Total profit greater than 100 % Newspapers like Forbes, CNet, WSJ and many other say that it is possible to have total profit of industry more than 100 %.

Examples:

  Apple and Samsung account for more than 100% of industry profits
  because other makers broke even or lost money

  Samsung and Apple combined to take 106% percent of the industry's
  profits. How do they take more than 100%? Well, it's a quirk of the
  fact that everyone else in the industry is losing money, or breaking
  even. Apple has 65% of the industry's profits, and Samsung has 41%.

Source: Business Insider: Apple And Samsung Have 106% Of The Smartphone Industry's Profits

I can't imagine that. Profit means positive value. Negative value is loss. Is it possible to count loss into total industry profit?

When I create an example like this:

  Company A - loss 100
  Company B - loss 100
  Company C - profit 10
  Company D - loss 10
  = total "profit" -200.

Company C would take -5 % of industry profits.
Company D would take 5 % ...

How it is possible that big newspapers use such irrational math?

I think this is just a reflection of the casual language of the popular press. What appears to be in dispute in this question is if profits can be negative or if that case is properly referred to as losses.  I agree that in many contexts this is at best a confusing usage. Most dictionary definitions are clearly implying that profits must be positive (e.g., "A financial gain", "Advantage", "Benefit", or "income in excess of costs").

I'm comfortable thinking about negative profits as being a synonym for losses. Industry profits can be thought of as the sum of profits and losses of individual firms just like a firm's profits can be thought of as the sum of profits and losses of subsidiary firms. That seems uncontroversial, at least if this total is positive but that doesn't seem like a standard usage. If we accept that profits can be negative and this aggregate is negative we have weird things happening like firms with positive profits making a negative fraction of total profits. Nothing wrong with the math here, but the math may be telling you something difficult to translate into words. If firm level profits were by definition positive then in @denesp's example Apple and Samsung's share of profits would be 100 percent.

The more technical term for a variable that when positive is profits and when negative is losses is Net Income. Wikipedia's page on Net Income says:

  In the context of the presentation of financial statements, the IFRS
  Foundation defines net income as synonymous with profit and loss.

I'd avoid the confusion and not refer to losses as negative profits. Instead I would use the term Net Income.  Let's fix that Business Insider quote to use net income:

  Apple and Samsung account for more than 100% of industry net income
  because other makers broke even or lost money

  Samsung and Apple combined to take 106% percent of the industry's
  net income. How do they take more than 100%? Well, it's a quirk of the
  fact that everyone else in the industry is losing money, or breaking
  even. Apple has 65% of the industry's net income, and Samsung has 41%.

Now let's use @denesp's example:

Company / Net Income
Apple / \$53
Samsung / \$53
Other / -\$6
Total / \$100

Everything looks good to me here after the change in terms.


Corporate Finance Book Recommendations Can you share your recommendations for corporate finance if you have any?

A slightly more detailed version would be better:

For example, something like:

Advanced Undergrad: Varian's $\textit{Intermediate Microeconomics}$
Ph.D. 1st-year: MWG's $\textit{Microeconomic Theory}$

An example for corporate finance would be lke:

MBA-level:  ~
Corporate Finance Theory-specific:  ~
Ph.D. level: ~

Thanks!

frequent Corporate Finance textbooks are

MBA level -

  Berk and de Marzo (2017) - Corporate Finance, 4th ed. (Stanford)

Advanced undergrad and corp theory-specific -

  Grinblatt and Titman - Financial Markets & Corporate Strategy

PhD 1st year level -
strictly articles from Journal of Finance, Review of Financial studies, etc, that empirically look at mergers and acquisitions, corporate governance, debt-shareholder agency problems, etc. Seminars on the subject typically have participants play discussants of existing recent work.


Why do we use mostly two-tailed Student's t-statistics to find if an explanatory variable is significant in a regression? Let's assume a standard regression model:
$$y=\beta x+u$$
We'd like to test if an variable $x_j$ is relevant in the respect of the model.

t-statistics: $$t=\frac{\hat{\beta_j}-b}{SE(\hat{\beta_j})}$$

Most of the time, we assume the following two-tailed hypothesis:
$$H_0: \beta_j=b, \ H_1: \beta_j \neq b$$

My question is, why don't we use more often the one-tailed hypothesis to test the significance of a variable $x_j$? When I read papers and write assignments I hardly ever come across one-tailed t-tests in the respect to regression models. Why?

We'd like to test if an variable $x_j$ is relevant in the respect of the model

means that we want to test its "statistical significance", so the null hypothesis is  $$\text{H}_0 : \beta = 0$$

(by the way, historically, that's why it is called the "null" hypothesis: a hypothesis of "null"-zero- effect).

The $t$-statistic for this test is

$$t=\frac{\hat{\beta_j}}{SE(\hat{\beta_j})}$$

Using a two-tailed test does not constrain us as regards the sign of the coefficient (the direction of the effect, if it exists). It may be positive or negative. If it is positive, the null if it is rejected, will be because the $t$-statistic takes a large positive value. But if the effect is negative, then the $t$-statistic will take a high negative value. So we want to test against either case, and this is why we use a "two-tailed" test.


Identifying Households in Panel Data For my thesis, I'm working with a panel dataset that tracks individuals within households. To allow for merging, the dataset provides the household ID in the previous round for each individual. What I'm wondering is how to decide based on this information that a household is "the same" in two different rounds of data collection.
As I'm looking at the socio-economic outcomes of various farming practices, my concern is that there are unobservable household characteristics that influence the farmer's returns to farming but then also correlate with various household outcomes, such as education, assets, etc. To control for this, I'd like to use household fixed effects.
However, I'm wondering which features of a household would justify me saying "this is the same household and these effects are fixed over time". For instance, I don't want to use the household head because if the head changes and every other aspect of the household is identical, the impacts on farming outcomes shouldn't be too large. My question hence is whether there are commonly accepted characteristics for identifying households over time?
For reference, the data I'm using is the Living Standards Measurement Survey - Integrated Surveys on Agriculture, by the World Bank.

I don't get it, you say you have a household ID. If it is the same across waves, then it is the same household over waves. Most likely the people who collected the data will be able to answer the question of whether this is the same household better than you.
The exact definition of who constitutes a household is necessarily arbitrary and varies to some degree between countries. However, the large surveys tend to gravitate around a definition that includes everybody who has lived in the same "building" (whatever that may be) and at least one meal together for more than 6 months in the last year. Then they tend to narrow this down e.g. excluding workers being paid while living there etc. You can check out a questionnaire from a large survey, for instance Living Standard Measurement Survey by the World Bank, at the beginning of the household roster they usually have a section describing who is part of the household.
Households will be given a unique id the first time they are interviewed and followed using this id over the subsequent waves (there is no kind of probability-based matching going on). To re-identify the same households, surveys will use address / GPS, name, and contact details.
In sum: There is really not much point worrying about this. Just group based on hhid and spend your time on other problems in the data (which likely there will be enough of ;))


Question on Open Market Operations I have a very basic question on the somewhat cryptic Open Market Operations. So, from what I understand, say the Fed wants to reduce interest rates (increase monetary base). It will firs buy a bond by writing a cheque to the security holder, the cheque is deposited in a commerical bank, which then deposits this in the central bank. The reserve account of the central bank is credited, and the commercial bank is then able to use these "excess reserves" to create fresh loans- injecting money in the economy.
I guess my question is: when the issuer of the bond (say the government) has to pay back to the security holder (central bank), wont it decrease money supply then?

Thanks!

Your question reversed the order... If the central bank issues the bond, its gets money (thus decreasing the amount of money in the system),
it pays the money back throughout a certain period of time, then money goes back to circulation
But by that time, the central bank can issue new bonds and repeat the process

Essentially, central banks manages a balance
Of course the catch is it pays money it *invents *...


Rationalizability or Nash equilibrium In the 'chicken' game, (stop,go) and (go, stop) are the two pure strategy Nash equilibrium profiles. If the game is played once, does it make more sense to use the rationalizability solution concept? The way I think of it is, that if we can't correctly predict the rival's action, then we might end up not coordinating. Thus, although Nash equilibrium gives a sharper prediction, rationalizability is more realistic to use, given that crashes do happen!

Any thoughts?

Rationalizability is a solution concept that is robust to small misspecifications about the information players have, as well as the payoffs of their actions. I do think it provides reasonable predictions in situations where Nash equilibrium requires coordination or precise beliefs about other players actions or beliefs, especially when players have not had enough opportunity to learn the game. This is a long way to say that I agree with you regarding the chicken game. Though I believe that the most common outcome is that all players decide to wait longer than necessary, that's the main coordination problem that signs, lights, and rules solve, I  believe.


Is there an index that describes how differentiated a product is? Some products are commodities, identical regardless of the producer. Others are very different an people look for specific brands of whatever they buy. Has anybody built any sort of index of how differentiated a product is or an industry? This would be very useful for research purposes!

For an industry, one indicator used by competition authorities is the Herfindahl–Hirschman Index measured as $\displaystyle \sum_{i=1}^N s_i^2$ where there are $N$ firms in the sector and the market share of the $i$th firm is $s_i$.

This gives a value of $1$ when there is a single firm and $\frac1N$ when the $N$ firms have equal market shares, but also deals sensibly with cases where different firms have different shares.

For a particular product, you could look at something else like comparing its price elasticity of demand when other similar products do not change price, perhaps comparing this with the price elasticity of demand for all the similar products together.  An undifferentiated product should have an almost infinite price elasticity as all demand will shift if there is any relative price change.


Why does the Loanable Funds Market model use the real interest rate instead of the nominal interest rate? As far as I understand, the majority of loan contracts specify a nominal interest rate, NOT a real interest rate. So a hypothetical loanable funds markets would have people suggesting potential borrowers different loan contracts with different nominal intrerest rates.

As Brian mentions in his comments this is because in standard models when people make intertemporal decisions they take inflation into the account. Take the following example:
You have a rational utility maximizing person who makes decision between consuming $\\\$100$ today or saving it for later consumption. For simplicity lets assume discount factor is exactly 1 (this is unrealistic as clearly consumption in the future should be valued less than in present but it just simplifies the math). Now let us suppose the nominal interest rate $i$ is $10\%$ and the question is should you save in this example or not?
Well the correct answer here is we dont have enough information to say yes or no. If inflation is $5\%$ the person should save those $\\\$100$ if inflation rate is $15\%$ the person should not do that because in the former case you will increase your resources in real terms after accounting for inflation and in the latter you will have less of them. Hence nominal interest rate can be any arbitrary number $x$ between $[0,\infty)$, but  without knowing what inflation rate is we cannot say whether you should consume today or tomorrow (i.e. whether you should supply loanable funds to the market or not).
What matters is the real interest rate which is the nominal interest rate less the inflation rate since by Fisher equation: $r \approx i - \pi$ where $r$ is the real interest rate, $i$ nominal interest rate and $\pi$ inflation.
This covered just the supply side of loanable funds but the demand side would work in similar way. A business person would want to take loan for projects should analogously work with real returns and interest rates when evaluating business proposals.
I would not call this handwaving as Brian does which is commonly used in science to refer to addition to the model that are ad hoc or do not follow from assumptions of the model in consistent fashion but Brian is correct in saying that it is not necessarily exactly how reality works.
For example, it is documented that people can suffer from behavioral bias 'money illusion' which is cognitive bias to think of money in nominal, rather than real, terms. Under a 'money illusion' what the nominal rate is also matters in non-trivial ways. However, in simple macro 101 model which presents the classical view of  loanable funds market  where agents are rational forward looking and having rational expectations such money illusion does not exist.


Effects of tariffs on company How would Trump's tariffs affect an American multinational company? Would the goods that they produce be considered imports?

In the case of Apple at least, the answer is yes: this article dated August 31 2019 talks about the impact of the latest round of tariffs on Apple products produced in China.

  Apple, the largest U.S. technology company by market cap, has its products assembled in China by Foxconn and then ships them to consumers all over the world. The Apple Airpods, Apple Watch and accompanying Apple Watch bands and the Apple Homepod are all products subject to the higher tariffs beginning Sunday. The iPhone doesn’t appear to be impacted this round, but could be subject to tariffs that begin Dec. 15.

https://techcrunch.com/2019/08/31/apple-products-under-pricing-pressure-as-new-15-tariffs-drop-sunday/


What are the advantages of using demand systems estimation? How is this different from individual demand estimation for product? I don't understand how is demand systems estimation is different from simple demand equation estimation. What's the best resource for getting a history line of demand systems?

One advantage of considering demand systems is that more data is available for identifying the degree of income effects and substition between commodities. A second advantage is the possibility to impose and test microeconomic restrictions (which are between equations). An introduction to the estimation of demand systems with an historical perspective is provided by:
Berndt, E. R., 1991, The Practice of Econometrics: classic and contemporary, Addison Wesley.
More recent treatments are available in most econometric textbooks.


Why are capital gains not counted in a poverty status calculation? So according to the US census office capital gains are not included when trying to figure out if a household is living in poverty. Does this mean if I have one billion dollars in index funds that I am living off of am I in poverty(at least by this definition)?

I am I understanding this correctly? Is this considered a fault in the poverty status measurements? If this is an oversight is there some political reason for it?

Thanks in advance!

Most Exchange Traded Funds of stocks or bonds have an associated income (even those which reinvest or accumulate this income)

For example Vanguard Total Stock Market ETF has a dividend yield of $1.81\%$.  So multiplying by $\$10,000,000,000$, this would give an annual income of about $\$18,100,000$, a little above income poverty levels

On the other hand, people with a large amount of physical cash or gold coins and no income, and using some of this for living expenses, would count as being in income poverty, no matter how high their wealth.  But there are relatively few people like this


Measure of New Firms Born in Each Period In an economy with stochastic overlapping generations of firms, how do we easily understand the measure of new firms born?

The set-up states:

"In each period, measure $\rho\in(0,1)$ of new firms are born and are endowed with net worth $w_0>0$. Firms survive to the next period with probability $(1-\rho)$ and hence the measure of firms alive in every period is 1."

$\textbf{My Question:}$

What is the measure of firms?
How does the measure of firms relate to the firm's survival probability onto the next period?

I think I am not understanding clearly when the author uses these related but distinct terms. Can someone explain easily?

I don't know your model. Just some general input:

It seems that you have a model with a continuum of firms. That is, there are many firms and each one is strategically small. In mathematics, a measure of a set is a way to express the size of this set. Hence, the measure of firms alive represents "how many firms are active". The authors normalize this measure to 1. If it helps, think of 1 million individual small firms and think of "measure 1" as a normalization of this million.

In your model, each period a only a fraction $(1- \rho)$ of this measure 1 survives and additional measure $\rho$ of firms are "born". Thus, the total measure of firms remains 1.
Again, it may help to think of it as a normalization of a large finite number. For example, if $\rho=0.5$, only 500 000 out of 1 000 000 firms from last period survive and 500 000 additional firms are created such that there are still 1 000 000 firms.


Benveniste-Scheinkman condition gives derivative that still depends on the value function What I mean by the title is often, if we have a value function like
$$V(K,I) = \max_{K',I'} F(K') +\beta V(K',I')$$
the First order conditions will give us something that depends on the derivative of the value function, say $V_1(K',I')$. We deal with this by using the B-S condition to find $V_1(K,I)$, then advance it one period?

What about when $V_1(K,I)$ still depends on $V_1(K',I')$, though? (see $(*)$ below for an example, if you wish to skip the setup)

Here is an example. Note that subscripts denote derivatives, the number in the subscript being the input with respect to which we are taking the derivative. The exception to this is negative subscripts, such as $I_{-1}$, where the subscript denotes being one period in the past (in the case of $-1$). A prime denotes being one period in the future (t+1), and two primes denotes being two periods in the future.

Back to the problem: We have value function
$$V(K,I_{-1}) = \max_{I,K'',C}u(C) + \beta V(K',I) \\
\text{ s.t. } C+I \leq f(K) \\
\text{ and  } I =K'' - (1-\delta)K'
$$
So basically we have a simple model with investment that takes two periods to build. $f(K)$ is our production function.

The FOC's give
$$
u_1(c) = \lambda_1 \text{, $\lambda_1$ is the multiplier on the first constraint }\\
\lambda_2 = 0 \text{ multiplier on second constraint is zero} \\
\beta V_2(K',I) = -\lambda_1
$$
Now to get $V_2$ we use the B-S condition, which gives
$$V_2(K,I_{-1}) = \beta V_1(K',I)$$
because $K' = I_{-1} + (1-\delta)K$ from the second constraint. Use the BS condition again and get
$$V_1(K,I_{-1}) =f_1(K) + (1-\delta)V_1(K',I) \tag{*}$$, again, because $K' = I_{-1} + (1-\delta)K$

The previous equation is my question To reiterate, we use B-S condition to find the derivative of the value function, but in $(*)$ the B-S condition depends on what we are trying to find. How do we handle this?

I feel like $V_1(K'I)$ may just be $1$, or $0$, but then that makes $V_2$ be $\beta$ or $0$, which seems wrong...

I guess I will provide a basic answer for now, since I somewhat figured this out, and hopefully either someone posts a more complete answer, or I add more to this later.

Basically, we can use forward or backward iteration. For example, for forward iteration (given that we know the value of $V_1$ at $t=0$, call it $V_{initial}$), then we can use this to find the value of $V$ at time $t$ simply by using the equation over and over again. Or we can work backwards, if for some reason forward iteration doesn't work (at this moment I'm not sure why/how this is different then forward). I don't think the solution is too neat, but it works.

Or, if we are at a steady state, then $V_1(K,I_{-1}) = V_1(K'_I)$, so we can simply rearrange the equation and solve.

Again, this isn't a great answer, but it is correct.


Optimal strategy in a single-agent choice problem under uncertainty Consider the following single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{v})$. First, let nature draw a realisation $v$ of $V$ from $P_V$.
Then, let the decision maker choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the decision maker gets a payoff $u(y,v)$.

For example, suppose that $\mathcal{Y}\equiv \{1,2,3\}$. $V$ is a $3\times 1$ random vector, $V\equiv (V_1,V_2,V_3)$. $P_V$ is the 3-variate standard normal distribution. $u(y,v)\equiv v_y$.

What is the definition of an optimal strategy for the decision maker in this setting?

I'm thinking about using "a sort of" Bayesian Nash equilibrium for a single-agent setting, i.e., an optimal strategy is $P_Y\in \Delta(\mathcal{Y})$ such that, $\forall y\in \mathcal{Y}$ such that $P_Y(y)>0$ and $\forall \tilde{y}\neq y$,
we have that
$$
\sum_{v\in \mathcal{V}} u_i(y,v)P_V(v)\geq \sum_{v\in \mathcal{V}} u_i(\tilde{y},v)P_V(v)
$$
that is, in my example,
$$
\sum_{v\in \mathcal{V}} (v_y -v_{\tilde{y}}) P_V(v)\geq 0
$$

But maybe a pure strategy is what people use?

Is existence and uniqueness obvious (at least in my example with a normal distribution)?

Could you also provide a reference discussing definition, existence, multiplicity?

Your problem can be simply expressed as
$$\arg\max_{y\in\mathcal{Y}}\sum_{v\in\mathcal{V}}u_i(y,v)P_V(v)$$
Note that the subscript in $u_i$ is un-necessary. Furthermore, this is not a game so "mixed strategies" are only a solution if the maximum is not unique, but you should not worry about them  (more on that later).

First, note that the maximization problem is equivalent to the inequality you present. Presenting it as a maximization problem is more akin to how you solve a single agent problem, and circumvents the need to specify the quantifiers. However, strictly speaking,  it only characterizes "pure strategies", (a better term would be deterministic actions), but this is, really, without loss of generality. A probabilistic action is justified only when the maximizer is not unique, in which case the $\arg\max$ operator will already be giving you a set, and any mixture between the maximizers in that set would be optimal.

Besides being more clear, another benefit from presenting it as a maximization problem is that you can use standard theorems to guarantee existence and uniqueness, for example, if $\mathcal{Y}$ is finite then $\sum u(y,v)P_V(v)$ is continuous in $y$ (using the discrete metric) and Weierstrass ensures the existence of a solution. In general  you need a $\mathcal{Y}$ to be compact and $\sum u(y,v)P_V(v)$ to be  continuous in $y$. Uniqueness is a bit more tricky, but if $\sum u(y,v)P_V(v)$ is strictly concave with respect to $y$, then uniqueness is guarantied.

In your particular example given the finiteness of the action space, existence is trivial, but not uniqueness. Knowing that $P_V$ is normal is not enough since we don't know anything about the shape of $u(\cdot)$.


Which is safer in the long-term, diffuse debt or concentrated debt? I want to deepen my understanding of risks as posed by debt burdens among actors of an economy. After some literature review, it seems the matter is nuanced; points can be made for both sides. I'll present a few points simply for context.

Diffuse

If debt burdens are distributed more uniformly (diffuse), then credit can be extended to many SMEs and likely to benefit employment and other long run growth drivers. Also, less concentrated risk may spread out debt burdens and make servicing more manageable.

Concentrated

If debt burdens are concentrated, then regulators will be able to pursue more effective macroprudential policies. For example, entrusting asset management companies to take care of NPLs and free up bank resources so they may lend more.

Question

Is this a debated position in the broader field of economics or is there clear theoretical framework and/or empirical evidence for one being better than the other for "safe" long run economic growth?

I do not think this question can be answered.

Firstly, the statement “ For example, entrusting asset management companies to take care of NPLs and free up bank resources so they may lend more.” does not follow. There are some administrative advantages to having concentrated borrowers, but this statement does not capture them.

We cannot choose the distribution of borrower size without choosing the distribution of firm size. So this question is: should societies aim for concentrated industries, or break them up? This is a large question, for example, the study of anti-trust law. I don’t think one can summarise that area without just posting an opinion.

If we just want to look at historical examples of debt crises, we can find crises both with distributed and concentrated borrowers. Qualitatively, they will look different, but it will be a matter of opinion which will be safer. E.g. concentrated firms pose the risk of a large planning error. However, it is easier to intervene into one firm. Which is “safer”?


Keynesian model functions In The Keynesian economy,
I have the following model

Here, the production function $F(N,K) = Y/K = A(N/K)^{1.10}$ and by the labor demand function $F_N(K,N)= N/K= b_0(w/p)^{b_1}$.
right?
That is, what is the F(K,N)=? and $F_N(N,K)=?$
Here , I mean that $F_N(N,K)$ is marginal product of labor.

To get back to the original production function just multiply both sides with capital. Here:
$$Y/K = A (N/K)^{1.1} = F(N/K,1) \implies Y = AN^{1.1}K^{-0.1}  =F(N,K)$$
Also, the marginal product of labor here will be:
$$F_N'(N,K) =  1.1 A N^{0.1}K^{-0.1} = 1.1 A (N/K)^{0.1}$$


Johansen test explanation I am trying to understand the whole Johansen procedure via wikipedia and some other articles and I'm a noob in econometrics so there is a lot of notation and jargon that I think I am not familiar with.

https://en.wikipedia.org/wiki/Johansen_test

First off, the test itself. I'm confused by how we test for the cointegration vectors. Where do these vectors come from? Based on this

https://en.wikipedia.org/wiki/Error_correction_model

It states that

Step 1: estimate an unrestricted VAR involving potentially non-stationary variables
Step 2: Test for cointegration using Johansen test
Step 3: Form and analyse the VECM.

So is it that we first estimate a VAR, then run the trace or eigenvalue test on the pi matrix, then estimate the VECM? Then when we estimate the vecm, is that where, ultimately, the cointegration vectors come from? And do those ultimate cointegration vectors reside in the pi matrix? How do we know which cointegration vectors to use?

Also I need some slight clarification on the models.

I'll start with the VAR(p) model

I'm relatively confident that Xt is a vector of values at time t, et is a white noise term at time t. I'm not quite sure what mu, phi, and Dt are (although I've heard read it's some sort of "seasonal" term). I'm also assuming that the pi's are matrices that we need to estimate. I'm asking for clarification here but in general I think I understand this VAR(p) representation.

This is where I need some more clarification for the VECM model. What exactly are the delta's? Are they just the first difference i.e. (1-L)?

Sorry for the noob questions but I'm trying to learn this on my own and haven't found many good resources for johansen test and vecm in particular.

I think that the wikipedia steps are not completely correct. When you perform Johansen cointegration test you first have to pretest the data to find if they have the same order of integration. That is all variables have to be either $I(1)$ or $I(2)$ etc. (see Verbeek, 2008). There are some cointegration tests and models that relax this assumption but Johansen is not one of them. But otherwise the steps 1-3 are correct.

$\Phi D_t$ are indeed seasonal dummies. However, note they are actually not standard part of the model, they are added when you think series can be affected by seasonal trends and if you dont use some other seasonal adjustment technique or if you dont already have data provided with seasonal adjustment from the source.

$\mu$ is the constant or 'drift' term. It allows your regression to have different intercept than zero.

You are correct $\Pi$ are matrices of coefficients but dont forget about $\Phi$. Any coefficient you add there has to be estimated - I know its just nitpicking but still.

The co-integration vectors come from the $\Pi$ in the second equation. This matrix can be decomposed as $\Pi = \gamma \beta'$ where $\gamma$ is the matrix of weights with which the cointegration vectors enter the model and $\beta$ is actually the matrix of cointegrated relationships. I am not really exactly sure what you mean by choosing cointegrated vector from this matrix. Its not like you just somehow extract one of the vectors somewhere into some separate vector.

Afterwards the Johansen test basically tests the rank or the number of columns of the cointegrated matrix (With dimensions $k$x$r$ you test for # of $r$). Intuitively this works because you need one column to represent each cointegrated relationship (note there might be multiple cointegrated relationship at the same time).

The rank of the matrix is tested either by trace or eigenvalue test as you correctly mentioned. The null of trace test is $H_0: r \leq r_0$ against $H_a: r <r_0 \leq k$  and the max eigenvalue test has the same null but alternative hypothesis is slightly different $H_a: r = r_0+1$. However, this might be hard to read in plain English. In practice if you would have only two variables both tests would boil down to first testing 0 cointegrated ($r=0$) relationship vs 1 $(r=1)$, second less or equal to 1 cointegrated relationships $(r\leq 1$) vs 2 ($r=2$), third less or equal to 2 coint. rel. $r\leq 2$ vs 3 ($r=3$). You dont really pick from these vectors although the number of cointegrated vectors can affect functional form of VECM, but with just two variables you are looking for $r=1$, if your test finds there should be more you are missing some important variables in your model.

Also yes $\Delta$ is the first difference $\Delta y_t = y_t - y_{t-1}$. Also including first differences assumes that the variables are all $I(1)$ if they are $I(2)$ you need at least second differences. The reason for that is that outside the cointegrated lagged variables the 'contemporaneous' variables must be stationary otherwise your ECM will be biased.  The whole justification of including level variables in ECM is that as long as there is a cointegrated relationship between those their combination will be stationary and their coefficient will be super consistent, but this does not extend to the variables that are added to the model outside the cointegrated relationship.

References:
Verbeek, M. (2008). A guide to modern econometrics. John Wiley & Sons.


Instrumental variables vs control function: Which approach and why to handle endogeneity? I'm curious if anyone here can summarize the differences between the I.V. and control function approach to handling endogeneity. I think endogeneity is usually addressed using the 2SLS or IV approach and, having briefly studied the control function approach, am unsure what the CF approach offers that 2SLS and IV approaches do not offer.

Can anyone help me clarify here? When would I use a CF approach rather than 2SLS, what does this approach offer that the other two do not etc.

I am a newbie with metrics so any and all input is appreciated.

The Control Function Approach (proposed by Heckman-1979 if I remember correctly) , is similar to an IV strategy in spirit, except that it can be used in a non-linear setting, such as Probit/Logit models etc. For instance, if you have endogeneity in a logit model, you cannot just run a 2SLS procedure as the first step is a linear projection of your x on your z, and the second step is nonlinear. There are two ways of handling endogeneity in nonlinear models: control function approach and a special regressor approach.


Where can I find statistics for numbers of 16-20 year olds in the world? Where can I find statistics about how many 16-20 years old people do live in the world?

If there is no such statistics, how about people in this age-range in europe+amerika?

That's not really an economics question, so it's probably off-topic. However, the US Census Bureau seems to track this here over time by sex.


Risk neutral probability for each of 3 states I need help to find the risk-neutral probability for states 1,2 and 3

I have two stocks: A and B.

The price of A today is 180 and in a year it will be worth 288 (S1), 180 (S2) or 120 (S3);

The price of B today is 100 and in a year it will be worth 94(S1), 134(2) or 54(S3)

The annual rf rate is 2%

I did the following for stock A:
up : 288/180-1= 0,6
down :120/180-1 =-0,3333

Finding prob for s1: (2- (-33,33))/ 60- (-33,33)) = 0,3785

Then tried to solve for p2 and then p3:
PV= (0,3785* 288 + p2*180+ (0,6215 -p2 )* 120)/1,02= 180

These probabilities don't add up when I try to find the price of stock B, so they are clearly wrong. Can you tell me how to find the right probabilities?

I cannot really follow your formulas, what is the logic behind them?

Seems to me there is no way to divine three state risk-neutral probabilities from 1 financial instrument's prices. The equation
$$
p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 = 180
$$
is underdetermined, there are infinitely many solutions to it. Now if you solved the equation system
$$
\begin{align*}
p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 & = 180 \\
p_1 94 + p_2 134 + (1 - p_1 - p_2) 54 & = 100
\end{align*}
$$
there you have two unknowns and two equations, so usually this would admit exactly one solution.

P.s.: The numbers in my example are not corrected for intertemporal differences, there has been no discounting.


The relation between econometrics and machine learning In my understanding the utility of econometrics versus machine learning is generally argued to be the research of causality.
If I'm not wrong, when we put aside experimental economics, the golden standard to prove causality is to use Intrumental Variables. But in practice it has been showed that finding a relevant IV is often not possible, and authors end up taking the best "IV" they can find.
Problem: This variable has a high chance to not in fact be exogenous, but we can not verify it when the IV is dependant of variables like happiness, for example. Since we have no way do verify those dependances, science stops here.
SO I am wrong to say that in most cases econometrics can not scientifically establish causality? In that case why using traditional econometrics instead of modern machine learning techniques?

I.)  2 Principles of econometrics can potentially be useful compared to Machine Learning.
(see  Hal R Varian 2014 Paper : https://pubs.aeaweb.org/doi/pdf/10.1257/jep.28.2.3)

A.) As you suggest the search of causality is one advantage but unlike what you think, even if causality sometimes could be tricky to measure it remains very useful and functional.

But first,  when you suggest that Instrumental variable is the only available tool which work for most of the cases and allow causal inference, i think there is a few more techniques that could  still be applicable  to measure causality in response to a treatment, a manipulation, or an intervention and still be relevant (unlike natural experiment as you precise because of its restricted direct applicability to most of the cases) in most of the situations such as:

● explicit experiments
● regression discontinuity
● difference in differences
● structural estimation

For instance you can investigate causality with theses techniques or instrumental variables because even if those techniques are subject to bias and correlation issues (all the more since big data because the increasing size of datasets limits the usefulness of instrumental variable methods, depending on instrument strength and level of confounding), it still give a hint to investigate causality.
In a way that, for instance, it enables the detection of conditional counterfactual (an if-clause which is contrary to fact) while also check the existence of potential selection bias.

What is a Conditional counterfactual:

If it is raining, then he is inside.  : Rain = Indicative Variable

If it were raining, then he would be inside. : Rain = Conditional Counterfactual variable

Hence, the better predictive model you have for the counterfactual, the better you will be able to estimate the causal effect.
Thus, even though a predictive model will not necessarily allow one to conclude anything about causality by itself, such models may help in estimating the causal impact of an intervention when it occurs.
Because it can highlights a bunch of conditional counterfactual which could be use as potential causal Variables to run the test on, in order to  eventually determined causality in a given dataset to do good decision making (clean up from confounding issues). If not, at least, it gives you some clues to conduct a deeper investigation to understand the problem : anything wrong with the theory ? anything wrong with my econometric model ? anything wrong with my data ?.

For more insight about the pro cons of differents causal inferences techniques in econometrics :

https://www.jstor.org/stable/pdf/44234997.pdf?refreqid=excelsior%3A18bacfd86299dcf19e7f1f13d9c52022

B.) Except causality, model uncertainty is another advantage of Econometrics compared to ML.

Probabilistic foundation of econometrics is a strenght in a way that it allows interpretability of most of the models and their parameters (avoiding black box phenomenon) and give a quantisation of uncertainty (with confident intervals ) .
The goal is usually to show that the estimate of some interesting parameter is not very sensitive to the exact specification used : how an estimated parameter varies as different models are used.This question illustrate a simple form of model uncertainty.
In this period of “big data,” it seems strange to focus on sampling uncertainty, which tends to be small with large datasets, while completely ignoring model uncertainty, which may be quite large.
One way to address this is to be explicit about examining how parameter estimates vary with respect to choices of control variables and instruments.

II.) On the contrary Machine Learning techniques could  also be useful for data analytics in social Science.

A.) Parameters Selection, Model validation methods in ML can improve traditional econometrics models

Researchers in machine learning have developed ways to deal with large datasets and economists interested in dealing with such data would be well advised to invest in learning these techniques.
For instance, Web Mining methods could discover new usuable explanatory variable.
Cross validation should identify a non-linear effect or a forgotten cross effect.
Model Validation should detect when a model is mispecified and thus allows a better specification of an econometric model and on the whole reduce omitted variable bias and error.
As an example, recent litterature in finance focus on Garch models (traditional times series models) improved with Neural Network to better predict volatility and asset prices.

For more insight about the usefulness of ML in econometrics check out this paper by Arthur Charpentier : https://arxiv.org/pdf/1708.06992.pdf

B.) Causal Modelling began to be a concern and a field of research in ML. Which means that in the long run ML could potentially overcome its own flaws ( like being exclusively focus on the fit ) and outperform Econometrics.

Some Theoretical computer scientists, such as Pearl (2009a, b) have
made significant contributions to causal modelling in computer science (by extension machine learning):

(see:  Causal Inference in Statistics: A Primer Wiley, 2016  Judea Pearl et Al)

Pearl defines counterfactuals directly in terms of a "structural equation model" – a set of equations, in which each variable is assigned a value that is an explicit function of other variables in the system. Given such a model, the sentence "Y would be y had X been x" (formally, X = x > Y = y ) is defined as the assertion: If we replace the equation currently determining X with a constant X = x, and solve the set of equations for variable Y, the solution obtained will be Y = y. This definition has been shown to be compatible with the axioms of possible world semantics and forms the basis for causal inference in the natural and social sciences, since each structural equation in those domains corresponds to a familiar causal mechanism that can be meaningfully reasoned about by investigators.

However, it appears that these theoretical advances have not as yet been incorporated into machine learning practice. Except in some recent research paper :

http://www.nasonline.org/programs/sackler-colloquia/documents/athey.pdf

As a Conclusion, in my opinion and as things now stand, Econometrics as more valuable input in social science than what machine learning could bring. So it is still adequate to use econometrics to conduct data analysis instead of ML techniques.


Which are some proven methods to compare alternative economic theories? Based on this question about alternative schools of thought, this question about untested economic theories, and the rather unsatisfactory answer (imo) to this question about which is the "most correct" growth theory at the moment, it would be great to learn more about how to compare economic models or theories. Anyone familiar with this literature, which I think belongs to philosophy and methodology of economics?

I have rudimentary ideas here, like Occam's razor and Friedman's defense of unrealistic assumptions, but this is deeply unsatisfactory for such an important question (imo).

There is one theoretically sound methodology to do what you are asking.  When you have multiple models to compare, the only choice is Bayesian model selection and Bayesian model averaging.

Bayesian methods have several properties that you would want.  Any statistic created is intrinsically "admissible," and "coherent."  The disadvantage, which does not exist for Frequentist methods, is that it is built on inductive reasoning rather than deductive reasoning.  An article on this can be found at https://plato.stanford.edu/entries/induction-problem/

Induction only allows you to test models you have, but cannot test models you do not have and so will always be incomplete.  If you reject a Frequentist null hypothesis, then you reject all theories except the null.  The problem with this happens when the model is complex and cannot be turned into a pure binary hypothesis.  It is worse when more than one model rejects its null.  If that happens then, Frequentist theory is silent on what to do.

A statistic is admissible if there is no less risky way to create it and it is coherent if you could place a fair gamble on it.  The idea of a parameter is more flexible in Bayesian thinking than Frequentist.  A model is a parameter in Bayesian thinking.  As it is a parameter, you can construct a probability statement around it.  Bayes theorem naturally rewards better fitting models, while penalizing models for more structure.  It is a natural Occam's razor with a mathematical interpretation.  It also simply sets aside Friedman's defense as unnecessary.

One of the better axiomatic constructions for Bayesian statistics is Cox's.  You can find it at

  Richard T. Cox. The Algebra of Probable Inference. 1961. Johns Hopkins University Press. Baltimore, MD

It assigns "plausibilities" to logical assertions.  Usually as odds or probabilities.  Surprisingly, it is worth reading.  If you are doing model or theory comparison, then, more so, this is the case.


How long does it take for shocks to propagate through the price system / market? Has anyone calculated or theorized about how long it takes prices to come to some kind of equilibrium after a market shock has happened? How has this changed over time?

An example might be a spike in the price of oil. If oil independently spikes from 50 dollars/barrel to 100 dollars/barrel in a day, how long will it take prices (food prices, car prices, wages, etc) to find equilibrium again?

This is likely only part of the story, but I found an interesting lecture by Milton Friedman that shows a graph of the quantity of money per unit of output (likely GDP) graphed alongside a normalized consumer price index (normalized so that the index intersects with the other line in 1970).

The graphs all clearly shows changes in the consumer price index lagging about 1-3 years after discontinuities in the money supply.

https://www.youtube.com/watch?v=THAaIZmxfNA&t=11m2s

In this other graph, you can see the lag is even longer than that, at anywhere from 4 years between 2008 and 2001 to 6 years from 1985 to 1992:

3-4 years is probably a reasonable average estimate for how long it takes for money created by the Fed to "trickle down" to the average consumer in the form of inflation.


Estimating price elasticity of demand I have 25 quarterly observations and I want to estimate price elasticity of demand. I intented to use GMM-IV estimator. However, I read that it is not good for small samples. What can you suggest me? Please keep in mind that I am not econometrician by profession.

Unfortunately for your case all GMM estimators suffer from finite sample bias. In the presence of an endogenous explanatory variable the GMM estimator is only consistent.

There's not much you can do to remedy this issue, except collect more data. Check for consistency of sign and magnitude of estimates between OLS and GMM, as well as sensitivity to instrument choice (if your model is over-identified).


"manufacturing sector" vs. "manufacturing industries" From a tutorial

  The main sectors of the economy are: Primary sector, Secondary / manufacturing sector, Service / ‘tertiary’ sector, Quaternary sector

which uses the word "sector", so does Wikipedia.

While some other posts, e.g. the Cambridge Dictionary, and hq.nasa use another word "industry". Which one is the professional term?

It depends how industry and sectors are defined.  In a few settings they could be used to mean the same thing.

However, usually, the term industry refers to a much more specific group of companies or businesses, while the term sector describes a large segment of the economy.

Industry can be used to define the group of organizations or firms that are engaged in the production of related goods or services.  The industry name will depend on the type of service or product made.     Sectors describes the division of the whole economy, which can consist of business groups(multiple industries) which are engaged in related or similar services or products.

Industry are a cluster of different firms while a sector are a group of different related industries.

Since an industry involves a particular process the scope it is more narrow while for a sector it relates to business activities which makes its scope wider.

https://www.wallstreetmojo.com/industry-vs-sector/


Network goods in competitive markets vs. socially optimal network size In networks economics, the socially optimum equilibrium is bigger than the one in perfect competition (see for instance, Easley and Kleinberg 2010, Ch. 17).

Why is that so? Could you provide me with an example?

There are two sorts of network effects a network good might have. It might have consumption externalities (such as with traffic) or it might have positive network externalities (such as with operating systems). The latter case is the case that your link discusses. I don't see a discussion of perfect competition in your link, but if you imagine a model in which consumers choose a level of some good with a positive network externality to buy at some price, equilibrium production should be less than socially optimal for the same reason that there is underproduction in perfect competition with any positive externality -- at the margin, consumers don't take into account the value buying the network good brings to other consumers.


What is banking compliance I'm not sure if I'm using the term correctly but there's something that's referred to as compliance in banking. Am I correct to understand that it's the process of obeying the laws and regulatory guidelines for financial institutions in general? Or is there a more specific meaning to the term?

What would be two or three common examples of such compliances?

You are using it correctly. It means the processes and protocols created by banks or those imposed to them by authorities. If the bank also has some quality certificate, like iSOs, them this imposes new compliance costs and processes.


How is it possible that all currency exchange rates are fixed w.r.t. to each other? I wondered if it would be beneficial to exchange a currency through one or more "in-between" currencies.

For example: instead of trading Russian Ruble (RUB) to Euro's (EUR), one would trade from RUB to US Dollars (USD) and then to EUR.

Neglecting exchange costs, I expected that there would be some difference in the resulting amount of money.

I tried this out in an Excel sheet, using data from Floatrates.

I first imported an xml of all the RUB exchange rates (column C in the Excel).
Then I imported an xml of all the EUR exchange rates (column G in the Excel).
By inverting the latter I found how much EUR every currency was worth (column H in the Excel).
Then I multiplied the column C with column H to find the resulting value.

What I found is that the resulting amount of money stayed the same, regardless of which currency I went through.

How can this be? Is my method of calculating the resulting amount of money perhaps incorrect?

International currency markets are highly liquid, with near-instant transfer of knowledge between trading centres.

This means that any arbitrage opportunities tend to get resolved within seconds.

To put it simply, the thing you tried to do, of trading two currencies via a third, is something that lots of real-world currency traders, and their automatic trading programs, are trying to do all the time; so any discrepancies are quickly mopped up by the market.


Arrow’s Impossibility & Validity of Social Welfare and Pareto Improvement Analysis Arrow’s Impossibility states it is impossible to formulate a social ordering (Social Welfare Function) without violating some desirable conditions ( one of the “No dictatorship”, “ Pareto Efficiency”,  “Independence of Irrelevant Alternatives”, “Unrestricted Domain” and “Social Ordering”). However, there are many (prestigious) academic papers assuming some (utilitarian) Social Welfare Function,the most popular Welfare Function in related literature, to draw some conclusions, For example “Redistribution through Markets” and "Monetary Policy with Heterogeneous Agents:
Insights from TANK models".
I'm wondering, having Arrow impossibility in mind (There is no valid Social Welfare Function), how one can wholeheartedly believe social welfare improvement claims? and why famous economists and Journal referees think that such analysis are justifiable?
Would you mind helping me? Maybe I did not get implications of Arrow impossibility Theorem.

Let $X$ be the nonempty set of alternatives, $\mathcal{P}_X$ the set of preference relations on $X$ and $N=\{1,\ldots,n\}$ a finite set of agents. Under the universal domain condition, Arrow's theorem concerns functions from the set $\mathcal{P}_X^N=\underbrace{\mathcal{P}_X\times \mathcal{P}_X\cdots\times\mathcal{P}_X}_{n\text{ times}}$ to $\mathcal{P}_X$. Utilitarianism does not just depend on preferences alone, and can therefore not be formulated in this setting.
Still, one can formulate Arrow's theorem in terms of utility functions. So let $\mathcal{U}_X$ be the space of real-valued functions on $X$, interpreted as "utility functions." I'll explain the scare-quotes later. We are now looking at functions from $\mathcal{U}_X^N$ to $\mathcal{P}_X$. Since utility functions determine preferences, we can do now the same things we could do before and more.
Now there are two ways to formulate Arrow's condition of independence of irrelevant alternatives in the new setting; adapting the other conditions is straightforward.
The first form is:
IIA1: The function $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ satisfies IIA1 if for any two profiles of utility functions $u=(u_1,\ldots,u_n)$ and $u'=(u_1',\ldots,u_n')$, and any two alternatives $x,y\in X$,  such that $u_i(x)\geq u_i(y)$ if and only if $u_i'(x)\geq u_i'(y)$ for all $i\in N$, we have $x\phi(u) y$ if and only if $x\phi(u') y$.
Note that IIA1 has implicitly two components. The first says that the social ranking of two alternatives depends only on the utility values for these two alternatives. The second says that only the preference ranking matters, but not the "intensity." If we only keep the first, we get
IIA2: The function $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ satisfies IIA2 if for any two profiles of utility functions $u=(u_1,\ldots,u_n)$ and $u'=(u_1',\ldots,u_n')$, and any two alternatives $x,y\in X$,  such that $u_i(x)-u_i(y)=u_i'(x)-u_i'(y)$ for all $i\in N$, we have $x\phi(u) y$ if and only if $x\phi(u') y$.
IIA2 is actually weaker than requiring that the utility for both alternatives are the same in both profiles; only the differences need to be the same.
Now, with the other axioms suitably reformulated, IIA1 implies that all such functions $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ must be dictatorial, while IIA2 is compatible with utilitarianism.
Now, so far this was a purely mathematical issue. In particular, we treated utility functions as being somehow objective things and not just representations of preferences. They also measure well-being in a way that is comparable across agents. So utilitarianism uses information that is not included in purely positive models of economic behavior, in which only preferences matter. Now, one can discuss how sensible it is to compare the well-being of different agents and what information one needs for certain SWFs. There has been a research program starting in the late 1970s of discussing the exact informational assumptions one uses for such comparisons of well-being. A nice introduction to the topic is given in the 1998 book "Theories of distributive justice" by John E. Roemer. Roemer directly relates all this to Arrow's theorem, so his book will give you a very extensive answer.


What happens with corporate deposits when a bank goes bankrupt? When a bank goes bankrupt the money of the private individuals who saved in that bank are lost but a part of the money are guaranteed by the government.
But say for example Apple has 10 billion dollars in an account at Bank of America and BoA goes bankrupt.
What happens to Apple then? Can they get their money back?

In simplified terms a bank or shadow bank (non-bank financial intermediary operating a balance sheet similar to a bank balance sheet) holds a mix of cash, marketable securities, and investment portfolio on the asset side of the balance sheet. The bank or FI issues a mix of equity, insured liabilities, and uninsured liabilities on the liabilities and equity side of the balance sheet. If the bank or FI is forced to write-down the value of assets in the investment portfolio then the creditors holding equity are forced to take the first loss and the creditors holding uninsured liabilities take the remainder of the total loss. Sometimes uninsured liabilities convert to equity in a new venture carrying the good assets or there is a bailout via government recapitalization effort in a systemic crisis.
These are FAQs describing the resolution of IndyMac Bank via the FDIC:
https://www.fdic.gov/resources/resolutions/bank-failures/failed-bank-list/indymac-q-and-a.html#Uninsured

If it is determined that you have some uninsured funds, the FDIC will mail you a Receiver Certificate.  This certificate entitles you to share proportionately in any funds recovered through the sale of the assets of IndyMac Bank.  You will eventually recover some of your uninsured funds.  Of course, you will receive immediate full payment for your insured amount by transfer to IndyMac Federal Bank or from your broker.  In addition, the FDIC will pay uninsured depositors an advance dividend of 50% of your uninsured deposit.

Large cash investors seek secure liquid investments and cannot rely only on insured deposits for this purpose. See Institutional Cash Pools and the Triffin Dilemma of the U.S. Banking System by Zoltan Pozsar (2011):
https://www.imf.org/external/pubs/ft/wp/2011/wp11190.pdf

Institutional cash pools prefer to avoid too much unsecured exposure to banks even through insured deposits. Short-term government guaranteed securities are the next best choice, but their supply is insufficient. The shadow banking system arose to fill this vacuum. One way to manage the size of the shadow banking system is by adopting the supply management of Treasury bills as a macroprudential tool.

Warren Buffet testified to the Financial Crisis Inquiry Commission (FCIC). There is an audio copy online. As I recall he describes having to make an $8 billion payment in the fall of 2008 during the financial crisis. He said Treasuries were the best option at the time to ensure the cash would be available to make the contractual payment obligation.


What does "writes a check to Congress in exchange for them" mean in this context? On page 192 of the 5th edition, George Edward Griffin begins to describe
the "Mandrake Mechanism", chapter ten, in "The Creature from Jekyll Island".

First, the Fed takes all the government bonds which the public does
not buy and writes a check to Congress in exchange for them. (It
acquires other debt obligations as well, but government bonds comprise
most of its inventory.)

I don't understand it.  The government issues bonds.  Some people buy it.  Some bonds are not bought.  How does Congress get involved in this?  Is it the case that Congress buys all those people did not buy?  With what money does Congress buy?  Is Griffin saying that the Fed itself ``writes a check to Congress'' so that Congress can afford it?
Update. The question is requesting not a profound understanding of the process, but only the process itself.  If Griffin doesn't describe it properly, I would consider an excellent answer the mere description of this first step of the process that Griffin tried to describe.

The US Government has $\\\$X$ in spending and raises $\\\$Y$ in revenue.
Usually, $X>Y$. So, the US Government (or more specifically the US Treasury) has to cover this $\\\$(X-Y)$ difference (or funding requirement/shortfall).
It does so by issuing bonds (or notes or bills) worth $\\\$(X-Y)$. Pretty much anyone is free to buy these bonds, including the public and the Fed.
So "writes a check to Congress in exchange for them" probably simply describes the act of the Fed buying some of these bonds.

The buying and selling of US Government bonds by the Fed is called open market operations. See e.g. Investopedia or this Fed webpage for how this works.

Three clarifications:

The US Congress doesn't have any direct official/technical/legal role in any of this, except insofar as it determines spending ($X$) and revenue ($Y$). So, it's not really correct to say that the Fed "writes a check" to Congress. We should instead say that Fed "writes a check" to Treasury. (Of course, we could quibble and say that Congress and the Treasury are all part of the US Government, so "writing a check" to the Treasury is "really the same thing" as to Congress.)
This writer writes, "the Fed takes all the government bonds which the public does not buy." This statement may suggest incorrectly that the government is free to sell as many bonds at whatever interest rate it likes. But this is incorrect. Instead, what happens is that the Fed targets certain interest rates. The government must then adjust its interest rates—so as to meet its $\\\$(X-Y)$ funding requirement and, at the same time, the Fed's targets.
"Writes a check" is not strictly speaking correct, since nowadays this is all done electronically with keystrokes rather than any physical writing of checks.

By the way, I've never heard of this writer, but Wikipedia calls him a conspiracy theorist and states that The Creature from Jekyll Island (1994) "promotes false theories about the motives behind the creation of the Federal Reserve System." So I wouldn't necessarily believe anything written by this writer.


Does the term 'marginal' refer to the last consumer? I was reading this answer on this website which talked about how MB = P at allocative efficiency. "Why does allocative efficiency occur when P=MC rather than MB=MC"

In this answer, it is stated that the last (i.e. marginal) consumer who buys will be the one for whom the benefit is just equal to the cost.

I'm a bit confused about the use of the world marginal. I read online that it means "one more", but isn't that different from the use of the word marginal above? And how is MB = P, isn't marginal benefit referring to many points along the curve?

Yes marginal refers to the last or one more unit or person. I don’t see there any contradiction if you have 2 people in room and third person joins it’s both the last person and one more person.

If you are looking for a precise definition then in economics the concept of margin is connected to the first derivative (instantaneous rate of change) of function at some point. For example, with utility function $U=q^2$ marginal benefit of consuming 10 units of $q$ would be $U’(10)=2(10)=20$.

Also, demand curve is an aggregate of demands of individuals who all might derive different marginal benefit of consuming the good, the point of the answer is that the marginal customer, last customer to participate in the market, will have marginal benefit of consuming good exactly equal to price (since if $MB<P$ the person would just not participate in market).


mixted strategy that assigns positive weight to a pure startegy that is dominated This problem is  of Fernando Vega Redondo(Economics and the theory of games)

Exercise 2.1
Let G be a game in strategic form. Prove that, for every player
$i\in N$, every mixted strategy $\sigma_{i}\in \Sigma_{i}$ that assigns positive weight to a pure strategy $s_{i}\in S_{i}$ that is dominated can be itself always be improved by another strategy $\sigma_{i}'$.

This is if $s_{i}\in S_{i}$ is strongly dominated for some $\sigma_{i}$$\in \Sigma$
with $\sigma_{i}(s_{i})>0$, $\exists$ $\sigma_{i}'\in \Sigma_{i}$ such that $\forall$$s_{-i}\in S_{-i}$$: \pi_{i}(\sigma_{i}',s_{-i})>\pi_{i}(\sigma_{i},s_{-i})$.

Q1: why they said that affirmation is obvious?

*I would like to know if anyone could tell me how to build that mixed strategy that dominates that strategy that assigns positive probability, since it is not obvious to me.

Thanks!

The short answer is that a mixed strategy $\sigma_i$ that uses with positive probability a dominated pure strategy $s_i$ can always be improved by excluding $s_i$ from its mixing support and redistributing its "probability weight" to one of its dominating strategies.

More formally, suppose that strategy $s_i'$ dominates $s_i$ (you can think of $s_i'$ as a pure or mixed strategy itself, this does not make any difference for the argument; for simplicity I suppose $s_i'$ is a pure strategy), that is $\forall s_{-i}\in S_{-i}: \pi(s_i',s_{-i})>\pi(s_i,s_{-i})$.

Also, suppose that strategy $\sigma_i$ plays strategy $s_i$ with positive probability $p_{s_i}\in(0,1)$. Build strategy $\sigma_i'$ as follows: identical to $\sigma_i$ but replace $s_i$ with $s_i'$ , that is play $s_i$ with probability 0 and play $s_i'$ with (additional) probability $p_{s_i}$.

By linearity of the payoff function you have, $\forall s_{-i}\in S_{-i}$:
$$
\pi(\sigma_i',s_{-i})=p_{s_i}\pi(s_i',s_{-i}) + (1-p_{s_i})\pi(\Lambda_i,s_{-i}) > p_{s_i}\pi(s_i,s_{-i}) + (1-p_{s_i})\pi(\Lambda_i,s_{-i}) = \pi(\sigma_i,s_{-i})
$$
where $\Lambda_i$, loosely speaking, denotes the residual combination of strategies played in $\sigma_i$, apart from $s_i$, inherited by $\sigma_i'$ too.


Exchange economy find core Find Pareto optimal allocations and the core for the following economies
Consider an exchange economy with two agents and two goods. Let consumption sets are the nonnegative orthant. Agent 1’s utility is represented by $x+y$ and agent 2’s utility is $xy$. Endowments are (e,0) for agent 1, and (0,e) for agent 2 with  > 0.
When I find MRS and equalize them, I am going to find $x_2=y_2$  This is interior point (contract curve. ). By feasibility constraint, $x_1=y_1$.
Its core is given as follows. Why?

How can we find (determine) core? What it’s appropriate calculation to find core? For example, why we assume $e/2$?

Core Allocations are Pareto efficient allocations that must satisfy individual rationality i.e. these allocations must yield at least as much satisfaction to the individuals as their respective endowments. In the given question, core allocations are represented by line connecting $(e/2, e/2)$ to $(e,e)$.
Dashed line on the left graph is representing the set of efficient allocations and on the right graph is representing the Core allocations.


Why is it that poverty incidence is high in the agricultural sector? I read this article earlier, and the "experts" there say:

  "The incidence of poverty is very high in agriculture because of its low productivity."

  "The incidence of poverty is very high in agriculture because of its relatively slow growth."

But why do they say that? How about you? What do you think is the cause? To me, they keep saying that:

  "[T]he share of agriculture in GDP, and not the demand for agriculture per se, which declines as income per capita increases"

But why would you want to be more productive/produce more goods when people in general are going to have a pretty much similar demand for the product ('cause it says there that demand for agri products don't necessarily decline)?

Well I think that agriculture poverty is there because farmer can't store its product and sell that when prices are good as his product is perishable. With this he has to remain busy throughout the year either harvesting or in production and has no control over markets.


Definition of Absolute Risk Aversion In its Wikipedia article, absolute risk aversion is defined as
$ARA = -\frac{u''(c)}{u'(c)}$. However, I have alternatively seen absolute risk aversion
defined as half the decrease in consumption that an investor is willing to accept to avoid a gamble $\varepsilon$, where $E[\varepsilon] = 0$, $E[\varepsilon^2] = 1$, and $\varepsilon$
is independent of consumption:
$$
U(C - ARA/2) \equiv E[U(C + \varepsilon) \mid C].
$$
Using the latter definition, how can I show that $ARA = -U''(C) / U'(C)$?

Set $y \equiv c+\varepsilon$. So $y$ represents changed consumption around and "close" to a given level $c$. Take a 2nd-order Taylor expansion of the function $E[u(y)\mid c]$  around $c$, which is treated as fixed since we condition on it :

$$E[u(y)\mid c] \approx E[u(c)\mid c] + E[u'(c)(y-c)\mid c] + E[\frac12u''(c)(y-c)^2\mid c]$$

But $y-c = \varepsilon$
so

$$E[u(y)\mid c] \approx E[u(c)\mid c] + E[u'(c)\varepsilon\mid c] + E[\frac12u''(c)\varepsilon^2\mid c]$$

Due to the conditioning, and the independence of $\varepsilon$ from $c$ the expected value distributes and applies only to $\varepsilon$:

$$E[u(y)] \approx u(c) + u'(c)E[\varepsilon] + \frac12u''(c)E[\varepsilon^2]$$

Since we assume

$$E[\varepsilon] = 0 \Rightarrow {\rm Var}(\varepsilon)=E[\varepsilon^2] =1$$
we obtain

$$E[u(c+\varepsilon)] \approx u(c) + \frac12u''(c)  \tag{1}$$

Now consider $u(c-\frac 12 ARA)$, with $ARA \equiv -\frac{u''(c)}{u'(c)}$, and take a 1st-order Taylor expansion in this case, again around $c$:

$$u\left(c-\frac 12 ARA\right) = u(c) + u'(c)\cdot (c- \frac 12 ARA - c) = u(c) - \frac 12 u'(c)\cdot ARA $$

Using the definition of $ARA$ to replace it we obtain

$$u\left(c-\frac 12 ARA\right) \approx u(c) - \frac 12 u'(c)\cdot\left (-\frac{u''(c)}{u'(c)}\right)$$

$$\Rightarrow u\left(c-\frac 12 ARA\right) \approx u(c) + \frac 12 \cdot u''(c) \tag{2}$$

The right-hand sides of equations $(1)$ and $(2)$ are equal therefore, approximately, so  are their left-hand sides or,

$$E[u(c+\varepsilon) \mid c] \approx u\left(c-\frac 12 ARA\right)$$

which is valid as long as $ARA$ is defined the way it is. QED.

If the variance of the gamble is not unity but $\sigma^2 \neq 1$, then the more general equation is

$$E[u(c+\varepsilon) \mid c] \approx u\left(c-\frac {\sigma^2}2 ARA\right)$$


Is possible that, in a choice structure $(B,C)$, $C(b) = \emptyset$, for some $b \in B$ It is a very simple question but with some implications. I just start reading Mas-Colell and it's not clear for me if it is possible a choice structure, $(B,C(.))$, where $\exists b \in B, C(b)= \emptyset$.
For example:
$B=\{\{x,y,z\},\{x,y\},\{x,z\},\{y,z\},\{x\},\{y\},\{z\}\}$
$C(\{x,y,z\}) = \{z\}$
$C(\{x,y\}) = \emptyset$
$C(\{y,z\}) = \{z\}$
$C(\{x,z\}) = \{z\}$
$C(\{x\}) = \emptyset$
$C(\{y\}) = \emptyset$
$C(\{z\}) = \{z\}$
is a valid choice structure?
Mas-Colell defines choice structure and choice rules in section 1.C, page 10 on oxford press 1995 edition.
I first though about it because I think it would have some interesting implications, but it was false. Anyway, the question remains. Thanks!

One usually rules this out by assumption, as Mas-Colell, Whinston, and Green do in their textbook (). The goal is often to try to characterize preferences that guarantee (at least for finite choice sets) the existence of choices from a revealed preference point of view. There is also the methodological point that if you confront a decision-maker with a choice problem and chooses none of your options, then you haven't really included all options in the first place.
That being said, one can still study the mathematics of possibly empty-valued choice rules, and this has been done. One can find some material within this more general context in the book [Aleskerov, Fuad, Denis Bouyssou, and Bernard Monjardet. Utility maximization, choice and preference. Vol. 16. Springer Science & Business Media, 2007.]


Intuition behind the positive slope of LM curve : 'Income must increase'? Source: by John Floyd PhD  (in Economics; U Chicago)

  The intuition behind the positive slope of LM is as follows:
  An increase in the interest rate reduces the demand for money
  and an increase
  in income increases it [the demand for money].
  To [...] [equalize] the demand for money equal to a constant money supply
   as the interest rate rises
  and we move along the LM curve,
  the level of income must increase.

Would someone please explain the bolded? It sounds too imperious and presumptuous:
how MUST a rise in interest rate increase consumers' level of income?
Eg, employers can still refuse to increase salary. Then income would NOT increase?

The intertemporal consumption equation is :

$$Y_{1}+\frac{Y_{2}}{(1+r)} = W$$

where $W$ is total wealth and $Y$ is revenue.

When interest rate rise investiment drops and savings rise, so lets say someone choses to save all his revenu from period 1 (this is just to calculate his total wealth), then in period 2 his total wealth will be :

$$Y_{1}(1+r)+Y_{2} = W(1+r)$$

So as you can see his total wealth in 2nd period grows when interest rate grows

Also we can isolate $Y_{1}$ :

$$Y_{1}= W-\frac{Y_{2}}{(1+r)}$$

And you see that when interest rate rises your revenu in period one rises.

I am not sure if this is a valid answer but it seems correct


Problem of generated regressors I use data (adv. labor and wage) through a nonlinear model to recover the unobserved data (use fsolve in Matlab to solve this system of equations) and run a regression of labor (or wage) on these unobserved data and other observed data. The regression equation that I run is the reduced form of the nonlinear model. Is this a problem?

Two eigenfuctions:
$\lambda AX = X$ and $\lambda BY = Y$, A and B are given, L and W are the elements in A and B.

I use matlab to solve it for x and y.

I can get the regression equation $L = ax + by + cz + \epsilon$ from these two eigenfunctions.

Edit1:

Based on the additional information you provided I still believe that my recommendations are valid and that you should analyze your missing data, and attempt to fill it before using a regression to create imputed values.

If you believe the following assumptions about your imputation regression and data analysis to be true, then you are probably in good shape:
1. The specification is correct and based on economic theory in the literature.
2. You correctly identified your missing data as MCAR or MAR.
3. You correctly ran your regression in matlab. (Note: I use SAS and Stata so I'm not able to provide specific advice on how to do your regression.)

Original Answer:

If you are using a regression to fill data gaps in another regression you may want to take a step back. By using estimated values alongside actual values you may run into problems when you try to interpret results. Imputation via regression does not store any way to analyze the precision or accuracy of your imputed values. Your final model may be over identified in that your final regression will not account for the uncertainty in the imputed values.

Data gaps - which it sounds like you are dealing with - are a thorn in the side of econometric analysis. When dealing with data that has significant gaps in it, it is very important to ask why these gaps exist. If it is because of bad data collection on the part of something or someone you can influence, you should call that person or company to find out why there is a gap in their data. Chances are that they either know and have a solution, or had no idea and need to find a solution. Of course sometimes they may not be supporting the data set anymore, or in a rare subset of cases simply don't care that their data is bad.

For those times when you absolutely cannot fill data gaps you have several options. One, which is sometimes advisable if you have significant resources at your disposal (ex. a Fortune 500 company) you can collect the data yourself and make a dataset that works for your specific needs (or pay someone to do it for you).

If you can't fill the gaps, and you can't collect the data yourself, you are stuck with a set of lesser options. The first step is to indent oft the nature of your missing data.

MCAR - Missing Completely At Random
If X is your variable of interest, and contains missing data, and the reason that the data is missing is independent of the non-missing data and independent of other data in the dataset, then your missing data is MCAR.
MCAR data is pretty harmless, and is common in big datasets. If it comprises less than 5% of your data, you can usually just delete the observations. If it is greater than 5% you will probably want to use imputation (more on this below) to fill the gaps. So if your data were MCAR, your strategy of using regression imputation is actually applicable. However, you should be aware that regression imputation isn't a sure fire strategy. As I mentioned above it can lead to over identification.

MAR - Missing At Random
The difference here is that the reason the data are missing is independent with the other data in the variable of interest but may depend on other data in other variables.
For MAR data you should use multiple imputation.

NON-IGNORABLE DATA
This type of missing data isn't missing at all. For example, say you want to find out if your son is dating a girl at school. So you give him a questionnaire. It asks: "Are you dating a girl at school?" He leaves the question blank. That would tell me: Yes he may be dating a girl at school.

For non ignorable data, the reason they are missing is dependent on the othe data in the variable. In our scenario, my son's non response was actually a form of implied response.

Non ignorable data should not be imputed nor should they be ignored. Every effort should be made to fill them. If you still cannot fill them you should consider switching data sets or, if that is not feasible, at least make detailed note of these missing data in your research.

IMPUTATION TECHNIQUES
For more details on imputation techniques, I recommend consulting your statistical package's documentation.

BASIC IMPUTATION METHODS(for MCAR only):
Hot decking - This is essentially just swapping in a randomly selected similar value from the non-missing data. I don't recommend this technique. It can lead to false conclusions and is statistically unsound.

Mean substitution - Replace missing values with the mean of the non missing data. This is better than hot decking but not by all that much. It is however an option if you just need an early idea of what your model results might be (ex. for informal paper proposals to colleagues).

Regression imputation - This is basically what you did. It has issues with overidentification but is not all that bad in the scheme of things assuming you are using a good model for these estimates.

Multiple Imputation (best option for MAR) - The most rigorous form of imputation requires that you draw your imputations from many datasets, using a particular procedure such as the one from Donald Rubin's An Overview of Multiple Imputation.

Many statistics packages have imputation methods programmed in already as they are quite technical to set up.

Overall, you have to balance your amount of missing data, type of missing data, time constraints, and need for highly accurate results, and make a decision on what you want to do. As in the rest of economics, this decision is a trade off.


Shouldn't national Stock Indices proportional to GDP of the country? I was looking at Japan: Japan's GDP trajectory and Nikkei 225.
It seems from 2011 - 2015 the GDP contracted but the Nikkei has gone up. Would there be a good explanation for this ?

One of likely many contributing factors is the definition of GDP.  Consider the humble iPhone, produced by Apple contractor FoxConn. FoxConn is headquartered in Taiwan, while Apple is headquartered in Cupertino, California, USA.

Apple owns the intellectual property that goes into the iPhone, and so iPhone sales to American citizens support growth in Apple stock.  But because the iPhone is produced offshore, when an American citizen buys a cell phone that buoys the stock of an American-domiciled company, it does not contribute to American GDP - it contributes to Chinese, or Taiwanese, or wherever the factory happens to be, and in fact subtracts from American GDP.


Will China's GDP be greater than the USA's and Europe's combined? A newspaper said in 2080 China can be a developed country, with a bigger population. Does that mean China's GDP is going to be bigger than the USA's and Europe's combined?

Also, is this possible even though China's land area per person is smaller?

Of course this is is possible. China is a very large economy currently and it is likely quite a way away from reaching its production frontier. It has lots of room to grow.

A lot can happen in 65 years and any forecasting this far into the future will be subject to a high degree of error. Don't forget that the US and Europe are dynamic as well. The current world economic climate may be in China's favour now but in 65 years who knows. Maybe everything we will need can be 3D printed an large scale manufacturing won't exist.

It is hard to look at The Next 100 years. It attempts to speculate on what can happen. *Spoiler*, the author does not expect the 21st century to be China's.


Alternative to linear regression I'm third year economics student and all econometrics we had so far and basically all empirical studies in economic subjects we had so far are linear regression. Is there any alternative, can anyone suggest any reading material or direction in which I can explore?

There are numerous directions to go which start moving you beyond ordinary least squares (OLS), linear regression. The universe of statistical methods is large!

Two books that I particularly enjoyed are Econometrics by Hayashi and Elements of Statistical Learning by Hastie et. al. Looking back at your question, these books may be too advanced. But maybe not. An easier version of the latter is An Introduction to Statistical Learning (and may be interesting from the perspective of a broader exposure to data science than just econometrics).

Hayashi's Econometrics introduces a variety of methods through the lens of GMM and with an eye towards time-series econometrics.
Elements of Statistical Learning is a modern classic of the statistics, machine learning literature. It's great for opening your eyes to methods outside of traditional econometrics.

Some examples beyond ordinary least squares...

Maximum likelihood estimation (MLE)

You must know this if you continue doing statistics. It's a ubiquitous workhorse.

If you can specify the likelihood function then parameters of the likelihood function can be estimated by maximizing the likelihood function. In certain special cases, (eg. linear regression with conditionally normal error terms) the OLS estimator is the MLE estimator. You've undoubtedly encountered MLE estimation before if you've estimated a logit model. MLE is all over physics, engineering, and the sciences.

There are issues with applying MLE to economic models though. Often we know that an overall economic model is false. A model will imply certain facts that are entirely fallacious. Forcing the model to match the data in a maximum likelihood sense may not induce a useful choice of parameters. GMM can be used as an alternative approach to selectively test certain predictions of a model.
GMM is another broad method for estimating parameters based upon moment conditions that in expectation should be zero. Hayashi's book Econometrics develops ordinary least squares regression, instrumental variables, maximum likelihood, and other methods as special cases of GMM with different moment conditions. OLS can be thought of as GMM using the orthogonality condition of the regressors and the error terms. MLE can be derived as GMM on the score.

A John Cochrane ode to GMM is here.
Matching methods for estimating causal effects are common in certain areas of economics.

The idea is to match a treated entity with an untreated entity based upon observable characteristics. A widely used technique for example is propensity score matching
There are all kinds of variations on classic linear methods:

The idea here is to start with ordinary least squares but then to bias coefficient estimates towards zero to reduce overfitting and improve out of sample prediction.

Ridge regression
Lasso

References

Fumio, Hayashi, 2000, Econometrics

Hastie, Trevor, Robert Tibshirani, Jerome Friedman, 2009, Elements of Statistical Learning

James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani, 2017, An Introduction to Statistical Learning


intertemporal utility maximisation Adam's consumption period 1 and 2 are denoted by $c_1$ and $c_2$ respectively. His utility function is $U(c_1,c_2)=4c_1^{0.5} + c_2$
Ben earns an income of \$3 in period 1 and \$3 in period 2, regardless of the level of inflation.
a.  Suppose there is no inflation and the interest rate is 5%. How much will Ben consume in each period?
b.  Now suppose that inflation rate rise to 100%. How much will Adam consume in each period now?

For part a, I took $c_2= -1.05c_1 + 6.15$ as the budget line and equated it to
MRS of his utility function. I managed to obtain $c_1 = 3.63$ and $c_2= 2.341$.

However, I am unsure of how to do part b as I obtain a negative answer for $c_2$ after doing the same steps as part a. The only differenece was that I took into account inflation for part b.

The purpose of part b is probably to illustrate what happens during hyper inflation: people spend there money immediately while they get anything for them.

Note that there is probably an implicit constraint that $c_2$ cannot be lower than 0, so the answer would be $c_1 = \$6, c_0 = \$0$.


How do companies with shareholders close up shop? How can a company with shareholders stop operations without being acquired or going bankrupt?

How is a company with shareholders that is closing up shop able to give shareholders money back for their shares if the shareholders invested more money than the value of the company's assets?

If a company with shareholders has closed up shop in the past without being acquired or going bankrupt, what are some well-known examples?

We need to distinguish between private corporations, and public. Private corporations are largely free in what they do, public corporations listed on stock exchanges have to follow various procedures.

For any corporation, there’s no guarantee that you make a profit, so what price you bought it at makes no difference.

If a company wound up its affairs, it would make a final cash distribution to shareholders, and that would be it. The cash would either be a dividend or a return of capital, depending upon the equity levels of the company.

The problem is that do this, the company would have to sell all assets simultaneously. That is normally difficult to do, and so it is much more usual for the entire firm to bought out (after selling some assets).

Furthermore, having a listing on the stock exchange is itself valuable. It would be far more common for a company to pursue a “reverse takeover”: the listed company appears to take over a private firm, but the private firm actually buys out the existing shareholders. This way, the private firm saves the expenses of seeking a listing on the stock exchange. Just shutting the firm down would probably be viewed as a suboptimal course of action, and might expose the directors to lawsuits.

What is more likely is that companies largely fail, and end up having to shut down most of their operations. After awhile, they will not meet listing requirements to remain on the stock exchange, and the shares are delisted. The shares still exist, but it is very hard to find buyers.

If you want to find examples, you could just do a web search looking for the terms “delisted companies” and the name of a stock exchange that you are interested in. The following link is an example:
TSX Venture Exchange listing changes


When should a company's revenue and share price grow faster or slower than the economy? I was listening to a famous investor discuss an investment of his in an interview. He had bought shares in a bank that was quite successful, though it was a small bank. The bank is based in a highly under-developed country with a massive population. This country's economy has a high rate of growth (about 8%).

The investor made an interesting comment. He said,

  "of course, the bank is going to grow at 3x the growth rate of the
  country's GDP."

He didn't offer any further reasoning for this. And it wasn't clear whether he meant the bank's revenues or its stock price are going to grow at 3x.

However, this led to me thinking about an interesting issue that I had never considered before.

There must be some kind of model which helps us predict whether a company's (1) revenue and (2) stock price will grow faster or slower than the overall economy.

What are some of the variables and considerations in this model, and in what direction do they act (e.g. do they grow or reduce (1) or (2))?

GDP is the sum of wages and capital income. In most developed economies, 2/3rds of GDP is wages (income and other compensation) and the remainder is capital income (owner occupied rents, interest and dividends mostly). This ratio is difficult to measure precisely but the general consensus is that it has been mostly but not perfectly stable over time.

Why Labor’s Share of Income Is Falling By Jared Bernstein
Therefore, if the economy $Y$ grows at a rate $g$ and the capital income is roughly $Y/3$ then it too must be growing at rate $g$. If discount rates $r$ are also constant this implies that wealth is $\frac{Y}{3(g-r)}$ which also grows at rate $g$. Now it turns out that discounts rates are not constant and have been in long term decline, and as long as this is the case the value of the capital and land assets of the economy can indeed grow faster than output does. The following figure, which depicts financial assets relative to GDP, doesn't fully capture this issue (it omits non-financial assets like much of real estate) but it is a decent approximation.

All of which implies that capital income like other quantities roughly proportional to output, must grow at the same speed as output. Assets can grow at a different rate in the short to medium term than output, but that only if there are long term changes in the discount rate. But most firms are very small and even the largest ones are small relative to the whole economy. Firms are being founded and failing all the time, so there is plenty of room for individual firm to grow faster or slower than the economy as a whole.


Does the money's "Store of value" function actually exists? I'm aware of the following four functions of money:

Medium of exchange.
Measure of value (in dollars, pounds, etc.)
Standard for exchanging goods (guaranteed by the Fed)
A Store of value (as an asset).

AFAIK, the last one didn't exist according to Classical Economists, but only John Maynard Keynes brought it later in his theory.

I want to understand whether this function really exists? I'm more inclined to agree with the Classical Economists -

I've created a simple analogy to understand this. Suppose there is a small village where there are N residents who trade only in barter. Now, you introduce a new bank in the village called Banko and a currency called pebbles. (For simplicity, lets assume that Banko is both a central bank and also lends/borrows money).

Now in order to leave barter and migrate to currency system, the villagers will have to either sell their goods to Banko, or borrow pebbles from them. So, there is a "give and take" or exchange of pebbles against goods.

But at the end of the day, if you consider the overall macro-economic position, the pebbles made no difference in value at all.  The villager's NET ENDOWMENT is same as it was before, the pebbles just facilitate transactions and used to count their assets now.

So, my question is, if money doesn't make any difference to the REAL value of goods and services, how can it be considered an asset?

What is value? Ultimately, value flows from the utility people get from consuming goods and services.

Now, suppose, in your example world, I am a farmer and it is Autumn so I have the product of a large harvest. Here are three things I could do with it:

Consume (and thereby enjoy the value of) my entire harvest immediately. But this would leave me hungry in the winter.
Store some food for later: Put some of the grain I produced into a silo and take it out of the silo to eat in the winter.
Sell some of my food to the bank in return for pebbles. Then, later in the winter, use those pebbles to buy back some food when my supply has run out.

Notice that points 2 and 3 are essentially equivalent (they result in the same distribution of consumption across time). In the former, I am directly storing food. In the latter, I am using the pebbles to store the value associated with the food until I want to claim that value back.

Although the store of value is really just an abstract implementation of the physical storage of the goods generating that value, this distinction becomes important in more sophisticated economies. Storing value rather than storing the goods that generate that value is attractive for a few reasons.

Many goods and services are hard to store. For example, as an economics professor it is impossible for me to "store" lectures during the autumn for use in the summer when I can't find any students to teach. How do I avoid starving during these long, studentless summer months? I take the (hopefully positive) value my lectures produce during the autumn and store this value in money. I can then claim this value back during the summer when I need it.
Storage of physical stuff takes up a lot of space and is expensive. Storage of money is much less so. Similarly, transporting stuff is expensive. With a bank card it becomes unnecessary to transport money at all.
Storage is often wasteful because it involves stuff sitting around doing nothing when it could be generating utility for someone.


Multi-product profit maximization I was just thinking about the question

  How does a firm choose the optimal input vector as to maximize profits when the firm produces more than one final product

More precisely if $\mathbb{R}_{\geq 0}^{\ell}=\{x \in  \mathbb{R}^{\ell} : x \geq 0\}$, the production function would be
$$
f:\mathbb{R}_{\geq 0}^{\ell}\rightarrow \mathbb{R}_{\geq 0}^{n}, \quad n\geq 2
$$
For simplicity we can assume the firm operates in a competitive market so that the price is given.

My question is, how does the firm choose the input vector $x \in \mathbb{R}_{\geq 0}^{\ell}$ as to maximize profits?

I did some research and apparently the type of problem this is a “multi-objective” optimization problem, and it appears to be an active field in engineering. Some of the methods to solve this kind of problems use pareto-dominance style arguments.

I wanted to know if anyone could clarify the specific question of multi product profit maximization, and more generally I would also appreciate any literature or books on this topic (multi objective optimization), written for economists, as I have been searching in google and every book I find appears to be written for engineers.

I suspect that a treatment of this topic from an economics point of view would involve some discussion of social choice theory.

Any help is most welcomed!

At the end of the day, the firm seeks to maximize profits, which is a one-dimensional object. So the firm simply solves
\begin{equation}
\max_{\mathbf x}\;\mathbf{p}\cdot\mathbf f(\mathbf x)-c(\mathbf x)
\end{equation}
where $\mathbf x\in\mathbb R_+^\ell$ is a vector of $\ell$ inputs, $\mathbf p\in\mathbb R_+^n$ is a vector of $n$ prices for the final output, $\mathbf f:\mathbb R_+^\ell\to\mathbb R_+^n$ is the production function mapping $\ell$ inputs to $n$ outputs, and $c:\mathbb R_+^\ell\to \mathbb R_+$ is the cost function. This is a standard unconstrained optimization exercise that any first year grad student in economics should know how to solve. The standard textbooks --- MWG, Jehle and Reny, Simon and Blume --- have sections that cover the techniques.


Why is bartering uncommon in modern countries? Recently, I witnessed two acquaintances of mine engaging in bartering, where one traded his services for the services of the other. Both services had about the same value. By trading them directly, they didn't pay VAT or tax on income, as no money or even goods were involved here. That wasn't the reason for the barter, but it's an interesting aspect.

I'm not pushing this behaviour, but it makes me wonder why there's not more people doing this, as it is an easy way to save money by leaving out money and thus taxes. AFAIK, this is legal, or at least a 'grey area'. I suppose in most countries some law exists to prevent this happening on a large scale, but it would be impossible to enforce it everywhere or on a small scale. Especially since goods can still be valued in terms of money, but it is much harder to do so with services. And their presence and traceability is volatile, unlike with goods.

I live in a rather small village where locals often (have to) make use of each others service, often requiring the service more than once. Taxes and VAT are high here in the Netherlands, so it would make a significant difference. I'd expect them to barter the service to save money, but they just pay by the bill. I did witness two people bartering once, but never before. I haven't heard it to be common in other modern countries either. Why is it not more common?

The main likely reasons why barter is not more common are:

The inconvenience of having to find another party who both offers what you want and wants what you offer.
Even if such a party can be found, the possible complexity of negotiating a "fair" transaction (eg I'll do your electrical job if you'll clean my windows monthly for the next 3 months).
I don't want to encourage it, but for those seeking to evade tax, the comparative ease in many circumstances of doing so via a normal monetary transaction, but with payment in cash.

A further point is that most countries with VAT have a significant "VAT threshold", that is, a minimum value of annual sales at which businesses are required to register for VAT and charge VAT (where applicable) on their sales.  In the UK, for example, the threshold is currently £85,000. So an individual could provide services not exceeding that amount in any year and would not have to charge VAT.  If two such individuals were to enter into a barter arrangement, there would be no VAT saving. The Netherlands is atypical in that its VAT threshold is unusually low: 1,345 Euros (see here, table on p 2).