Do the donated NFL losing team shirts destroy local economies? I've recently become aware of the fact that the NFL prints both championship teams' shirts in advance to satiate incredible demand -- but instead of destroying the losing teams' shirts like they used to, they instead donate them to developing countries. But the thought of flooding these local economies with free shirts, which doesn't create jobs, GDP, and hurts local clothing business, seems like it would be terrible for the economy. Is this indeed the case?

My answer is no, it doesn't hurt the local economy. This seems like an example of the "Broken Window Fallacy":

Suppose, when nobody is looking, I break all of the windows in my neighbour's house. This creates a bunch of work for a glazier who might otherwise have been idle. So it seems like, by going round smashing things up we can provide a boost to the economy. Surely this must be wrong!?

Indeed, the problem with this logic is that we have to think what would happen if I had not broken the windows. My neighbour, not having to incur the expense of replacing his windows, would have some extra money to spend on a hair cut or a new bicycle, creating work for a barber or a bicycle maker. If the windows are broken then those guys miss out.

So the glazier earns some extra income and gains; the barber earns less income and looses. But these two things "cancel out": this is just a transfer of wealth from one person to another rather than the creation of new wealth. My neighbour, on the other hand, definitely loses because he doesn't get a hair cut any more. So overall, the broken windows are a net loss to society.

The reason I think this is relevant to your question is that we could reframe it as: "if we burnt the shirts [broke the windows] instead of giving them away for free, wouldn't that create work for a tailor and therefore make society richer?"

I think the answer is no because, freed from the need to buy a shirt, people have more income to spend on food, educating their child, taking care of their health, etc. All of these things create jobs in the local economy to replace the lost jobs in the clothing industry.

It is true that the tailors lose out at the expense of the farmer, teacher, or nurse. But over long time horizons people will adjust to this by training to do the jobs that pay.

tldr; Destroying real resources may create winners and losers, but is almost always bad for society overall.


performance as utility function / price? I am trying to define a 'model' for the perceived performance of a product based on this definition:

  Presumably, customers take both price and quality
  into account as they form an overall evaluation about
  a product's performance. To avoid a confounding of
  the two, each was measured in light of the other—
  perceived performance is thus measured by price (given
  quality) and quality (given price). (Fornell, 1992)

I thought that a utility function might be a good way to describe the quality of a product, so ended up with:

$$P(x) = \frac{u(x)}{c}$$

Where the perceived performance P of a product x is equal to its utility function u(x) divided by its total price c.
Is this more or less ok? Or should I include the price variable in a different way, such as e.g.:

$$ P(x) = u(x,c)$$

...or even simpler

$$ P(x) = u(x)$$

As cost might be assumed to be part of the trade-offs of the function? Any ideas or help with the notation or function would be really appreciated.

There is a long tradition in economics (some 50 years worth of research) called "Hedonic Price Analysis".

It consists of "allocating" (through regression estimation) the price of a product to its various qualitative features ("characteristics") quantified, that presumably are valued by the consumers, i.e. those that are utility-enhancing. These characteristics are either quantified (e.g. Computer RAM measured in GB), or "1/0" dummy variables (the characteristic exists/it doesn't). Note that in the related empirical studies actual transacted prices are used, i.e. prices that the consumers have accepted to pay.

The estimated coefficients from the regression provide the monetized "willingness to pay" per unit of characteristic (or per existence of it). Once you have them, you can extrapolate outside the sample to find the total "willingness to pay" for a product $i$, say $W_i$, by multiplying its characteristics by the marginal willingness to pay for each characteristic.

In essence, in this way you transformed something that is considered ordinal and un-measurable (utility) into a monetized equivalent (using the "revealed preference" approach, since you used prices that the consumers actually paid).

If you then compare this magnitude to some advertised price of this product, $P_{ai}$, then a ratio like

$$\text{Prf}_i = \frac {W_i}{P_{ai}}$$

could be your "Performance" index. If for example it is higher than unity, it means that the product "is a good bargain" for the consumers, because the monetary value of the utility they will get out of it, is higher than the money they will have to pay to acquire it.

NOTE: Hedonic Price Analysis is not just something that academic economists do. In many nations, official Price Indexes are for several years now adjusted for quality, using the methods developed by economists throughout the years.


What is the link between econometric theory and applied work? I just don't understand how the theory, particularly pertaining to parameter identification is relevant when conducting applied work. Why must we present a sound identification strategy when carrying out an applied project.I hope this is clear. Thank you.

Edit: I guess I wasn't very clear in my question. I understand how identification strategies work in applied papers. I was more interested in how, say, somebody introduces a new estimation method in an academic paper. Then they propose the conditions on which the estimated parameters are identifies using this method. There is usually some talk of the dimensions of some parameters, and technical things like that. I just wanted to know why these kinds of concerns are relevant when one wants to apply said estimator empirically.

I am not exactly sure what you're asking. So...I hope we both understand identification strategy to mean the same thing. If so:

The identification strategy serves as the bedrock upon which you build your argument.

For example, consider some policy implemented by a school district that focuses on improving academic outcomes for low performing students. Say this school district sets a cutoff of a 1.5 GPA. Any student at or below this GPA is placed on academic probation. This assignment to AP triggers a set of responses designed to increase a student's academic success. However, these responses are time consuming. Now, suppose at the same time that a neighboring school district with a similar GPA distribution does not implement this policy.Suppose students can move freely between the two districts.

There are several ways to analyze the efficacy of such a policy. I will consider two.

Assume researcher A decides to investigate the efficacy of the policy by simply conducting some simple trend analysis. Let's suppose he decides that it is reasonable to determine the efficacy of the policy by evaluating the amount of students with a "low" GPA both before and after the policy implementation. This seems like it could serve as an intuitively sound approach.

Let's assume now that when the school implements this academic probation policy that half the students placed on academic probation leave and begin attending the neighboring school to avoid the time demands associated with the academic probation.

The trend analysis by researcher A likely now overestimate the efficacy of the academic probation program. This is because there will be a large decrease in the number of students with a low GPA. However, this will be largely due to those students leaving the school and not because the policy helped raise their GPAs.

Suppose researcher B uses a difference-in-difference approach using the neighboring school district that does not implement the academic probation policy as a counterfactual (where the similar trends assumption holds). His identifying assumption and research strategy will allow him to identify and control for the change in composition brought about by the policy implementation.


Your errors aren't the same anymore.  For example, instead of writing $Y = \beta_1 + \beta_2 X + U$, you're actually writing  $Y = \alpha_1 + \alpha_2 X + \alpha_3 X^2 + V$.  There is no expectation that they should be the same.

In other areas, the trouble is that the error term is likely correlated with your regressors.

Fear not: running wages on education is a common specification.  Searching for something like

  wages education endogeneity

should bring up a whole slew of papers that treat the topic.


Expected utility theory (Lottery notation) A wheel of fortune has outcomes $S=\left \{ 1000,100,50,20,0 \right \}$ as money prices. A consumer has the preferences
$$20\sim \left ( \frac{2}{100}\cdot1000 \oplus \frac{98}{100} \cdot 0 \right )$$
$$50\sim \left ( \frac{15}{100}\cdot1000 \oplus \frac{85}{100} \cdot 0 \right )$$
$$100\sim \left ( \frac{45}{100}\cdot1000 \oplus \frac{55}{100} \cdot 0 \right )$$
What does this notation mean? I figure that the direct sum is just showing that the percentage for winning $1000$ in the first equation is $2/100$ and $98/100$ for $0$. But what does the $20$ denote? Is this some utility and how would that be read with the indifference sign in front of it? Never seen the notation like this before.

$$
\left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right)
$$
is the lottery where you get $1000$ with probability $2/100$ and $0$ with probability $98/100$.
The expression
$$
20 \sim \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right)
$$
usually says that the decision maker is indifferent (in terms of preferences) between taking the lottery and having $20$ for sure.


A maximization program expressed by per capita variales - Ramsey model I have a problem to be maximized with the following program

$$max\int_{0}^{\infty}\left\{ u\left(C\right)+\beta S^{\chi}\right\} e^{-\rho t}dt$$
s.t

$$\dot{K}=AK-C$$
$$\dot{S}=\left(1-S\right)S-\gamma AK$$

where $K$ is physical capital $S$ is the natural capital. All capital letters represent aggregate variables. The economy enjoys the natural capital as a positive amenity and the natural capital is negatively affected by the physical capital accumulation by a rate of $\gamma$.

I am trying to express this program by variables per capita since I would like to have constant steady state values. Note that the population grows at rate $n$. ($L(t)=L(0)e^{nt}$ with $L(0)=1$) It is easy for physical capital.

I will have something as

$$\dot{k}=Ak-nk-c$$

However, for the natural capital, the logistic growth of the natural capital and the amenity $S^{\chi}$ complicate the issue.

Is there a way to express all the maximization program by variables per capita? I would really appreciate if you have some hints to give or some solutions. Or if it is not possible to express by per capita variables, I would be happy to know the reason. Thanks!

Take the differential equation for the natural resource. I will write $S \equiv sL$ where lowercase is the per capita magnitude. So

$$\dot{S}=\left(1-S\right)S-\gamma AK$$

$$\implies \frac {d(sL)}{dt} = (1-sL)sL - \gamma AkL$$

$$\implies \dot sL + s\dot L = sL - s^2L^2 -\gamma AkL$$

Divide by $L$ to get

$$ \dot s + sn = s - s^2L -\gamma Ak$$

and using $L=e^{nt}$ and re-arranging we get

$$ \dot s = (1-n) s - s^2e^{nt} -\gamma Ak$$

Setting this equal to zero, we have a quadratic polynomial in $s$

$$\dot s = 0 \implies e^{nt}s^2 - (1-n) s + \gamma Ak=0$$

We see that the leading term increases as $t$ increases. The only way to maintain equality with zero, and constant $s$, is if $k$ decreases as time passes (and this will be only temporary since $k$ cannot turn negative).

So it appears that you cannot have a "steady state in per capita terms" for all variables here, consumption, capital and the natural resource.
Most probably you can maintain consumption and capital at constant steady state values while the per capita, at least, natural resource depletes to extinction (but you have to show that this is optimal with respect to intertemporal utility maximization).


Two Players Different Strategies in infinitely repeated game I am vexed by a Game Theory question regarding infinitely repeated games. Usually, strategies such as Grim Trigger or Tit-for-Tat are discussed in a context where both players have the same strategy.

However, I am wondering how to proceed when two players play different strategies, say Player 1 plays Grim Trigger and Player 2 plays Tit-for-tat.

My question is whether there exists any $\delta\in(0,1)$ for which a profile of {Grim Trigger, Tit-for-tat} constitutes an equilibrium.

The stage game payoffs are as follows:

\begin{array}{|c|c|c|}
\hline
& C & D \\\hline
C & 3,3 & 0,4 \\\hline
D & 4,0 & 2,2 \\\hline
\end{array}

Just so that we have a common understanding of the strategies you mentioned:

  Grim trigger: "Play $C$ unless the outcome of any previous stage is not $(C,C)$; in that case, play $D$."

  Tit-for-tat: "Start by playing $C$, and then play the action used by the other player in the previous stage."

Therefore, on equilibrium path, the outcome $(C,C)$ will be observed, and each player gets a discounted payoff of
\begin{equation}
3+3\delta+3\delta^2+\cdots=\frac{3}{1-\delta}.\tag{1}
\end{equation}

Suppose player 1 who adopts grim trigger deviates in the first stage only. The outcome in stage 1 would be $(D,C)$, and the outcome in all subsequent stages would be $(D,D)$ according to the strategy profile. Thus, player 1's payoff from the deviation would be
\begin{equation}
4+2\delta+2\delta^2+\cdots=4+\frac{2\delta}{1-\delta}.\tag{2}
\end{equation}
$(1)$ and $(2)$ imply that a one-stage deviation is not profitable for player 1 if $\delta\ge\frac12$.

Suppose player 2, who adopts tit-for-tat, deviates in the first stage only. The outcome in stage 1 would be $(C,D)$, and $(D,C)$ in stage 2, $(D,D)$ in all subsequent stages. Player's payoff from the deviation would be
\begin{equation}
4+0\delta+2\delta^2+2\delta^3+\cdots=4+\frac{2\delta^2}{1-\delta}.\tag{3}
\end{equation}
$(1)$ and $(3)$ imply that a one-stage deviation is not profitable for player 2 if $\delta\ge 1-\frac{\sqrt 2}{2}$.

Note that $\frac12>1-\frac{\sqrt2}2$. Hence, as long as $\delta\ge\frac12$, player 1 using grim trigger and player 2 using tit-for-tat can be sustained as a Nash equilibrium of the infinitely repeated prisoner's dilemma. However, that the strategy profile is not a subgame perfect equilibrium (for a similar reason why a symmetric tit-for-tat strategy profile is not subgame perfect).


What is the formula for how many more I need to sell in order to justify raising cost? I need to know the name and formula for this problem.

I have widgets which have a certain lube content to them.  We currently are selling the widgets for 3.00 each.  The lube costs .50 cents for each widget, but we are considering a change in the lube content from the current state of 1 oz.  We currently are making sales of 1000 widgets per year.  I want to know how many more widgets we would have to sell in order to justify a given increase in the lube content of a widget.  I am not asking how many more we would sell if we increased the lube content or what the revenue would actually be.  I am merely asking what the number of additional widgets sold must be in order to justify a given lube content increase.

Price of widgets is constant.

What is the name of the formula I am seeking?  What is the formula I am seeking?  What is the answer?

This is an algebra problem. My answer is quite similar to snorams, but without the numbers plugged in (which hopefully makes it a bit easier to see what is going on). Let

$p$ be the price of a widget
$c_L$ be the old cost for producing one widget (i.e. with a low amount of lube)
$c_H$ be the new cost for producing one widget (i.e. including the high amount of lube)
$n_L$ be the number of units sold when you use a low amount of lube
$n_H$ the number of units solve when you use a high amount of lube.

With a low amount of lube, your profit is $(p-c_L)n_L$ (in words: you earn a margin of $p-c_L$ on each unit, and sell $n_L$ units).

With a high amount of lube, your profit is $(p-c_H)n_H$.

You would like to know when is it true that the higher lube content is more profitable, i.e.

$$(p-c_H)n_H\geq(p-c_L)n_L$$

Now, to find the number of units, $n_H$, at which the change results in break even, we can simply solve the inequality:

$$n_H\geq\frac{(p-c_L) n_L}{(p-c_H)}$$

Let's look at an example:

price of a widget is $p=3$.
unit costs of everything but the lube are $1$.
cost of low amount of lube is $0.5$
cost for high amount of lube is $0.6$
low-lube sales are $1000$.

We calculate the unit cost as $c_L=1+0.5=1.5$ and $c_H=1+0.6=1.6$.

Now we plug these numbers into the formula

$$n_H\geq\frac{(p-c_L) n_L}{(p-c_H)}$$

$$n_H\geq\frac{(3-1.5) 1000}{(3-1.6)}$$

$$n_H\geq\frac{1500}{(1.4)}$$

$$n_H\geq1071.42.$$

So if you sold at least 1072 units (i.e. an extra 72 units) then the change would pay for itself.


Statistical Learning and Econometrics The principal goal of Econometrics is to find causal effects between economic variables. But the growing technology level seems to show us that Big Data and Statistical Learning result in a tradeoff with econometrics, because of according to some professors (and professionals),  the Statistics field, once is applied in economics, must be guide by economic theory because its alternatives find  correlations (Not inference).

Given the above, is it possible to find causal effects between economic variables with Statistical Learning methods as alternatives to Econometrics?

Thanks!

(Pardon forerrors, I'm not a native English speaker)

As @markleeds suggested, Judea Pearl made a significant contribution to the field of Causality.

His work (though not innovative) explains that causality cannot be represented by joint probabilities. In fact, even econometric models can't model causality.

So, we have to use Rubin's Causal model, Granger Causality, Difference-in-Differences or whatever, to explore causality.

So, the answer to your question is yes and no. We can use statistical learning techniques in combination with causal frameworks, but they alone can't provide causality rigorously.

For example, I came up with a very simple solution to use special impulse-responses (in my case, one-hot vectors like $(1,0,0), (0,1,0), (0,0,1)$) to clear out the partial effects of input variables to the outcome in Neural Nets. Link here: http://ravshansk.com/articles/irf-ann

But all of these are just hacks and don't quite have a rigor. Judea Pearl argues that we must have a new notation for causality or use graphical models whatsoever.


Empirical explanation on divergence between productivity and wages since the 70s There is a well know graph, that shows a tight link between productivity and wages, until the 70s, and a break of the link since:

There's a lot of possible explanations for this, I keep hearing more of them:

Tax cuts to corporations, that changed the profits distribution
detachment from the gold standard
Rise in global trade and the consequential drop in the US industry power in the economy
The loss of power of labour unions

There is a good article here

I actually thought of a possible reason myself: the decrease of competitiveness of the labor market as a result of mergers and acquisitions or the drop in power of labour unions, as said

High wages in today's hi tech industry can support this explanation in the opposite way

Is there any empirical evidence to support any of these theories?

This is sort of my field, tough I focus more on education. One explanation that seems to fit the data is that the american labor market now is incorporating benefits and variable pay, so wages should decouple from productivity if total compensation doesn't - and voi lá, that's whats happening.

  One reason, perhaps, is that the character of the productivity acceleration
  changed circa 2000. Prior to that date, studies have suggested
  that the more important effect was an increasing ratio of capital
  to labor (capital deepening) as businesses substituted relatively
  less expensive information technology and communication equipment
  for labor. Since 2000, some studies suggest that the more
  important factor has been a re-engineering of business practices,
  which has increased the “skill bias” in the labor market, that is,
  the premium paid for higher levels of technical, professional, and
  managerial education and experience. For employers, variable pay
  solves, in part, the problem of monitoring the performance of such
  workers: Unlike traditional factory and retail service workers,
  whose hours at work are relatively easily monitored, many skilled
  professionals work at varied locations and times of day. Finally,
  increasing reliance on variable pay also perhaps is a type of risksharing
  arrangement between businesses and workers. If the recent
  productivity trend slows, reducing or curtailing variable pay may
  be less offensive to professional workers than reductions in base
  salaries.


What is the classic paper that derives borrowing constraints from asymmetric information? In reading the Wikipedia article about the "financial accelerator," I read this

  Firms’ ability to borrow depends essentially on the market value of
  their net worth. The reason for this is the familiar story of
  asymmetric information between lenders and borrowers. Lenders are
  likely to have little information about the reliability of any given
  borrower. As such, they usually require borrowers to set forth their
  ability to repay, often in the form of collateralized assets.

So, I believe that models with a financial accelerator, such as Kiyotaki and Moore (1997), assume a credit constraint. I am looking form papers that derive these credit constraints. That is, where do they come from? Could someone provide the reference?

Papers that I can think of are, for example, Hart and Moore (1994), which derives constraints that are driven by asymmetric information and the inalienability of human capital, or Myers and Rajan (1998), which derives debt capacity from inefficient liquidation and the ability of the manager to engage in asset substitution. What are the classic papers on this topic?

I don't know if you refer to the extensive margin (some borrowers not being able to get credit) or to the intensive margin (one borrower not being able to get as much credit as (s)he wants). If you are referring to the former, one of the theoretical papers for borrowing constraints on markets with asymmetric information is the following one:

Stiglitz and Weiss (1981) AER's paper on Credit Rationing in Markets with Imperfect Information.
http://www.jstor.org/stable/1802787?seq=1#page_scan_tab_contents

The argument is very interesting. If lenders cannot distinguish between high risk and low risk borrowers, then the ''market-clearing'' (loosely speaking) equilibrium interest rates will be such that identical (to the observer) consumers can be rationed out.

Other good papers are Bester's (1985) AER's paper on Screening vs. Rationing in Credit Markets with Imperfect Information. http://www.jstor.org/stable/1821362?seq=1#page_scan_tab_contents

Finally Chapter 16 on the Handbook of Monetary Economics by Jaffee and Stiglitz may help you.


Lucas Critique and RE: Did one arrive before the other? Hi All: I have no formal background in economics but I've been teaching myself RE for the last 3 or so years when I get the chance. I know the history and parts of the literature pretty well but it's clearly a vast field so some things are still quite fuzzy. The following question may help me with a confusion I've had for a while but am only asking to bother now. ( fortunate to find this list. Didn't know about it early on when I started in RE ).

The Lucas Critique was in 1976 and gives examples to show that the standard and well known keynesian approach to econometrics is  not terribly useful from the standpoint of policy. Once a policy changes, expectations can change and keynesian econometrics didn't handle that.

My question is: Did the Lucas Critique

A) represent the start of what is termed the RE revolution ( in order to combat the critique )

OR

B)  was RE already in place and well known  before the Lucas Critique ?

Basically, I'm trying to understand whether the Lucas Critique could have had the title

"Keynesian type econometric models need to be replaced by a new econometric technique called RE".

I am aware that Muth actually came up with the basis for what later became RE
but I'm not counting that as the start, atleast for the moment. ( no disrespect to Muth ).  I'm asking whether was RE was already in play say
after 1970 but before the Lucas Critique ? It seems like it was as far my
reading goes ( Lucas has famous RE type papers from 1972 ) so, then I'm wondering why was the Lucas Critique such a big deal in that case ? Didn't everyone know it already since RE was already around. Thanks for any insights-references etc.

                                                      Mark

While this probably doesn't fully answer your question, at least it might be able to serve as a start.

I think the biggest conceptual difficulty here is regarding what exactly the heart of the Lucas critique is, and why it was so influential. Before he published his seminal 1976 work a large body of macroeconomists were primarily engaged in attempting to investigate policy analysis by estimating fundamental parameters of a model (at the time, since Keynsian analysis was in vogue, it generally focused on things like the marginal propensity to consume or save, or estimating consumption functions for use in policy counterfactuals, etc.). How this was done was by using aggregated historical economic data to try and back out the parameters of interest.

The Lucas Critique points out that this effort cannot be guaranteed to be internally consistent. That is, all of the observed, aggregate historical data being used was actually the output of a complex system. When considering how a change in policy might impact welfare, the economist cannot assume that these parameters will remain unchanged.

Critically, this is true independent of whether the model is using rational expectations. No matter the macro modeling assumptions, the internal optimization that the agents are "doing" inside the model must be grounded in actual fundamentals of economic theory. That is to say, there's no theoretical reason why one consumption function should be used by an individual over any other (for example). Historical data, then, can only give insights into what the optimal decision making behavior was at the time it was observed.

There's actually a pretty good example on the Wikipedia page for the Lucas Critique in the last section (behavior surrounding Fort Knox). Rational expectations would suggest that individuals are maximizing some, known objective function for whether they should try and rob the vault or not. Given that we don't see many attempts, we can then say that, given individuals are following rational expectations and are maximizing some consistent utility function, in general, the perceived costs outweigh the benefits. The Lucas Critique, on the other hand, says that we cannot use the fact that we don't observe many attempts at robbery now to suggest that Fort Knox spend less on security. While RE theory says people aren't robbing because the costs outweigh the benefits, the Lucas Critique says we cannot use the fact that we don't observe many arobbery attempts at this level of security to say anything about how robberies will change if we change security levels. All we know is that, in this particular set of circumstances, we observe some equilibrium behavior (few robberies). To make policy statements, we have to base our models on some fundamental theory- one of the most prominent being a form of RE. But ultimately, whether we believe RE holds or not, the Lucas Critique still points out that we must base our predictions in some theoretical foundation, and not only use aggregated outcome data.

Sorry that was kind of long winded and sloppy- I'll try to tidy it up later if it would help!

EDIT:

  Is it fair to say that the LC is a criticism of keynesian models because they didn't include expectations of any kind rather than just not rational ones?

Broadly, that's exactly it. Keynseian models at the time needed lots of aggregated parameters (again, like marginal propensities, etc.) to come up with conclusions on how different policy changes, or economic shocks, might impact the economy (or industry, or group of people, etc.). The problem is, Keynsian analysis (again, at the time) requires these parameters without any theoretical context. Why should the marginal propensity to consume be X? What would cause that to change? These questions require the microfoundations that we've seen spring up in the macro field- models that attempt to examine what determines important macro parameters, and how changes to those underlying causes will change the parameters of interest. Not only did Keynsian models not include expectations (specifically forward looking variables), they tended not to include any theoretical basis for the parameters they estimated.

  And, as long as the included expectations can be connected to either A) the policy rule or B) the underlying stochastic process ( if there is no specific policy rule sometimes they seem to do this), the model is then internally consistent.

While these are certainly two good microfoundations that an economist can use to build a macro model, model consistency (again, using the broadest possible definition) can rely on an even weaker basis. For example, you could have a model of the macro labor market where labor supply is completely divorced from any sort of rational or forward looking behavior at all. You could base your model on something like:

Every period, all agents wake up and flip a fair coin. If the coin is heads, they go to work. If it's tails, they go back to sleep until the next period.

Of course, this premise is completely divorced from reality- coin flipping has nothing to do with the decision of whether to go to work. But it would at least provide a clear theoretical basis (even if absurd) for why the labor market behaves the way it does. Instead of just observing "last period, the labor market participation rate was 60%, thus, we should take that as some underlying parameter" (like the classical Keynsian methodology might), you'd say "I see last period there must have been an inordinate number of coins that landed 'heads.'" Your theory would give you a specific reason for believing the participation rate was 60% (the proportion of heads), a prediction for what it would be next period (50%, since that's the expected number of heads you'd expect to see next period), and a reason ** why ** it should be 50% (again, because everyone's flipping coins).

Moreover, using the coin-flip model of labor markets, you could consider how a policy change should affect behaviour. How would a doubling of wages change turnout? In this case, we'd predict it wouldn't have any impact (still doesn't impact the coin probabilities). And while that might seem absurd, at least you can very clearly explain why there will be no change, using some economic reasoning. The old Keynsian models couldn't provide these types of explanations, other than "because the last time wages doubled, X happened to participation." While that might be nice to know, it cannot be considered a useful answer, since there might have been any number of other things causing the wage change to happen the last time, that might not be at play now (and which might have been even more important in driving the participation changes).

Rational expectations, then, is just one useful, generalizable, easily applicable mechanism for how to bring microfoundations into these macro models.

Again, hope that helps!! Let me know if I can clear anything up!!


When can one drop time subscripts? Example from Angrist and Kugler (2003) Not the first time I am asking myself, but in this paper they actually start with a time dependent maximisation problem and then drop all time subscripts.
Background: They have profit maximisation problem that reads
$$
\Pi = \phi^{t} \sum_{i=0}^\infty \left \{f[\theta_{g}(N_t,I_t)] - w_{Nt}N_t - w_{It}I_{t}-\lambda C_N N_{t-1}\right\}
$$
where $\Pi$ is profits, $\phi^{t}$ is discount factor, $f[\theta_{g}(N_t,I_t)$ is output, $N_t$ and $I_t$ is different type of labour, $w_{Nt}$ and $w_{It}$ the respective wages, and $C_N$ a firing cost for the $\lambda$ workers that are replaced in each period.
Steps: They write "Adjustment costs are linear and there is no aggregate uncertainty, so time subscripts can be dropped and the objective simplified to be:"
$$
\Pi = (1-\phi)^{-1} [f(\theta_{g}) - w_{N}N - w_{I}I- \phi \lambda C_N N ]
$$
Question: I only knew this step with the Bellmann equation were you simplify the infinite horizon problem into basically a timeless problem of finding the optimal policy function, but here it seems to be literally just dropping the dimension like that. Does anybody know more generally when one can simplify a problem like this? This question interests me also more generally because I see many papers writing basically immediately version 2 of the problem and not considering optimizing the sum of the profits through time.
Reference: Angrist, J. D., & Kugler, A. D. (2003). Protective or counter‐productive? labour market institutions and the effect of immigration one unatives. The Economic Journal, 113(488), F302-F331.

Because "adjustment costs are linear and there is no aggregate uncertainty", the FOC for $N_t$ is $$f'\theta g_N(N_{t}, I_{t}) - w_N = \phi \lambda C_N$$. Notice that this is exactly the same form for each period. The same is for $I_t$. This means that a firm will choose the same labor inputs in all periods. In other words, the firm gets into the steady state immediately in this dynamic optimization problem. This is not the case when there is aggregate uncertainty so that $w$s can vary over time or the adjustment costs are nonlinear, e.g. $C_N(N_t-N_{t-1}).$
As a result, the profit at each period is constant. And thus the present value of the discounted profits is $$\Pi= \frac{\left[f\left(\theta_{g}\right)-w_{N} N-w_{I} I-\phi \lambda C_{N} N\right]}{1-\phi}$$.


What is Money Market Rate? I am looking at the imf interest rate data page for Sweden right now. There are two different interest rate indicators named "Central Bank Policy Rate" and "Money Market Rate". I know what policy rate is but I have no idea about money market rate. Their monthly values are really similar but not exactly the same. What is "money market rate"? If I would have used money market rate instead of policy rate to compare different interest rates of different countries would that be sound?

The money market rate is the interest rate at money market which is market where highly liquid assets are traded (see more about this at Investopedia). It is called money market because in economics very liquid assets can also be considered to be 'broad money.'
The rate at money market rate and central bank policy rate will be similar because the policy rate influences all other interest rates in economy and has very large influence on money market since the price at which banks can borrow from central bank will directly affect the price of funds at money market.
Regarding your question:

If I would have used money market rate instead of policy rate to compare different interest rates of different countries would that be sound?

Well that depends on what is your research question. Depending on specific of your research using central bank interest rate might be more appropriate or less appropriate or you might consider using one as a proxy for other.
For example, in some research about transmission of monetary policy it will be important to distinguish between them because money market interest rate is one of the channels through which the central bank's policy (set with its policy rate) is transmitted to economy (e.g. like in Abbassi & Linzert, 2012).
In some other research where this transmission mechanism is of no interest you might use as a simplification only money market rate, or vice versa skip the money market rate and use policy rate directly as money market rate will be some function of policy rate.
Whether such simplification is justified is research specific. For example, in economics risk free rate and government bonds rate are completely distinct rates but if you can justify saying that it is virtually impossible for Germany to default so its bond rate will have very little risk and will be almost as good as risk free rate then you can use it in a research instead of risk free rate. However, while that justification might fly now, it would be dismissed in first half of 20centruy when Germany had large default risk.  The point is that although they are not same you can depending on context justify one as approximation of other (science is in the end all based on approximation - even in physics you don't deal with exact results and infinite precision).


Set Theory Properties of the Budget Constraint In Microeconomic theory, the budget constraint is defined by 4 distinct properties:
It is

Bounded
Closed
Convex
Non-empty

The 1. 2. and 4. are very straight forward and the benefits in terms of analysis are apparent. My question is with respect to convexity.

Convexity of a set implies that any line between two points is the set is also within the set (i.e. take any two points say $q$ and $q'$ that both satisfy the constraint, then any linear combination of the $q$ and $q'$ also satisfy the constraint)

My question is, what is the benefit or added utility to this property holding?
I immediately think of a budget constraint that, rather than exhibiting diminishing marginal returns, exhibits increases returns in a kind of learning sense: As more of one good is produced, the production becomes more efficient as the producer is learning. Whether or not this is a realistic circumstance, it begs the question.

What happens if convexity is violated? or rather, what can we no longer do?

Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier.

Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed here and here.

Here is also a short, concrete example of something for which we need convexity of the budget set:

Suppose you want to prove that Marshallian Demand $x(p,m)$ is single-valued for every (larger than zero) price ($p$) and (non-negative) income ($m$) combination. That is to say, a Marshallian Demand Function exists, since without single-valuedness we would have a Marshallian Demand Correspondence (i.e. we'd have multiple good combinations as part of the optimal demand choice for a single given price and income level), which can be harder to work with. This is a pretty nice result that we (almost) always work with in undergraduate texts. For that you need strictly convex preferences and a convex budget set $B(p,m)$. The proof is as follows:

If two points $x_1$ and $x_2$ are both elements of the demand $x(p,m)$ at given prices $p$ and income $m$, i.e. if $x_1, x_2 \in x(p,m)$, then by definition we must have indifference between the two points, that is $x_1 \sim x_2$. For a single-valued demand function two different points $x_1$ and $x_2$ cannot exist  for single given level of prices and income, i.e. they would have to be the same point. In other words  $x_1 = x_2$. We prove it by contradiction. Suppose $x_1 \neq x_2$.

For all $\lambda \in (0,1)$, we have $\lambda *x_1 + (1-\lambda) *x_2 \in B(p,m)$ by convexity of $B$. By strict convexity of preferences we have $\lambda *x_1 + (1-\lambda) *x_2 \succ x_1$. However, this contradicts the fact that $x_1 \in x(p,m)$ in the first place. Hence, $x_1 = x_2$.

So, in words, some combination of $x_1$ and $x_2$ is better than just $x_1$ at the given price and income level. But such a combination is also attainable at that given price and income level because of the convexity of the budget set. This is where the budget set convexity assumption is important. Furthermore, if for a given price and income level there is a point (in this case a combination of $x_1$ and $x_2$) that is both strictly better and attainable, then this contradicts the original starting point that $x_1$ is a point in the demand function for the given price and income level.

As a side note, I think it is a bit misleading to talk about producers in your post when talking about budget constraints. If we are looking at a budget constraint, we should be discussing consumers.


Can a country survive and prosper without international trade or investment from overseas? To what extent can a country be 100% independent from the rest of the world? Is there any country like that in the world? Maybe North Korea, but North Korea receives supports from China.

Besides allowing countries to take advantage of comparative advantage, as mentioned, an open economy can engage in intertemporal trade.

This possibility is useful because it allows a country to smooth consumption over time, even when domestic output fluctuates. For instance, in good times the economy can save (consume less than actual output, lending to foreign countries), allowing the economy to consume more than actual output in bad times.
This in fact is welfare improving relative to autarky (in which consumption in any period must equal current output (to simplify assume there is no investment), when we assume that consumer's preferences are convex.

Therefore, I don't know if a country can survive/prosper being a closed economy, but certainly there are many advantages for an economy that engages in international trade.


Can implicit costs be negative? I think that conventionally implicit costs are positive but if for example we consider a negative interest rate as an implicit cost, the implicit cost would be negative. Is this true? Can implicit costs be negative?

We can define a (positive) cost to be simply a negative benefit.

In which case, a (positive) benefit is simply a negative cost.

So yes, in general, costs can be negative.

Note though that in economics, it doesn't usually make sense to speak of costs in isolation or benefits in isolation. Instead we usually look at the net benefits (i.e. all benefits minus all costs) or net costs (all costs minus all benefits).

So to use your interest rate example, usually when one buys say a 10-year bond, the benefit one gets in return is the positive interest rate. However, countries like Germany and Japan have issued bonds with negative interest rates. So, it seems silly that anyone would buy such bonds -- with negative interest rates, one is clearly getting a (negative) benefit and hence incurring a cost. However, this cost must be compared to other costs -- for example that of safely storing some sum of money. When the proper cost-benefit analysis has been made, we will usually find that for the buyers of these German and Japanese bonds, the net benefits are positive (or equivalently, the net costs are negative).


Macroeconomic effects of Profit Maximizing Federal Reserve The Fed makes money through its open market operations. I am considering a scenario in which the Fed, instead of trying to achieve goals like controlling inflation, tries to maximize its profits in real terms.

What exactly would such a policy mean- would it be equivalent to the Fed targeting a certain variable like inflation or output? In the event of a supply or demand shock, what would the Fed do?

The Fed also conducts discount window borrowing. If pure profit maximising were the goal, they might only provide reserves through the discount window. They might even shake down weaker banks.

If confined to buying only Treasury securities, they would just need to maximise the real rate differential between currency and the nominal rate (plus, shrinking their balance sheet so that they no longer need to pay interest on reserves). The maximum level of the real policy rate that is sustainable depends upon your view of the natural rate of interest. If the belief is that the actual real policy rate cannot greatly depart from a natural rate, we would see little difference in policy.

Buying bonds and keeping rates low works to lock in profits once, but the profits will dry up as bonds mature, and are replaced by low-yielding ones.


Does China subsidize its exports? And if so, to what extent?

Also, how does the extent of China's subsidies compare internationally?

(I understand this is a broad question but I can imagine that some economists somewhere have come up with some convenient measures or indices for this purpose.)

World Trade Organization (WTO) rules prohibit export subsidies and China joined the WTO almost 15 years ago (in December 2001). As a condition of membership, China had to eliminate export subsidies in the late 90s. So, strictly speaking, China can't subsidize its exports.

However, Dani Rodrik considers that China subsidizes them indirectly through the exchange rate by maintaining an undervalued currency (here and here). Rodrik adds two interesting thoughts. First, the undervalued currency boosts China's growth and this, in turn, is good for the world's recovery and the alleviation of poverty. Second, China could maintain its growth without trade imbalances if it could introduce industrial subsidies to offset a rising yuan. So, it be would better to subsidise tradables directly than to subsidise them indirectly through the exchange rate.


discount factor, function, and rate Consider an exponential discount factor $\delta\in(0,1)$.

Similarly, consider an exponential discount $\textit{function}$: $g(t)=\delta^t$.

Then, is defining the discount $\textit{rate}$ as below a correct way of defining one? My intuition is that the discount rate represents the rate at which the discount function decays:

$\rho=\frac{\frac{-dg(t)}{dt}}{g(t)}.$

Given a discount function $g(t)$, the discount rate is the rate at which the discount function declines over time. If time is discrete, then the (possibly time-varying) discount rate is
\begin{equation}
\rho(t)=-\frac{g(t)-g(t-1)}{g(t)}.
\end{equation}
If we divide each period into $n$ equal intervals and let $n\to\infty$, then we get the discount rate in continuous time as
\begin{equation}
\rho(t)=-\frac{g'(t)}{g(t)},
\end{equation}
which is the same as your expression.

The (possibly time-varying) discount factor is defined as
\begin{equation}
\delta(t)=\frac{g(t)}{g(t-1)}.
\end{equation}

With exponential discounting, the discount rate is constant, i.e. $\rho(t)=\rho$ for all $t$, and the (discrete time) discount function is given by
\begin{equation}
g(t)=\frac{1}{(1+\rho)^t}=\delta^t
\end{equation}
where $\delta=\frac{1}{1+\rho}$ is the constant discount factor.

The continuous time analogs are
\begin{align}
g(t)&=\mathrm e^{-\rho t}\\
\delta(t)=\delta&=\mathrm e^{-\rho}
\end{align}
Again, these are obtained by dividing each time period into $n$ equal intervals and observing that $\lim_{n\to\infty}(1+\frac{\rho}{n})^{-nt}=\mathrm e^{-\rho t}$.


Which economist suggested solely printing money to borrow and not have bonds? I remember reading this really interesting idea by some economist that a government need not issue fixed-term debt such as bonds, notes or bills. Instead all it could do is print money and use it to pay for everything, thereby borrowing.

It would repay by levying taxes, which it would vary based on the rate of inflation. I. e. if the rate of inflation would be too high, it would increase tax rates to bring it down and vice versa.

My question is -- who was this economist? I need to cite this idea and need a name or better yet a citation to a paper.

I think that Stephanie Kelton is an economist who has mentioned this before. From my research, it is based on the MMT. Some practitioners of the MMT believe that the government should not issue bonds, but can instead print money. I think that I may have heard Kelton mention this a few years ago, but I am not that into the MMT. I would do some research on the MMT however, and Kelton, and that can likely lead you to the answer you're looking for.


Difference between Negotiable Instrument and Financial Asset So here are the two pages I was reading to try and understand :

Negotiable Instrument
and
Financial Asset

Confusion started as both pages use Certificates of Deposits(CDs) as an example.So a CD is both a financial Asset and a Negotiable Instrument ?

Considering the other examples of negotiable instruments like cheque, draft,money order, traveler's cheque I understand that these instruments are basically used to make payments from one person to another or access money from your own account.

And financial assets like stock,bonds,or bank deposits are well, just assets, that is your money kept secured or invested somehow in some place and you might earn a profit from it regularly or use it to bear various expenses. Basically it's your property.

But then, how can CDs be both? Are they financial assets that can be transferred to other people if deemed necessary? Because those articles clearly mention one thing that Negotiable instruments can be transferred. Then bonds,stocks etc can be Negotiable Instruments too? Can't we have someone else as a nominee in those?

This is very confusing please explain. Thanks.

CDs are money market instruments and hence are financial asset. There are traditional and negotiable CDs.

Traditional CDs are like term deposits in banks, and are not tradable. Liquidation before maturity entails penalty charges.

Negotiable CDs can be traded in the secondary market (referred to as money market). The original holder can 'liquidate' them at prices below the face value in the market. At maturity, the holder of the negotiable CD receives the face value of the security from the issuing bank.


Does education have any value for most people's work? Let me first say that I am not an economist. However, I was having a discussion with one of my friends who claimed that he had absolutely no gain from his education other than it actually being the ticket for him to get a job. He argued that he hardly ever used any of the stuff he had learned during his education. So while it was a formel requirement for him to have some kind of education to gain entry at the firm where he is currently working, he sincerely believed that his education did not make him a better worker in any way.
I must admit I was a bit offended. I love my own job and I maybe also have some idealistic notions on behalf of education that it can both make us better as humans as well as help us work in better ways. However, I think that what most annoyed me was that I could see he had point and still felt that economists must have some kind of counter argument against this notion that education is basically valueless. So my question is:
Do there exist in economics a serious discussion along these lines?
and furthermore, if that is the case,
What are the main arguments on behalf of those arguing that education certainly has its economic merits?

Yes, this debate actually exists in economics. There are two main arguments why education increases wages:

Theory of Human Capital: This is theory that posits that education is an investment that is virtually analogous to investment in regular capital goods. You pay for your education (pay for capital), education makes you more productive (and wages critically depend on productivity) hence afterwards you get higher wages (return to your human capital). This theory was first developed by Becker (see Becker, 1980), it got a bit more nuanced since then but since you state you are non-economist I will not go more deeper into the arcana.  This is basically 'your' view of education.

Signaling/Screening Theory: This theory posits that education is a sort of screening device (signal) for employers about your innate ability and does not necessarily builds up your human capital. If you are unable to pay attention, memorize stuff, be able to coherently reason you will not get good grades and graduate and likely not be a good employee either. This theory was developed by Spence (1973) and later expanded Stiglitz (1975) - also the same caveat as above applies. This is basically 'your friends' view.

Now the above theories are not mutually exclusive as they can be both true at the same time. Education can be both a screening device and productivity enhancer. In fact empirically most of the time they both will be present. This being said it is an empirical question which one dominates and answers will differ from field to field. I think it is fair to say that most empirical evidence shows that education increases wages via human capital accumulation, but this is not true for every field.
For example, Hussey (2012) shows empirical evidence that when it comes to MBA degrees most increase in graduate wages was due to screening not human capital (although returns to human capital are present too). Again I think most economists would say that education contributes more to wages via human capital accumulation but the example above illustrates that this is not universally true and has to be empirically tested. It might very well be that your friend is just from an industry where screening is more important than human capital.


Why do banks loan at APR < 6%, when they can invest that principal in ETFs? I screenshot the picture beneath from HSBC HK's website. Scroll down a quarter of the way, and click on 'Rates' to download that PDF. I ask just about loans under my green horizontal line.

Canadian ETFs with MERs < 0.15% like  XUS (iShares Core S&P 500 Index ETF) can probably outstrip an APR of 6%.

Thus why would a wealthy bank loan at APRs < 6%?
Conversely, why don't they charge more than 6%? Why not charge the opportunity cost of loaning, that is, the APR gained from investing in a relatively safe ETF like XSP?

Furthermore, HSBC probably has PhDs in math like James Simons who can invest in alternative investments that can yield a higher rate of return for the same risk.

XSP is riskier than the preferential rate personal loan. The Standard and Poors 500 is a large composite of U.S. equities, hedged (short USD, long CAD). XSP combines the equity market volatility and the USD / CAD volatility but it tries to offset currency value change by holding a hedge. Some of the volatility is offsetting; some is additive. The equity composite risk is acceptable for many investors if they intend a long investment period, for example, 30 years. The USD / CAD risk is acceptable for some investors if both the Federal Reserve and the Bank of Canada manage to achieve roughly equal inflation rates over the long investment period, but XSP's manager aims to minimize this risk. For many investors, the risk of XSP is only acceptable if a long investment period is intended. However, the preferential rate personal loan is likely for a shorter period. Over that shorter period XSP is expected to be riskier than the personal loan. The preferential rate personal loan is made after checking employment history, income, assets, and other indications of ability to meet obligations. A bank's business model is to own loans; not huge amounts of an equity index. Bank regulators force banks to keep risk below a certain level.

Incidentally...

If XSP is held in the investor's taxable account, keep in mind that the XSP manager sometimes returns capital to the investor which lowers the investor's average cost. The XSP manager sometimes reinvests cash into more equity which effectively means it raises the investor's average cost. Unless you plan to invest extremely large amounts, you might find keeping track of these things to be not worth the effort.

Addendum:

There was a request for information about XUS. You avoid the "currency hedging risk" inherent in XSP by choosing XUS instead. Currency hedging risk is the risk that the fund manager's hedging activity imperfectly eliminates your exposure to the USD which is inherent in the underlying asset which is based on the S&P 500. By choosing XUS instead, you are accepting "currency risk". That is the risk created by exposure to the USD which is inherent in the underlying asset which is based on the S&P 500.

In this risk chart they show that XSP has both kinds of risks, because the hedging is imperfect so XSP still has currency risk. By implementing hedges, the manager aims to make that currency risk less than what is in XUS.


How much better off would the average person be if we took away enough money from the top 1% to make them no richer than the average person? In my hypothetical situation, the USA population takes enough money away from the top 1% so that they are no richer than the average person. That money is then redistributed to the lower and middle class in the country. How much better off would the lower and middle class be if this happened?

I assume the following:

We redistribute wealth (not income) from the top 1% to the bottom 99%
Everybody among the bottom 99% receives the same amount.

Source for my data: World Income Database

In 2014, the average personal net wealth among the top 1% was around \$11 million in 2017 prices. The total wealth to be distributed is hence (3.3 million people in the top percentile $\times$ 11 million) = \$36.3 trillion ($3.63\times10^{13}$).

This means that everybody among the bottom 99% receives around \$111,000. How much this makes you better off depends on your initial state. Concepts of low- and middle-class are somewhat ambigious. According to the WID data, net personal wealth among the bottom 50% is essentially zero, and \$205,000 among the 50 to 90th wealth percentile.

EDIT:

Please note that this is a purely mechanical and unrealistic exercise. For starters, it ignores legal and administrative hurdles. A major concern is also that a substantial share of the private wealth of the top 1% is invested in productive firms which employ millions of people. What happens  to their employees if you expropriate the owners of these firms?


Statistics of links between public debt and recessions When the debt of a country gets above a certain level, is there a bigger chance for an economic crisis, recession or even civil unrest?
I know things are different today.
But did anyone bother to study the relation between debt levels and such problems along history?

Yes actually many people bothered to study this link. To be more specific the studies are between debt-to-GDP and economic growth (but recession is defined as negative economic growth - more narrowly by NBER as two consecutive quarters of negative growth). Here is a brief summary of the recent literature:
Rogoff-Reinhart (discredited) work:
As mentioned in the title the work of  Rogoff-Reinhart is now discredited due to several embarrassing mistakes they made in their analysis but any contemporary mention of literature review on this issue would be incomplete without mentioning them as they arguably brought attention to this topic.
Rogoff-Reinhart research is non-technically summarized in their book This Time Is Different: Eight Centuries of Financial Folly. They did quite a comprehensive study of a historical records and also very meticulous collection of historical data on government debt statistics. Analyzing this data they argued that there was a 'magic number' of debt to GDP ratio of $90\%$. They argued that below this cut-off debt did not had much effect on GDP growth but above it it started to have a negative effect.
However, it was later during replication discovered that Rogoff and Reinhart made several 'embarrassing' but probably unintentional mistakes in their analysis and replication of their study found that their result virtually disappeared when these mistakes were corrected. If you are interested in knowing what the mistakes were you can read this Economics.SE answer).
Post Rogoff-Reinhart Research
However, while Rogoff-Reinhart work was discredited it brought much needed attention to this issue. Further studies that paid attention to cross country heterogeneity, endogeneity and avoided arbitrary debt-to-GDP brackets suggest that the correlation between debt-to-GDP and growth is actually still negative but the issue is also much much more nuanced than suggested by original Rogoff-Reinhart work with their 'magical' threshold. It is also still up to debate what exactly is the nature of causality here and it is agreed that even if there is a causal link the relationship is definitely non-monotonous.
A nice survey of post Rogoff-Reinhart literature can be found in    Panizza and Presbitero 2013. Some important papers that you should read if you are interested in this include Minea and Parent (2012), Egert (2015), Panizza and Presbitero 2014 or Baglan and Yoldas (2013). There is also this VOX$^{EU}$CEPR article which makes also short summary of this area. You can find more papers in sources cited therein or by searching for keywords from those papers.
When it comes to civil unrest I would recommend Politics.SE or some sociology forum as such issues are not normally studied by economic profession or if then in a interdisciplinary way.


Basic question: why do markets grow? I apologize if this is gonna be very basic, but I am a beginner. My question is: why do markets (expressed as indices) grow? Is it because money is constantly moved from the real economy into stock markets, but the ingoing flow surpasses the outgoing flow?

Suppose no money can flow in or out a market, besides what is already in. Would that market (or a stock index that represents that market) just reach a dynamical equilibrium? I would say yes, but please correct me.

The general answer is no, a closed market with only money circulating within the market, and that money never leaving it or entering it, cannot increase its aggregate price level - provided that the quantity of goods being sold on that market also remains constant. Price wise however, it can display variations over time, so share values can change - even if the money circulating that creates them doesn't.

The main reason market prices grow and shrink in the real world is that not only are new companies always being listed, but money is entering them and sometimes leaving them at different rates. In a typical western economy for example, the money supply is continuously growing due to bank lending, a percentage of that growth it being invested in the stock market through pension funds, and so there is a linkage between monetary growth and the market.

Consequently any time some kind of conduit between lending and stock market purchases gets created, things tend to get exciting. cf. 1929 US Crash.


Compute the average income and the average utility I don't know how to compute the average income and the average utility for the second case. It is confused because I have to consider two densities $[0,0.5]$ and $[0.5,1.2]$.Can anybody tell me how to compute this?

Thank you in advance!

Edit: $\bar {z} =$ the average income, and $z_i$ = the income of an individual $i$. $U(\cdot) =$ utility function for all.

These are not two densities, but one density defined on the interval $[0,1.2]$, taking two different values. The average income is then the integral of the density function, i.e.

$$
\int_0^{0.5} 1 x \, dx + \int_{0.5}^{1.25} \frac{2}{3} x \, dx = \frac{9}{16} = 0.5625
$$

In order to obtain the average income, you do the same with the Utility and $\bar{z} = \frac{9}{16}$.


Why is overlapping generation model not a main model in macroeconomics? It seems that overlapping generation model has far more realistic assumptions than ordinary Ramsey-Prescott RBC model for use in macroeconomics. Why is Ramsey-Prescott ones, even New Keynesian variants, are more popular than overlapping generation ones? Is this just due to simplicity, or is there any fundamental problem with the consequences of overlapping generation model?

Complexity issues are always a factor in choosing model frameworks.

In addition, my feeling is that historically the impressive possibility brought to light by the OG models was that an economy could be "dynamically inefficient" (=being characterized by over-accumulation of capital). After some empirical studies provided evidence that the main western economies did not appear to over-accumulate capital, there was perhaps an impression that the model, while structurally more realistic, had not "that much to offer" to compensate for its added complexity...

Not so, according to Philipe Weil, who celebrated the model's first 50 years of existence in the paper
Weil, Philippe. 2008. "Overlapping Generations: The First Jubilee." Journal of Economic Perspectives, 22(4): 115-34.  (freely downloadable).

The paper is an enjoyable non-technical read, full of compacted information, knowledge and insights.  A quote (bold my emphasis):

  "Samuelson’s stroke of genius was to construct a model that makes economies in which the first welfare theorem always holds, absent
  externalities and distortions, look like quite a special case. It is
  hard to escape the conclusion that the features of the overlapping
  generations model are the norm, rather than the exception: after all,
  can we seriously argue, once we understand what “new” means, against
  the realism of a model that rests entirely on the assumption that “we
  live in a world where new generations are always coming along?” In
  that respect, it is not the overlapping generations model, with the
  wealth of interesting issues it raises and its rich welfare
  properties, that is a simple toy model, but rather the competing
  workhorse of modern macroeconomics, the Ramsey–Cass–Koopmans model
  that assumes that no “new” generation ever comes along as future
  agents are all part of pre-existing families. Barro’s (1974) famous
  paper on debt neutrality and Weil (1987) make it clear that such a
  model emerges only if parents love their children (or future
  immigrants) enough to leave all of them positive bequests. This
  condition is very restrictive, and therefore the Ramsey model, with
  its long-run interest rate that always exceeds the growth rate, is
  literally quite extraordinary compared to the overlapping generations
  model."

This is the rallying cry -in his paper Weil details many aspects and issues raised by the model, and the insights that it can provide.


Demand Curve of Foreign Exchange My high-school Macroeconomics textbook states that 'There is an inverse relationship between the price of foreign currency, or exchange rate, and demand for foreign exchange. When the exchange rate rises, demand for foreign exchange falls and vice versa'
Is there any particular reason for making the exchange rate the dependent variable in this scenario? I believe if I take demand to be the independent variable; that is- when demand rises, the foreign exchange rate rises. This case exhibits a direct and not an inverse relation and, hence, creates a contradiction.
All in all, I want to know if there is any procedure to decide what becomes the dependant variable. Or both of the cases are distinct and valid in their own.

At market both price (in this case exchange rate) and quantity are endogenous ('dependent') variables.

You are confusing two completely distinct phenomena. Movement along demand curve and shifts in demand curve.

When exchange rate increases demand will be lower because we are moving along given demand curve - but increase in exchange rate does not cause shift in a demand curve.
In your second example when you talk about 'demand rising' you must be implicitly talking about shift in demand curve otherwise the rest of the paragraph would not make sense.
So  you are actually talking about two distinct concepts. When we talk about movements along demand curve even if you reverse the relationship and 'make the exchange rate dependent variable' there will still be inverse relationship between the two.  For example, if the relationship between demand for currency $(D)$ and exchange rate $(E)$ would be given by:
$$D= 100 - E$$
Then 'making exchange rate the dependent variable' would mean you have to solve $E$ which would give you:
$$E=100-D$$
As you can see the relationship is still inverse.
However, you are not doing the above when you talk about 'making exchange rate dependent variable' and then asking what would happen when demand raises - you are really talking about what happens when demand shifts right which in this case would happen if in the equation $D=\alpha-E$ the parameter $\alpha$ which used to be in the above example  $100$ would increase.
See the graph below from Mankiw's Principles of Economics that showcases the distinction between the shifts in demand and movement along demand. The graph is not about exchange rate market but you could as well just change y-axis label to exchange rate and x-axis label to quantity of currency and the point would be the same.


When people talk about “Companies Hiding Money Overseas” to avoid taxes, what does this really mean? Presuming the consequence is holding cash somewhere else: I'm really interested in the objective effect of this from a global value creation perspective. I'm less interested in the money itself and more interested in how it impacts the profile and efficiency of creating value.

Capitalistically speaking, does it have the effect of making our economy less efficient?

Does it add value or efficiency anywhere else e.g. the countries who act as host for the cash?

Is the holding of cash a negative effect on value creation?

Re: main question.

It can range from a) companies fraudulently charging above/below market prices among departments/subsidiaries in order to fraudulently lower tax liabilities in a particular jurisdiction, to b) companies retaining profits and associated capital in the foreign country that the profits were earned in.

Most often, it refers to locating headquarters in a lower-tax jurisdiction regardless of whether the corporation has a substantial presence in that jurisdiction for its main area of business activity.

The other questions are all very complex and can have extremely numerous caveats and/or special cases, etc. The questions are also prone to predictable divisions along ideological lines.

However, here are some general points relating to each:

1) An empirical point of reference on the role of taxation in governmental capacity to address market imperfections: Somalia's tax-to-GDP ratio is about 10% (too low), while the tax-to-GDP ratio of the USA is above 30% (too high or too low?) and that of France approaches 50% (commonly viewed as too high). If all profits go overseas and are not taxed domestically, this could reduce the ability of government to correct market imperfections.

2) Does it add value elsewhere? Here you could enter into debates about what does it mean to "add value" (the most obvious assumption would be that you refer to "value added" as determined by monetary amounts representing value at different stages of production) or different views on "efficiency" (e.g., profit maximization of firms as compared to social welfare maximization in a society). I think you'd really have to get a handle on the main positions of the debate about those concepts, in addition to their most commonly used statistical indicators, in order to be sure of what exactly you're asking and/or want to ask.

3) Ignoring useful theories about things happening instantaneously and always equaling out in a way that is optimal across all uses: a) liquidity is needed both to undertake any investment opportunity that arises (including buying rivals for the purpose of enjoying monopoly profits or possibly for cost savings) and also to protect against liquidity shocks such as those related to a major fall in the value of capital assets across an economy (e.g., 2008-09); b) the case of holding a lot of cash should be compared to opportunities which exist for them to deploy liquidity to earn profit - for example, corporations should not be expected to engage in explicitly redistributive zero-value digging and filling of ditches (leaving specifically redistributive questions to the larger tax pool via government, which can more easily re-capture the value of human capital growth), and it is not desirable to "excessively" promote companies to deploy cash holdings in a pro-monopoly anti-competition manner.


Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$ I am interested in an economic interpretation for the ratio of partial derivatives of a demand function $Q_1(p_1, p_2)$, which is
\begin{equation}
t=\frac{\frac{\partial}{\partial p_1}Q_1(p_1, p_2)}{\frac{\partial}{\partial p_2}Q_1(p_1, p_2)}.
\end{equation}
Assume the goods are complements so that both partial derivatives are negative.

This has the same form as the MRS in consumer theory, but the intuition does not readily apply.

I have been thinking about this as the ratio of demand sensitivities: if $t>1$ then the demand for good 1 is more sensitive to changes in $p_1$; if $t<1$, then the demand for good 1 is more sensitive to changes in $p_2$.

I would also say this is somewhat related to the cross price elasticity of demand, but the formula is not the same.

Any interpretations or intuitions would be great!

One interpretation I can offer. The demand function can be expressed as:

$$Q_1 = Q_1(p_1,p_2)$$

Let us take the total differential:

$$dQ_1 = \frac{\partial Q_1(p_1,p_2)}{\partial p_1}dp_1+\frac{\partial Q_1(p_1,p_2)}{\partial p_2}dp_2$$

Assume that $Q_1$ remains unchanged with respect to a change in prices. This implies that $dQ_1=0$. Solving the equation:

$$\frac{d p_2}{d p_1} = - \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$

The above expression implies the following (since both partial derivatives are negative):

$$t = \left|\frac{dp_2}{dp_1}\right| = \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$

$t$ can be interpreted as follows: if good 1 became more expensive, how much would the price of good 2 need to change (decrease) such that the demand for good 1 remains unchanged.

Mathematically, $t$ represents the magnitude of the slope at ($p_1,p_2$) on the level curve when we fix $Q_1$.


The question is related to topic consumer surplus Suppose the demand function is 12-3p=q and price falls from 3 to 1. How much is the change in consumer surplus?

Demand function in your question is given as

Q=12-3P

In order to find consumer surplus,we first find inverse demand function from it

Q=12-3P
12-Q=3P
12/3-Q/3=P
4-Q/3=P or
P=4-Q/3

At P = 3 ,from inverse demand function, we get Q= 3

Since intercept in the inverse demand function is 4 and base is 3,we can calculate consumer surplus by using the formula for area of a triangle as ab/2 i.e 4x3/2=6

Similarly,when price falls to 1,we get value of Q=9

Here,area of triangle becomes 9x4/2=18,which is the new consumer surplus

Subtracting 6 from 18, we get the change in consumer surplus as 12

So the answer to your question is 12.


How do Sharks make money on Shark Tank through equity alone? When the Sharks on Shark Tank buy equity in private companies, how do they eventually make money by purchasing only equity?

For example, if the company makes a lot of money for a few years but then goes out of business, does the Shark lose all of their money if they don't sell their equity share to another private investor before the company goes out of business even if the company was very profitable for a few years?

Getting cash out is no different for sharks on Shark Tank than any other investors in private equity: they can make money back from dividends that are yielded by profits; or by selling on the shares to someone else; or by asset-stripping the companies - i.e. selling off their assets, and closing down the trading activities of the company.

As with most other equity investors, they own shares in a mix of companies. So to make money overall, they don't need to make money on every single company. They can accept that some will do badly (often losing them their whole investment), and some will do well, but overall, they expect to make a profit on the portfolio.

Indeed, in general, for investing in small companies with private equity, the expectation might normally be that most investments will yield the investor a 100% loss, but the few that do make a profit, make such a big profit that the overall bottom line is positive.

However, the sharks do have the huge advantage that every one of their companies has received significant free tv advertising, and has had the public endorsement of a supposed "smart" investor, which tilts the odds in their favour.


Difference type of crowding out effect in economics In macroeconomics I came across two type of crowding out. First, in IS LM model where in classical case, when LM curve is vertical increase in government expenditures have no increase in output i.e. full crowding out.
Secondly, in AS AD model due to increase in government expenditures when AD curve shift in classical case i.e. Vertical AS curve there is no effect on output. What's the difference between these two cases of full crowding out.

A vertical LM curve means that any change in government expenditure will result in a change of the interest rate which exactly offsets the initial change of public demand (private demand increases/decreases).

A vertical AS curve indicates the level of long-run output. If AD shifts, all that this will create is a change in prices in the long-run. This is also called the Classical Dichotomy: real variables are independent of monetary variables.

So, both diagrams tell the same story about crowding out: changes of private expenditure offset changes of public expenditure. The diagrams just highlight different parts of the same mechanism.


Forecasting vs econometrics, how is regression used differently? I have this homework question:

  Q: Suppose that a marketing company is seeking to identify customers interested in their product.
  They regress the number of sales in an area on the average education level in that area. Does the coefficient likely represent a causal effect? Why? Is this a problem? Explain.

  A: The coefficient is not likely to represent a causal effect, it
  merely shows how the number of sales and average education move
  together. That relationship could be causality, reserve causality, an
  omitted variable could be determining both, or the two could be
  simultaneously determined. This isn’t a problem because the company
  isn’t trying to cause people to have more interest, they just want to
  predict if they’re interested or not.

So now I want to know, how is a regression used differently in a forecast vs a causal analysis?

If I'm looking for predictive power vs causality, how might the regression models be different, if at all?

Or how might the results be interpreted differently?

In forecasting, we just want to predict what will happen. So assuming that there is a certain degree of stability in the relation, so that the relation is valid outside the sample, we estimate how past sales co-vary with education level (or whatever), and we say "so if we move into New Area "X", and apply the same sales strategy, since in "X" the educational level is "Y" we predict that our sales will be "Z", given the estimated co-movement relation".

In an econometric analysis, we certainly are interested in estimating the same relation, but if we want to perform also causality analysis, identifying assumptions become necessary and specific models arise, (like the "Differences-in-Differences" approach for example), designed to detect and assess causality.

By the way the first sentence in the "A" part of the question is "wrong in the other direction": the phrase "is not likely to represent a causal effect" is inaccurate: it is "as likely" to represent one, as it is not, since basic regression is "agnostic" about causal effects, it is just about detecting and estimating correlation.


Strategic form representation of extensive form games Proposition. Every finite extensive form game is associated with a unique strategic form representation.

I think this proposition is true. But how do we prove it rigorously?

I rely on the definitions from Chapter 2 of the Handbook of Game Theory, Volume 1, by Sergiu Hart.

If I understand you correctly, the proposition can be re-written as

Proposition. For, every finite extensive form game $\Gamma^E$, there exists a single
 strategic form representation $\Gamma^N =[I^N,\{S^N_i\},\{u^N_i(\cdot)\}]$ (up to relabeling of the agents) such that

$I^N = I^E$,

and for all $i\in I^N = I^E$,

$S^N_i = \{$ pure strategies of $i$ in $\Gamma^E$ $\}$ ,
and $u^N_i(s) = u^E_i(c(s))$, where $c(s) $ associates every profile of pure strategy in $\Gamma^E$  with a terminal node of  $\Gamma^E$ resulting from the pure strategy profile $s$.

I think 1. and 2. are obvious. There only remains to show 3, which is equivalent to proving that $c(s)$ is a function, i.e. every profile of pure strategies is associated with one and only one terminal node in $\Gamma^E$. This follows directly from the fact that the pure strategy of some player $i$ is a function selecting one and only one possible action from every information set.

Start from the root node, $r_0$.
By definition of a game in extensive form, the nodes are partitioned between the agents.
Thus $r_0$ belongs to some agent $i_0$.
By definition of the game in extensive form, the nodes of $i_0$ are partitioned into information sets.
Thus $r_0$ belongs to some information set of $i_0$, say $H_i^0$.
By definition of the pure strategy $s$, for any node in  $H_i^0$, $i_0$ always choses a single successor, among the possible successors at $H_i^0$.
Let this successor be $r_1$.
Then $r_1$ must be the successor of $r_0$, and the next node on the path.
Now consider $r_1$.
By definition of agame in extensive form, the nodes are partitioned between the agents.
Thus $r_1$ belong to some agent $i_1$ (where $i_1 = i_0$ is allowed).
$\vdots$
Repeating the argument as many time as needed, because the game is finite, we necessarily reach a single terminal node $r_* = c(s)$.


How long can a population grow once it has reached the replacement fertility rate? The Ministry of Health and Family Welfare answered in the upper house to a question that India has a fertility rate of 2.2 (link here). The replacement fertility level is usually given at 2.1 to 2.2.

These sources suggest that India's population is projected to grow at least until 2060 (2050 in some sources): Quartz, The Hindu.

The Quartz article even cits UN population projects and says:

  Even under the instant-replacement fertility variant, with the country’s fertility assumed to fall immediately to 2.1 births per woman, India’s population would reach 1.9 billion by the century’s close.

This source defines the replacement fertility level as:

  Replacement level fertility is the level of fertility at which a population exactly replaces itself from one generation to the next. In developed countries, replacement level fertility can be taken as requiring an average of 2.1 children per woman.

What I don't understand is: how can India's population be projected to grow until 2060 if it has already reached replacement fertility level in 2017? A 'generation' is usually 25 years. Even if we take it as 30 years, India's population should stop growing in a couple of decades. How does the UN project that it'll grow until 2060?

Population size isn't just about birth rate.

It's also about death rate and net migration.

If the health of India's population continues to improve, then life expectancy will increase, and population size will continue to grow until life expectancy stops increasing, assuming zero net migration.

Net inward migration would also increase the population size.


How successful is the Islamic banking model? Under Islamic law (Sharia), any interest is considered usury and is prohibited. This makes banking in Islamic ruled countries difficult as the normal way for banks to create profit is through lending money at a rate of interest. Rather, Islamic banks work by profit sharing or buy-sell method (e.g. when asking for a mortgage, the bank buys a house and sells back to the borrower at a higher price).

My question is: how successful is this model of Islamic banking? Does it make more money? Are there any advantages that this model gives not present in the regular banking doctrine?

In Ernst & Young 2016 Report we read (p. 17) that for the period 2010-2014 the Average Return on Equity for "Participation Banks" (this is the secular name of Banks under Sharia) was $12.6$% compared to $14.5$% of the "comparable conventional" banking. Although the ROE is lower, it is not that lower.

We also read that in the same period (p. 10), the assets of International Participation Banks grew by $16$% yearly on average.

It looks like a profitable way of doing business with strong growth.

From a philosophical point of view one should ask: how bankers in Participation banks view what they do?

Are they viewing their business model as essentially a way to cheat the religious doctrine-turned secular law, and everything they do they "translate" it into "what was the interest earned" or, after accepting the constraint that they won't do banking business in the traditional way, they explore different ways of making available funds earn a return?

The two examples offered by the OP are very useful: first, "profit sharing" makes a Participation bank more like a joint venture enterprise -and it reminds us of Ancient Greek bankers (they were the first bankers), whose bulk of business consisted of funding ships to travel and trade around the Mediterranean Sea, for a share of the profits.

Second, when a Participation Bank, instead of loaning an amount of money for a house-buy against a mortgage, buys the house and sells it to the interested party at a higher price (to be paid gradually through installments), it may appear as just a trick to avoid calling it "loan with interest", but a large construction firm that constructs houses and then sells them with a long-term payment plan, does nothing different.

So it appears that Participation banks use very standard ways to do business, only, they are ways that we usually associate with other sectors of the economy, and this is why they may appear "strange" for the banking sector.

As regards the pros and cons of Islamic Banking, there are many studies, for example

Beck, T., Demirgüç-Kunt, A., & Merrouche, O. (2010). Islamic vs. conventional banking: Business model, efficiency and stability. The World Bank.

where the abstract reads

  This paper discusses Islamic banking products and interprets them in the context of financial intermediation theory. Anecdotal evidence
  shows that many of the conventional products can be redrafted as
  Sharia compliant products, so that the differences are smaller than
  expected. Comparing conventional and Islamic banks and controlling for
  other bank and country characteristics, the authors find few
  significant differences in business orientation, efficiency, asset
  quality, or stability. While Islamic banks seem more cost-effective
  than conventional banks in a broad cross-country sample, this finding
  reverses in a sample of countries with both Islamic and conventional
  banks. However, conventional banks that operate in countries with a
  higher market share of Islamic banks are more cost-effective but less
  stable. There is also consistent evidence of higher capitalization of
  Islamic banks and this capital cushion plus higher liquidity reserves
  explains the relatively better performance of Islamic banks during the
  recent crisis.


Is monetary expansion the same as quantitative easing Is monetary expansion policy the same as quantitative easing (i.e the buying of US treasuries by Federal reserve which injects more USD into the economy)?

It is not. Monetary expansion is increasing the money supply (through whatever method).

Quantitative easing increases the money supply / injects liquidity, but  also reduces privately hold stocks (and hence can reduce risks from the banks' balance sheets).


Does it mean that the value of currency decreases? Higher interest rates offer lenders in an economy a higher return relative to other countries. Therefore, higher interest rates attract foreign capital and cause the exchange rate to rise.

Does to rise means the value of currency decreasing?

This is a constant source of confusion in international macreconomics. There are two ways to define the exchange rate.

$$E = \frac {\text {home}}{\text{foreign}}$$

Here, we express "units of home currency per unit of foreign currency" (or basket of foreign currencies). In this definition
$$E \uparrow \Rightarrow \text {home currency depreciates}= \text {value} \downarrow$$
so it can be confusing at times, since usually the object of study is a specific "home" currency.

The other way is the reciprocal

$$S = 1/E = \frac {\text{foreign}}{\text {home}}$$

Here we measure "foreign currency units per unit of home currency", and so
$$S \uparrow \Rightarrow \text {home currency appreciates} = \text {value} \uparrow$$

Some argue that this second expression is more "natural" since it expresses the "price" of home currency in terms of foreign currency, i.e. "what is the price of 1 euro ? 1.22 USD", "what is the price of 1 USD?  0.82 euro".

But not even the notation is standard (for example, the wikipedia article uses the symbol $S$ to denote the exchange rate that we denoted by $E$).

One way to avoid confusion is to write the various relations with subscripts, and use just one symbol for the exchange rate. Using $S$ as this symbol, the subscript $f$ for "foreign" and the subscript $h$ for "home" we have

$$S_{h/f} = \frac {\text {home}}{\text{foreign}},\;\; S_{f/h} = \frac {\text{foreign}}{\text {home}}$$

For the specific question, since interest rates are in focus, it appears that the underlying law is the Uncovered Interest Rate Parity (UIRP), where

$$1+i_h = (1+i_f)\left(\frac {S_{h/f, t+1}}{S_{h/f,t}}\right)$$

$$\Rightarrow_{\text{approx}}\;\; i_h = i_f + \Delta s_{h/f,t+1}$$

or

$$i_h= i_f - \Delta s_{f/h,t+1}$$

where $s$ denotes the natural logarithm, and $\Delta$ is the first-difference operator. By looking at the UIRP, we see that if the home interest rate is higher than the foreign interest rate, $i_h > i_f$, it must be the case that $\Delta s_{h/f,t+1} >0$ to satisfy the relation. $\Delta s_{h/f,t+1} >0$ is expected percentage change in the "home per foreign unit" exchange rate. So we say that the home currency is expected to depreciate...

But hey, intuition appears to say otherwise:
Assume $i_h$ rises above $i_f$. Then foreign investors have a motive to deposit their funds in the home economy, since the interest is higher. But in order to do so, they increase the demand for the home currency, since they need to convert their funds in home currency units. Assuming constant home money supply, this leads to an increase in the value of home currency compared to other currencies. But it does that now so mathematically it decreases the current exchange rate, $S_{h/f,t}$:

$$i_h > i_f \Rightarrow S_{h/f,t} < S_{h/f,t+1} \Rightarrow \Delta s_{h/f,t+1} > 0$$

In other words a higher home interest rate increases the value of the home currency now, creating expectations for home currency depreciation in the future.


Production Set: Not satisfying Free Disposal Assumption I saw the figure which satisfies the free disposal assumption in Mas-Colell, Whinston and Green (1995), but wondering if there is a figure that DOES NOT satisfy the free disposal assumption? Any leads will be great.
Thanks!

This is something you can figure out yourself. If you have not yet tried, I encourage you to do so.

Draw any set $H \in \mathbb{R}^n $. Select any point $x$ of set. Is the set of all points $y$ "under" this point $x$, that is
$$
\left\{y\in \mathbb{R}^n | y << x \right\},
$$
a subset of $H$?
If no, free disposal is violated.
If yes, remove any of these points from $H$. The reduced set now violates free disposal.
For a figure of such a set, you can consider the Mercedes logo:


Recommended ratio of sells of software of basic and advance edition Our company is selling one software in two edition: - cheap one with basic features - expensive one with basic and advance features

According to sells statistic, more than 90% of people are buying the expensive one edition. I would like to know if the sells model is suitable. Isn't the expensive variant too cheap? I know that Microsoft is selling Windows Office in Home edition and Professional edition, but they don't sell more than 90% of in Professional edition.

It seems to me that such product configuration is different in software development, because production price of creating software copy is same for cheap and expensive edition (which differs for example from automotive where production of luxury car is ore expensive).

Is there some recommended ratio in sells for cheap and expensive edition of software to maximize revenue?

I ask my colleague about it and he answered: "Isn't it good that we sell the highest edition at most?" It is reasonable argument?

I like this question because it provides an opportunity to showcase how one can use the simplest of mathematical analysis, full of simplifying assumptions, and still get something useful/applicable.

So let's call "advanced" and "basic" the two versions of the software. Each customer buys one unit of the one or the other. Let's assume/approximate the demand function as linear and let's assume away cross-price and competition effects.

Assuming linearity is acceptable as a local approximation.

Assuming no competition effects is to assume either that the software is seriously differentiated from the competition, or that we examine the case with competition's prices fixed.

The worst assumption is the one assuming away intra-cross-price effects: we do expect that if we raise the price of the Advanced version, potential customers may turn to look to the Basic version (and, possibly, but less strongly, vice versa).

Anyway, with these assumptions we have the demand functions

$$Q_A = a_A - b_AP_A,\;\;\; Q_B = a_B - b_BP_B$$

and the Revenue function

$$TR = P_A\cdot Q_A + P_B\cdot Q_B = a_AP_A - b_AP_A^2 + a_BP_B- b_BP_B^2$$

Interpretation of the coefficients comes later. Maximizing this function with respect to the two prices is straightforward, and one gets

$$P_A^* = \frac {a_A}{2b_A},\;\;\; P_B^* = \frac {a_B}{2b_B}$$

Then optimal revenue share for the Advanced version can be shown to be

$$S_A^* = \frac {a_A^2}{a_A^2 + (b_A/b_B)a_B^2}$$

Now, $a_A$ is the maximum number of customers that we would have if we gave the Advanced version for free, and likewise for $a_B$. These quantities are routinely estimated/assessed by businesses. The beta coefficients are trickier, because they reflect the marginal loss of customers as we increase the price by one currency unit. But again, a business can and does make educated guesses on this matter also (perhaps considering bigger increases than one unit), based on past experience, and knowledge of the market and the potential customers.

So we see that the abstract and simplistic $S_A^*$ can be applied to an actual situation, and of course we can form some intervals by saying "$a_A$ is from... to..." etc. and so get a sense of where the optimal customer mix may lie.

The first step to make this a bit more realistic is to include an asymmetric intra-cross-price effect: namely, if we increase the price of the Advanced version, potential customers may turn to the Basic version, but if we increase the price of the Basic version we lose potential customers to the competition. In this case the demand function of the Advanced version remains as before, but the demand function of the Basic version becomes

$$Q_B = a_B - b_BP_B + c_BP_A$$

It is a bit more complicated but the final expressions are still workable. One of course would here have to assess the magnitude of $c_B$ also.