. This will lead us to an optimal $K_j^*$ as a function of the structure we will impose here. Then, comparative statics with respect to $K_j$ will turn into comparative statics with respect to the factors that affect the determination of $K_j^*$, which may very well prove to involve also $\mathbf q, w$ and even the other parameters in $(6)$, changing in this way the comparative statics results obtained in a partial equilibrium setting.

b) We also need to take into account any macroeconomic identities that characterize this economy, something along the lines of $mY_j \equiv ...$ where the right hand side will be determined by the assumptions we make related to capital, but also, for example, by whether we will assume that the economy is closed or open, or partially open to the outside economic system.

So, apart from being more complicated as a model, it may also lead us to different conclusions than partial equilibrium analysis.


The Backward Bending Supply Curve, Asymmetric Information, and Monopoly This recent question about when monopoly is undesirable made me curious about when monopoly is actually desirable? I was reminded of an argument for the efficiency of monopolies when moral hazard effects create a backward-bending supply curve in competition, like in healthcare perhaps. But, for the life of me, I cannot remember exactly how this was justified or if I am just imagining the whole thing. Is anyone familiar with arguments like this?

I know you can create a backward bending marginal cost curve and then with the usual linear marginal revenue curve, you might have MR=MC at a greater quantity than where demand intersects marginal cost, but I don't understand how the backward bending supply is also a a marginal cost curve--then the curve isn't even a proper function.

Information is a funny thing, so I wouldn't be surprised if incomplete information made for strange results.

Edit: I am thinking of pictures like these. I am just having a hard time reconstructing the jump from a backwards bending supply or demand curve (I understand these--this is often taught in 101), I just don't understand the MB/MC interpretation below where it seems like you might get a higher equilibrium quantity with monopsony or monopoly.

For these to make sense, MC and MB can't be functions of quantity unless I have a marginal cost in each direction? As in I could sell $q_1$ at a high price a low price, so am I approaching from above or below.

This definitely is not a complete answer. But I can imagine the case of increasing returns to scale. Or natural entry barrier. Increasing returns to scale is often discussed in international trade theory, but it can be also relevant here.

Example: think about commercial aircraft industry, dominated by Boeing and Airbus. Aircraft industry is seen as exhibiting increasing returns to scale, so monopolistic power those companies have (well, it is oligopoly, but that does not change the conclusion) may actually lead to cheaper quality aircraft.

Now there are people who really believe that all monopolistic powers are undesirable, but I believe those people are only minorities.


What is an unconditional model for a time series variable? If I am being asked to do an unconditional analysis of a time series variable, lets say GDP starting in 2000, what model am I supposed to estimate?

So the simplest response would be that an unconditional model is a model that does not include any other stochastic regressors. Lets say your variable of interest is $Y_t$ then a conditional model would be

$Y_t = \alpha + X_t\beta + v_t$

where you are implicitly conditioning on $X_t$ (i.e. treating it as fixed/exogenous). In time series $X_t$ can also include lags or transformations of $Y_t$. In the example that you gave, you could think of inflation (or lags of GDP) as potential conditioning variables.

So then, given this, what is an unconditional model? Well it is a model that does not include $X_t$ and instead only has deterministic terms (such as a constant or a trend). That is to say in the context of GDP, GDP growth doesn't depend on (i.e. isn't conditional upon) another variable but rather grows at a constant rate (or has a linear trend).


is univerisity's scholarships a price discrimination? Many universities allocate financial aid to undergraduate students on the basis of some measure of need. Does this practice reflect charity or price discrimination? If it reflects price discrimination, do you think it lies closer to first-degree discrimination or third- degree discrimination?

Scholarships certainly count as price discrimination. Charging different prices to different groups of consumers for the same good is third degree price discrimination, which is what occurs with school sponsored scholarships. First degree price discrimination occurs when a firm charges a different price for every unit consumed, and while theoretically possible for a school to do, is not done in practice and would require a much more complex pricing scheme than that offered by scholarships in the usual sense of the word.


Should publishers of scientific papers such as Elsevier be regulated by the government? To get a pdf copy of a paper published in Elsevier is quite difficult. You need to pay a high price. Publishers like Elsevier used to have good reasons to charge people a lot: before the invention of internet it was expensive to make those papers accessible.

However, now the cost to make those papers accessible is very low. The marginal cost of sending out a pdf copy is zero. The fixed cost of administration expense is extremely low compare to the publishers' stunningly high profit. What's worse is that the creators of those research papers do not get paid from publishers at all most of the time.

According to the most basic theory of economics, the consumer surplus is maximized if and only if the price is equal to the marginal oppotunity cost. So if we want to benifit the society as a whole, we should set the price of those pdf papers to nearly zero. So the copyright laws today actually unfairly benefit Publishers at the cost of the interest of the whole society.

So should the government force publishers of research papers to make their price more reasonable? Is there any other solution to this problem?

I am an ESL learner. please condone my error if it was a language problem. Thank you.

According to the most basic theory of economics, the consumer surplus
  is maximized if and only if the price is equal to the marginal
  opportunity cost.

You are mixing agents in using these concepts. Consumer's surplus (or utility) is independent of the producer's marginal cost. Paraphrasing Investopedia's explanation, the consumer surplus or utility is maximized when the purchase of an additional unit equals the marginal benefit the consumer obtains from that additional unit.

  Should publishers of scientific papers such as Elsevier be regulated
  by the government?

No. Doing so could prompt the publisher to reduce the quality of its product or --in an extreme scenario-- close operations altogether. The existence substitute publications which are available for free further weakens the need for governmental intrusion in a private entity's decision-making.


Concave utility functions corner solution explanation I seem to not be getting this. Could someone explain me the mathematical way to show a concave utility function [like (ax^2+by^2)] subject to a budget constraint has a corner solution. I get the economic intuition behind it but not the mathematical part behind it.

Intuitively, one can see that the corner solutions lie on the highest possible indifference curve(for concave preferences). Thus, utility obtained will be highest on those points. The following figure(courtesy-Google Images) might help one to visualize the above statement.

Mathematically, it is a bit tough to visualize whats going on. However, the Kuhn Tucker conditions can tell you whether a corner solution will exist or not(note that concavity/convexity of preferences are of no use while using Kuhn-Tucker Theorem, however one needs to check for differentiability of the objective function/constraints). First of all, according to Weierstrass' Theorem, it is guaranteed that an optimum point for $U(x_1,x_2)$ exists in the budget set $B(p_1,p_2,w)=\{(x_1,x_2):p_1 x_1 + p_2 x_2 \leq w\}$ (only if $U(x_1,x_2)$ is continuous in the given domain,i.e; the budget set). This tells us that we cannot rule out any solution/optimal point where either $x_1=0$ or $x_2=0$, or both. Now, to check whether a corner solution exists or not, we can proceed as follows. Consider the maximization problem
\begin{gather*}
\max\ U(x_1,x_2); \\ s.t.\ p_1x_1+p_2x_2\leq w;\ x_1\geq 0, x_2\geq 0.\\
\end{gather*}
The Kuhn Tucker Lagrangian is given as
\begin{align}
L = U(x_1,x_2)+ \lambda(w-p_1x_1+p_2x_2)+ \mu_1 x_1+ \mu_2 x_2.
\end{align}
The Kuhn Tucker conditions are given as-
\begin{gather*}
\frac{\partial L}{\partial x_1}\leq0\tag{1}\\ \frac{\partial L}{\partial x_2}\leq0\tag{2}\\ \frac{\partial L}{\partial \lambda}\leq0 ,\ \frac{\partial L}{\partial \mu_1}\leq0\ ,\frac{\partial L}{\partial \mu_2}\leq0\tag{3}\\ \lambda\frac{\partial L}{\partial \lambda}=0\tag{4}\\ \mu_1\frac{\partial L}{\partial \mu_1}=0\ ,\mu_2\frac{\partial L}{\partial \mu_2}=0\tag{5}
\end{gather*}
The inequality signs in conditions $(1)$ and $(2)$ can be replaced with the equality sign, if the constraints are binding(you can read more about binding constraints/constraint qualifications from Simon and Blume).
However, the given problem of corner solutions can be understood from the conditions in $(5)$, which say that $\mu_1 x_1=0$ and $\mu_2 x_2=0$. These equations tell us that either any one(or both) of the $\mu_i=0$ or any one(or both) of the $x_i = 0$. By considering all the above cases one by one, we can figure out whether a corner solution will exist or not.
For example, if the utility function is given as $U(x_1,x_2) = x_1 + \ln(x_2)$, the solutions obtained by using Kuhn Tucker theorem are-
\begin{gather*}
    x_1=
\begin{cases}
    \frac{w}{p_1} - 1,& \text{if } w>p_1\\
    0,              & \text{otherwise}
\end{cases}\\
x_2=
\begin{cases}
    \frac{p_1}{p_2} ,& \text{if } w>p_1\\
    \frac{w}{p_2},              & \text{otherwise}
\end{cases}
\end{gather*}
The corner solutions in this case are obtained whenever $w\leq p_1$.


Why do we have autarky price of mud pies equal to 0.5 of cheese, while it must be zero? (It's assumed that we work with the Ricardian model)
Absurdtopia is an autarky. Production of 1 mud pie requires 1 labor, while production of 1 cheese requires 2 labor. We know that price of 1 mud pie in terms of cheese under these conditions will equal to (labor needed to produce 1 mud pie)/(labor needed to produce 1 cheese)=1/2=0.5 cheese. Or in other word, 1 mud pie has price equal to the opportunity cost of its production, in this case 0.5 of wine. It's also called "Autarky price"

But there is one strange thing about this price. It completely ignores demand and utility of customers. Consequently, it is possible to imagine such result when this price will false. In our case it's obvious, nobody in their right mind would agree to give 0.5 of their hard-earned cheese for absolutely useless mud pie. Thus price of 1 mud pie in terms of cheese isn't 0.5, it's strict zero. It seems like our formula miserably failed to take into account utility function and demand of consumers, that it's inadequate.

In Ricardian model prices depend also on consumer utility and demand. Without that it would be impossible to solve the model no matter what the goods are. However, this is usually not explicitly mentioned in bachelor level textbooks where all models are for better or worse described in overly simplified form as not to load too much on most students. In fact you always have to make an assumption on demand in Ricardian model. Standard undergraduate model assumes that utility is identical for everyone home and abroad and such that the demand derived from utility function is homothetic across countries. Of course, you could also make some own 'unrealistic' assumption on utility and demand by assuming lets say country only wants one good or just some fixed quantity of the good etc. Under some assumptions it could be even impossible to actually solve for the price. However, if you for some didactic purposes want to assume there will always be 2 industries no matter what the demand is then price can be still determined even without any explicit demand.

The reason why you can say that relative prices are equal to the ratio of productivity is that assuming perfect market, then the prices will have to be equal to the marginal cost of production which has to be in equilibrium equal to marginal product of the factor of production (in this case only factor is labor). As long as you impose restriction that both industries exist prices have to be such that they equal the ratio of productivity. If this would be violated then all workers would just go to the sector where they can get higher wage  for the same effort in terms of labor (here there is no economic profit all accounting profits are paid out as wages). For example, if productivity is such that 1 cheese = 1 labor and 1 mud pie = 2L then if the relative prices $\frac{p_{ch}}{p_m}$ are anything else than ratio of the productivity $1L/2L$ the other sector would completely collapse and could not exist anymore, violating the assumption both sectors exist.

But again all of above is extremely simplified version of Ricardian model. In actual Ricardian model you dont have just 2 goods but $n$ number of goods, you have wages utility and much more. Following PhD level textbook by Feenstra, the set up of Ricardian model includes:

Continuum of industries indexed by $z\in [0,1]$ with productivity in each industry being $a(z)$
The utility is given by $U = \int_0^1 \ln c(z) dz$, and since consumption has to be equal to the income this utility function is subject to the constraint $c(z)= wL \implies c(z)= p(z)z$, where $z$ would be some good consumed (remember these are all distributions, its not just 1 good but potentially infinite amount of goods since $z \in [0,1]$ and there are infinite numbers between 0 and 1). The demand is derived from utility function by solving the consumer optimization problem (that is how to spend resources in a way to maximize utility).
To clear markets in autarky prices are set $p(z) =wa(z)$ where $w$ is the wage.
Next you would assume the all above would also hold for foreign country and then let them trade (I might have also skipped some minor auxiliary assumptions - for full treetment see Feenstra (2015) chapter 3.).
Once countries open up to trade, balance of trade is assumed so: $\frac{w}{w^*} = \frac{z}{1-z}\left(\frac{L^*}{L}\right)$, where star * indexes the variables from the second country. Once the trade opens up the prices will be given not by national supply and demands but by world supply and demand which could be derived from the above assumptions.

Now, I wont post the full solution to this model but due to the budget constraint prices also depend on what the utility, and by extension what demand is (since demand is always derived from utility). Hence, in Ricardian model there is absolutely nothing strange about price determination - its a part of the whole system and each part makes quite a lot of sense if you think about it individually.

To sum it up, the full treatment of Ricardian model does and not only that but also has to include both utility, demands, consumption, wages etc. So your description of Ricardian model in your question does not make it justice. In a proper Ricardian model the question does not make sense because the prices actually depend also on utility and demand - you cant really make models in economics without including utility somewhere in one way or another, explicitly or implicitly. However, dont expect to learn all this at once. The purpose of the simplified version of Ricardian model that does not include all of the above explicitly is to make it easy for students to follow, but as a result the simplified Ricardian model will have many 'plot holes' and if you just try to rigorously examine the simplified Ricardian model it will never hold up - it misses too many important parts but again thats just for didactic purposes - not because the 'full' Ricardian model does not cover these.

References:
Feenstra, R. C. (2015). Advanced international trade: theory and evidence. Princeton university press.


Should minimum wage be regulated for developing countries I live in a developing country, where unemployment is at roughly 25%. There is a large divergence in population whereby skilled workers are typically active within in the manufacturing, finance, tourism, retail  etc. sectors, however the bulk of the unskilled population typically would/could find employment in other sectors such as Manufacturing(again), Mining, Agriculture, construction and the like.

The problem is, within a highly unionised country, minimum wage is set to first world standards whereby its mostly cheaper to export our raw materials, and buy back finished products from China, India etc.

Is there a timing factor associated with implementation of minimum wage? surely there are natural supply vs demand factors at play in determining minimum wage. would it not be better to reach a critical mass in the labor force whereby skills, infrastructure & techniques are at par with international competition.

A welfare or efficiency enhancing minimum wage can be justified in several theoretical settings. For example, if there is monopsony (a single employer), the employer can internalize the effect of his labor demand on the labor market price (the wage) and therefore holds down both labor demand and wages. A minimum wage can therefore increase both employment and wages. In a developing country with a dominant industry like resource extraction or tourism controlled by a small number of connected families, this might be a decent model and therefore an industry specific minimum wage perhaps could work. Another theoretical motivation, if an industry has inelastic labor demand, due to pricing power in the good markets and low substitutability of other inputs for labor (maybe French wine would be an example of this), then the minimum wage could raise wages without great inefficiencies or reduced employment.

Of course economists dispute if those settings hold in practice. Emigration, competition in factor or product markets, and labor market competition across industries can lead to competitive wage setting even in monopsonistic industries. And minimum wages may work, and yet we might not have a good model for why. Or they might be helpful in the short-run but harmful in the long run. The empirical evidence of the short-run costs and benefits of small minimum wages changes does not show they are particularly harmful, even if we have good reason to think that the long run costs and the costs of large minimum wage increases would be significantly larger.


Is profit margin the same as net profit margin? I'm quite confused by these two terms because different websites define them differently.

Profit margin is profit after the cost of sales. If you review an income statement, you will see revenue, and then typically a data field called “cost of selling goods”. Revenue - Cost of sales is profit margin.

Net profit margin is Revenue - all expenses. So this includes taxes, interest expense, administrative expenses, and ALL expenses. An easy way to remember this is that profit margin is simply revenue - the cost of selling products, and net profit margin is revenue - ALL costs.


How can I learn about economics? I'm a civil engineering student and I'm interested in economics and I would like to learn everything about it. I would like to learn everything from banks to the stock markets and everything there is to learn about. I always read articles about stocks and the market but I feel really ignorant. Can you recommend me some introductory books or some learning ressources?
Thank you in advance.

I would recommend Tim Harford's "The Undercover Economist" for an easy way to get some exposure to how economists think, without having to go through all of the dry academic mathematics etc.

To get a sense of the shape of the economics discipline, you might also like to look at "Economics: A Very Short Introduction" by Partha Dasgupta.

Lastly, if you really want to get into the discipline and you want a slightly more formalised/academic treatment then any good introductory textbook should do. N. Gregory Mankiw's "Economics" is a popular choice.

The above describes how you might go about learning about economics. But from your question, it seems more like you are more interested in learning about finance, which is the discipline that encompasses the study financial markets, etc. I learnt finance from a book called "Introduction to Finance" by Lawrence J. Gitman and Jeff Madura, but there doesn't seem to be a recent edition so perhaps someone else can recommend a more recent introduction to the subject.