Question about Social Welfare Function and Social Profile What are the meanings of a social welfare function and social profile? How are they related?

In its most general formulation, a social welfare function is just a utility function representing the preferences of "society as a whole" (or the preferences of a hypothetical "benevolent social planner" who makes decisions for the society).
Let $X$ be some space of "social outcomes".  (Social outcomes could be anything.  But if you want a concrete example, then suppose that social outcomes describe allocations of resources to individuals.  If there are $N$ individuals, and there are $M$ distinct types of resources, the an allocation is an ($N \times M$)- dimensional vector, so in this case $X$ would be some subset of $\mathbb{R}^{N\times M}$.)  A social welfare function is just a function $W:X \rightarrow\mathbb{R}$;  heuristically, if $x$ is some element of $X$ (some "social outcome") then $W(x)$ measures the "total welfare" or "social value" or "overall desirability" (or whatever) of $x$.  Thus, the benevolent social planner should aim for policies that maximize the value of $W$.
The key question, of course, is how to define $W$.  One way to do this is to somehow build $W$ out of the preferences or utility functions of individuals.  Suppose there are $N$ individuals, each of whom has a preference order $\succeq_i$ on $X$.  This collection of preference orders $(\succeq_1,\succeq_2,\ldots,\succeq_N)$ is an (ordinal) social profile.  (This partly answers your second question.)  A big part of the modern theory of social choice and welfare is about defining systematic, principled ways of defining $W$ based on the profile $(\succeq_1,\succeq_2,\ldots,\succeq_N)$.
Typically, purely ordinal preference information about the individuals is not sufficient. (This is one way of reading Arrow's Impossibility Theorem --I won't get into the details here).  So we might want to endow each individual with a utility function $u_i:X \rightarrow \mathbb{R}$ (which is more informative than just a preference order).  A collection of utility functions $(u_1,u_2,\ldots,u_N)$ is a (utility) social profile.  (This is the rest of the answer to your second question.) We  then seek to construct $W$ out of $(u_1,u_2,\ldots,u_N)$.
For example, given a social profile of utility functions $(u_1,u_2,\ldots,u_N)$, we could define $W(x):=u_1(x)+u_2(x)+\cdots+u_N(x)$ for all $x\in X$ ---this is the utilitarian social welfare function. Or we could define $W(x):=\min\{u_1(x),u_2(x),\ldots,u_N(x)\}$ for all $x\in X$ ---this is the egalitarian (or Rawlsian) social welfare function.  Or (assuming all utilities are strictly positive), we could define $W(x):=u_1(x)\cdot u_2(x) \cdots u_N(x)$ for all $x\in X$ ---this is the Nash social welfare function.
Different social welfare functions have different properties, which may be more or less desirable, depending on your beliefs about moral philosophy and social justice.  Much of social welfare theory can be seen as developing a mathematically precise way to think about moral philosophy and social justice, by understanding the properties of different social welfare functions and the tradeoffs between these properties.  For example, axiomatic characterization theorems begin with a list of axioms (each encoding a property corresponding to our intuitions about rationality, morality, justice, etc.) and show that there is a unique social welfare function that satisfies all of these axioms.


Empirical studies on network equilibria There have been a number of papers in the literature (pioneered by Sanjeev Goyal, Jackson, Kranton, amongst others) emphasising the role and importance of network structure in economics, the equilibrium arising therein and the relationship between the topology and other graph properties and the equilibrium.

Such networks arise in a number of cases, for example, research collaboration, social networks, crime networks, etc. My question is what published empirical studies have tested the hypothesis presented by the theory? Do such studies also find that the graph properties are critical, as predicted by the theory? Or is network effects in economics a theory-only area?

Because the notion of pairwise stability is so important in the theory of network formation and equilibrium, there has been a few attempts to test it in experimental setups. I know at least two papers:

Myopic or farsighted? An experiment on network formation, by Georg KIRCHSTEIGER1, Marco MANTOVANI, Ana MAULEON and Vincent VANNETELBOSCH.
Departures from Best Response in Market-like Link Formation: An experimental study, by Margherita Comola and Marcel Fafchamps.

The first is a discussion paper. A word of warning regarding the second : the version of the paper that is linked is very much a draft. You might want to have a look to get an idea of what the authors plan on doing, but don't rely on it too much. I attended a talk where Fafchamps presented the paper this week, and he told us that they initially got the wrong data from their experiment. They only realized it rather recently and had to revise the whole paper. Because the linked draft is from February 2014, there are chances it is still based on the the old data (I did not read it).

For some field experiments on networks in development economics, you might want to have a look at other papers by Fafchamps :
http://web.stanford.edu/~fafchamp/index.html. I am not sure these explicitly test equilibrium concepts though, but they might.


Which values to assign to a quantified dummy variable I am working on a data set from kaggle (https://www.kaggle.com/spscientist/students-performance-in-exams) about how student performance relates to some explanatory variables, such as if the school provides lunch, gender etc.

r/dataanalysis - Which values to assign to a quantified dummy variable
I have changed all the variables to quantative dummy variables and the variable I am concerned about in this post is educ_par (education parents). It can take on 6 different values, which refer to:

if both parents have a bachelors degree

if some went college

three if both of them have a masters degree

if both have an associates degree

highschool

if some have a highschool diploma

Now since, I only have dummy variables my teacher suggested me to at least include one quantitative variable. So that I can interpret it better. Here I want to change educ_par into years of education by parents. However, I run into one main problem here.
Since this is a fake dataset and the variables are not that clear I do not know what values to assign. For example, given the value 5, meaning both parents have a highschool degree, and 6, meaning only some have a highschool degree. In case with 6 I do not know what education the other parent has.Also, imagine both went to highschool, which may mean a total of 12+12= 24years of educatin by both parents, whereas if some went to college, this may mean a 15+0 years of education by both parents, since we do not know what the other parent did. In turn this could mean that assigning values like this having a highschool diploma from both parents affects the grade of their children more than if only one parent has a masters degree.
So my question is how should I quantify this variable? To not run into the problem I am facing above. May I just leave out the cases in which only some parents have a specific degree just to simplify it?
Also how does including a quantitative instead of the dummy variable effect any of my regressions? Or may they just be the same?

It's generally not good practice to go from categorical/dummy variables to quantitative variables, as csilvia noted, because categorical/dummy variables have less information than their quantitative analogs. None of the variables in your dataset are quantitative except for math, reading, writing, and total, which I'm assuming are outcomes of interest, not inputs.
If your teacher wants you to include quantitative variables, I'd go back to the original dataset and see if there's anything quantitative included there rather than trying to make a categorical variable fit as a quantitative one.


Max Utility Function and Finding associated demand curve I have a max utility function, therefore;
U(x,y)= max(2x,y)
and I am trying to find the demand function x = x(x , y , ), note this function cannot be differentiated. I am familiar that the utility function states that it is best to have x=0 and all of the good y, or vice versa. So I've been trying to solve with the budget line making one x=0 and then again y=0 but I am unsure what to do from this point?

Working so far:
p1x+p2y=M
when y=0
x=m/p1
U= 2m/p1

&
when x=0
y=m/p2
U= m/p2

So now I have two equations in terms of U

From your formula for $x$ when $y=0$, you should be able to find $U$ in terms of $M$ and $p_x$ when $y=0$.  Similarly, $U$ in terms of $M$ and $p_y$ when $x=0$.  The key then is to find the critical price ratio at which, to maximise $U$, the switch needs to occur from $y=0$ to $x=0$.

Can you take it from there?


Stationarity vs weak dependence I am doing an undergraduate course in econometrics where we are using the text Introduction to Econometrics by Dougherty. While going through time series, it was mentioned that one of the necessary assumption to justify OLS, is weakly dependence, which then states that this concept will be covered in more detail in next chapter, which after completing is primarily about stationarity and no mention about weak dependence. I am now wondering if weak dependence and stationarity are similar or related?
In a particular textbook, Introduction to Econometrics by Wooldridge, he states that weak dependence and stationarity are different concepts. While he states that weak dependence is a necessary assumption for OLS, stationarity is not but however needed for the Central Limit Theory to hold. And Wooldridge also defines an integrated process as highly dependent process that can be transformed into weakly dependent. Whereas Dougherty (and Gujarati) links integrated process with stationarity.
Could someone clarify the concepts of stationarity vs weak dependence and how they fit into OLS and also the definition of integrated process.

Weak stationarity and weak dependence are complementary conditions.
A weakly stationary time series $y_t$ has an underlying statistical process which is time-invariant.
This is characterised by three conditions:

$$E(y_t)= \mu$$
$$E((y_t-\mu)^2)=\sigma^2 <\infty$$
$$Cov(y_t,y_{t+s})=Cov(y_t,y_{t-s})$$

Condition (1) indicates that a weakly stationary series has an underlying (or long-run equilibrium) unconditional average.
Condition (2) indicates that the unconditional variance of a weakly stationary series does not change through time and remains within a certain range/finite.
Condition (3) indicates that a weakly stationary series has stable (symmetrical) dynamics between periods, which only depend on the number of periods, rather than time itself.
That is to say, Condition (3) states that a weakly stationary process we can assume a stable relationship between certain periods, regardless of whether it's the year 2000 or the year 3000. Time plays no role, only the relationship across periods.
Weak Dependency:
Weak dependency ensures mean reversion back to an underlying process.
Weak dependency states that the correlation between $y_t$ and future periods $y_{t+h}$, tends to zero sufficiently quickly. And with periods increasing to infinity, dependency should tend to zero.
$$Corr(y_t,y_{t+h})\rightarrow 0, h\rightarrow\infty$$
Importance in the Context of Time Series
With time series data, our central objective remains to estimate the magnitude of change in Y terms of linear unidirectional causation from X.
That is to say, our goal is to take a sample time series and fit a line which best explains the change in Y given X characteristic of the underlying population.
Yet, in the time series context, conducting estimation representative of the underlying population is undermined by a failure of i.i.d conditions
An evolving (non-stationary) time series, entails that observations will not be drawn from an identical distribution and parameter estimates will not convey a constant linear (time-invariant) relationship on Y given X;
Furthermore, given that time series data is necessarily non-independently distributed (i.e. current realisations dependent on past realisations), then again parameter estimates will no longer convey information on the underlying population.
It follows that two mutually dependent properties must be upheld to ensure the i.i.d conditions known as:

Sufficiently fast weak dependency – random variables should exhibit
finite memory/asymptotic independence;

Weak stationarity – the underlying statistical properties (mean,
variance, and covariance) should be time invariant as if drawn from
an identical distribution.

Note that both conditions are mutually dependent, given that sufficiently quick finite memory is a requirement for a process to exhibit mean-reversion back to some underlying population.
Moreover, the combination of weak dependency and weak stationarity ensure that a property known as ergodic stationarity is upheld.
Essentially, ergodic stationarity implies that with a quick return to an identical distribution, then realisations are representatives of the greater process, and averaging is relevant.
Further, weak stationarity is important in terms of the exogeneity condition to control for a causal relationship, given that if contemporaneous exogeneity holds in one period; then by weak stationarity, it should hold for all periods.
Also, ergodic stationarity ensure that the process mimics a random independent sample required to uphold LoLN/consistent estimation/CLT are valid.
Finally, weak stationarity overall avoids the problem of so-called spurious regression in which two mutually trending series are wrongly identified to have a causal relationship.


Why U.S Import from China is a big deal? Economics books often quotes import as a borrowing

Why it is borrowing since the importer pays to exporter for the goods they have received. Lets say, a US company is importing apparel from China.

You are right. There is nothing in "importing" which is intrinsically related to "borrowing". If economics books often quotes this, it is because they make an accounting-like shortcut (under assumptions of balanced public accounts) that is representative of the USA-China case.

On the one hand, USA has a net negative trade balance with China, i.e. USA is "net-importing" from China.
On the other hand, the "U.S. debt to China is $1.2 trillion as of August 2017".

The accounting-like shortcut consists of saying that USA debt to China is actually used to import goods from China. This is not the factual reality, but an accounting one. The real story behind is that actually, when importing goods from another country, a U.S. private company is not doing so from a local one who would pays taxes and thus generates tax income for the U.S. government, which would in turn allows it to contract less debt and so on...


What is the Walras law vs first welfare theorem As far as I know, both of the first welfare theorem and the Walras law are closely tied to the invisible hand. what is the difference between them?

thank you very much for your help

Walras' law describes market equilibria conditions, which states roughly speaking that if there exists within an exchange economy market equilibria for $n-1$ good markets, then the last good market $n$ is also in equilibrium. Thus, we have a market system with $n$ equations and $n-1$ independent variables, where the last market is linear dependent from the other $n-1$ markets.

Now, the first Welfare Theorem of Arrow states that for every exchange economy with convex preferences, every Walrasian equilibrium is Pareto efficient.

The implication of the first Welfare Theorem is that we need no central planner to achieve a Pareto efficient outcome in an exchange economy. The Pareto efficient outcome is based on decentralized decisions within the exchange economy. If we now introduce transaction costs and incomplete information into the system, then it should be obvious that every decentralized economy is doing better than every centralized one due to the information problem.

Update:

I forget to mention some interesting implications of Walras' law. The law reveals to us that we have only one degree of freedom. This implies that a central instance can only fix one price level, for instance, that of the numeraire good, the others prices of the markets are determined by the market forces. Hence, a central instance has no control on them.

To make this argument more precise, let us apply this law to a central bank policy. We have only one degree of freedom, the central bank can now fix either the interest rate of the money market or a price index, but not both. If it fixes the interest rate, it has to accept the price levels and therefore the inflation rate, and vice versa for the price index. This also implies that a central bank cannot simultaneously targeting the price index and the unemployment rate of an economy.


perfect bayesian nash equilibrium is simply nash equilibrium Is it true that for two player zero sum game, Perfect Bayesian Nash equilibrium is simply Nash Equilibrium?

I am learning game theory and our lecturer does not explicitly cover it.

The quick answer is Yes.

A Bayesian Nash equilibrium (BNE) is Nash equilibrium (NE) generalized to situations where there is uncertainty over players' payoffs; that is, games with incomplete information. For example, in the following zero-sum game, $t$ is a "state variable" that may take values $\epsilon>0$ or $-\epsilon$, each with probability $1/2$, say. Then the equilibrium you solve for would be a BNE.

On the other hand, if $t$ is not a random variable (or a degenerate random variable, as the jargons would have it), e.g. $t=0$ with probability $1$, then the equilibrium would simply be a NE.

In other words, a NE (of a complete information game) is a special case of a BNE (of a game of incomplete information, where there is only one payoff-relevant state, so to speak).

$$
\begin{array}{|c|c|c|}\hline
  & H & T \\\hline
H & 1+t,-1+t & -1+t, 1+t \\\hline
T & -1,1 & 1,-1\\\hline
\end{array}
$$

Adding perfection to BNE puts more restrictions on the players' off equilibrium beliefs about their opponents' strategies. In the case of matching pennies (the game above with $t=0$), which I assume you have in mind when saying "zero-sum games", this consideration wouldn't matter.


Is it worth while to get a master's in math (or applied math) before pursuing phd in economic theory? I'm planning to apply to phd programs in economics and I'm hoping to eventually work in academic research on economic theory, social choice theory, welfare economics or related areas. I'm wondering if doing a masters in math before applying to an economics phd would be worthwhile? Seems like it might be nice to have the math stuff down really solid, so I can focus on economics in grad school.

By the time I finish undergrad (in U.S.), I will have taken the following math courses (one semester each): Multivariable calc, linear algebra, abstract algebra, real analysis, topology, optimization, probability-statistics, basic numerical analysis with matlab, differential equations.
Econ courses that I'll have taken: year of micro, year of macro, year of econometrics, year of graduate micro.

Edit: My undergrad major is philosophy and also a joint major math-economics.

The following answer is based mostly on gut-feeling and I urge you to read some of the papers found in the last section to base your decision on something quantifiable.

Coursework (-)
Having an additional degree in math will be completely irrelevant in the coursework, where your current courses will be sufficient preparation. Indeed, a game theory/advanced econometrics summer course may be more useful as preparation.

Thesis Advisor (+)
The degree will make it easier for you to find an advisor who is willing to invest time in working with you. If you move from a philosophy/math/economics undergrad degree into a top-school economics Ph.D. you will most likely compete for the attention of supervisors with students from Europe and Asia who already have Master's degrees in some field. Some research from the last section of my answer suggests that in most departments a lot of resources are spent on the very best students.

Research (+++)
The math degree will be helpful in your research. From the fields you listed as your interests a Master's degree in math does not seem like a bad idea. All fields mentioned are very math-heavy and often use math that is not taught at undergraduate level. However, if you later decide to work in other fields such as macroeconomics, your math Master's will not help you much.

Job Market (+)
Your decision will also affect your job market prospects. The fields you have outlined have a fairly small job market compared to let's say applied econometrics or macroeconomics. If you do not stand out from the crowd it may be difficult to find a position as assistant professor. Thus, having a math master's may help you write an outstanding job market paper. However, you will be older when you enter the market. For very few employers, this may be a secondary criterion.

Tenure (+)
When you obtain an assistant professor position after your Ph.D., you will need to write a number of strong papers within several years. If you at that point need to acquire additional math skills this will slow down your research in this crucial time.

Lifetime income (--)
You will most likely lose at least one year of income from doing the degree. If your degree is two years or 1.5 years, you may at most be able to make up for one year via a shorter Ph.D. by being more productive. It is not unlikely that you will lose 2 years of income. However, doing the Master's may allow you to enter better Ph.D. programs, write better papers, and achieve a better job placement, earning you more money. But this effect is hard to estimate.

Readings
Be sure to check this website to make the right strategic decisions. There is a lot of research on what makes Ph.D. students in economics successful.


Differentiation of the value function in Burdett Mortensen (1998) I'm currently making my way through Burdett and Mortensen's classic paper of on the job search. What should be an easy task of finding an expression for the reservation wage is made slightly more complicated by the presence of the max operator.  We are faced with the following Bellman equation for the value of a job paying a wage $w$. The bellman equations are standard. The value of a job paying $w$ consists of the wage $w$ plus the expected gain from searching and finding a better job discounted by the probability a job offer comes along $\lambda_1$ plus the loss of due to becoming unemployed when the job is destroyed at rate $\delta$. The value of unemployment $V_0$ consists of unemployment benefit $b$ plus the expected gain from becoming employed discounted by the probability an offer comes along $\lambda_0$. Note the probability an offer is made is different depending on whether someone is already employed or unemployed. The distribution of offers is given by $F$ \begin{equation} rV_1(w)=w+\lambda_1\bigg[\int \max\{V_1(w),V_1(\tilde{x})\}-V_1(w)\bigg]\;dF(\tilde{x})+\delta [V_0-V_1(w)]   \end{equation} \begin{equation}rV_0=b+\lambda_0 \bigg[\int \max\{V_0,V_1(\tilde{x})\}\;dF(\tilde{x})-V_0\bigg]\end{equation} Since $V_1(w)$ is increasing in $w$ and $V_0$ is independent of it we know a reservation wage exists such that if $w>R\implies V_1(w)>V_0$, $w<R\implies V_1(w)<V_0$ and $V_1(R)=V_0$. Standard arguments (integration by parts) shows that \begin{equation} R-b=(\lambda_0-\lambda_1)\int_R^\infty V_1'(\tilde{x})[1-F(\tilde{x})]\;d\tilde{x} \end{equation} from here I would like to take the derivative of the first equation and solve for $V_1'(w)$. However If I use Leibniz integration rule I need the integrand to be differentiable. The max of two continuous functions is usually not differentiable where they are equal so I have a problem. If I assume that I integrate over all $\tilde{x}\geq w$ then $V_1(\tilde{x})\geq V_1(w)$ (wage offers that will be induce a worker to switch jobs) and the result follows by Leibniz rule. But there are wages in the distribution that won't be accepted and this derivative wont hold. The derivative is \begin{equation} V'(\tilde{x})=\frac{1}{r+\delta+\lambda_1(1-F(\tilde{x}))} \end{equation} I imagine I'm missing something but I'm not sure what. If anyone could give me any advice I would really appreciate it.

When you take the integral of a $\max_{\{\cdot\}}$ operator, I think you have to split the integral into two separate integrals with different supports on them.

Even if your value function is complicated and there is no differentiability, you only need continuity for an existence of a solution to solve the optimization problem.


Forecasting future currency exposures I have a data set (about 60 observations for each variable) for monthly operating income for a Japanese company denoted in EUR, USD, and JPY, and their respective currencies. I believe there is linear relationship between the currencies and the respective operating income denoted in the currencies. However, the first differences in operating income and currencies are random.

What is the best way to forecast the exposure of the future short-term month's operating income?

I have translated the foreign income to JPY and added them together with their domestic operating income, and then use moving average, weighted moving average and trend line to forecast the next month (into the future).

I have also used the first differences/increments of the operating incomes and their respective currencies to make them stationary, and to find the regression and coefficient. However, I don't know how to use the results to forecast.

Are there any other models that produce better outcomes for this fairly limited sample set?

I believe that you may be looking at a spurious relationship. You are looking at the same data transformed by a scaling factor (the currency) and regressing it on itself. Of course they will be related. If they weren't scaled, they would be perfectly co-linear.

What you really want to know is if there is an effect of changes in the relative value of Yen to USD or Euros on the operating income of the company.

You should start by visualizing this relationship by graphing it. Graph the relative value of Yen in your time series, then graph operating income in your time series. Do they move together? Look for structural breaks. For instance, they may have only begun to move together in the last year, that can really mess up an econometric model if you don't account for it.

After you have finished analyzing it graphically, I suggest reading up on co-integration and the Engle-Granger method. Some literature that are good places to start are: Stock and Watson(1988), and Engle and Granger(1987).

This part of time series econometrics enables you to work with variables that are NOT STATIONARY. One problem with first-differencing such as you mentioned in your question, is that it throws away the very relationship you want to know about. The Engle Granger method enables you to analyze the relationships that matter economically, and is still statistically sound.

The world of time-series economics is a lot of fun, but make sure you stay grounded in the economics behind your data. Don't ignore the little things in your data, they may be the most meaningful. Don't be surprised if the relationship you thought was important actually is not important. That tells you just as much as if it is!

There's so much to talk about with the fascinating area you are delving into but I hope this can be of interest to you.


Minimum Wage effects on Adjacent Cities When a city like Seattle raises the minimum wage to 15/hr, how does it affect the labor market in adjacent cities?

Are there examples, studies or accepted practices to address: Difficulty staffing entry level positions when labor pool can work across the street for several dollars/hr more?

A very famous study in this direction is Card and Krueger (1994).
They look an increase in the minimum wage in New Jersey in 1992. While New Jersey raised the minimum wage from USD 4.25/h to USD 5.05/h, the minimum wages remained at $4.25 in adjacent Pennsylvania.

You should have a look at the subsequent research.


Nominal vs real growth of monetary base Can someone clarify the difference between nominal and real growth of monetary base? I believe it is the policy of some central banks to target for example 0% nominal growth in the monetary base. What would an example of 0% real growth target be - nominal growth of say 2% but inflation also of 2%?

Are there advantages or specific motivations of designing policy according to one vs the other?

For this question, its useful to work off of the equation of exchange.
Let's start with the equation:

MV=PY

M = the money supply, V = Velocity of Money, P = Price level, Y = real output

Log both sides

ln(MV)=ln(PY)

Split the terms up according to log properties

ln(M) + ln(V) = ln(P) + ln(Y)

Take the derivative of each term so that we have

dM/M + dV/V = dP/P + dY/Y

to define each of these, dM/M = the rate of change of the money supply(nominal), dV/V = the rate of change of the money velocity, dP/P is the inflation rate, and dY/Y is the real GDP growth rate.

Real Money growth is but a simple algebraic manipulation by doing as such:

dM/M - dP/P = dY/Y - dV/V

Real Money growth = dY/Y - dV/V

Real money growth must be correlated with real GDP growth. The money supply can only be said to have increased in "real" terms if the quantity of goods and services that the money supply can purchase has increased.

So we can see that if we wanted to have 0% real money growth, we would simply need to have the rate of change of the money supply be equal to the inflation rate, or alternatively the real GDP growth rate be equal to the rate of change of the money velocity.

However, we can also see clearly that a nominal money supply increase of 0% is likely undesirable. If dM/M is just 0, then a positive value on the right side, which is to mean that real GDP is expanding faster than money velocity is increasing, the result will be deflation.

As to policy implications, central banks usually no longer target monetary aggregates, as they tend to be unreliable and volatile. Central banks these days target interest rates or alternatively there is some interest in targeting a certain growth of the nominal GDP, which is just the MV of the MV=PY equation (or P*Y if you like). The main disadvantage of interest targeting is that the central bank can be said to have "run out of ammo" in a sense when it pushes interest rates down to the zero lower bound. This is to mean that the target rate can not go below zero, so the stance of the Fed cannot become more expansionary past this point. This is important as the stance of the Fed is derived from the position of the target rate relative to the natural rate of interest. If there is a shortfall in spending in the economy, and the natural rate of interest falls to zero, then The Federal Reserve can be said to have no power in this situation. Even if it were to drop interest rates to zero, the effect would be a neutral stance, which is highly undesirable in a recession. Recommended reading on the subject: https://www.brookings.edu/blog/ben-bernanke/2017/04/12/how-big-a-problem-is-the-zero-lower-bound-on-interest-rates/


Derive the Demographic Structure in the Steady State I am reading a paper with following description on the demographics in their model: "... each (representative) agent lives for $T$ periods ... We assume that each individual has $e^{f}$ children at age $B$. Since we consider only steady states, we need to derive the stationary age distribution of this economy associated with this fertility rate. Our assumptions imply  $N(a, t)=e^{f} N(B, t-a)$ and $N\left(t^{\prime}, t\right)=0, t^{\prime}>T$. It is easy to check that in the steady state $N(a, t)=\phi(a) e^{\eta t}$, where $\phi(a)=\eta \frac{e^{-\eta a}}{1-e^{-\eta T}}$ and $\eta=f / B$ is the growth rate of population."
I have no idea how is this steady state calculated?

I don't know the paper nor the notation, so I am just guessing here. I gues $N(a,t)$ is the number of agents of age $a$ at time period $t$.
Let's follow the number of age $B$ accross generations:
$$
\begin{align*}
N(B,t) &= e^f N(B, t- B),\\
&= e^{2f} N(B, t - 2 B),\\
&= \ldots,\\
&= e^{f t/B} N(B,0),\\
&= e^{\eta t} N(B, 0).
\end{align*}
$$
Then using this in the definition of $N(a,t)$, we have:
$$
N(a,t) = e^f N(B, t- a) = e^f e^{\eta(t - a)}N(B,0).
$$
Next, I assume that at period $0$ there is a mass of size 1 but nobody lives longer than $T$ periods, so integrating across all ages:
$$
\begin{align*}
&\int_0^T N(a,0) da = 1,\\
\to &\int_0^T e^f e^{-\eta\, a}N(B,0) da = 1,\\
\to &e^f N(B,0) \left[-\frac{e^{-\eta a}}{\eta}\right]^T_0 = 1,\\
\to &e^f N(B, 0) \left[1 -e^{-\eta T}\right] = \eta,\\
\to &e^f N(B, 0) = \frac{\eta }{1 - e^{-\eta T}}
\end{align*}
$$
So subsituting into the expression for $N(a,t)$ gives:
$$
N(a,t) = \eta\frac{e^{-\eta a}}{1 - e^{-\eta T}} e^{\eta t}
$$


How does one deal with trending variables in Linear Regression? I have a linear regression model with two independent variables that takes the form:
$$y_{it} = \beta_{0} + \beta_{1}x_{it} + \beta_{2}z_{it} + u_{it}$$
where $u_{it}$ is the error structure.

I want to know if I have data on all of $y,x,z$ how it would effect my estimation of the parameters if they were all increasing over time?

Context: I am running a panel regression analysis using a Fixed Effects model. All of my variables are trending all upwards. I am concerned that I will have biased results but not sure how this would enter into my model or if there are ways to account for this.

Depends on a trend. Trends come generally in two categories:

Deterministic trend - this can be controlled for using various methods. For example, in panel regression you could include time fixed effects that would correct for an effect of each time period on all firms and hence should control for a trend in data. This would look like this: $$y_{it} = \beta_{0} + \beta_{1}x_{it} + \beta_{2}z_{it} + \gamma_t+ u_{it}$$ Alternative approach is to just include time as separate variable $t$ which will be a series that will just increase by 1 in each time period this would look like this $$y_{it} = \beta_{0} + \beta_{1}x_{it} + \beta_{2}z_{it} + \beta_3 t+u_{it}$$This would control for any linear trend in the series, you could also make the trend quadratic but quadratic trends are quite rare.
Stochastic trends (unit root) - if there is stochastic trend in your data you generally can’t use the variable in standard regression models. You can test for presence of stochastic trend by some unit root test for example fisher test is quite popular for panel data, but it’s always context dependent. In case your variable contains unit root you can’t use it directly in most regressions but you can still transform it by taking first difference and provided the first difference does not contain stochastic trend (after you test it again) you can use the differenced variable. So in this case if all variables would contain stochastic trend you would want to run model like this: $$\Delta y_{it} = \beta_{0} + \beta_{1} \Delta x_{it} + \beta_{2}\Delta z_{it} + u_{it}$$Alternatively, sometimes variables can be cointegrated, in such case this indicates that variables tend to move together because they tend towards some long run equilibrium, you can still include them in their levels in cointegrated regression for example in panel version of fully modified OLS or other similar model, or alternatively build an error correction model where you would have both the first difference to capture short run dynamic of the model and level variables which capture the long run equilibrium.


How does Perfect Elasticity imply infinite quantity for any higher price? I reference Principles of Microeconomics, 7 Ed, 2014, by NG Mankiw. The graph above is from p 100.

  [p 99:] ...supply
  is perfectly elastic. This occurs as the price elasticity of supply approaches infinity
  and the supply curve becomes horizontal, meaning that very small changes in the
  price lead to very large changes in the quantity supplied.

Please explain the red box? How's it true? The graph itself shows that at \$4 exclusively, all quantities $\ge 0$ are supplied. The graph depicts nothing for prices above \$4? (Footnote: I tried this)

"Calculus at infinity" is a useful tool in Economics also, when we have managed to represent mathematically an economic situation. For example, when we model an intertemporal utility maximization problem with an "infinite" planning horizon, we only do it because it simplifies the mathematics considerably, while the approximation error is usually negligible for our purposes, since due to discounting, a finite-horizon setup would not produce any additional insight of value. We do not really argue that economic agents have "infinite" planning horizon.

The convention to talk about "infinite supply" belongs to the same category -but in this case it does not appear to help in anything.

The reason that the Economics discipline exists is the finiteness of resources. Supply of a good or service, or anything else related to Economics cannot be infinite - so whenever we use the term we usually mean "very large"...

...and the excerpt the OP quotes from the textbook does exactly that: it talks about "price elasticity approaching infinity... meaning that very small changes in price lead to very large changes in quantity supplied". This passage takes a mathematical concept, and correctly translates it in meaningful economic terms.

More formally, "any quantity" does not mean "infinite quantity" because "infinity" is not a number, so it cannot represent a quantity (and let's not confuse quantity with the concept of cardinality used in set theory). The expression $x\geq 0$ does not include "$\infty$". $x$ belongs to the real numbers, and $\infty$ is added to the real numbers to create the "extended real numbers" - it is not a real number itself.

Perhaps by "logical extension", if all non-zero real quantities are supplied at USD $4$, then what is left for higher prices is only "infinity". But this has no economic meaning -apart from the approximate one already mentioned in the passage quoted.


Effects of change in tax on output Suppose the government imposes a tax on each unit of output
produced of $t. What is the firm’s new profit maximizing level of
output?
The question suggests the use of comparative to determine how a change in tax affects a change in output. The firm would operate under perfect competition conditions.
I started to answer it this way.
Before Tax:
π = revenue - costs
π = pq - C(q)
dπ/dq = p - C'(q)
Set dπ/dq for profit-maximisation
0 = p - C'(q)
p = C'(q)
p = MC
After Tax:
π = revenue - costs - tax
π = pq - C(q) -tq
dπ/dq = p - t - C'(q)
Set dπ/dq for profit-maximisation
0 = p - t - C'(q)
p = t + C'(q)
p = t + MC
Next I would try to find out how quantity changes with tax t by taking derivates? Any ideas?

you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:
$$\pi (q,t) = p q - C(q) - tq$$
However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have $q^* = q(t)$.
Now by envelope theorem:
$$\frac{\pi^*(t)}{dt} = \pi'_2(q^*(t),t) = - Q^*(t) $$
So the optimal level of output given the tax $t$ will depend inversely  on change in profit when the tax rate changes or  $-\pi'_2(q^*(t),t) = Q^*(t)$.


Consumer Econometrics: Identifying nature of goods I have some data on household income and goods purchased. The first three columns and rows are:

Income    Good1    Good2  Price1  Price2
  100       4        12      1.5    2
  140       18       12      1.5    2
  200       9        3       1.5    2

I'm interested in understanding how I would use regression analysis/econometrics to understand the nature of the goods in question (i.e are they normal, ordinary, inferior, etc).

There are few things that need to be clarified before we want to understand the nature of goods.

First concept is elasticity: which is just change of one variable in response to change in another variable, here increase/decrease in demand for good as a result of change in its price or consumer income.

Mathematically:

Price (point) elasticity: $\varepsilon_{price} = \frac{P}{Q_{d}} \frac{\partial Q_{d}}{\partial P}$

If certain good has $|\varepsilon_{price}| < 1 $, it is inelastic meaning that any change in price will have low impact on your quantity demanded (think about drugs, which are very inelastic). Opposite works for $|\varepsilon_{price}| > 1 $.

Income elasticity: $\varepsilon_{income} = \frac{I}{Q_{d}} \frac{\partial Q_{d}}{\partial I}$

The idea here is the same as in case of price elasticity but now income will have and impact on your demand.

In example you provided the prices are constant, therefore we cannot measure price elasticity of demand. However, it is relatively simple to calculate income elasticity of demand. We can either calculate it by hand or use simple OLS model. For sake of convenience I will use the latter (as it also was what you asked for).

I create simple linear model where my dependent variable is the demand for good x and independent variable is income. The result will be just measuring unit impact. In case you are interested in measuring % change you have to take logarithms on both sides.

$$demand_{i} = \alpha + \beta_1 income_i + \varepsilon_i$$

Good 1 (normal):

            Estimate Std. Error t value Pr(>|t|)
(Intercept)  5.31579   20.82775   0.255    0.841
income       0.03421    0.13674   0.250    0.844

Good 2 (inferior):

            Estimate Std. Error t value Pr(>|t|)
(Intercept) 22.89474    6.24832   3.664     0.17
income      -0.09474    0.04102  -2.309     0.26

As you can see there are only 3 observations so the results are not very robust but should give you main idea. Furthermore, as there is not enough data we can only hypothesize about characteristic of a good, there is more information needed to distinguish between normal and luxury goods.

Nevertheless, current data shows that demand for first good increases as income increases. The case is opposite for the second good.

EDIT: As noted by @luchonacho in order to generalize into income and price elasticity we would have to include price into our regression. The resulting model would look as follows:

$$demand_{i} = \alpha + \beta_1 income_i + \beta_2 price_i + \varepsilon_i$$

Now one has to be careful with interpretation of the coefficients. $\beta_{i}$ shows and effect of 1 unit increase in income\price on the demand for good (elasticity), keeping all other variables constant (ceteris paribus).


Social Welfare and Pareto Optimality in a Bayesian Game In a normal form game with two players, we call a joint strategy $s=(s_1,s_2)$ a Pareto optimal outcome if for no joint strategy $s'$, for all players $i\in\{1,2\}$, we have $$u_i(s')\geq u_i(s)\qquad(1)$$ and also at least for one player $i$ $$u_i(s')>u_i(s).\qquad(2)$$

The social welfare of $s$ is defined as $$\sum_{i=1}^{2}u_i(s).\qquad(3)$$ If the social welfare of $s$ is maximal, then the joint strategy $s$ is a social optimum.

I want to use these definitions in a Bayesian game where the utility functions are defined as $u_i:A\times\Theta\to\mathbb{R}$, where $A$ is the action set and $\Theta$ is the type set.

Can we simply replace $u_i(s)$ in $(1),(2),(3)$ with $u_i(a,\theta)$? Or we have to consider the expected utility of a player?

How should we write the definitions if the game is in the ex-ante stage?

While it is a bit unusual to describe a strategy profile as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of such games as follows. Recall that in a Bayesian game, a pure strategy is a function $s_i:\Theta_i\to A_i$.

  A strategy profile $s(\theta)=(s_1(\theta_1),\dots,s_n(\theta_n))$ is
  ex ante Pareto optimal if  \begin{equation} \mathbb E_\theta [u_i(s(\theta);\theta)]\ge \mathbb
 E_\theta[u_i(s'(\theta);\theta)],\quad\text{$\forall i$ and $\forall
 s'(\theta)$} \end{equation} where the above inequality is strict for
  some $i$. [$\mathbb E_\theta$ means taking expectation over the state
  vector $\theta$.]

  A strategy profile $s(\theta)=(s_1(\theta_1),\dots,s_n(\theta_n))$ is
  ex post Pareto optimal if, given a particular realization of states $\bar\theta=(\bar\theta_1,\dots,\bar\theta_n)$,
  \begin{equation} u_i(s(\bar\theta);\bar\theta)\ge
 u_i(s'(\bar\theta);\bar\theta),\quad\text{$\forall i$ and $\forall
 s'(\bar\theta)$} \end{equation} where the above inequality is strict
  for some $i$.


Understanding the Choice Rule in MWG I am reading the Microeconomics Theory book by MWG, and I am having a tough time interpreting what things mean to a real life example, so any help would be appreciated.
For example, it gave this.  Suppose X = {x, y, z} and B, the budget set, = {{x, y}, {x, y, z}}.  It says we can define a choice rule so that C({x, y}) = {x} and C({x, y, z}) = {x, y}.
I am having trouble understanding what C({x, y, z}) = {x, y} means.  Does that mean that when faced with the choice of x, y, and z, the person is indifferent between x and y?  It doesn't mean he chooses both right?  It just means he will choose either or so it's random which one of the two he chooses?
Furthermore, how can we even have a situation where if the consumer is choosing between x and y, he'll always choose x, but then when he is faced between choosing x, y, and z, he will choose either x or y?  Can someone give me a real life example of how this would be possible?  That usually helps me understand better.
Thank you so much!  I appreciate any responses.

I am having trouble understanding what $C(\{x, y, z\}) = \{x, y\}$ means.

MWG already explain this on p.10:

When [$C(B)$ contains more than one element], the elements of $C(B)$ are the alternatives in $B$ that the decision maker might choose; that is, they are her acceptable alternatives in $B$. In this case, the set $C(B)$  can be thought of as containing those alternatives that we would actually see chosen if the decision maker were repeatedly to face the problem of choosing an alternative from set $B$.

[H]ow can we even have a situation where if the consumer is choosing between $x$ and $y$, he'll always choose $x$, but then when he is faced between choosing $x$, $y$, and $z$, he will choose either $x$ or $y$? Can someone give me a real life example of how this would be possible?

Consider a voting situation as follows. Individuals $x$, $y$, and $z$ are potential candidates for the position of a department chair. A five-member committee is to select the chair by voting via simple majority rule. Let's label the committee members $1$ to $5$ and suppose that they each have a private ranking of the three candidates as follows ($\succ_i$ is member $i$'s ranking):
\begin{align}
x\succ_1 y\succ_1 z \\
y\succ_2 x\succ_2 z \\
z\succ_3 x\succ_3 y \\
y\succ_4 z\succ_4 x \\
x\succ_5 z\succ_5 y
\end{align}
Suppose that each member will vote for his/her top-ranked candidate. Then when only two candidates $x$ and $y$ are under consideration, $x$ will get three votes (from members $1$, $3$ and $5$) and thus always be chosen. But when all three candidates are being considered, $x$ and $y$ will each get two votes ($x$ from $1$ and $5$, $y$ from $2$ and $4$) and thus both be "acceptable" candidates.


How does fuel tankering save airlines money The aviation industry practice of fuel-tankering has been in the news recently. As the BBC describes it: "planes are filled with extra fuel, usually to avoid paying higher prices for refuelling at destination airports". The weight of the extra fuel can significantly increase the fuel consumption and emissions of the planes.

How can this be cost effective? This seems to be essentially air-freighting fuel to the destination, but I thought air freight was only cost effective for goods that are perishable or are required with a low lead time, neither of which seem to apply to aviation fuel.

Airlines typically have a "good" contract for fuel with some supplier for their base airports (they buy a LOT of fuel and hence, they are able to negotiate higher discount). When a plane flies to a small(er) airport (e.g. where the airline goes 2-3 times a week), there they won't have a discount, so they would have to pay more. Therefore, it might be a case that it is more efficient (or profitable) to take on board more fuel (even if it means that the plane will burn more fuel) than to buy fuel for a higher price, i.e. they might prefer to burn more of cheaper fuel than burn less but of more expensive one. I am talking about the case of domestic flights: the base price of fuel might vary quite a bit (you can easily google fuel prices for different airports in various countries). So if it is a case for domestic flight, a similar observation could possibly occur for international flights.

In general, airlines do not produce a lot of economic profit. That is why, in order to maximize their profits, airlines do everything they can (one of these actions is fuel-tankering). They of course do not take into account the negative externality they create by burning more fuel (the extra social cost). So that they end up providing access amount of the activity (more than social optimum).

They could stop doing so but their profits (if occurring) would be lower. Next question could be - whether passengers would like to pay more so that airlines stop doing it? You can probably guess the answer to that.

Oh and in the second link provided by @E.Sommer, this, you have someone who actually has some real world experience on that matter:

"I only saw this happen operationally when we were going to be very light and the outstation fuel was very expensive. Usually our loads were high enough that we were struggling to get all 50 passengers, bags, and especially jumpseaters on board with the required fuel load. In that case it would not be economical to take more fuel because we'd have to take fewer passengers." ~ @casey


What is Negative Interest Rate? I want to know about Negative Interest Rate. What it is? Is it good or bad? How it affects economy and thus general people?
I have been reading these words a lot lately in business news...

An interest rate is the percentage that is paid for borrowing money, and that one gets for lending. If it is negative that means you have to pay so that someone takes your money for a while. So how could that even work, why should anyone lend money under such conditions? Normal savers would just keep their money with them, even if that bears the risk that it might get stolen more easily than from a bank. What can however happen is that the real interest rate is lower than zero and people still keep their money at the bank. The real interest rate is the (nominal) interest rate minus inflation. In times of high inflation this can easily happen, and as you see here in the world bank data it frequently happens in many countries: http://data.worldbank.org/indicator/FR.INR.RINR

In times of crisis and low inflation central banks can even lower the short term nominal deposit interest rate below 0 in order to give incentives to commercial banks to lend out money that because of mistrust they would otherwise rather just store at the central bank and wait for better times. There is no consensus if this is good or bad and if it works and how it affects the economy and the people. The European Central Bank adopted this strategy since mid 2014 because of the severe crisis, and the Japanese central bank followed in January 2016.(WSJ) The idea is to stimulate the economy, which should lower unemployment. It should also make the currency unattractive, so its value should decrease compared to others and that makes exports easier as long as other central banks do not do the same. Some say this goal can be achieved better with the negative interest rate than the goal to stimulate the economy via increased lending of commercial banks, and especially in smaller economies like Denmark, and Switzerland, the mechanism seems to have helped to protect companies from getting too much harmed by currencies getting too strong too quickly.(ST)
In March 2016, the European Central Bank even started to offer commercial banks a negative interest rate for loans, rewarding lenders.(NYT)

There are also bonds that have negative interest rates.(BBC) This phenomenon needs some additional explanation and has its own question here.


Under what conditions should the External Rate of Return (ERR) be used instead of Net Present Worth (NPW) and vice versa? I have just read the concept of both Net Present Worth (NPW) and the External Rate of Return (ERR). This is what I have understood of these two values:

NPW is the net discounted value of all the cash flows that occur throughout the life of project, while the discount rate is the MARR -- a rate that I have decided as a minimum rate of return for acceptance of a project. If NPW > 0, accept the project else reject.

ERR is the rate of return of the project when I reinvest the cash produced at a rate RR (Reinvestment Rate). This rate may be different than the ERR itself. If ERR > MARR, accept the project, else reject it.

I may have understood wrong, please correct me if I am wrong.

So both of the methods are used to make accept/reject decisions for the projects. Certainly there are perks of using one over the other, otherwise there would not be two distinct methods. Then under what conditions is one preferred over other? Or is one of the methods always better than the other in general? Why?

Investment protects are inherently risky, and when calculating either of these numbers many assumptions need to be made. From projecting what future inflows and outflows will be, to deciding what rate of return to use to discount these projects payoffs. As far as I know, usually both measures are calculated and considered when making a decision since they give a slightly different information.

For example, the NPW needs to assume some rate at which to discount payoffs, in your example you call it the MARR, the choice of such rate can be controversial, while the calculation of the ERR does not need to assume any rate. For example I can show you a project with a very high NPW, but this high number may only be due to the fact that I chose a very low MARR, this will be evident because the ERR will necessarily be low. The two measures give a more complete picture. Likewise, I can give you a project that has a very high ERR, but the actual level of the value of the project can be tiny, this will be reflected by the fact that the NPW will be a small positive number.

If the MARR is not controversial, the king of all measures is certainly the NPW, as it literally tells you how valuable is a project. But there are many ways to tinker with the NPW and make it look better so complementary measures give you a better picture of the quality of the project.


Negative expected value; risk neutral choice Suppose there are two options: (1) take a gamble with 50% chance you win \$100 and 50% chance you lose \$110 or (2) don't take the gamble at all and win/lose nothing. Would the risk-neutral take the gamble or not?

The definition of a risk-neutral individual is someone who is indifferent between a guaranteed amount and the expected outcome of the gamble, which is -\$5 in this case. But since \$0 is only guaranteed amount does that mean all individuals who would take the gamble are risk-seekers, since \$0 is greater than the expected payoff?

There are three type of individuals : risk averse, risk neutral and risk loving.

Individuals evaluate risky prospects such as to maximize the expected level of their utility. So, an agent is risk averse if, at any wealth level $w$, he or she dislikes every lottery $Z$ with an expected payoff of zero, $EZ = 0$, so that : $Eu(w + Z) =< u(w)$

Risk neutrality : $Eu(w + Z) = u(w)$
Risk love : $Eu(w + Z) = u(w)$

Assuming that $Eu(w + Z) = EZ$ (expected value of the lottery $Z$) => Risk neutral agent.

$Eu(w+ Z) = EZ = - 5$

$Eu(w) = 0$

An risk averse agent won't take the gamble. A risk neutral agent won't take the gamble. We don't know for the risk loving agent, depends on his utility function.


What is a Labor Rate Index For an report (professional not academic) I was asked for the Labor Rate Index of my region, which is a data point I've never been asked for before.

After some googling I've found the term referred to in a small number of papers (listed below) and oddly they seem to be from military organizations. Often times when I'm asked for a datapoint that uses the word "index", it comes from a proprietary report that costs money. I apologize if that's the case. Anyways, here are some references I've found to a Labor Rate Index with no explanation of what it is:

INFLATION AND ESCALATION BEST PRACTICES FOR COST ANALYSIS: ANALYST HANDBOOK

  The term Constant Price (CP$) may be used to refer to costs normalized
  with an escalation index. Constant Price indicates what a narrowly
  defined basket of goods or services would cost had it been fully
  produced and purchased in the base year. Examples of Constant Prices
  include contractor labor rates normalized with a labor rate index;
  aircraft unit costs normalized with an aircraft index; and fuel costs
  normalized with a fuel price index. Constant Prices remove both
  inflation and real price change from observed dollars, returning the
  base period’s price.

Inflation & Escalation

  “Constant Price” (CP) may be used to refer to costs normalized with an escalation index

  – Constant Price indicates what a narrowly defined basket of goods
  would have cost in a base year

  – Examples of Constant Price include contractor labor rates divided by
  an labor rate index, aircraft unit costs divided by an aircraft
  index, etc.

  – CP addresses market forces (supply and demand) with respect to goods
  and services

  – CP is the normalization used for estimating purposes

A STUDY OF SHIP ACQUISITION COST ESTIMATING IN THE NAVAL SEA SYSTEMS COMMAND

  (2) A Labor Rate Index Provides Similar Allowance For Inflation In The
  Labor Portion Of The Estimate Since 1970, the Branch has produced an
  annual guidance table for use by estimators in selecting labor bid
  rotes. Data sources for this task are some 80 union-management
  agreements. Bureau of Labor
  Statisticsstudies,historicalbids,U.S.DepartmentofCommercesta- tistics,
  naval shipyard employment figures and other sources. This guidance
  promotes consistency in all estimates and is intended to narrow ihe
  margin of error over previous personalized methods.

Anyways, as you can see the term is never defined or sourced. I've tried reaching out to the company to find out why they asked for this, but I'm working through a consultant and so many degrees of separation make it hard to get nit-picky questions answered.

The labour rate index is actually synonymous with a more popular term, the labour cost index.

In the same way that the consumer price index measures inflation of consumer prices, the labour cost index measures the cost of purchasing labour. For example, 12 dollars could buy you more labour 30 years ago than it can now. Inflation indexes like these can be used to normalize historical data. That way we can compare changes and trends across years without worrying about labour cost being inflated.

https://nso.gov.mt/en/nso/Sources_and_Methods/Unit_C2/Labour_Market_Statistics/Pages/Labour-Cost-Index.aspx

https://www150.statcan.gc.ca/n1/en/pub/71-586-x/71-586-x2001001-eng.pdf?st=mrJ1RXhQ


Basic Reason of Why the Money Supply Needs to Increase? Hello, all. So I am trying to gain a thorough understanding of why the Fed, or any other Central Bank, needs to keep pumping money into the economy as it grows.  My question is not asking why inflation happens, but rather focuses on the money supply. I grasp how increasing the money supply tends to increase inflation, and how, on the other hand, the Real GDP increasing tends to decrease inflation (all this is bundled up in the MV = PY equation) and other higher-level economic concepts, but why does the money supply need to increase at all in the first place?

I am essentially asking what would be wrong with keeping it the same as the economy grows or contracts.
The colorful analogy found here helped me understand it a bit more, but I am still thoroughly confused.

Actually, the expression more money supply is not clarifying at all (i) which effect is desired and (ii) what really occurs, whence your (really natural) question.

What is implicit behind this expression is that commercial banks will have an easier access to (re-)financing. If commercial banks play their role of Money multiplier -- which is not that obvious since they may prefer to clean their balance sheets --, they will basically be more prompt to grant credits. Put differently and using an abusive but pedagogical term: they will be psychologically more prompt to waste money, and thus to stimulate demands here and there.

Since growth is very often conceived as being demand led, more demands (of goods or whatever) at a constant level of supply (of those goods or whatever) in turn means inflation, until levels of supply, here and there, adjust and create jobs, generate dividends, etc...

To conclude and summarize

  Basic Reason of Why the Money Supply Needs to Increase?

Because one needs commercial banks to be more relax and play their role.

  what would be wrong with keeping it the same as the economy grows [...]

The economy cannot really grows without leverage.

  [...] or contracts

Bubbles may form and explode sooner or later in this case, which is nothing desirable.

Adjusting the money supply is a difficult balancing exercise for central banks.


What does it mean by 'intensive form'? I encountered this phrase while reading up on growth models - that is to work with a function in intensive form. What does it mean when a function is in intensive form?

Thanks!

The intensive form of the production function is derived from the following. Let us assume a production function, F, with inputs of capital, K, and labor and technology, AL. Thus, output, $Y = F(K,AL)$. Assuming constant returns to scale, i.e. $F(cK,cAL) = cF(K,AL)$, we can say $F(K,AL) = ALF(K/AL, 1)$. The intensive form of the production function, f, takes the argument k = K/AL. k is ratio of capital to labour, or how much capital is there per effective unit of labour. $f(k) = Y/AL = F(K/AL, 1)$, or the output per effective unit of labour.

The intensive form is relevant because although the production function may have constant returns to scale, each individual input may exhibit diminishing returns. In other words, if you increase both labour and capital, this will increase your output proportionally, but if you increase only capital, your output will rise proportionally less. It is also easier to express interest rates and wages in the intensive form. For more, you can read any graduate level macroeconomic textbook. (I've used Romer's Macroeconomics.) Hope this helps!


What are the specific assumptions of MRS? I know MRS is independent of prices and compares the ratio of goods which the consumer will exchange one good for the other good. But is the marginal rate of substitution affected by the consumption of those goods or not? It seems trivial but can't get my head around it.

Different consumer preferences will lead to different properties of the consumer's willingness to trade one good for another.

For instance, suppose that consumers preferences are linear in the consumption of either good:
$U=\beta_x x + \beta_y y$
where $\beta_x$,$\beta_y$ are positive constants

The marginal utility of $x$ would be $\beta_x$ and the marginal utility of $y$ would be $\beta_y$, and the MRS would be $\frac{\beta_x}{\beta_y}$, also a constant. This tells us the consumer would be willing to trade between $x$ and $y$ at a constant rate.

An alternative would be consumption of each good entering utility in a multiplicative fashion (Cobb Douglas), for instance
$U=xy$. The marginal utility of $x$ would be $y$ and the marginal utility of $y$ would be $x$ and so the MRS would be $\frac{y}{x}$, which indeed would depend on consumption. In this example, the MRS decreases as consumption of $x$ rises. As the consumer enjoys more $x$ he is willing to give up less and less $y$ for another unit of $x$.

So it's not entirely trivial whether the MRS depends on consumption, whether it does or not depends on how you model consumer's preferences.


What is the difference between solving for utility maximization separately or aggregately? I derived the Walrasian equilibrium for an economy of two consumers with their respective utility functions (u1, u2) and initial endowments but later on I am asked to maximize (u1+u2).

Does anyone know how this changes the problem ? Like, what is the difference in both approaches when it comes to the optimal allocations ?

Under what constraints is $u_1+u_2$ maximized? The exercise may be about the First Welfare Theorem: a competitive equilibrium is Pareto efficient. In this case, maximizing the sum may be designed to highlight that the equilibrium you found also maximizes the sum of utilities.

In general, the maximum of a sum of functions need not equal the maximum of any of these functions.


How to achieve the best outcome by a single statement in this game? This game is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which contenders practice brinksmanship to their own advantages. It goes as follows:

Anderson, Barnes, and Cooper are to fight a gun duel. They will stand close to one another, so that each can kill one of the others in one shot or deliberately miss. The first to fire will be chosen at random, and they will rotate in the order Anderson, Barnes, and Cooper, each firing one shot at a time.

If there is more than one survivor after a number of rounds, one of the contenders will be chosen at random and forced to kill one of the others, and this will be repeated if there is still more than one alive after a few more rounds.

Before the duel starts, Anderson may make any statement, followed by a statement from Barnes, and finally one from Cooper. They will adhere to the following rules:

Statements are irrevocable. A contender may not act to contradict his statement.
He will act in his own best interest when it does not conflict with Rule One.
He will act randomly when it does not conflict with Rules One and Two.

There are referees to ensure that the rules are followed. If a contender commits himself to a mixed strategy (for example, to miss with a probability of 1/3), the probability will be determined objectively (by tossing dice, etc.).

Q1: What statement will Anderson make? What's his best strategy and his probability of surviving?

Q2: If three contenders are to make their statements in the order of ACB, what would be the best statement for Anderson?

Q3: If there're more than three contenders, does this game become simpler or more difficult? Can we say anything about the $N$ contenders case for $N\gt 3$?

Notice that if no one makes any statement, no one will shoot, and everyone has a surviving chance of 1/3. If only Anderson is allowed to make a statement, he can guarantee near certain survival by making this statement to B and C: "If you don't kill each other at your first opportunities, I will kill the first of you who fail to do so at my first opportunity; otherwise, I'll shoot at the survivor of you with 1% chance of missing."

In the book, a suggested best solution for Q1 is for A to say:"If B does not commit to unconditionally shooting C, I will shoot him." The argument is that B has no choice but to accept, because refusing A's proposal would result in certain death for B. But that is clearly erroneous! Because B can say: "If C doesn't promise to kill A at his first opportunity, I will kill C at mine; if C does, I will kill A at my first opportunity and shoot at C with 1% chance of missing if he kills A first." By refusing B's proposal C has 2/3 chance to survive (1/3 for A shooting first and 1/3 for C shooting first); by accepting he has 2/3+0.33% chance (1/3 for A shooting first, 1/3 for B shooting first and 0.33% for C shooting first). So C will accept. Then A is doomed, bound by his own inconvenient statement.

A caveat, this is not really an answer, more of an extended comment--that's why I've put it as community wiki. This is a cool game, one of commitment.

I'll try to formalize the problem. This is a first pass, so people should edit as they see fit.

It's probably easiest to solve the case where, $T$, the number of rounds before a forced execution is $1$. Let each player get $1$ if they shoot someone else and $-1$ if they are themselves shot.

There are two stages of the game, the commitment stage and the shooting stage. The sequential stage is (fittingly) sequential move and the commitment stage is not really a game persay, since the players are automata (their strategies are predetermined). I'll return to this shortly.

An easy result: there is an equilibrium in which rules 2 and 3 are rendered superfluouous since each player can commit to every possible contingency.

There is a source of exogenous randomness, a random state of the world $\Theta = \Theta_{1} \times \Theta_{2}$, $i = 1,2$, where $\Theta_{i}$ is the set of permutations of $\left\{A,B,C\right\}$. These are the possible orders in the shooting portion.

The set of actions for player $A$ is $\mathcal{A} = \mathcal{A}_{1} \times \mathcal{A}_{2}$ where $\mathcal{A}_{i} = \left\{B,C,\emptyset\right\}$ for $i = 1,2$ (corresponding to "shoot $B$," "shoot $C$," and "shoot nobody," respectively). The sets of actions for the other two players, $\mathcal{B}$ and $\mathcal{C}$, are defined analogously.

A (mixed) commitment strategy of $A$ is a mapping $$\sigma_{A}: \Theta \times \Sigma_{B} \times \Sigma_{C} \to \Delta\left(\mathcal{A}\right)$$
where $\Sigma_{i}$, $i = A, B, C$ is the set of all (feasible) mixed strategies. An example of an infeasible strategy for player $A$, say, is one that does not have $\emptyset$ played with probability $1$ when the state and mixed strategies of the other players are such that $A$ has been shot for certain.

Note that I am using the fact that the shooting portion of the game can be viewed as a instantaneous (random) outcome of the commitment portion, which is the real game.

I'm not quite sure about the commitment strategies for $B$ and $C$, due to the sequential move nature of the commitment stage. I suspect that that does not matter, and that they are defined analogously. I'm not completely sold on this and open to being convinced otherwise.

Each player's payoff is defined in the obvious way.


Utility function that generates a demand curve which will have an U shaped MR curve This is based off an answer given by @Ubiquitous in here: Can marginal revenue be increasing?

The solution he proposed involved a MR curve that sloped down, then up and then down. His equation for the demand curve (for example) to generate such a MR is $P(q)=1−q+3q^2−3q^3$ (see comments below his answer).

My question is:

Is there an Utility function that can generate a demand curve which gives, say an U-shape looking MR?

Note: I am not looking for an utility function that replicates Ubiquitous's MR.

Let $v(x) = \int  1 − x + 3x^2 − 3x^3 \text{d}x$.
The quasi linear utility function
$$
U(x,y) = v(x) + y
$$
generates the inverse demand curve given by Ubiquitous if income $I$ is large enough and $p_y = 1$, as
\begin{align*}
|MRS(x,y)| & = p_x \\
\\
v'(x) & = p_x \\
\\
1 − x + 3x^2 − 3x^3 & = p_x.
\end{align*}
Hence it will also result in the same $MR$ for $x$.

P.s.: I have not checked if $U(x,y)$ is convex, so second order conditions may need to be examined here. Note that $U(x,y)$ need not be convex everywhere, only above the $MR$'s U shape. Everywhere else it can be altered as we see fit.


Estimate function expression for TC and MC I have this

Labor (L)   0   1   2   3   4   5   6   7   8   9   10  11
Output      0   32  72  114 161 204 242 272 292 304 302 288

and I need to estimate function expressions for $TC$, $AVC$, and $MC$.

I have capital $K = 10$, wage $w = 30$, and rate $r = 50$.

I think that total costs are
$$
TC = wL + rK = 30 L + 50 \cdot 10 = 30 L + 500.
$$
and marginal costs are $MC = TC'(Q)$ but my $TC$ doesn't depend on $Q$. Is it $MC = TC'(L) = 30$ instead?

If I'm calculating the marginal cost ($MC_t = \frac{\Delta TC}{\Delta Output}$) in discrete quantities I get

Labor (L)   0   1   2   3   4   5   6   7   8   9     10    11
Output      0   32  72  114 161 204 242 272 292 304   302   288
MC          -   25  20  19  17  19  21  27  40  67    -400  -57

but these numbers seem very wrong.

As the OP corrected in the comments, using $K=20, w=800$, and $r=500$, so $TC = 10000 + 800L$, we get , in the $\{Q,C\}$ space

This looks pretty text-book like. The fact that there are levels of labor for which output is reduced, tells us that for these levels we have to employ more of the "fixed" factor, capital (which thus stops being fixed).
The function as is, with capital fixed at $10000$, stops having economic sense after $L=9$.


Is market failure constant? What properly defines it? My textbook defines market failure as when "the production or consumption of a good or service causes additional positive or negative externalities on a third party not involved in the economic activity". That being said, I'd like to ask, Don't all activities produce externalities? For example, the production of oil will always have negative externalities, no matter how the government intervenes. Thus the market will always fail.

Now perhaps I misunderstood. Perhaps it means whenever there are any net externalities (positive - negative), the market fails. Please tell me if this is right.

Furthermore, from what I've read, I've gathered that the market fails whenever social cost $\neq$ social benefit. But this is a different definition from the one I previously mentioned (the one only dealing with externalities).

Please tell me what market failure actually represents. Does it take into account social costs and benefits? Or only external costs and benefits? And if it only takes into account externalities, will some markets fail continuously (like consumption of oil)?

I have to intervene to say that market failure and externality are not the same thing. So I do not think it is at all correct to define market failure as

  when "the production or consumption of a good or service causes additional positive or negative externalities on a third party not involved in the economic activity".

Externalities are but one example of market failure. Market failure is more properly defined as any situation in which a market, left to operate without any intervention, fails to produce the efficient (welfare-maximising) allocation.

Sources of market failure include

Externalities: if there is a negative externality then there will tend to be too much of an activity from a social perspective—resulting in inefficiency.
Market power: if the market is not perfectly competitive then firms will tend to increase price above marginal cost to increase their profit. This results in consumers not buying the good even though they are willing to pay more than its cost of production—which is inefficient.
Information asymmetries: If one party in a transaction has an informational advantage over the other then s/he will try to exploit it to the counterparty's detriment. This, in turn will lead to transactions taking place where it would be efficient for them not to (or to mistrust and the failure to realise efficient transactions).
Missing markets: sometimes efficient trades don't occur because the market simply doesn't exist. For example, there is no market to insure against the risk that an unborn child will be born disabled and requiring a lifetime of care even though many parents and their children would like such insurance (an argument often used to for the existence of state-provided social security schemes).

To address your actual questions:

"Don't all activities produce externalities"? Yes, but many of these externalities are priced. For example, if I buy an apple then you can no longer consume that apple, which is an externality. However, this does not result in a market failure because the price mechanism in a competitive market ensures that I get an apple and you don't only if I am willing to pay more for that apple than you are. So the apples go to the people who value them the most, which is the efficient thing to do. Since we are doing the efficient thing, there is no market failure.

So, when should we worry about externalities? We should check whether the net effects can cancel each other out. For example, suppose that the private benefit of some action was lower than the social benefit, but that the private cost was also lower than the social cost by exactly the same amount. Then the net effect would be that MPB=MPC at exactly the same quantity where MSB=MSC. The private individual would then take the socially optimal action and there would be no market failure. A market failure only occurs if the externality is such that MPB=MPC at a quantity different to that where MSB=MSC. Only then will the behaviour of the private individual (whose optimal action is to equalise private marginal benefit and private marginal cost) differ from that which is socially optimal.

A note on marginal benefit and cost:

When performing this kind of analysis, We typically assume that the objective is to maximise to total social welfare (green line), which is defined as the difference between the total accumulated benefit of the activity (blue line) and the total accumulated cost (red line):

The marginal social benefit is the benefit society gains if we increase consumption by one unit. In other words, the MSB is given by the slope of the TSB curve. Similarly, the MSC (defined as the extra cost bourne by society if consumption increases by one unit) is equal to the slope of the TSC curve.

Now, we observe something interesting: the total welfare curve obtains its maximum at exactly the point where the slopes of the TSB and TSC curves are equal:

In other words, welfare is maximised when MSB=MSC. This is not a coincidence for this particular graph, but rather is a far more general property.

This is actually quite intuitive. Suppose that MSB > MSC. If we increased consumption by one unit then society would get MSB units of extra benefit and MSC units of extra cost. Since MSB > MSC, this results in an increase in total social welfare. Similarly, if MSB < MSC then we could reduce consumption by one unit and society would save more in costs than it would loose in benefits. So neither $MSB>MSC$ nor $MSB<MSC$ can be consistent with maximised social welfare. Only when MSB=MSC do we find that there is no way to increase welfare by increasing or reducing consumption.


What are the economic consequences of discounting deferred tax liability? I was wondering why the IAS prohibits companies from discounting their deferred tax items?

Are there any particular economic reasons for implementing such prohibition?

Thank you!

The answer is in the text of IAS 12, and basically amounts to "it's unreasonable to require it, and if we allow firms discretion in doing it, they will only do it when it's in their favor":

  The reliable determination of deferred tax assets and liabilities on a discounted basis requires detailed scheduling of the timing of the reversal of each temporary difference. In many cases such scheduling is impracticable or highly complex. Therefore, it is inappropriate to require discounting of deferred tax assets and liabilities. To permit, but not to require, discounting would result in deferred tax assets and liabilities which would not be comparable between entities. Therefore, this Standard does not require or permit the discounting of deferred tax assets and liabilities.


How are the price signals of stocks in the stock market centralized? We can look up the price of a stock, like that of Google, and it gives a particular figure. I would like to know how the price signal is "centralized". What I mean is: if the trades happen between individuals holding the stocks, then how is the overall trading price arrived at? Surely, the individuals who are on the buy side and the sell side would have to "register" the sale with the SEC or something? That's the way it happens with other tangible items, like houses and cars, etc.

The stock price on the market acts as a global signal, which efficiently moves financial capital around. Without it, we would be left to our best guesses, which might take too long to converge on a single number, if it ever did at all.

Think of the stock certificate as a physical object being traded around. It may be traded on more than one market. The centralisation you speak of depends on the source for the quote. The quote, generally (some exceptions), is a last trade price (a delayed last trade price). It is not global. I have trading software that allows me to see quotes with less delays, I can choose to see it only on certain markets or last trade from any market. I see different signals than you do.
The most direct answer to the question in the title is "they're not". There is no central, global signal. What you are likely referring to is just the (delayed) last trade price of the security on the markets that the quote service pulls from.


Comparing cost of living of one state with that of other states In the past I read something like this:

  If basic expenses for family went from 100€ to 200€ but its wage
  went from 200€ to 400€ it can expend more than before.

  Weighting the costs of living in state X, in the median wage in state X,
  is the best way to compare state X's COL with COLs of other states.

I assume that weighting COL in MW particularly is better than weighting COL in average wage (AW), because, although there are few families with extremely high wages, their wages bias AW extremely upwards, while MW is less effected from such extreme cases.

Is weighting COL in state X in MW in state X the most accurate way to compare that COL to these of other states?

Quality of life matters too. This sometimes appears in the labor economics literature as "compensating differential". So it's probably not enough to weight cost of living by median or average wage unless you can objectively determine that all other factors are identical (e.g, tax structure, quality of transportation networks, population density, greenspace, etc.).

These factors are in general not likely to be pecuniary, so they are usually treated as a residual - if MW/COL in state A is less than MW/COL in state B, then the difference must be due to improved amenities in state A.  This is obviously unsatisfactory, and as a result there's a lot of active literature in this area.  And they will, of course, be highly subjective.


How Does Keynesian Theory Address Supply Shocks, Like the California Drought? From what I understand about Keynesian theory, when we're facing deflation, lowering interest rates or increasing government spending/lending (when consumers are cutting back) helps stimulate economic growth through consumption (GDP = C + I + G + NE).  While a few loud voices (Schiff, the ZH guy, Martin Armstrong, etc) have been screaming about hyperinflation, Krugman has been right that we haven't seen it, even with the Federal Reserve's QEs.  It does seem like QE has helped move the economy along, even though I know this depends on a person's point of view; ie: Austrian economists won't agree.

What happens in the case of a major drought causing food prices to rise, as in double, triple or quadruple (See this post by Bill McBride. and now this post)?  Is the Keynesian solution here to raise interest rates => wouldn't that be a double whammy on the poor in that it would be more difficult to get a job and food prices would be rising?  What is the Keynesian solution to supply shock problems, like a major drought, impacting basic necessities, such as food prices?

To be more specific because of the comment, I am looking for the Keynesian solution to a supply shock, or major crises that involve a supply shock - such as a drought that causes food prices to rise?

In what follows, I'm going to be very hand wavy - take with a grain of salt.

If you think about it, a demand shock is like a scenario of multiple equilibria (given the fixed levels of wages and prices):

If everyone is employed, everyone has a lot of income, can spend that on goods, and the demand for goods implies a large demand for labor, making everyone employed
If few people are employed, total spending is smaller, so is demand for goods and labor, corresponding to only fewer people employed

The Keynesian solution can be thought of as forcing the first case to happen rather than the second.

If you really have a supply shock, that is, lack of a factor important to production, there is no indeterminacy here. There just isn't that much (oil) available, increasing its price. The RBC spokesperson of your favor would say

  The high price stems from scarcity; both the high price, the low production and high unemployment are efficient.

In the standard problem, rigidity of wages and prices create inefficiency, they can be dealt with. Here, there is no such inefficiency, and no inherent solution to a supply side shock.


Profit-maximization for a monopoly Between two countries, Richland and Poorland, with a strict ban on cross-border sales agreed with the Poorlandian government. The respective demand functions for both countries are:
$Q_{poor} = 10 - P_{poor}$ and $Q_{rich} = 14 - 0.5P_{rich}.$
The total cost, TC, in millions, depends on the total amount of units produced $Q = Q_{rich} + Q_{poor}$ and also includes 5 million of fixed costs.
TC $= 5+2Q + \frac{Q^2}{8}$
I. What is the profit-maximization strategy? Please indicate the prices and quantities, as well as the amount of social welfare generated.
II. Also are the prices above consistent with the rule of elasticities?
Can someone please help me with this?
$\textbf{My Idea:}$ I took the derivatives of the TC to find MC. I isolated $P$ for poor and rich and multiplied it by $Q$ to find each revenue function, then subtracted it from TC to get the Total revenue, then I took the derivative to find MR = MC; which is this the maximization for a monopoly. Is this path the optimal one? Can someone please help me solve these? Thank you for your time and help.

A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize
$$\max_{Q_p,Q_r} \quad P_p(Q_p)Q_p + P_r(Q_r)Q_r - TC(Q_p+Q_r)$$
$$\max_{Q_p,Q_r} \quad (Q_p-10)Q_p + (28-2Q_r)Q_r - 5-2(Q_r+Q_p) -\frac{(Q_r+Q_p)^2}{8}$$
The FOC gives you two equations with two unknowns, $Q_r$ and $Q_p$, which you solve for. Both FOC equations can be written in the form $MR=MC$. You then plug these $Q$s into the demand equations to get the prices. Next, you calculate consumer surplus and producer surplus and add them to get social welfare.


Finding data on (estimated) trade cost elasticities of imports I'm an undergrad student of economics, and I need to find estimates of the trade cost elasticity of the UK's imports for some homework. My instructors pointed me to the World Bank's databases, specifically the World Development Indicators, but I've not been able to find those estimates there. Google has not proved helpful either.

I'd be grateful if anyone could point me in the right direction.

Answering my own question --- although I don't have a a reference per se, I've been told that

if you have no idea at all, -5 is a good ,,first guess``, and
there is a paper (and this is what I don't have a reference for, sadly) where the author worked through all the available estimates and arrived at -4,25 as the ,,best guess``.