Bayesian vs. Frequentist Approach I hope this question is not too general in nature; I just wanted to ask the members of this community their opinion on the following question:  How can one choose between the Bayesian approach and the Frequentist Approach? In what context is one approach better than the other?
The reason I ask this is because whenever I attempt to answer causal empirical questions, I immediately think of trying to estimate Treatment Effects (for instance using difference in means, Difference-in-differences or matching estimators). On the other hand, when I wish to forecast, Ive heard that Bayesian Model Averaging Techniques often outpeform linear projections. On what basis can I choose between the two approaches? Thanks!

This is a more "practical" answer, vs a deeper theoretical one, or even a specific one.

Take it as "a broad answer to a broad question."

Also, since it is a "Bayes vs frequentist" issue, at least the last few suggestions must be taken tongue-in-cheek.

Step 1: Use simple models you understand to start. Lowering researcher error due to misapplication should always be a first-order priority.
Step 2: Learn more about each approach as much as possible. Implement the same estimator from both perspectives if at all possible, for purposes of really wrestling with what is going on at a deep level. Again, start simple. replicate things you already know work and are well established. (Can you write a basic regression from scratch using both approaches? This is much deeper than it may appear.) Get some good introductory texts. Don't be afraid of "lowly undergrad" texts for introduction. The "puppy book" on Amazon is good here, as is "Think Bayes" from Green Tea (both for initial intro to Bayes).
Step 3: Hopefully the earlier efforts will start revealing that some problems are more naturally approached in one framework vs. the other. When you start getting a good feel for this, re-examine the problem you want to address. Which approach seems appropriate to you now?
Step 4: realize that if you are using likelihood-based methods, Bayesian and frequentist are two sides of a similar coin. (Or to quote the ever-entertaining Cosma, "Bayesian estimation is a likelihood-based method in which the impact of facts and experience is blunted and smoothed by prejudice.," haha.)
Step 5: realize that there are even more varieties of estimation which don't neatly fit into previously-understood categories -- non-parametric, semi-parametric, semi-non-parametric -- and begin to despair that you will ever know or understand all there is to know about estimation, and which is the absolute best method for any one particular question.
Step 6: Enlightenment. Use the tool that you believe is best for the task, in your own honest opinion -- even if it is "best" based only on Step 1, which is simply "I use this because I understand it the best and I know I am not accidentally doing something extremely wrong."

So there you have it, "a broad answer to a broad question."

All that said, I really do like both Bayesian and frequentist approaches. A lot of things goes into deciding which to use. Some problems lend themselves well to one vs the other. Some audiences lend themselves to understanding one vs the other (recall that often effective communication of a concept can be very important). It's up to you to learn enough about both the methods and the problems you are trying to address to choose well.


How do I adjust for inflation? I will begin by saying that I have very little knowledge about economics, so don't make any presumptions.

My problem is that I want to calculate the actual value of gold in Swedish Krona(SEK) for the last 15 years, accounting for the overall inflation.
I have calculated some average prices as well as having researched the inflation rate of the Swedish Krona. The information is listed below.

Average Price/Gram
Year 2000: 81,21 kr
Year 2001: 89,90 kr
Year 2002: 96,54 kr
Year 2003: 94,65 kr
Year 2004: 96,89 kr
Year 2005: 105,63 kr
Year 2006: 144,82 kr
Year 2007: 149,54 kr
Year 2008: 179,83 kr
Year 2009: 236,83 kr
Year 2010: 279,83 kr
Year 2011: 324,20 kr
Year 2012: 363,68 kr
Year 2013: 299,28 kr
Year 2014: 275,83 kr
Year 2015: 315,10 kr

Yearly Inflation
Year 2000: 0,98%
Year 2001: 2,66%
Year 2002: 2,08%
Year 2003: 1,27%
Year 2004: 0,28%
Year 2005: 0,88%
Year 2006: 1,64%
Year 2007: 3,45%
Year 2008: 0,90%
Year 2009: 0,58%
Year 2010: 2,34%
Year 2011: 2,29%
Year 2012: -0,05%
Year 2013: 0,14%
Year 2014: -0,31%
Year 2015: 0,05%

Now, I request nothing more than a methodology. If you know of a somewhat accurate formula that would result in inflation adjusted values, please do share.
Please remember to also include an example of how you would make use of it. Thank you for your time and effort.

First it would be about the index used to take account inflation.

There are different indexes for calculating inflation, for example there could be by national prices for consumer or national prices for producer or other references as index calculations. Every index has its own methodology.

To take this example and crunch it with the numbers provided we'll assume the inflation was calculated with the National prices for consumer with a basic market basket of prices for swiss on the different periods and then the inflation was calculated by the change of values in the index for every period.

Taken this on account note that inflation is a percentual change of a wide selected prices or quantities change depending on the methodology.

So the inflation shown in your data comes from a change on index, the base index from wich the changes will be calculated generally has to be a period of realtive economic estability.

In this case we'll assume the base period as year 2000 because is the starting period of your data. The Index for base period will be 100.

  Year        2000
  Inflation   .98%
  Index       100

From there we will calculate the value of the other indexes.

To do this we aply the aritmethic formula

  (%Inflation+1)*(Prior year index)

    For example

  2001 Index will be = (1+.0266)*(100) = 102.66

  2002 Index will be = (1+.0206)*(102.66) = 104.77

  2003 Index will be = (1+.0127)*(104.77) = 106.11

The running totals are :

Having the index values then we can take the price of a commoditie or a service we want to take inflation on by account. The price with inflation is called "Nominal Price" and the one without inflation is "Real Price To do this we employ the arithmetic formula:

  (Nominal Price for Selected Year)*((Index of year selected)/(Index of base year))

    For 2000 Real Price = (81.21)*(100/100) = 81.21

    For 2001 Real Price = (89.9)*(102.66/100) = 92.29

    For 2002 Real Price = (96.54)*(104.77/100) = 101.15

The Running Totals would be

I hope the explanation was not that technical or that thedious, we can enter in more detail or less detail but i guess this would be right.

You can apply the same methodology for montly or quarterly basis

Now you can see than given the general level of prices on swiss economy 2012 was the Peak year on real prices for the selected period.

But 'ill add that given that we live on a world economy and the price of gold is fixed by international exchanges betweeen coutries a theoritical calculation of the real price involves currency exchanges and more theory.


influence of growing foreign demand for local products on the net exports in a fixed exchange rate regime Suppose we have a country $A$ with a fixed exchange rate regime.

Suppose that, suddenly, the foreign demand for $A$'s made products rose up.

What will happen to $X-IM$?

As far as I can tell $X$ grows and thus $GNP$ grows. Finally since $GNP$ grows and the exchange rate is fixed $IM$ grows. So, finally we can't say anything about $X-IM$?

Yes, $IM$ increases in the equilibrium, but, no, $IM$ cannot increase more than $X$ does.

Logical explanation: "$\Delta X < \Delta IM$ in the equilibrium" means that $A$'s GNP decreases. When GNP decreases, we have $\Delta IM < 0$ while $0 < \Delta X$ so we can't have $\Delta X<\Delta IM$. Contradiction! We can never have $X-IM$ decreasing in the equilibrium.

Mathematical illustration: Consider the model $Y=C+I+G+X-IM$, $C=a+bY$, $IM=c+dY$. Then the equilibrium is $Y = (a-c+I+G+X)/(1-b+d)$. When $X$ increases by $\Delta X$, $\Delta Y = \Delta X / (1-b+d)$ and $\Delta IM = d\Delta X / (1-b+d)$. Thus,
$$\Delta X - \Delta IM = \left(1 - \frac{d}{1-b+d} \right) \Delta X = \left( \frac{1-b}{1-b+d} \right) \Delta X > 0.$$


What purpose does general equilibrium serve in practice? What purpose does general equilibrium serve in practice?

How is it useful for people (since economics is supposed to assist politics, which is supposed to benefit people)?

Daron Acemoglu, in a paper called Theory, General Equilibrium and Political Economy in Development Economics, discusses the role of economic theory in empirical work in development economics, which investigates the causes of poverty and low incomes. He puts a special emphasis on general equilibrium considerations. He discusses why counterfactual analysis based on microdata that ignores general equilibrium issues may lead to misleading conclusions. In particular, depending on magnitudes of various effects, general equilibrium interactions can offset or even reverse sensible partial equilibrium conclusions.

The importance of general equilibrium effects in theory

Acemoglu considers three important types of general equilibrium effects, which are usually not estimated in partial equilibrium comparisons:

In response to large policy interventions or shocks, imperfect
substitution between factors and diminishing returns imply that
factor productivities and prices will change.
The same policy interventions or shocks can lead to endogenous technology responses.
There may be composition effects resulting from equilibrium substitution of some factors or products for others.

The importance of general equilibrium effects in practice

Consider the large increase in the relative supply of college-educated workers in the United States starting in the late 1960s. Given technology, this change in relative supply should have reduced the college premium. As is well known, the opposite happened in practice, and the college premium increased sharply from the late 1970s onwards. This was a consequence of the endogenous response of technology to the relative abundance of more skilled workers.
A large microeconometric literature shows that healthier individuals are more productive. On this basis, we would expect a greater aggregate productivity if health conditions improve. But one should take general equilibrium effects into account, since an increase in health conditions also increases population and longevity, and because of diminishing returns to capital and land, it may decrease labor productivity and may in fact reduce income per capita.

Other references

Townsend (2009) for a discussion of the role of general equilibrium analysis in development economics, with special emphasis on credit market issues;
Heckman, Lochner and Taber (1998) for a discussion of general equilibrium issues in the analysis of the effects of technology on wage inequality.


Macroeconometrics:How to measure capital depreciation? In an econometric sense, how do macroeconomists measure capital depreciation $\delta$ in the formula $K_{t+1}=K_t(1-\delta)+I_{t+1}$.

What data would I need and what kind of regression would I run?

I tried to empirically estimate the rate of depreciation a couple of years ago. I gave up! Following a large literature on growth, I assumed a constant rate of depreciation eventually. I did not have data for all the variables needed to estimate the model.

Take a look at Nadir and Prucha's modelling approach.
In chapter 7 of this book, the authors develop an accelerator model of investment. You may then calculate the depreciation rate using the parameter estimates of the model.


Where did the Greek cash go? Apparently (1, 2) Greece is running out of cash right now. I am wondering where the cash went.
One option that I can image is people withdrawing funds and taking them out of the country but this cannot be the core of the problem, can it?
The other possibility would be people just withdrawing money and saving it but eventually the money will have to be spend because people need to buy things so the problem might solve itself?
What am I missing here?

Relaying from @lumi's answer to this question (not a duplicate though) we see that

As @lumi writes

  ..."if we look at the Greek money supply figures, and we know that that there are essentially three ways for liability deposits to
  decrease as shown - loan repayment, loan write-off and cash
  withdrawals, we can deduce pretty easily that Greece is experiencing a
  massive outflow of cash. We can assume that some of it is cash
  withdrawals particularly recently, but some of it also debt repayment
  to debt owners outside of the Greek banking system."

For the last months, I would increase a bit the emphasis on "people withdrawing funds and taking them out of the country".


Is inequality increasing? Does it correspond to increasing social harm? I have a series of related questions.

My understanding is.

Economic wealth (ie. GDP) per capita is increasing around the world. We can refer to this as 'the pie is getting bigger'.
At the same time, inequality (the distribution of the pie) is also increasing.
Economic inequality is correlated with social harm, such as crime.

The questions I have is:

Is it correct to say that inequality is correalated with social harm? What do we mean by this?
Is inequality increasing consistantly around the world? Or is it decreasing in some places?
If it is increasing, are we also seeing a corresponding increase in social harm?

Regarding your first question:

Usually public economists assume that utility is a concave function of income. Therefore, the utility gain from one additional dollar of income is lower for a rich person than for a poor person. This is consistent with risk averse behavior of individuals which we often observe when individuals face risky choices.

If utility is concave in income and the total income is fixed, then the optimal utilitarian distribution would be that all individuals receive the same amount of income. Any deviation from this optimal income generates some loss in social welfare, which you may call "social harm". However, in many cases the optimal utilitarian distribution of income will still be unequal if a higher level of total income can be reached by allowing for inequality. Reasons for this may be asymmetric information or labor market incentives. Thus, higher inequality alone does not necessarily mean lower social welfare. But higher inequality with lower or equal income means lower social welfare in a utilitarian framework.

Disclaimer: notice that if your definition of social welfare is not utilitarian, none of this applies.

Extended reading: in case you are interested in this subject, I recommend you to read an introductory textbook on public finance or public economics.

For a more detailed survey of how social welfare and inequality are related, see:

Lambert, Peter J. "Evaluating impact effects of tax reforms." Journal of Economic Surveys 7.3 (1993): 205-242.


Is it right to derive social marginal benefit by adding individual prices instead of quantities? I come across a lecture material on market functions and externalities that makes me quite confused. Here's the setup: Two stores are located next to each other. If one installs a camera system in front of the shop to provide security supervision then it will cover both stores (hence both will be benefited).

The willingness-to-pay functions of these stores for that camera system look like this:

The question is to draw the social marginal benefit function $(SMB)$ of adopting the camera. This is how it is done:

What it does is to add vertically the prices for each level of quantity. Hence for the first part where $P_{WTP2} \ge  0$, $SMB=WTP1+WTP2$. Then after that point $SMB=WTP1$.

What I usually do and see for deriving total demand is to add up quantities for each level of price. Is it because now they are WTP and SMB functions that it's done the other way around?

Vertical summation of the individual marginal benefit curves is the correct way to find social marginal benefit if the camera system, so far as the two stores are concerned, is a public good. Horizontal summation is the way to find total market demand for a private good.
A good is a public good if it satisfies two conditions.  One is non-excludability: that would be satisfied if neither store can prevent the other from benefiting from the camera system.  The other is non-rivalrousness: that would be the case if neither store's use of the system limited the other's use of it.
Even if the camera system covers both stores, it could be that it does not fulfil both these conditions. For example, one store might control its use and limit or charge for the other's access to photos / film taken.  In that case the non-excludability condition would not be met.
Let's suppose however that the conditions for a public good are fully met. Let's also suppose that "quantity" of the camera system can meaningfully be defined (this does not seem straightforward). In that case the quantity must be the same for both stores, and the social marginal benefit at that quantity is the sum (graphically the vertical sum) of the marginal benefits to each store. The principle of vertical summation is implicit in the Samuelson condition for optimal provision of a public good, according to which the quantity provided is optimal where the sum of the marginal benefits equals the marginal cost of provision (Samuelson 1954, see equation 2 p 387 and explanation p 388).
It possibly makes vertical summation seem paradoxical when the vertical axis of the diagram is labelled P for price.  It would be more correct I suggest to label it MB for marginal benefit, as in this diagram illustrating the Samuelson condition.


What do game theorists think of behavioural economics? Though behavioural economics has been popularized over the past decade in both industry and pop culture, it seems that not much has changed with regard to the standard economic definition of  rationality.

This is because for the most part (in my view and experience with my professors) it seems to have been irrelevant or unconvincing for the game theorists who deal with very similar theory to the behavioural economists. This is where a "super-micro" theory of consumer and prodoucer behaviour is generated and analysis takes place.

Now im not a huge fan of behavioural economics, but points like discontinous time preferences and prospect theory are useful for adding a degree of realism to models.

This all being said, (except for a few articles on behavioural game theory) why has a large chunk of young game theorists decided to stay away from "behavioural" elements?

Disclaimer: My academic coming of age was in an environment where behavioral economics played only a minor role. My research is theoretical, both "behavioral and non-behavioral", economics.
I believe it is incorrect to say that game theorists (or economists) in general are not convinced by behavioral economics. You can easily see it is considered to be quite relevant by finding Thaler, Shiller, Kahneman and Selten (yes!) in this list. I also do not feel like young economists stay away from behavioral economics.
That being said, there are some (not necessarily good) reasons why many economists have a certain dislike for behavioral elements. One is that students often criticize standard models, taking a "behavioral approach" that in the end simply breaks down to "why should I study Bayesian Nash equilibrium if people are not clever enough to make such calculations anyway" -- this is a fair point, but if you want to have game-theoretic analysis or predictions you should base these on some closed system. I don't find "people are stupid and hence anything can happen" particularly illuminating. Luckily, this is not what modern behavioral economics does although some critics of "standard economics" seem to think this. Adding behavioral aspects often makes a model more complicated to analyze, which the same kind of critics then complain about again.
Behavioral economics often simply incorporates systematic "errors" (?) such as time inconsistency, loss aversion, social preferences, overconfidence, probability weighting and so on. This is reasonable because there is almost undeniable evidence of such biases.
An issue that I sometimes have with such models is that they don't deliver surprising insights. For instance, if you assume that a decision maker is overconfident and then find that this decision maker has a preference for riskier prospects than a "rational" decision maker without this bias, I am not very moved. The result just reflects the assumption. When I see the magician putting the rabbit in the hat, I don't clap when he pulls it out. I am not saying that every publishable result needs to be surprising, but it needs to add to our current knowledge. Serious economists know the limits of their models and are aware that changing the beliefs in some direction alters behavior.
By far not every behavioral paper is like that: many, in fact, expand our current knowledge and our theoretical toolbox tremendously. However, we then have many little toy models that are applicable in only some environments and not so much in others. In contrast, the rational benchmark is always applicable and usually comes with a normative touch: this is how utility maximizing agents are supposed to behave. Maybe it is just me, but I sometimes feel that this is also a point that many critics of "standard assumptions" actually mean: people should be more social and less egoistic and firms must not only maximize profits. That is, this normative aspect gets mixed with the analysis. However, you cannot blame behavioral economics for people having wrong perceptions about it.
Moreover, you usually cannot exploit homo oeconomicus. In contrast, the behavioral agents have systematic flaws by design and therefore they will be exploited. Behavioral agents often introduce a Dutch book. As a result, behavioral assumptions often only make sense in decision theory (i.e., individual decisions without interaction), but not so much in game theory (i.e, strategic interaction) because no equilibrium exists when a money pump is present. I guess what I am trying to say is that while some behavioral biases are reasonable and hard to deny, they -- if all their logical consequences are taken into account -- can still produce unreasonable predictions.A nice paper making such a point is the AER paper "Until the Bitter End" by Ebert & Strack discussed here.


In a bi-weekly payroll cycle, there are only ever 26 pay dates, correct? In a bi-weekly payroll cycle, there are only ever 26 pay dates, correct?  It never jumps to 27 or falls to 25?

I'm asking this question in a leap year, and I know that calculating the first pay date has to do with selecting a day of the week to run payroll on, and then selecting the first occurrence of that day in the year.

You mean 26 pay dates per year, right?

Then no, because a year is 365 days long and a 26 biweeks are $26 \cdot 14 = 364$ days long. Therefore any year that starts with a Friday (which is usually a pay date) should have 27 pay dates. However if Jan 1 is a holiday (depends on where you live), the paydate is probably moved to Dec 31 so the preceding year would have 27 pay dates.


Why can principal not be lost, if a bond fund is held for a period equal to its duration? [Source:] Which leads us to the key message for investors: as long as your time horizon is at least as long as the duration of your bond fund, you won’t lose any capital.
You’ve probably heard people say they prefer individual bonds to bond funds, because as long as they hold on until maturity, they won’t lose principal. Well, the same is true if you hold a bond fund for a period equal to its duration. You can be sure that XBB will not have a negative total return over any period longer than 6.3 years: any price decline from rising interest rates will be offset by higher coupons within that time frame. In fact, history suggests the recovery is likely to be more swift than that: even a three-year period of negative bond returns is extremely rare.

1. Are the bolded clauses true absolutely? To wit, capital loss is identically (and NOT just near) 0?
2. Why are the bolded true?

A bond is basically a contract between the debtor and the creditor, which roughly usually says something like

You, the creditor, agree to lend me \$10,000 today. In return, I, the debtor, will make the following payments to you (or, in the event that you sell this bond, to the new owner of the bond):

\$500 every year for the next five years.
An additional final payment five years from now to return your $10,000.

That the original invested principal is returned to the investor at the end of the bond's life is fundamentally written into the terms of the agreement. Thus, provided you don't sell the bond before its maturity, the terms of the bond guarantee that your original principle will be returned. This means that, under normal circumstances, capital loss will be zero if the bond is held to maturity.
Here are some reasons why capital loss might be non-zero for a bond investor:

The debtor defaults (for example, by declaring bankruptcy) and ceases making the payments required by the bond.

A minority of bonds have terms that require an amount different to the original invested principal be returned at the end of the bond's lifetime. In this case, the amount to be returned will be described in the bond, but might be less than the original investment.

or, of course, you sell the bond before its maturity date, and the amount for which you sell it is less than the amount you originally invested.

It is also important to remember that even if nominal capital loss is zero, inflation will erode the value of the invested capital over the lifetime of the bond. Thus, the real value of the returned principle will be lower than the value of the amount invested.


What are the benefits of reducing zeroes from currency (eg: Iran)? Recently, Iran decided to replace its old currency to a new one.

Quoted from Wikipedia

  A proposal has been agreed to by the Iranian parliament to drop four zeros, by replacing the rial with a new currency called the toman

I have started studying basics of economics very recently. And I assumed the previous and new situations to be something like:

Imaginary Situation 1

Value of each coin = $1
Total coins = 1000
Value = $1000

Imaginary Situation 2

Value of each coin = $10
Total coins = 100
Value = $1000

Since, there are less number of coins it might be easier to perform transactions. That's my assumption of the possible benefits of this scheme.

Question:

Is the assumption correct?
Any other benefits of such a scheme?

Instead of the word "benefit" I prefer to use the old Saxon word "bebebebebebebebebebenefit". Some people say using the shorter word has bebebebebebebebebebenefits, but I don't see any.

P.s.: coins and notes of extremely small value are usually removed from circulation. It is unlikely the average person would ever pay with 1000 coins.


Homothetic function versus straight Engle curve Homothetic functions can be defined as follow (Green, 1964, p.49):

  If two figures A and B are "homothetic", or "similarly placed" with reference to a point P [for example, the origin], then for any two straight lines PQR and PQ'R' through P, cutting A in Q and Q' and B in R and R', the ratios PQ/PR and PQ'/PR' are equal.

Additionally, consider the case where "each set of points, in output of commodity space, at which marginal rates of substitution are constant, is a straight line". This is, a straight Engel curve.

Are these two definitions equivalent, in the sense that each implies the other?

Or to put it differently, can we define:

an homothetic function as a function which yields straight Engle curves? and
a function with straight Engle curve as an homothetic function?

I am a bit confused, because in a text I'm reading both are used at different times, without mention their direct relation.

The answer to me seems to be yes. However, Maybe in special cases like when an homothetic function does not start in the origin, or where Engel lines are horizontal, the relation might break. If so, would these be the only exceptions, such that we could conclude that, for most of our purposes, they are indeed equivalent?

A general result using indirect utility functions, is that expenditure linear in income ("straight-line" Engel curves) is produced if and only if the indirect utility function has the "Gorman-polar form".

The original paper is

Gorman, W. M. (1961). On a class of preference fields. Metroeconomica, 13(2), 53-56.

...where we read that a linear Engel curve is produced when we have a homothetic utility function (Gorman does not use the word, he writes "homogeneous" and in a footnote, enhances that to "function of a homogeneous function", which is how a homothetic function is defined).

But the reverse does not hold:homotheticity is sufficient for linear Engel curves, but not necessary.

For example, linear Engel curves are also produced by a utility function that reflects "quasi-linear" preferences, $U(x,m) = u(x) + m$, which are not homothetic.

A homothetic function is properly defined mathematically as follows:

Let $f(\mathbf x)$ be a homogeneous function of some degree, and let $g$ be a function with non zero derivative. Then $g[f(\mathbf x)]$ is called a homothetic function. In economics, we usually require for the derivative of $g$ to be strictly positive, so that we remain inside economically meaningful territory.


How is momentum justified as a common risk factor? Momentum as a common risk factor?

This question is partly a follow-up to another question found here. In this other question it was noted in momentum is difficult to explain as a common risk factor in factor pricing models like the intertemportal capital asset pricing model (I-CAPM) or arbitrage pricing theory (APT). In these models, it is assumed that exposure to one of these factors represent exposure to some sort of undesirable risk. In this question, I'm trying to understand how to interpret exposure to momentum as exposure to some form of common risk. In particular, I'd like to know

Who was the firm to include momentum as a risk factor? What was the explanation?
It seems like momentum is often attributed to behavioral over- or under-reaction. (This could be irrational or maybe even rational-overreaction, I suppose---right?) Is there an interpretation that rationalizes momentum? (I mean one that gives an explanation where exposure to momentum is a bad thing.)

For some reference:

Jegadeesh and Titman (1993) review some explanations of momentum,
including overreaction to information, relation to size effect and systematic risk, short-term price pressure, lack of liquidity, delayed stock price reaction to common factors

The paper argues that the "relative strengths" premium (the strategy of buying past winners) is not due to exposure to systematic risk, cannot be attributed to "lead-lag effects that result from delayed stock price reactions to common factors," but that the evidence seems to be consistent with delayed price reactions to firm-specific information.

  Stocks in
  the winners portfolio realize significantly higher returns than the stocks in
  the losers portfolio around the quarterly earnings announcements that are
  made in the first few months following the formation date. However, the
  announcement date returns in the 8 to 20 months following the formation
  date are significantly higher for the stocks in the losers portfolio than for the
  stocks in the winners portfolio.

  The evidence of initial positive and later negative relative strength returns
  suggests that common interpretations of return reversals as evidence of
  overreactlon and return persistence (i.e., past winners achieving positive
  returns in the future) as evidence of underreaction are probably overly
  simplistic.

The Jegadeesh and Titman (1993) paper is usually considered Original Source, though I'm sure you could find something earlier that looks similar if you looked hard enough.

I don't think there is a satisfactory explanation. Momentum is not correlated with macroeconomic variables, it does not seem to reflect persistent exposure to other (known) sources of risk, and is driven almost entirely by the 7 to 12 months before stocks are chosen as part of a momentum portfolio. This all makes it extremely difficult to put a solid theoretical model to it that doesn't have a behavioral aspect.

Recently (mid 2009), there was a momentum crash in the market. Daniel and Moskowitz argue that momentum has rare "panics" periods like this, where crashes occur for momentum. So during normal periods investors may be being compensated for this risk. The question becomes "what are these crashes caused by?" The authors make the case that following a period of declining stock values, momentum portfolios will be long a lot of low market beta stocks, and short a lot of high beta stocks. If there is a sudden rebound in the market, momentum strategies will experience a crash. So this is at least suggestive of a direction for more formal modeling, but they ultimately suggest a behavioral interpretation may be necessary as well.

If you have not seen it, Jegadeesh and Titman (2011) review the evidence for momentum in more detail, and explore behavioral explanations for momentum, of which I have little knowledge.


How exactly is a Bertrand equilibrium defined? By 'given the price set by other firm' does this mean the firm knows its competitor's exact output? I had read 3 textbooks, but one describes that they have a precise expectation of what their competitor's output might be but this sentence here sounds as though the firm is informed about their competitor's exact output. Could someone explain? Thanks

What do we mean by "makes the best decision based on what the other player has decided to do"

Your question touches a little bit on the philosophical foundations for Nash Equilibrium. As you know, a Nash Equilibrium occurs when each player chooses the best response to the strategies chosen by others. In other, words, they act as if they know what the strategies of the other players will be, and play the best response accordingly—even if all strategies are chosen simultaneously.

Why does it make sense to model people behaving as if they knew everyone else's strategy? The whole point of looking for an equilibrium of a game is to try to find a way to 'solve' the game such that the solution gives a plausible prediction about how people might play the game. Suppose we have identified a way to compute the solution. If we can compute the solution then the players can compute the solution too (using the same method), and figure out what everyone's equilibrium strategy will be. If the players compute the solution and discover that their equilibrium behavior in the solution does not maximise their own payoff then they will refuse to play the solution! So any solution concept that (a) players can use to solve the game and (b) has one or more players behaving sub-optimally will be self-defeating. Flipped on its head, any solution concept that players are actually willing to go along with must necessarily involve them playing a best response—i.e must be a Nash Equilibrium.

One concise way to phrase this is that a player's strategy can be accurately predicted by the other players (using the same solution concept we are using the solve the game)! Note that this is slightly different to what you wrote in a comment ("the competitor's action can be accurately predicted"). We can use the equilibrium to predict strategies, but might not be able to predict actions (because the strategy might be a mixed strategy, meaning the action is random).

What is a Bertrand Equilibrium

Recall that to set up a static game of complete information we need to know (1) the players, (2) the actions, and (3) payoffs. To turn to your question, a Bertrand Equilibrium is just another name for the Nash Equilibrium of the following particular static game of complete information:

There are N firms (players)
Each firm's action is to choose a price, $p_i$
The payoffs are that the lowest price firm serves all consumers and earns $p_i-c$ per-consumer (where $c$ is the marginal cost). If $m$ firms tie at the lowest price then each of them serves $1/m$ of the consumers. firms that serve no consumers earn zero profit.

So, in a Bertrand equilibrium, using the principle of Nash, each firm acts as if it knows the others' price and chooses the best price in response.

Characterising the Bertrand Equilibrium?

To find the Nash ("Bertrand") equilibrium of this game, we have to find a set of strategies (prices) such that each firm is playing a best response (i.e., such that no firm could profit by changing its price).

Claim There is no equilibrium in which the lowest firm's price is bigger than $c$.

To prove this claim, suppose on the contrary that $\min p_i>c$. We will show that at least one firm can profitably change its strategy (and is therefore not playing a best response). Order the firms so that $p_1\leq p_2\leq p_3\leq\ldots\leq p_n$. There are two possibilities:

All firms charge the same price (so $n$ ties at the lowest price). Then $n$'s profit is $(p_n-c)/m$. But suppose $n$ reduced its price by some amount $\Delta$. Then it would have a price strictly below anyone else and earn profit $p_n-\Delta-c)$. If $\Delta$ is small enough, this is a strict increase in $n's$ profit.
$p_n>p_1$ so $n$ does not attract any consumers. $n$'s profit is zero. But $n$ could deviate to choosing a price of $p_1-\Delta$, yielding a profit of $(p_1-\Delta-c)$. Provided $\Delta$ is small enough, this is positive so $n$'s profit has increased.

Claim At least two firms must set $p=c$ in equilibrium. All firms earn zero profit.

To prove this claim, start by noticing that if some firm sets $p_i<c$ then at least one firm will make negative profits and would prefer to deviate to $p=c$ (thus guaranteeing zero profit).

Suppose that fewer than two firms set $p=c$. By the previous claim, we know at least one firm sets $p=c$. This means precisely one firm sets $p=c$, while all others set $p>c$. But then the firm that sets $p=c$ earns profit $(p-c)=0$, but could get positive profit by increasing $p$ (just a little so it is still the cheapest firm). Because there is a profitable deviation, we can't have an equilibrium with fewer than two firms setting $p=c$.

Lastly, we know the firms that set $p=c$ earn zero profits. We also know that no firm sets $p<c$ so any firm not setting $p=c$ must set $p>c$, meaning it attracts no consumers and also makes zero profit.


who is responsible as international court to solve economic conflicts between countries and is it obligatory? you know turkey was one of the 7members of F-35 joint strike fighter program..its engine is being manufactured in turkey.turkish government made all legal obligations to program and paid 35 billion dollars to budget pool so far but american congress decided to end delivery of 200fighters to turkey illegally by trump's decision last week..
so i heard a news on cnn turk this evening that turkey minister of defence is going to moscow to negotiate agreed terms to buy 300 SU-57 stealth fighter from sukoi instead next month..i think this will be important step of turkey's way to waited exit from NATO after S400 agreement of last year..it is called 'bricst-it" in turkey at the moment..it is possible to join to BRICS instead of NATO&EU for turkey..but in continue of news,there will be a government decision of turkey to prosecute united states government worth 1,8 trillion dollars sum plus with all turkey's army lost from F-35 program and legal compensations and interest rate of 20 years..ok here is my question..which international court may look at this case and is decision obligatory?

As far as I am aware, there is no "general" international court for economic affairs.
The ICJ (Interntaional Court of Justice) is the judicial organ of the UN, and it in fact decided few cases with economic background, but usually in the context of other questions of international law.
The closest to what you are looking for is probably WTO dispute settlement system, which is described here: https://en.wikipedia.org/wiki/Dispute_settlement_in_the_WTO
In theory, it is for WTO members only, but in practice, 96 % of global trade is covered under WTO.

Other possible court is the International Court of Arbitration: https://en.wikipedia.org/wiki/International_Court_of_Arbitration
It has no clearly defined jurisdiction, instead, the parties must agree to bring the case to it, but it is becoming increasingly popular.

Last, but not least, an ad hoc panel can be established specially for given dispute.


King-Plosser-Rebello Preferences: Scale leisure KPR preferences are given by

$$ U(c, l) = \frac{\left(cv(l)\right)^{1-\sigma}-1}{1-\sigma}$$

with concave increasing $v$ and $c$, $l$ denoting consumption and leisure. In the limiting case of $\sigma\to 1$, we receive the standard additively-separable preferences

$$ U(c,l) = \log c + v(l)$$

In the latter case, if I want to scale the relevance of leisure in total preferences, I can rewrite the preferences as $\log c + Av(l)$, and use $A$ for this matter.

How do I do that - scale leisure - in the general case?

If I rewrite the nominator as $(c A v(l))^{1-\sigma}$, it is not clear to me whether $A$ is scaling $c$ or $l$ (probably neither).

If you specify

$$ U(c, l) =\left(c\cdot [(v(l))^A]\right)^{1-\sigma}$$

then

$$\ln U(c,l) = (1-\sigma)\ln c + A(1-\sigma)\ln[v(l)]$$

...and $A$ can be seen as regulating the relative weight of leisure, for any $v(l)$.


Do I make correct conclusions about international trade (and ways to increase GDP) from the expedenture approach formula for GDP? (Please correct me if my reasoning is wrong somewhere)

Net Exports is a part of GDP and consequently its increase or a decrease also causes GDP to increase or decrease, assuming everything else stays equal. Higher import means lower Net Exports, while higher export means higher Net Exports.

Let's take two countries, country A and country C. Let's also assume that all internatinal trade happens between these two countries. If country C will increase its exports to country A, then GDP of country C, assuming everything else holds equal, will increase due to increased Net Exports part of its GDP equation. While for country A its GDP will decrease due to its decreased Net Exports.

From this it seems logical to conclude two things:

1.International trade is a zero-sum game from point of view of increasing/decreasing GDP.

2.If we want to increase our GDP, then among other things we must consider decreasing our import and increasing our export. For an example, we could outright ban some of foreign goods or raise tariffs, thus forcing our population to decrease consumption of foreign goods, this way increasing our GDP.

The only problem I see so far is that in the real world such policies can lead to trade wars, with other nations banning goods from our country and rasing tariffs on our goods, thus hurting our exports.

Your conclusion would be correct in pure endowment economy where there is no production $Y(L) =0$ (assuming only labor as factor of production) and everyone has the same endowments. In an endowment economy where there is no production and GDP is just dropped on both home and foreign country trade is indeed a zero sum game as $M$ subtracts from your endowment since $Y=C+I+G+X-M$ (Although its worth noting that in the long run trade must balance $X-M=0$ for reasons relating to exchange rate, savings, investments and movements of factors of production - but since it is not directly related to your question I wont go in depth into that)

However, real life economies are not pure endowment economies. Production and output $Y$ depend on how efficiently we can produce output. In real life trade allows countries to produce more i.e. have higher $Y$ and thus GDP than without trade.

Consider simple Ricardian model where you have 2 countries utopia and neutopia. Both have 100 labor $L$ avaiable. Assume there are two products wine $w$ and cloth $c$. Utopia can produce 1 unit of $w$ for 2 units of labor and 1 unit of $c$ for 1 unit of labor and neutopia $1w$ for $1L$ and $1c$ for 2$L$. Now if they dont trade with each other and if they both split their labor supply equally between wine and cloth Utopia gets to consume $25w$ and $50c$ and neutopia $50w$ and $25c$. Now if instead of autarky they decide to trade and Utopia will specialize only in cloth it can produce $100c$ and Neutopia if it specializes in wine can produce $100w$. Now both countries can trade lets say 50 units of wine for 50 units of cloth and now both countries have $50w$ and $50c$. Both countries expand their production possibility frontiers by engaging in trade and trade is not zero sum as both countries are better off.

Hence, as a result of comparative advantage $Y$(trade) > $Y(autarky)$. So once you allow in the identity $Y=C+I+G+X-M$ the output $Y$ to be increasing function of country's  PPF its not correct to say that trade is zero sum game.

Of course, in a classroom setting its fine to apply national identity just to some pure endowment economy where your zero-sum conclusion about trade is correct. Or you could even allow for some production function but assume all countries have exactly same production function and exactly same resources endowments and there can be no economies of scale etc. where there is no reason for trade. But you are making a mistake by using a simple identity that ignores comparative advantage, factor endowments etc. and just making conclusions about international trade in real world.


Utility maximization question setting up. Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$

    Assume the agent's income is $y=5$. The price of one unit of good $1$ is $p_1=1$. For each unit of good $1$ the agent buys, he qualifies to buy up to one unit of good $2$ at an additional price of $p_2=1$. In other words, to buy one unit of good $2$ the agent has to first buy one unit of good $1$. The agent must consume everything he buys. Using this information, sketch the feasible set. Is it convex? Derive the utility maximizing bundle.
    How does your answers to question 4. change if the agent does not have to consume everything he buys ("free disposal")?

For question (4)

The budget constraint is

$$x_1p_1+x_2(p_1+p_2)=y$$

My question is

1) my budget constraint is true?

2) what is the difference when budget constraint allow free disposal and not allow?

I don’t understand the part (5).

My solution also is posted.

Thanks a lot.

Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$

    Assume the agent's income is $y=5$. The price of one unit of good $1$ is $p_1=1$. For each unit of good $1$ the agent buys, he qualifies to buy up to one unit of good $2$ at an additional price of $p_2=1$. In other words, to buy one unit of good $2$ the agent has to first buy one unit of good $1$. The agent must consume everything he buys. Using this information, sketch the feasible set. Is it convex? Derive the utility maximizing bundle.
    How does your answers to question 4. change if the agent does not have to consume everything he buys ("free disposal")?

For Q 4 :

Utility maximization problem of the consumer is :

\begin{eqnarray*} \max_{x_1, x_2} & \ \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ \ x_1+x_2 \leq 5 \\ \text{and} &  \ \ 0 \leq x_2 \leq x_1  \end{eqnarray*}

Here is the constraint set of the consumer, along with a few indifference curves:

Observe that the constraint set is convex and the consumer does not spend all his income in optimum. His optimal consumption bundle is $(x_1, x_2) = (1,1)$.

For Q 5 :

Utility maximization problem (with free disposal) of the consumer is :

\begin{eqnarray*} \max_{x_1, x_2, b_1, b_2} & \ \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ \ b_1+b_2 \leq 5 \\  &  \ \ 0 \leq b_2 \leq b_1  \\ \text{and} &  \ \ 0 \leq x_1 \leq b_1, 0 \leq x_2 \leq b_2\end{eqnarray*}

Here $b_1$, $b_2$ denotes the amount of the two commodities bought by the consumer, and $x_1$, $x_2$ denotes the amount consumed. In this case, the consumer will try and maximize his consumption of commodity 2 $(x_2)$ by buying as much amount of commodity 2 $(b_2)$ as he can. Clearly, the solution to this utility maximization problem is
$b_1 = b_2 = x_2 = 2.5, x_1 = 0$.

For Q 4, here is a way to solve the optimization problem using Lagrangian method :

Given the utility maximization problem of the consumer :

\begin{eqnarray*} \max_{x_1, x_2} & \ \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ \ x_1+x_2 \leq 5 \\ \text{and} &  \ \ 0 \leq x_2 \leq x_1  \end{eqnarray*}

We set up the Lagrangian as follows:

$\mathcal{L}(x_1, x_2) = \dfrac{x_2}{(1+x_1)^2} - \lambda(x_1+x_2-5)  +\mu_1(x_1-x_2)+ \mu_2x_2 $

Necessary conditions for optimality are as follows :

$\dfrac{\partial \mathcal{L}}{\partial x_1} = \dfrac{-2x_2}{(1+x_1)^3} - \lambda + \mu_1 = 0$

$\dfrac{\partial \mathcal{L}}{\partial x_2} = \dfrac{1}{(1+x_1)^2} -\lambda - \mu_1 + \mu_2 = 0$

$x_1+x_2 \leq 5$, $\lambda \geq 0$ and $\lambda(x_1+x_2-5) = 0$

$x_1 \geq x_2$, $\mu_1 \geq 0$ and $\mu_1(x_1-x_2) = 0$

$x_2 \geq 0$, $\mu_2 \geq 0$ and $\mu_2x_2 = 0$

Solving the above system, we get

$x_1 = 1$, $x_2 = 1$, $\mu_1 = \frac{1}{4}$, $\mu_2=0$, $\lambda = 0$

Alternatively for Q 4, the problem can also be converted to a single variable optimization problem. This is because the individual will always consume equal amounts of $x_1$ and $x_2$ in equilibrium. Substituting $x_2=x_1$ yields :

\begin{eqnarray*} \max_{x_2} & \ \ \frac{x_2}{(1+x_2)^2} \\ \text{s.t.}  &  \ \ 0 \leq x_2 \leq 2.5  \end{eqnarray*}

Differentiating $\dfrac{x_2}{(1+x_2)^2}$ with respect to $x_2$ gives the following necessary condition

$\dfrac{(1+x_2)^2 - 2(1+x_2)x_2}{(1+x_2)^4} = \dfrac{1-x_2^2}{(1+x_2)^4} = 0$

which yields $x_2 = 1$, and the corresponding value of $x_1 = 1$.


Difference between Outflow of capital and Flight of capital I may sound stupid to ask this question but pardon me for that. What's the different between Outflow of capital and Flight of capital? I googled about both terms but couldn't find how they differ from each other.
Edit : Let me quote the definitions from investopedia.

Capital outflow is the movement of assets out of a country. Capital outflow is considered undesirable and results from political or economic instability.
Capital Flight is a large-scale exodus of financial assets and capital from a nation due to events such as political or economic instability, currency devaluation or the imposition of capital controls.

From these two definitions what I can conclude is that Capital flight is a large scale  capital outflow means these two are same things, the difference is just of the scale.Am I correct or is there any other difference ?

This is actually pretty tricky, as "capital flight" can be a contested term itself. Here's a distinction provided by Darryl McLeod in the Concise Encyclopedia of Economics, edited by David Henderson and provided online by the Library of Economics and Liberty:

  There is no widely accepted definition of capital flight. The classic use of the term is to describe widespread currency speculation, especially when it leads to cross-border movements of private funds that are large enough to affect national financial markets. The distinction between "flight" and normal capital outflows is thus a matter of degree, much like the difference between a "bank run" and normal withdrawals.

There's some discussion by Frank R. Gunter in the International Encyclopedia of the Social Sciences, edited by William Darity, Jr., about the fuzziness of the term "capital flight" as related to motivations or the nature of the transactions:

  Capital flight is generally defined as an outflow of funds from a country motivated by an adverse change in the country’s economic, political, or social environment. Some believe that this definition is too broad. They distinguish between outflows that reflect “normal” international diversification motivated by marginal changes in risk-adjusted returns and funds fleeing or propelled across national borders during a crisis. According to this view, only the latter category represents true capital flight. Related definitions restrict capital flight to short-term speculative outflows—“hot” money—or to an outflow of illegal transactions only.

So, it seems that scale, motivations (adverse events, crisis), and the nature of the transactions could all be distinctions to some degree between capital outflow and capital flight.

I hope this helps!


Dynamic optimisation Consider a simple dynamic consumption-saving problem. A solution can be characterised using a Lagrangian approach that generates a set of first order conditions and some boundary conditions.

An alternative approach is to set use a Bellman equation. What I don't understand is how come a Bellman equation approach does not seem to use information regarding the boundary conditions in the solution methodology? That is, a Bellman equation does have a tradeoff between time period t and t+1 but not some overall budget constraint does not seem to be represented.

The Dynamic Programming ("Bellman' Equation") formulation incorporates the terminal boundary condition ("transversality conditions") needed in case we use the Lagrangian/Euler equation formulation. The initial conditions are still needed in both approaches.

To see why, consider the problem

$$\text{max}_{\{x_t\}} \sum_{t=0}^{\infty} \beta^t U(c_t)$$

$$s.t.\;\;\; a_{t+1}= (1-r)a_t + w - c_t, \;\;\; a_0 \; \text{given}$$

Solving the budget constraint for $c_t$ and inserting into the objective function, we have compactly

$$\text{max}_{\{a_{t}\}} \sum_{t=0}^{\infty} \beta^t U(a_t, a_{t+1}),\;\;\;\; a_0 \; \text{given}$$

where we treat the stock variable next period as our decision variable.
Now for the optimal sequence, $a^*_{t+1}$ must solve the following problem:

$$\max_y \big[U(a^*_t, y) + \beta U(y, a^*_{t+2})\big],\;\;\; y\;\; \text{feasible given}\;\; a^*_t$$

In the words of Stokey and Lucas "a feasible variation on the sequence $\{a^*_t\}$ cannot lead to an improvement on an optimal policy".

Then we get the first order conditions (differentiating with respect to $y$ essentially)

$$U_2(a^*_t, a^*_{t+1}) + \beta U_1(a^*_{t+1}, a^*_{t+2}) =0$$

where the subscripts $1,2$ denote partial derivatives. This is a second-order difference equation, and so it requires two boundary conditions. We have already one of them, the given initial value of wealth $a_0$. We need one more, hence the terminal, "transversality" condition.

In contrast, using the Dynamic Programming approach we arrive at a first-order difference equation, and we do not need an additional boundary condition.

For the full exposition, see Stokey & Lucas p. 97-100.


Is car accident/theft a fair bet? A person with a current wealth of 100,000 who faces the prospect of a 25% chance of losing his or her 20,000 automobile through theft during the next year. Since there is no upside to this event and E(gain) = 0.25*(-20000) + 0.75*0 = -5000
This would mean we all take insurance against unfair bets in life. Is this conclusion correct?

You are confused about how insurance works, and thereby why people take insurance. When someone buys insurance against a certain even, s/he has to pay a fee (called a "premium"), which entitles the person to be compensated for the loss in case the negative event happens. Unexpectedly, insurance comes in many ways. For instance, the compensation can be of the total loss (in your example, 20,000), or could be partial. The size of the premium usually reflects this difference.

Now, in the example you give, what you have calculated is simply the expected value of no taking insurance, because (i) there is no premium, and (ii) if the bad event happens, the car owner losses the full amount. Naturally, no taking insurance against a negative event will always give you a negative expected value (your "unfair bet" conclusion), as long as that event has positive probability. That is precisely the reason why individuals are willing to pay for insurance.

How much is the person willing to pay? This depends on the preferences of the individual. For example, consider someone who is risk neutral (i.e. that is indifferent between uncertainty and certainty, if the expected value of the former is the same as that of the latter). In this case, the person is willing to pay no more than 5,000 for an insurance which fully cover the cost of the car loss. For any premium below 5,000 the individual would take the insurance.

The case is different for risk lover individuals, who would prefer an uncertain situation (no insurance) rather than a certain one (insurance). This individual would not be willing to pay more than 5,000 to get insured. Conversely, a risk averse is willing to pay more than 5,000, because s/he dislikes uncertainty.

For reference, see the entry on Wikipedia. See this pdf for a treatment from Economics.


Electricity as inelastic good Could be electricity considered as an inelastic good?

I am interested in the electricity delivered to final consumers in member countries of Nordpool Spot (Norway, Sweden, Finland, Denmark, Estonia, Latvia and Lithuania). I found that market prices in Nordpool Spot is not correlated with electricity consumption, therefore I explain this by considering, in this case, electricity as an inelastic good.

Is this reasoning correct?

It is. And it is not.

Electricity markets are generally not set up for the demand-side to do much active participation at all. So the short-run demand curve as seen in, for example, Nordpool Spot, is almost perfectly inelastic. Not quite, because there are some large industrial demands that exhibit some elasticity, and are exposed to the spot market. But a very large chunk of demand isn't exposed to the spot market at all; and a lot of the demand that is exposed, is highly inelastic - particularly to half-hour-by-half-hour changes.

That tells us a lot about the structure of the markets, and only a little about the actual underlying demand curve.

Until there's a lot of demand-side response (DSR), we've only got stated-preference studies and DSR pilots to go on, to estimate what the demand curve would look like in a well-functioning market: the evidence we have (e.g. from the Olympic Peninsula study), suggests there's quite a lot of potential demand elasticity out there, just waiting to be harnessed. Note that when short-run demand elasticity has been observed, it manifests in two ways: some demand will move between half-hours that aren't too far apart, when the price differentials incentivise it. Some other demand will just vanish altogether when prices rise.

If you want to try to distinguish the elasticity signal from all the noise in the existing market, then you need to take demand and prices for a particular half-hour slot on Tuesdays-Thursdays in one particular season across several years; and look at how weather-adjusted demand varies with price, while adjusting for any long-term trends.


Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008) A specialized question for those, who excel in monopolistic competition and modern trade theories. I am interested in a derivation of an aggregate demand function for a model of monopolistic competition presented in the (text-)book by Combes, Mayer and Thisse (2008).

Setup of consumer problem

Utility of an individual consumer: $$U_i = CM^{\mu}$$
Where $M$ stands for a composite manufacturing good defined a a sum of "manufacturing" goods consumed by a representative agent: $$M \equiv \left(\Sigma_i^n q_i^{\frac{\sigma - 1}{\sigma}}\right)^\frac{\sigma}{\sigma-1} $$
Budget constraint:
$$PM \leq y$$
Where $P$ is a composite price index for a composite manufacturing good defined as follows: $$P \equiv  \left(\Sigma_i^n p_i^{-(\sigma - 1)}\right)^{-\frac{1}{\sigma-1}} $$
Here comes the punchline (p. 57). Without any additional derivations, authors write the following:

  In this case, it is well-known that the aggregate demand function take the form: $$ M = \frac{\mu L y}{P} $$ My comment: $L$ stands for a total labor supply in a country.

Two questions:

This well-known result is totally unknown to me. Could someone kindly show its derivation (or maybe hint to a textbook) in detail?
Throughout the chapter, authors treat $y$ as an exogenous value albeit the model they present is in a general equilibrium setting. This implies that $y$ should be a sum of wages in the economy that feature in variable costs of firms: $C(q_i) = f + wq_i$. I suppose that they assume a perfect competition on a labour market among the identical workers and therefore the wage rates could be treated as fixed. Is my intuition true?

As we have discussed in the comments, the utility function used in the book is Cobb-Douglas: $U = CM^{\mu}A^{1 - \mu}$. The well known fact mentioned in equation (3.3) on page 57 - the one that you highlighted - is that the Cobb-Douglas utility function is homothetic - the share of total income spent on any of the goods is constant over all possible incomes. More specifically here a share of $\mu$ of the expenditure is always spent on manufacturing goods and a share of $1-\mu$ is spent on the agriculture good. $\sigma$ and $p_i$ only determine how the $\mu y$ units of income that the representative consumer spends on manufacturing output is allocated among the distinct manufacturing goods.
Although the output goods are different most such models assume that both the jobs and the employees are identical and there is a perfectly competitive domestic labor market. Hence the identical wage rate.

Note: $y$ is personal income and $L$ is the number of employees, thus the total income of the economy is $yL$. (E.g. the last sentence on page 56).

Thus the total expenditure $E = yL$ - the total income, as the model is static. The expenditure on manufactured goods is $\mu E = \mu yL = PM \Leftrightarrow M = \frac{\mu yL}{P}$.


Does National Savings equal zero in a closed economy? Please give an economic intuition Context:

$Y -(C+I+G) = X - M$

In a closed economy, $(X-M) = 0$

Notations

Y --> Income/Output

C,I,G --> Consumption, Investment and Government expenditure respectively.

X --> Exports

M --> Imports

I want an economic intuition how is it possible that aggregate of all savings in a closed economy equals 0.

In a closed economy, savings equal investment. Your equation actually shows this. We have:

$Y−(C+G) - I = 0$.

Note that savings, by definition, is just equal to the production $Y$ that is not consumed. Here we have consumption by private citizens $C$ and consumption by the Government $G$. Therefore, savings $S$ is:

$S = Y - (C+G)$

and therefore:

$S=I$

So national savings can only be zero if investment is zero.

Here is a concrete example. You don't spend all your income and save money in a bank. People who want to invest (e.g. to start a business or buy a tractor) go and borrow that money, which is then invested.

In an open economy, there is no reason for savings to equal investment. Nevertheless, the two are often strongly correlated in reality, which is known as the Feldstein-Horioka Puzzle. This is/was one of the major puzzles in international economics and a large literature has been devoted to explaing it.


What is the advantage and disadvantage of fiscal illusion policy in terms for the financial market? What is the advantage and disadvantage of fiscal illusion policy for the financial market?

Fiscal illusion suggests that when government revenues are not completely transparent or are not fully perceived by taxpayers, then the cost of government is seen to be less expensive than it actually is.

Too long, didn't read: the market does not care.

Colombia's Pension Reform: Fiscal and Macroeconomic Effects, by Klaus Schmidt-Hebbel, states black on white (p 22) that

  issuing explicit domestic debt for paying off implicit government debt does not have first-round macroeconomic and financial effects

on the condition that the financial markets see through the debt swap. Fiscal illusion aimed at tax payers does not have any impact on tax payers, but the same Schmidt-Hebbel notes that the market could be fooled too.

Peek and Wilkox looked into it and conclude that there is no data showing that fiscal illusion ever had an impact on interest rates, which leads me to conclude that, despite KSH's prudence, there is no (first hand) effect of fiscal illusion on the financial market. Of course, it can have secondary advantages such as leading the market to assume more political stability by raising popularity, or leading the market to distrust you, but these are corollaries of the effects of fiscal illusion on the people rather than pros and cons of fiscal illusion to the market.


Uncertainty in an unfair gamble If a risk averse person is given the option of a certain amount of 2000 or playing a lottery game giving him 10000 with 25% probability, and 500 with 75%, then what would he do ?.

The bet here is not a fair one, so wouldn't his choice depend on the degree of his risk aversion ?. Can somebody please help me out.

So let's decided that decision A is taking the certain sum and decision B is taking the risky sum, where $\bar{w}$ is current wealth and $\mathcal{U}$ is a utility function where $\mathcal{U}'>0$ and $\mathcal{U}''<0$, then you have two potential outcomes in utility.  The first is $$\mathcal{U}(A)=\mathcal{U}(\bar{w}+2000)$$ and the second is $$\mathcal{U}(B)=\frac{\mathcal{U}(\bar{w}+10000)+3\mathcal{U}(\bar{w}+500)}{4}.$$

Given only this information, the decision depends upon $\bar{w}$, the shape of $\mathcal{U}$, the payoff outcomes, and the probabilities involved.  If you wanted to determine how sensitive a decision would be to change, you could treat the currently fixed factors as variable factors one at a time.  To provide an example, you could create a new variable $\pi$ so that option A would become $$\mathcal{U}(\bar{w}+\pi).$$

You could also do this to substitute for $\bar{w}$, the other payoffs, or the odds.  If you added the assumption that the actors were indifferent, then you could also use the Envelope Theorem to look at local sensitivity.


Exogenous Money Supply In the first 10 seconds of the following video: https://youtu.be/anZ58gZcxqk,
A claim is made that in the Exogenous Money Supply Model, Money Supply is not determined by interest rates. The person further adds that the money supply is independently determined by the central bank.
My question is that isn't the central bank only responsible for determining the interest rates? I know there are other tools such as open market operations that can be used to control the money supply in an Economy. But, If the central bank is determining the money supply, shouldn't it then be a function of interest rates.

My question is that isn't the central bank only responsible for determining the interest rates?

No, central banks do not even have directly responsibility to adjust interest rate. Depending on which country we talk about they will have responsibility to keep prices stable, high employment and bank regulation and often combination of the above. In order to follow their mandates they will often fiddle with interest rates but typical central bank has many instruments at its disposal. For example, they can also increase money supply or even adjust reserve requirements that also affect money supply (Fed recently set them to zero in their effort to increase inflation).
As mentioned above  most of the time contemporary central banks adjust interest rates on their reserves that influences interest rate in the economy and let the new money supply be determined by what the interest rate is but I don't know of any central bank that would not be allowed to directly increase $M$ if they would want to in some way. Also, note even if central bank sets itself some goal achieving some interest rate without stating any goals in terms of money supply, since interest rates depend on what $M$ is, they in principle could fully control interest rate just by varying $M$. They might want to use more tools because relying only on $M$ would be more difficult but it is possible.
In fact the open market operations that you mentioned are one way how central bank can directly increase money supply in the economy and the open market operations are not that scarce. These are becoming increasingly common, for example you can see from the graph below what an effect QE (which is not fundamentally much different from OMO expect in its scale and that it is part of holistic strategy that includes other tools) had on increase in monetary base in the US (first round of QE started in 2009 you can easily spot it by the shar increase in monetary base), so that is non-trivial  change in $M$ (technically $M0$ but $M$ consist of base money as well) that wasn't created with interest rates.


Interpretation of Difference-in-Differences Regression Results when Only the DID Coefficient is Significant I have a standard DID regression of the form:
Y= β0 + β1*[Time] + β2*[Treatment] + β3*[Time*Treatment] + ε
where Time is a dummy equal to 1 for period after policy change and Treatment is a dummy for the treatment variable.
Based on my results β0, β1 and β2 are all insignificant. However, the difference-in-difference estimator β3 is significant. How can I interpret these results? Does that mean that I cannot say anything about the exact behavior of the treatment and control pre or post period other than the fact that there is a significant difference between them in the period after treatment? Is there still value in this DID regression if all other coefficients are insignificant?

I would heed the advice in the comments.
In settings where policy adoption is standardized and you have multiple pre- and post-treatment time periods, you could also interact your treatment indicator with separate time dummies specific to treated and untreated units. In the comments you indicate you observe multiple units $i$ across months $t$ from 1990 to 2005. Your model would look something like this:
$$
y_{it} = \gamma T_{i} + \lambda_{1} (T_{i}*\mathbf{I}_{t = \text{Jan-2000}}) + \lambda_{2} (T_{i}*\mathbf{I}_{t = \text{Feb-2000}}) + \lambda_{3} (T_{i}*\mathbf{I}_{t = \text{Mar-2000}}) + \lambda_{j} (T_{i}*\mathbf{I}_{t = j}) + \epsilon_{it},
$$
where the $\lambda_{j}'s$ represent post-exposure effects after the policy goes into effect. Note, to estimate all post-period effects up to period $j$ you must create a 'month-year' variable to distinguish, for example, between January in 2000 and January in 2001. To do this, concatenate categorical indicators for month and year together and append it to your data frame. Interacting $T_i$ with individual post-treatment time dummies results in separate estimates for each post-exposure period (e.g., Jan-2000, Feb-2000, Mar-2000, ... , Dec-2005, etc.). This approach is helpful for many reasons. First, treatment may grow or fade over time; the policy might be perceived immediately upon implementation, or with delay. Due to its flexibility, this approach allows you to assess the relevant time dependencies. Second, if there is a treatment withdrawal period, you can estimate effects in those periods beyond policy/program termination; effects might persist even in the absence of treatment.
I admit this results in 72 interactions which might be more than your data can handle. Notwithstanding this concern, it is still a useful way to model effects in the post-treatment epoch. You can still estimate unit and time fixed effects in this setting. As already noted, the unit fixed effects will absorb $T_i$ and the time fixed effects will absorb your 'month-year' main effects. But I wouldn't worry about this. Your interactions (i.e., $T_i \times \mathbf{I}_{t=j}$) should be your focus.


Current House Demand and Supply Home prices are at an all time high so according to supply and demand, that means suppliers will supply more homes due to the high prices. Demand for a house on the other hand is also at an all time high. I really am confused on why demand is high when prices are high. From my understanding of demand and supply when supply is low, demand will be high and so will prices. When the price is high, supply is also high but the demand is low.
There are some articles saying supply is low which is why home prices are at an all time high. But that doesn't make any sense to me since home prices are very high right now so it incentivize suppliers to make more to provide for the buyers. Also, bidding wars are very common when supply is low, but there is lack of bid wars happening in the housing market.
My questions are:
Are the principles behind demand and supply pointless in this scenario?
How come demand for a house is very high but the prices are high and the supply seem to be relevantly adequate?

The principles of demand and supply are always operating in the background so they are not pointless.

The magnitude of the effect that increase in price has on increase in supply is given by the price-elasticity of supply for housing. If the supply is inelastic then it takes large increases in prices to increase the quantity supplied. In a an extreme case when supply is perfectly inelastic it would not respond to price at all (but this is completely consistent with supply and demand theory - so it would not make it 'pointless', you just always have to take into account the parameters of supply and demand before trying to predict what will happen).
Empirical research (Green, Malpezzi & Mayo 2005) actually shows that housing supply is in many metropolitan areas inelastic so that is one explanation.

In many places government puts heavy restrictions on building of new housing. Especially in historic cities (e.g. Amsterdam) it is excruciatingly difficult to actually build a new housing. Even outside historical cities many places will be heavily zoned. If government restricts supply then of course it will not be able to increase. Also research shows that this is one of the leading causes behind high housing prices (see this empirical study by Glaeser & Gyourko (2002) that examines the effect of strict zoning on prices in Bay Area in US).

You should also not confuse movements along demand or supply curve and changes in demand or supply. Changes in prices produce movement along demand curve. If demand curve is fixed then increase in price (excluding extreme cases such as perfectly inelastic demand or Giffen good) will lead to lower quantity demanded. However, no matter what price is demand can always shift to the right (for example due to change in preferences) which will make people demand more at any price.
If there would be no changes in demand or supply the price should remain constant. The high prices are likely due to the fact that demand for housing is increasing as economies grow and people are having on average higher incomes (since owning a house is likely normal good), and the prices increase because this demand outpaces supply. Of course, that is just gross generalization every city/area will be different. It is not even universally true that house prices increasing in every single area even though on average they do there are areas where they fall (see IMF global housing watch).


Data source for average effective personal income tax rate? Anyone know a source for the average effective personal income tax rate for the US. By state also would be great.

Definition: average effective income tax rate = (Total income tax collected)/(Total income earned)

Possibly a source for the data to do the calculation.

For a lot of things, including average effective (Federal) tax rates, the Urban Institute/Brookings Institution Tax Policy Center is a great place to start. Look at the table that reads "Average Individual Income Tax Rate," as the first one also includes payroll, excise, and corporate taxes.


Why was Modigliani-Miller so relevant and innovative when it came out? The Modigliani-Miller theorem*, a foundation of modern corporate finance, basically states that, in a no-frictions world, two identical firms have the same enterprise value, regardless of their financial structure.

Given a firm A, whose liabilities are 50% equity, and 50% debt, and a firm B which is financed 100% by equity, the value of firm A and the value of firm B are the same.

Isn't it just obvious? What am I missing?

* -
Modigliani, F.; Miller, M. (1958). "The Cost of Capital, Corporation Finance and the Theory of Investment". American Economic Review 48 (3): 261–297. JSTOR 1809766.

People, particularly business leaders, seem to remain confused about this issue even today. At the core of is the question Is equity finance expensive?. We certainly observe in the data that the realized returns on firm debt are much lower than the realized returns on firm equity. Does this mean that firms have too much equity?

If equity capital always costs 9 percent per year and debt always 3 percent then a firm would be worth more if management kept equity to a minimum and used as much debt as possible. But in the real world there are lots of complications that might make debt cheaper than equity. Monitoring costs are real, default is costly, tax wedges are large, equity is riskier, and contracts are incomplete. It may be that factoring in these costs, which are hard to measure, provides a complete explanation for the differences in return between debt and equity. Or maybe not. If observable characteristics fail to explain this difference does this mean that firms should take on more debt? Or does it mean that these characteristics are poorly measured?

Modigliani-Miller provide a clear way of thinking about this question. In the absence of frictions, as you shrink the amount of equity and offset it with more debt the required returns adjust to leave the firm's cost of capital unchanged. But even so, there is no problem with return on debt being lower than the return on equity. So in and of itself, there is nothing to learn from the lower cost of debt compared with equity. This is a surprisingly robust result:

  The two Modigliani-Miller theorems hold good, irrespective of
  individual differences between shareholders' valuations of risk,
  leverage effects, durability of loans, etc. The logic of the theorems
  rests in fact upon the assumption of perfect markets, namely that a
  shareholder can always, through his own borrowing or lending, compose
  his asset portfolio as he sees fit and that he can, without costs,
  give it the composition he desires with respect to risk, leverage,
  etc. If for instance the risk level of a firm's assets is increased,
  the shareholders can neutralize this by lowering the risk of other
  assets in their portfolios.

Press Release: 15 October 1985 THIS YEAR'S ECONOMICS PRIZE AWARDED FOR PIONEERING STUDIES OF SAVING AND OF FINANCIAL MARKETS

This is an extremely valuable insight. The explanation is also clear and powerful. So good and powerful that it is hard to see the world any different once you hear it. In one of the Sherlock Holmes stories, Holmes says "The world is full of obvious things which nobody by any chance ever observes." In TV tropes this is called Seinfeld Is Unfunny:

  It wasn't old or overdone when they did it. But the things it created
  were so brilliant and popular, they became woven into the fabric of
  that show's genre. They ended up being taken for granted, copied and
  endlessly repeated. Although they often began by saying something new,
  they in turn became the status quo.

Additionally, it is hard to remember from the perspective of the present how devoid of theory was corporate finance in the period before Modigliani and Miller. Here's a quote from an award ceremony speech related to the Nobel award:

  Until the latter part of the 1950s, no viable theory of corporate
  financing of investment, debt, taxes, and so forth had been developed.
  It was not till Modigliani and Miller presented their theorems that
  more stringent theorizing began to appear in this field. By treating
  financing decisions within the framework of a theory of
  financial-marketplace equilibrium, Modigliani and Miller provided the
  general guidelines for continued research in this area.

Award Ceremony Speech: Presentation Speech by Professor Ragnar Bentzel of the Royal Academy of Sciences


Calculating expected tenure of a customer In his How To Project Customer Retention, Fader computes the expected tenure of a customer according to $$E = \sum_{t=0}^{\infty}S(t)$$ where $S(t)$ is the Survival Function.

From a purely mathematical standpoint, expectation is calculated via $$E[X] = \sum_{x \in X} xP(X=x).$$ Hence, I would expect expected tenure formula to be $$E = \sum_{t=0}^{\infty}tS(t).$$

Can anyone shed some light on this?

I am going to use a $t$-continuos scenario, and then discretization follows naturally.

Call $f(t)$ the distribution function of the variable $t$. The survival function is just

$$
S(t) = \int_t^{+\infty}f(u)~{\rm d}u \tag{1} = 1 - \int_0^t f(u)~{\rm d}u
$$

It is pretty clear from here that

$$
\frac{{\rm d}S}{{\rm d}t} = -f(t) \tag{2}
$$

and that $\lim_{t\to\infty}S(t) = 0$. Now consider the integral

\begin{eqnarray}
\require{cancel}
\mathbb{E}[t] &=& \int_0^{+\infty} t f(t) ~{\rm dt}
\stackrel{(2)}{=} -\int_0^{+\infty} t \frac{{\rm d}S}{{\rm d}t}~{\rm d}t  \\
&=& -\int_0^{+\infty} \left[\frac{{\rm d}(t S)}{{\rm d}t} - \cancelto{1}{\frac{{\rm d}t}{{\rm d}t}} S\right] {\rm d}t \\
&=& -\cancelto{0}{\strut{t S}\big\rvert_{0}^{+\infty}} +  \int_0^{+\infty}S(t)~{\rm d}t
\end{eqnarray}

So in summary

$$
\mathbb{E}[t] = \int_0^{+\infty}S(t)~{\rm d}t \tag{3}
$$

Now you can move to discrete time and represent this in the form

$$
\mathbb{E}[t] = \sum_{t = 0}^{+\infty} S(t) \delta
$$

for some small number $\delta$. In the paper you cite, they choose $\delta = 1$


Pareto efficiency and maximization of sum of utilities I'm trying to understand the relation between maximizing sum of utilities and finding Pareto efficient allocations. According to https://econ.ucsb.edu/~tedb/Courses/UCSBpf/pflectures/chap2.pdf (page 13), concavity plays an important role in whether maximizing the sum of utilities will help obtain the efficient allocations or not. Could someone explain this further?
A similar question is posted at Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?, but the counterexample in this answer seems contradictory to the result stated in the above document.
Specifically, my questions are:

When can we maximize the sum of utilities to find all the efficient allocations?
What criteria can be used to locate all the efficient allocations in corner/edge cases?

Finding Pareto efficient allocations via social planner's problems is a special case of scalarization in convex optimization.
Suppose agents' utility functions are  $u_i:\mathbb{R}^n \rightarrow \mathbb{R}$, for $i = 1, 2, \cdots, m$, and the feasible allocations are given by $g(x) \geq 0$ for some $g:\mathbb{R}^n \rightarrow \mathbb{R}^p$.
The general result is as follows.
Sufficiency under general conditions
If an allocation $x$ solves the social planner's problem with strictly positive social weights, then it is Pareto. This is true with no assumptions on utility functions and feasibility constraint.
In other words, if
$$
x \in \arg\max_{g(x') \geq 0} \sum_{i = 1}^m \lambda_i u_i(x'), \mbox{ for some } \lambda \in \mathbb{R}^m_{++}, \quad (*)
$$
then it is Pareto optimal.
This is easy to see. If $y$ Pareto-improves $x$, then
$$
\sum_{i = 1}^m \lambda_i u_i(y) > \sum_{i = 1}^m \lambda_i u_i(x)
$$
for any  $\lambda \in \mathbb{R}^m_{++}$. It is also easy to see that zero entry in $\lambda$ cannot be allowed---it would make the statement false.
The converse is not true in general---this sufficient condition is not necessary. There are Pareto allocations that do not solve $(*)$. This is because the set of achievable utility values
$$
\mathcal{U} = \{ (u_1 (x), \cdots, u_m(x)); g(x) \geq 0 \}
$$
is in general not a convex subset of $\mathbb{R}^m$. The social planner's weights $\lambda$ corresponds to certain supporting hyperplanes of $\mathcal{U}$. If $\mathcal{U}$ is not convex, its Pareto frontier cannot be recovered by varying $\lambda$.
However, there is a partial converse.
Necessity under concavity
Assume $u_i$'s and $g$ are concave. If an allocation $x$ is Pareto, then it solves a social planner's problem with nonnegative social weights.
In other words, if $x$ is Pareto then
$$
x \in \arg\max_{g(x') \geq 0} \sum_{i = 1}^m \lambda_i u_i(x') \mbox{ for some } \lambda \in \mathbb{R}^m_{+}. \quad (**)
$$
This is true because of convex sets and separating hyperplanes.
This is a partial converse because it's necessary but not sufficient. There are allocations that solve $(**)$ but not Pareto.
Comment
Even under concavity, $(**)$ is a only a partial converse of $(*)$. Notice the gap between strictly positive and nonnegative social weights.
To recover the Pareto frontier, the general procedure is to first find solutions to $(*)$. Then check the solutions to $(**)$ case-by-case for Pareto optimality. Alternatively, one can also take the closure (the limit points) of solutions to $(*)$.


Continuous preferences definition Suppose consumption set $X=R_{+}^N$. According to the definition preferences are continuous if for any $x\in X$ sets $(y\in X:x\succeq y)$ and $(y\in X:y\succeq x)$ are closed. It is clear that preferences represented by utility function $u(k_{1},k_{2})=k_{1}+k_{2}$ are continuous. But now lets take some bundle x=(1,1). Then set $(y\in X:y\succeq (1,1))$ is not closed as it approaches infinity. My question is how to handle with this contradiction?

It appears that you are conflating a "closed interval" with a "closed set". Certainly, (multidimensional) closed intervals are closed sets, but the concept of a closed set is wider.

A closed set is, well, only closed, and not necessarily bounded.

I understand that the everyday-meanings of the terms "closed" and "bounded" can indicate that "being closed is a stronger property than being bounded" (i.e. that being closed implies being bounded), but this is not true in mathematics. Neither property implies the other.

Yes, a set can be bounded and yet it may not be closed. Standard example: Consider the set of rational numbers  in $[1,3]$ . This is a bounded set. Now consider the sequence of rational numbers

$$x_{n+1} = \frac {x_n}{2} +\frac {1}{x_n},\;\; x_0 = 3$$

The sequence remains in the $[1,3]$ interval, but its limit is $\sqrt 2$, which is not rational. So this set of rationals in $[1,3]$ is bounded but it is not closed, because it does not contain a limit point of a sequence that can be constructed in it, stay in it, except its limit point.

So, what "closed set" means then in mathematics? There are various ways to look at it:

A closed set is a set whose complement is open.
A closed set is a set that coincides with its "closure".
A closed set is a set that contains all its limit points.

Try them out mentally and stick with what is more intuitive to you.


Is ongoing economic growth possible without additional non-renewable inputs? Can ongoing economic growth occur without use of additional non-renewable inputs? That is, any additional inputs would only be from renewable resources: all non-renewable resources would be reused, recycled or removed from the economy.

I don't think anyone knows how to "make" an economy grow, but there's a simple channel through which economic growth can occur under fixed non-renewable inputs: innovation.

Economic growth does not require increasing use of inputs; if there's profit to be gained through more efficient use of fixed quantities of inputs, or by increasing output of renewable resources, people will have an incentive to figure out how to do so.

Regarding whether fixing a minimum price of a nonrenewable resource is a good idea: possibly, but likely no. Typically, the most efficient ways to reduce the use of a particular input are to either charge the marginal social cost of the use of that input as a tax (allowing quantities to vary freely, which you disallow in your scenario), which incidentally has the effect of setting a minimum price but which requires that you know the marginal social cost of the use of that resource, or to do cap-and-trade, where you fix the quantity available and let market prices determine the most efficient allocation. Setting a minimum price in a cap-and-trade scenario can result in inefficient underuse of the resource.

If you're interested in the theory of the price of nonrenewable resources, you might want to start with Hotelling's Rule.


Shapley value with large number of players I'm a complete statistics rookie but I'm looking for some general advice on workarounds if I'm trying to calculate the Shapley Value in games featuring many players.

i.e. If I have 20 players, I need to run all combinations (n!) which is 2,432,902,008,176,640,000.

Is there a solid method of solving the problem by taking a sample of the combinations?  What do I need to look out for in order to make the results of using a sample as accurate as possible?

The sampling approach rigorously would look like this. For each player $i$, we want to estimate the expected marginal contribution, where the expectation is taken over a the subset of players that precede $i$ in the permutation ordering. So for each $i$, we do the following.

Let $X_i$ be a random variable equal to the marginal contribution of $i$, when we draw the permutation of players randomly. Let $\mu_i$ be the true Shapley value of $i$ (which we do not know). Then $\mathbb{E} X_i = \mu_i$. Now, if we sample many independent copies of $X_i$ and average them, this average, call it $\bar{X}_i$, should be very close to $\mu_i$, and the closeness is given by e.g. Hoeffding's inequality, which says

  Let $K$ be an upper bound on any marginal contribution (so they are always between $0$ and $K$). Let $m$ be the number of independent copies of $X_i$ that we draw. Then
  $$ \Pr[ | \bar{X}_i - \mu_i | \geq \epsilon ] \leq 2e^{-2m \epsilon^2 / K^2} . $$

So for each $i$, we sample $m$ permutations randomly and calculate the average marginal contribution $\bar{X}_i$.

For example, if $K=100$ and we want an accuracy of $\epsilon = 0.01$ and a probability of failure of $2e^{-50}$, then we need $2m\epsilon^2/K^2 = 50$, so $m = 25K^2/\epsilon^2 = 2.5$ billion. So we need to sample $2.5$ billion permutations to "guarantee" accuracy of $0.01$, except with a miniscule $2e^{-50}$ chance of failure.

As comments have mentioned, there are big improvements for many special cases.


First-price auction with discrete values Exercise 4.5 in Vijay Krishna's Auction Theory:

  Suppose that bidder 1 always has value $X_1=2$, while bidder 2's value, $X_2$ is equally likely to be $0$ or $2$. Find equilibrium bidding strategies in a first-price auction. (Note that since values are discrete, the equilibrium will be in mixed strategies.)

I don't understand why it is that since values are discrete, the equilibrium will be in mixed strategies. Is there some rule/principle to this?
How do I set up for the problem? Since equilibrium are in mixed strategies, I can suppose that player 1 bids $b_1,\ldots,b_k$ with probabilities $p_1,\ldots,p_k$ respectively. But that seems too general to make any progress.

The suggested pure strategies do not create an equilibrium, because if $X_2=X_1=2$, there is a tie. Whichever tie-breaking rule is employed, one of the players would want to deviate to $\beta_i=1+\epsilon$ to avoid the tie and win with certainty. If tie breaking is always in favor of the second player, player 1 deviates to $0+\epsilon$.

Discrete values in this setting do suggest a mixed strategy equilibrium (non-degenerate), because the probability of a tie in valuations is non-zero. This is why the pure strategy equilibrium from the continuous case is missing here.

The more general case of this problem is in Doni & Menicucci (2013).


Can a monopoly supply negative units to one of its two markets? A question I have seen in my microeconomics textbook is as follows:

Consider a monopoly supply to both a domestic and foreign market, where the market demands are:

Yd(pd) = 20 - 2pd, hence pd(yd) = 10 - Yd/2

Yf(pf) = 20 - pf, hence pf(Yf) = 20 - Yf

Total cost = .5Y^2 +20

Total output, Y = Yd + Yf

We are assuming the monopolist cannot price discriminate and therefore pf = pd = p

We are asked what price should the monopolist now set for its output? What quantity will it sell to each market?

To solve this question i first calculated the new total market demand, or aggregate demand by summing both the foreign and domestic markets.

D(p) = 20 - 3p + 20 - p = 40 -3p, hence p = 40/3 - Y/3

I then calculated total revenue and marginal revenue.

TR = (40/3)Y - (Y^2)/3 , hence MR = 40/3 - 2Y/3

I then calculated marginal cost.

MC = Y

I then solved for Y where MR = MC and substituted the value into the market price function.

Y = 40/3 - 2Y/3, hence Y = 8, p(8) = 32/3

My problem arises when i substitute this value for P into the domestic market's output function to calculate how many units will now be supplied to the domestic market.

Yd(8) = 20 - 2(32/3) = -4/3

My interpretation of this is that the monopoly is purchasing 4/3 units from the domestic market to sell into the international market, however this concept is not familiar to me. Having a negative answer for units does not sit well with me. Any guidance as to something i did wrong in calculating the question or an economic interpretation of the solution would be much appreciated! I am very confused.

Marginal cost is only guaranteed to be equal to marginal revenue in optima where $Y > 0$ (and when the profit function is concave, but in your case it is). If
$$
MC(0) > MR(0)
$$
the producing the first unit results in a loss, and a profit maximizing monopolist would not produce at all (if the profit function is concave).

An example: I am willing to buy golf ball sized diamonds from you. I am willing to pay 5 dollars for the first, 4 dollars for the second, 3 dollars for the third and so on, ending in 0 dollars for the sixth and any further units. How many golf ball sized diamonds will you sell me?


Does GNI change once it has been published? From what I understand, GNI is published for the last year once the year has ended

I am trying to have a look at the GNI for Korea for the past couple of years

The official GNI is published by Bank of Korea on their website

10 days ago I checked the GNI, and the latest published GNI was for 2016, and it was 3198.40 * 10000 Won (about $30000)

Today I checked the GNI again and the system shows the GNI for 2016 and 2017 as:

2016: 3212.40 * 10000 Won (higher than previous value of 3198.40)
2017: 3363.60 * 10000 Won

Why is the GNI for 2016 different after 10 days? How does this work? Does this mean the GNI for 2017 as published now is not final either? Can either of these GNI values change again?

GNI is GDP plus net primary income from abroad

Any such numbers are subject to revision as more information is obtained; in many countries such numbers are typically revised every three months as part of a National Accounts and Balance of Payments update, with more substantial revisions annually.  Korea's latest update was published on 28 March

The fact that you did not see 2017 data the first time and did see 2017 data the second time was a pointer to these two readings being across such a revision point

Such numbers are typically never final, and potentially keep on changing, though after long periods revisions to numbers then tend to be affected more by methodology changes than by the arrival of new information


Economic Order Quantity w/ no set up costs Is it possible to modify the EOQ model to work in a purchasing environment when ordering costs are effectively $0$?

The classic EOQ model is:
$$
Q=\sqrt{2aK/h}
$$

with $a$ being demand, $K$ ordering costs, and $h$ the holding costs. If $K$ is $0$ it yields an order quantity of $0$. Is there any optimal way of determining the order quantity when there are no order costs?

The total ordering cost are

$TC(Q)=P\cdot a+\frac{a\cdot K}{Q}+\frac{h\cdot Q}{2}$

If K=0 TC becomes

$TC(Q)=P\cdot a+\frac{h\cdot Q}{2}$

P\cdot a is a constant. Let denote it as c.

$TC(Q)=c+\frac{h}{2}Q$

You can see that TC is a straight line with a slope of $\frac{h}{2}$ and a domain of $Q \in \mathbb R^+$. The marginal cost is

$\frac{dTC}{dQ}=\frac{h}{2}$

It is a constant. Therefore there is no single optimum order quantity.

The store has a capacity. This is the upper limit of the order. If you have still some goods in the store the maximum order is the capacity (c) minus the goods in the store (x).

Therfore you can order bewtween $0$ (exclusive) and $c-x$ (inclusive) goods. The total cost of inventory will be always the same for a given demand in a period.