Why does the inflation not follow the money supply? To my understanding classical economic theory tells us that inflation occurs when the money supply is increased faster than the economic growth.
Lately there has been a sharp rise in the rate of money supply, but GDP has not increased the last year:
M2 money supply and GDP for USA

However there has been no corresponding inflation (consumer price index) increase:
M2 money supply and CPI for USA

Why is this? Does the money accumulate somewhere, if so where?

This is because classical economic theory absolutely does not say that:

inflation occurs when the money supply is increased faster than the economic growth.

Even in most basic 101 models inflation depends on variety of factors. Inflation is determined by changes in price level which in turn depends on what the money market equilibrium is. This is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY \implies \ln P = \ln M + \ln V− \ln Y$$
Where M is the money supply, V velocity of money, P price level and Y output. In the log-linearized version the effects of right hand side variables can be interpreted as % changes.
Even in this most simplest model your statement "inflation occurs when the money supply is increased faster than the economic growth" would only hold if velocity of money would be constant but in real life it is not. As you can plainly see from the data provided by Fed the velocity of money dropped drastically since the 90s which offests most of the increase in money supply during that period.

Lastly, the model above is still just the most simplistic 101 textbook money market model. In more realistic and complex models there are more variables that matter. For example, inflation expectations are absolutely crucial. Inflation will not move drastically if inflation expectations are well anchored (e.g. see Castelnuovo et al 2004; Kose et al 2019 or Lejsgaard & Grothe 2014 and sources cited therein for more info on the concept).
In addition, as famously shown by Krugman (1998) what matters is not just the actual change in money supply or other quantities but what are peoples expectations about changes in money supply and so on. An increase of money supply that is not credible to be permanent and expected to be quickly reversed will have no effect on inflation as if it would never even happen. Furthermore, Krugman also showed that when nominal interest rates are at zero lower bound (ZLB)  even large monetary expansion won't necessary be inflationary. The reason for that is that if you would want to go significantly below ZLB the peoples will strictly prefer to hold cash and any increase in money supply will be completely offset by drop in velocity of money.
There are also further nuances to this but full overview of modern scholarship on this topic would be well beyond scope of SE answer. I recommend you to have look at Monetary Theory and Policy by Walsh or Woodford Interest and Prices and sources cited therein for state of the art models and more information on the matter.


Why is consumer surplus the area under the curve? So I understand that the definition of consumer surplus is the difference between what the consumer pays and what the consumer would have been willing to pay. I just don't understand why the consumer surplus is the area of the triangle, as shown in the diagram. Is it representing all the possible consumer surpluses?

This is quite fundamental, so I'd encourage you to look up your textbook as well... but here's a short explanation.

Imagine that there are $n$ consumers in the market. You sort them by their willingness to pay from high to low. So the first consumer is willing to pay very high, and so on. Let's say the willingness to pay (sorted) is 12, 10, 9, 7, 5, 4, 2. This is represented by the downward-sloping demand curve.

Let's say the price is \$8.

The first consumer's surplus is $12-8 = 4$.
The second consumer's surplus is $10-8 = 2$.
The third consumer's surplus is $9-8 = 1$.

The fourth consumer onward wouldn't buy the item, since their willingness to pay is lower than the price. Their surpluses are all zero. They are the ones on the right of the equilibrium quantity.

The sum of the consumers' surplus is 4 + 2 + 1 = 7, which is given by the area under the demand curve and above the price the consumer pays.

Hope that helps!


Example of a game with no Nash equilibria but at least one correlated equilibrium In this answer there is the offhand remark

  Of course, a game with no Nash equilibria may have a correlated
  equilibrium, but I'm not aware of any simple examples where this is
  the case.

Can anyone provide an example of such a game?

The game should allow for mixed strategies, otherwise correlation would be meaningless. Because of this finite games are out, since their mixed extensions always have a Nash equilibrium.

There are three players, 1,2,3, and the action spaces of players 1 and 2 are both $[0,1]\times\mathbb{N}$, the action space of player 3 is $[0,1]$.
The generic action of player 1 is written as $(x,m)$, the generic action of player 2 is written as $(y,n)$, and the generic action of player 3 is written as $(x')$.

The payoff of player 3 is simply $1$ if $x'=x$ and $0$ otherwise.

The payoffs of players 1 and 2 are more complicated. The payoff of player 1 and 2 is both $2$ if $y=x\neq x'$, and both $-2$ if $x=x'$. In the case that $y\neq x\neq x'$, the player with the highest number in the second coordinate gets a payoff of $1$ and the one with the lower number gets $-1$. If they pick the same number, they get both $0$.

This game has no Nash equilibrium. Suppose there were one. If player 1 would not randomize in equilibrium, player 3 can match them for a lousy payoff of $-2$. By randomizing the payoff will be higher, but it will also be the case that with positive probability $y\neq x\neq x'$, since the behavior of all players is stochastically independent. But then both 1 and 2 have to play the game of picking a higher number and that game has no equilibrium even in mixed strategies (if mixed strategies must be countably additive).

But there is a correlated equilibrium in which the behavior of 1 and 2 is given by a distribution on the diagonal $D=\{(x,y)\mid x=y\}$ without mass points, and player 3 plays anything stochastically independent. Player 1 and 2 can clearly do no better and there is no way for 3 to match 1 with positive probability.


What are some good graduate-level econometrics books for someone with a strong mathematics background? Related: Book recommendations on empirical methods in economic research and econometrics?

I would like to focus mainly on graduate texts in Econometrics. From the question above, I gather that Wooldridge's text is nice.

In terms of "strong math background," I did my undergrad in Statistics, consisting of two semesters each of linear algebra, real analysis, and abstract algebra, along with a measure-theoretic probability course.

What are some econometrics texts that you would recommend for me? I learned some intro econometrics in a course which taught from Studenmund but was extremely bored.

"Adult" Wooldridge is great intro to various microeconometrics topics.

For time series, Hamilton's Time Series and Lutkepohl's Introduction to Multiple Time Series Analysis are both nice, though Hamilton is a bit dated and Lutkepohl is more focused.

As far as more foundational, rigorous material, Herman Bierens has a short Introduction to the Mathematical and Statistical Foundations of Econometrics. Gourieroux and Monfort have a whole flock of graduate econometrics texts, but to quote an anonymous reviewer, they are "in the French tradition of excellent precision and terrible pedagogics," though they have their champions.


Cash Flow and present value Question First: for year 0, I do not know why the cash flow = $- 1000 / ( 1 + 0.10)^0 = 909.01$

Second: for year 1, if cash flow = $30,000-5000/(1+0.10)^1 = 25 454 $

How is "Cash Flow" column useful in getting the "Present Value" column?

The table appears to have a bunch of errors in it.  In particular, the cash flow column seems wrong in every instance.  I don't blame you for having problems understanding.

For your first question, the cash flow in time 0 is wrong.  $1.1^0=1$ as we know.  The "end of year 0" means "right now," so there should be no discounting at all.  The cash flow should be -100,000, as indicated by the correct value in the present value column.

For your second question, cash flow is discounted at the required rate of return to get the present value.  In this case, presumably

$$
\frac{25,454.55}{(1+0.1)^1} = 23,140.50
$$

Why does that not equal the present value column?  Because the table was set up by someone who was confused.  The quantity they put in the present value is simply the discounted value of \$30,000 - \$5,000.

$$
\frac{30,000-5,000}{(1+0.1)^1} = 22,727.27
$$

To me that seems like a reasonable solution to the problem text.  The next period's cash flow should again be (\$30,000-\$5,000) and its present value should be

$$
\frac{30,000-5,000}{(1+0.1)^2} = 20,661.16
$$

Anyway, the present value column seems right given the text of the problem.  The "cash flow" column was done by someone who was as confused as you are, if not more so.

Probably you got this already, but to solve the problem you simply take the sum of everything in the present value column.

-------------- EDIT -------------

Now that I look at it again, I bet they just forgot to put parentheses around the (30000-5000) in each entry.  And left off a few zeros in the first.  Shabby, but not as bad as I was originally thinking.


Doesn't the concept of marginal utility speak to a cardinal utility function? When we differentiate the utility function with respect to some input $x_i$, we get a number that tells us how "fast" the utility function is changing at some point with respect to $x_i$. Doesn't that mean that when we compare marginal utilities, we are comparing with something that is based on the structure of the numeric output of the utility function, which shouldn't be the case since it's ordinal?

I know I'm missing something here intuitively, but I can't seem to figure out what it is.

To the pure ordinalist who believes that preferences are purely ordinal, the concept of marginal utility (MU) has no meaning. (And a fortiori, the concept of diminishing MU also has no meaning.)
However, the concept of the marginal rate of substitution (MRS) does have meaning.
In the course of our work, we may compute something that we call MU. But to the pure ordinalist, any such number found has no meaning in and of itself.

Example. Say an individual's preferences $\succsim$ over two goods $A$ and $B$ can be represented by the utility function $U:(\mathbb{R}^+_0)^2\rightarrow\mathbb{R}$ defined by $$U(A,B)=AB.$$
The intermediate microeconomics student may then carry out these computations:
$$MU_A=\frac{\partial U}{\partial A} = B.$$
$$MU_B=\frac{\partial U}{\partial B} = A.$$
$$MRS = \frac{MU_A}{MU_B}=\frac{B}{A}.$$
The above says that if, for example, my current bundle is $(A,B)=(200,1000)$, then  $$MU_A=B=1000\text{ and }MU_B=A=200.$$ However, these two numbers have no meaning whatsoever.
The only number that has meaning is the MRS: To get another unit of $A$, I'm willing to give up (approximately) $$MRS=\frac{B}{A}=\frac{1000}{200}=5\text{ units of }B.$$
To the pure ordinalist, the above reasoning is completely legitimate, so long as one assigns meaning only to MRS. What's illegitimate is to assign any meaning to $MU_A=B$ or $MU_B=A$.
The pure ordinalist knows that if $\hat U$ is a strictly increasing transformation of $U$, then $\hat U$ is also a utility representation of $\succsim$. So, for example, if $\hat U:(\mathbb{R}^+_0)^2\rightarrow\mathbb{R}$ is defined by $$\hat U(A,B)=2AB,$$ then $\hat U$ also represents $\succsim$.
However, with $\hat U$, our computations seem to differ slightly from before:
$$M\hat U_A=\frac{\partial \hat U}{\partial A} = 2B.$$
$$M\hat U_B=\frac{\partial \hat U}{\partial B} = 2A.$$
$$\hat{MRS} = \frac{M\hat U_A}{M\hat U_B}=\frac{2B}{2A}=\frac{B}{A}.$$
Any conclusions we arrive at with the new utility representation $\hat U$ are the same as before.
If again my current bundle is $(A,B)=(200,1000)$, then  $$M\hat U_A=2B=2000\text{ and }M\hat U_B=2A=400.$$ However and again, these two numbers have no meaning whatsoever.
The only number that has meaning is the MRS: To get another unit of $A$, I'm willing to give up (approximately) $$\hat{MRS}=\frac{2B}{2A}=\frac{2000}{400}=5\text{ units of }B.$$
The quantity MU by itself has no meaning. Confusion arises only when one attaches meaning to MU and wonders how for example it is that $$M\hat U_A = 2MU_A,$$ and what the above equation means. (Answer: It means nothing.)
Out of convenience, the intermediate microeconomics student will often compute something called $MU_A$ and $MU_B$ and these can often be evaluated as actual numbers. But on their own, these numbers have no meaning (to the pure ordinalist). Only the ratio of the two quantities has any meaning: $$MRS=\frac{MU_A}{MU_B}.$$

Some quotes. Hicks (1939):

We have now to undertake a purge, rejecting all concepts which are tainted by quantitative utility, and replacing them, so far as they need to be replaced, by concepts which have no such implication.
The first victim must evidently be marginal utility itself. If total utility is arbitrary, so is marginal utility. ...
The second victim (a more serious one this time) must be the principle of Diminishing Marginal Utility. If marginal utility has no exact sense, diminishing marginal utility can have no exact sense either.

Dittmer (2005, emphasis added):

Many introductory microeconomics textbook authors derive the law of  demand from the assumption of diminishing marginal utility. Authors of intermediate and graduate textbooks derive demand from diminishing marginal rate of  substitution and ordinal preferences. These approaches are not interchangeable;  diminishing marginal utility for all goods is neither a necessary nor sufficient condition for diminishing marginal rate of substitution, and the assumption of diminishing marginal utility is inconsistent with the assumption of ordinal preferences.


Does the Marshallian demand function always include prices and income? I have the following utility function:

$$U(x_i)=x_1x_2+x_3$$

with budget constraint:

$$p_1x_1+p_2x_2+p_3x_3\leq I$$

I use the Kuhn-Tucker method to find the optimal choices of the Utility maximization problem.
My equations are:

$$x_2-\lambda p_1+M_1=0$$
$$x_1-\lambda p_2+M_2=0$$

$$1-\lambda p_3+M_3=0$$

$$p_1M_1=0$$
$$p_2M_2=0$$
$$p_3M_3=0$$
$$p_1x_1+p_2x_2+p_3x_3-I=0$$

$$(M_1,M_2,M_3,\lambda \geq 0)$$

When I set $M_1=M_2=M_3=0$ (Lagrangian case), I got the optimal solutions for $x_1,x_2$ as:

$$x_1=\frac{p2}{p3}$$ and $$x_2=\frac{p_1}{p_3}$$

How could I construct a Marshallian demand function in this case? The optimal solutions haven't got the I (income) variable.

Is it correct to define a Marshallian demand function for good $x_1$ as:
$x_1(p,I)=\frac{p_2}{p_3}$?

  Marshallian demand function (named after Alfred Marshall) specifies
  what the consumer would buy in each price and income or wealth
  situation, assuming it perfectly solves the utility maximization
  problem

These are proper Marshallian demand functions, even though Income does not appear in them. This is due to specific form of the utility function (and the candidate solution of all goods being purchased at strictly positive quantities). It emerges that there is no income effect for goods $x_1$ and $x_2$ - optimal uncompensated demand does not depend after all on the level of income, but only on the relative prices.

The reason why Marshallian demand is defined as it is, is to make clear that it does not include any kind of "income compensation" as Hicksian demand does. But this does not preclude a case like yours, which again, depends on the form of the utility function.

Thinking economically, what goods can you think of whose demand may realistically not depend on the level of income but only on relative prices? Answering this question would be useful in order to map the mathematical expression of the utility function to real-world economic phenomena.

Also, note that the specific utility function points to "quasi-linear" frameworks, where the good that enters additively, essentially functions as Income itself.


Why is basis point defined as "1/100 of a percent"? Shouldn't it be defined as "1/100 of a percentage point"? If you take "1/100 of a percent" verbatim, it could mean the following :
If 1% increases to 1.01%, that is a 1% increase, which, according to the definition "1/100 of a percent", could mean an increase of 100bp.
Instead, the above case means that an increase of 0.1% points is equal to an increase of 1bp, not 100bps.
Defined "1/100 of a percent", "percent" could mean a ratio increase, instead of a point increase. So the definition itself is ambiguous.

The way in which basis points are almost invariably used is as a spread relationship (at least in fixed income). That is, it is the difference between two yields.
Yields are quoted as percent. (If you wish, percentage points.) Most spread relationships involve yields that are close to each other, so they are multiplied by 100 - so we get basis point spreads.
A change in yield is the spread between the yield now and where it was previously, so it is once again a difference, and so there is no ambiguity in the meaning. If a yield goes from 1% to 1.01%, it is a $100(1.01 - 1.00) = 1$ basis point rise.
Some people might quote low yields in terms of basis points. E.g., they might say that a bond yielding 0.1% has a “a yield of 10 basis points.” This is an non-standard usage of basis point, since it is referring to an absolute yield.
(The only exception I can think of for the usage convention I outlined is for management fees that are low - e.g., an index fund has “a management fee of 10 basis points.” However, since people think of that as a reduction in the return by 10 basis points, it slips through. Equity investors might use basis points for intraday returns. Since they do not quote spread trades in basis points, they have less reason to distinguish the terms.)


Why is ATC tangent to Demand in a Monopolistic or Mopolistically Competitive Firm Making Zero Profit? I understand that the Average Total Cost is equivalent to demand in a zero profiting monopoly or monopolistically competitive firm in order for there to be zero profits (when ATC = P at a given quantity), but why is ATC tangent to demand instead of intersecting demand at a point when ATC is downward sloping?

See:

A profit maximiser will aim to seek to maximise the difference between total revenue and total cost, providing this is non-negative

If there is a point at which this is positive, then this is equivalent to a point at which average total cost is below the demand curve, for example slightly to the right of $Q^*$ in your second diagram. At this point the firm will be making a profit, contrary to your assumption of zero profit.  So your second diagram does not illustrate the zero-profit case
Alternatively, if the demand curve is always strictly below the average total cost curve then all situations lead to a loss, so not a zero profit
That leaves the case where the demand curve touches the average total cost curve but does not cross it, as in your first diagram.  At this point there is zero profit and no opportunity to increase it.  If the curves are smooth enough, you might describe this as the average total cost being tangent to demand


Do privately owned banks ever indefinitely lend money to their owners? By privately owned bank, I mean specifically a bank that is not registered as a corporation with publicly tradable shares.

Does it ever happen that such banks create money by lending it to their own owners? The owners could simply borrow the money indefinitely, and spend it on luxury goods, keeping the bank in debt until they die, at which point they don't care anymore.

e.g. if the bank starts out for the first time, and obtains 1 B in cash deposits which it has as reserves, it could have a 10% reserve ratio, and lend 900 B to the owners, which they use to party and buy yachts. The other 100 M they keep as reserves.

Assuming a bank run doesn't occur, this can be stable indefinitely.

Does this ever happen? examples? or why not?

e.g. if it does not happen, is that because it is illegal, or are there incentive reasons intrinsic to banking why it would never happen?

Bank regulations vary by country, but it is a safe bet that they would frown upon having such a concentrated loan portfolio. Furthermore, there are presumably restrictions on lending to related parties, but how that would be defined would definitely vary by jurisdiction.

By doing a quick search on the internet, it was possible to find this FDIC letter about concentration risk. Link to FDIC page On that page is a link to a PDF which indicates that concentration risk is defined as 25% of capital. Unless the owners can get others to buy capital of the bank (which is unlikely if the only business plan of the bank is to lend money to the owners), the owners can only lend themselves back 25% of the money they put in before the regulators shut them down. (And in practice, I doubt that regulators would even let them get close to 25% of capital for related party lending.)

From a business perspective, it's not as if deposits can be attracted for free. The bank has to either pay interest for deposits, or else has to offer services (branches, ATMs, lending) which requires employees and cost money. A bank could not survive without having a lending portfolio that is paying a market rate of interest.

Finally, the bank needs equity, provided by the owners. Any sweetheart lending deals the bank offers the owners would reduce the profits on their equity. If the bank does not have a credible business plan, it is not going to attract other investors to supply capital needed for regulatory purposes.

The Savings and Loan Crisis in the United States offers a lot of examples of the breakdown of the regulation of small banks, but even then, the self-dealing was generally hidden.


Inability of a long-term bond market intervention to contribute to achieving a peg In «The Federal Reserve's Large-scale Asset Purchase Programmes: Rationale and Effects» 2012 paper, there's the following sentence:

  According  to  the  pure  expectations theory of the term structure,
  targeting three separate interest rates should succeed if and only if
  the private sector's expectations of the path of the short rate are
  aligned in a way that precisely   generates   the   targeted
  configuration   of   rates. In this environment,  for  a  given
  expected  path  for  the  short-term  rate,  direct  intervention in
  longer  maturity  Treasury  markets  could  not  contribute  to
  achieving  the  peg.

Why is it that intervening in the long-term market would not help achieve the peg?

Any help would be appreciated.

The issue isn't so much that intervening in the long-term market wouldn't help, but rather that under rational expectations all the return rates on government bonds are determined by the same process, and thus have to be consistent with each other and converge to the same rates. What D'Amico et al. seem to be suggesting is that the Federal Reserve tried to enforce mutually inconsistent rates on bonds with different maturities, such that there was a sequence of operations on bonds with short-term maturities (what I think they're calling "an expected path for the short-term rate") which would yield a return either above or below that of bonds with different maturities.

What it seems that they're really saying is that, under rational expectations, the FED would have had really just one policy instrument: the interest rate at any time. Bonds with different maturities just had to conform to that rate. But in practice, the FED tried to enforce a set of different rates, one each for each kind of maturity, which the authors think it's doable because the variance of bond prices is lower than what would be expected under rational expectations (hence the "the slightest fluctuation in expectations of the short-rate path would immediately imperil the peg" bit).


What are some good repositories for economic data Economic data is a very broad concept. It can include discrete preference relation data-sets as well as extensive time series data.

But it is important that theories are tested against data, and the earlier this happens in the development of an economic model, or of a new economic hypothesis, the more beneficial will be to the researcher, providing early signals on how good or bad a match with the real world his theoretical endeavors are. Economics being a Social Science, it is also important to have plentiful of different data sets, since testing a theory on just one data set does not provide strong enough evidence of the model's usefulness or uselessness.

So I am posting this question with the hope that it will be gradually enhanced with answers and it will become a point of reference for Economic data of all sorts for this community.

My offering: The World Bank for example has very useful information on trade lanes. Other repositories:

-Data links hosted at the
Econometrics
Journal

-Data archive of the
Journal
of
Applied
Econometrics

-Supplemental material for
Econometrica

Be more specific on what you need.
Quandl would be a pretty general source which hasn't been mentioned yet.
For macro data the St. Louis Fed is pretty good and thorough. Eurostat for European data. historicalstatistics.org for historical data.


Why not 2cov in this question? Okay, So this is the whole question. It's a bit far. But i just have one question about it:

So in question c) we have the equation for the sharpe ratio as:
$$\frac{E(w_1R_{1A} + w_2R_{2A})}{\sigma_c}$$

Where

$$\sigma_C = w_1^2 var(\beta_1 R_{1t}) + w_2^2 var(\beta_2 R_{2t}) + 2w_1 w_2 Cov(R_1R_2)$$

We have the $2cov$ here, but in e) question. The right answer is that they not have the $2cov$ here. Otherwise it is the same equation.

Why?

/

Because as your covariance-variance matrix shows the covariance is 0 between them.

The question e asks you to use the “true” covariance matrix from d presumably.

It’s not that you exclude it, that’s sort of misinterpretation, it is just zero ($cov(R_1,R_2)=0 \implies 2w_1 w_2 Cov(R_1R_2) =0$) and per convention zero is usually dropped from equations.


Homothetic Production Technologies Can someone suggest a good resource on homothetic technologies and what properties they imply about cost function, profit function, input demands, output supply etc? Also is it possible to have inferior inputs with homothetic technology?

The classic reference on this topic (as suggested by my professor) is the book;

Theory of Cost and Production Functions by Ronald Shepherd.


Induced technical change vs. directed technical change I am trying to get a better grip on the literature on directed technical change and induced technical change. In other words, I am trying to understand if these concepts are the same and used interchangibly in the literature to refer to endogenous technical change.

Also, can anyone please point to a good review of the literature on this topic(s)?

Directed technical change is the relatively recent name for what it was was previously called Induced technical change.

Informal discussion about the endogenous direction of technical change was first discussed (but not "microfounded") by Hicks (1932) and Fellner (1961). Yet, it was Kennedy (1964) who first proposed a formal model about endogenous technical change (previous examples like Solow (1957) assumed exogenous technical change).

The more recent denomination of directed technical change was proposed by Acemoglu (2002). This author has plenty of material about the topic:

these very comprehensive slides (also here, and why not, see all the material he shares here).
The 2002 paper on the subject
Chapter 15 of his book on economic growth

You can also see more about the historical literature on induced technological change here and here.


Is there an equation for compounding interest while adding to the principle amount supposed I invested \$5000 at 10 % per annum , but at the beginning of each year I add another \$5000. So I will have (\$5000 + \$500) + \$5000 = \$10500 at the beginning of the 2nd year and compound that at 10% and so on.
So in short suppose I invest \$5000 at 10 % per annum and add \$5000 at the beginning of each year , how much would I have after 30 years. Is there a simple equation to work that out?
P.S. I already solved it using excel but was wondering if there was a simpler formula

Let $A_t$ be the amount of money at the beginning of year $t$ and let $r$ be the interest rate. Assume that each year, we invest an additional amount of $Q$ Then we can recursively write this as:
$$
A_t = (1+r)A_{t-1} + Q.
$$
So the amount at time $t$, $A_t$ is the amount last year $A_t$ plus a percentage $r$ of $A_t$ plus $Q$. We would like to find a closed form solution for this problem, i.e. remove the $A_t$ from the right hand side.
Notice that, by recursion, also:
$$
A_{t-1} = (1+r)A_{t-2} + Q.
$$
As such substituting for $A_{t-1}$ in our original equation gives:
$$
A_t = (1+r)^2 A_{t-2} + (1+r)Q + Q.
$$
Next, substituting out $A_{t-2} = (1+ r) A_{t-3} + Q$ gives:
$$
A_t = (1+r)^3 A_{t-3} + (1+ r)^2 Q + (1+ r) Q + Q.
$$
There's a pattern here. $A_{t-j}$ gets multiplied by $(1+r)^j$ and then terms are added: $Q$, $(1+r)Q, (1+r)^2 Q, \ldots, (1+ r)^{j-1} Q$.
We conjecture therefore that the solution will be of the form:
$$
A_t = (1+r)^t A_0 + Q \sum_{j = 0}^{t-1} (1+ r)^j
$$
We can show that this is indeed the solution by induction. For $t = 1$ we have:
$$
A_1 = (1+ r) A_0 + Q.
$$
which is true.
Also:
$$
\begin{align*}
A_t &= (1+r)^{t} A_0 + Q \sum_{j = 0}^{t-1} (1+ r)^j,\\
&=(1 + r)\left[(1+ r)^{t-1} A_0 + Q \sum_{j = 0}^{t-2} (1+ r)^j\right] + Q,\\
&= (1+ r) A_{t-1} + Q.
\end{align*}
$$
Now, to finish up we can simplify the geometric sum:
$$
\sum_{j = 0}^{t-1} (1+ r)^j = \frac{(1+ r)^{t}- 1}{r}
$$
As such:
$$
A_t = (1+ r)^t A_0 + Q \frac{(1+ r)^{t} - 1}{r}
$$


Why there are no famous software/IT companies in Europe? (unlike USA) Both USA and Europe have highly developed economies with famous car, aircraft, food, oil etc. companies.
However, when it comes to IT -

USA has Google, Facebook, Amazon, Dell, Cisco, Microsoft, Red Hat, HP, Apple etc.
Europe has... nothing? (Nokia is dead)

Why?

In Europe there is typical more regulations and more and higher taxes on profits, financial transactions, and on products and services. This means there is less incentive for entrepreneurs to create their businesses since they face for example higher yields and higher interest rate coupons, environmental regulations, state bureaucracy which increases the cost and decreases the incentive (profit). Since most of innovative companies are generally start-ups such as Facebook was, there is higher cost of funding once these companies are considered riskier. In Europe that cost is further increased by the tax system for example.   There are also other factors such as education quality that differs. Europeans generally regards the US universities to be more prestigious and this is a decisive factor that makes lots of Europeans, typically the brightest to migrate to the US for further studies and higher wages since Americans can afford to pay more since there is less state interventionism.
I hope I gave you some ideas of what might be going on. In economics there is no definitive answer so keep on investigating and asking other people. Sorry for the lack of data.


Why is Walras equilibrium inefficient when we are dealing with public goods? I know that when we have public goods we have that:

$$MRT = MRS_a + MRS_b$$

Though I fail to understand why does this makes Walras equilibrium inefficient.

Thank you very much for your help!

In a competitive market for a private good (y) individuals may consume different quantities but the equilibrium condition requires that: $$ \frac{\frac{\delta u^{i}}{\delta y}}{\frac{\delta u^{i}}{\delta x}} = MRS^{i}_{yx} = MRT_{yx} \; \forall \; i $$

In the case of a public good (g) individuals may have different MRS but consume the same amount of the public good. Then for a given amount of a public good, the marginal benefit is the sum of individual marginal rates of substitution. Therefore, the optimal allocation must satisfy: $$ \sum_{i} \frac{\frac{\delta u^{i}}{\delta g}}{\frac{\delta u^{i}}{\delta x}} = \sum_{i} MRS^{i}_{gx} = MRT_{gx} $$

You can easily see that both conditions are not the same. In the case of an economy with only private goods, the benefit to
society of the last unit of a private good provided (expressed as the
willingness to forgo units of another good) is equal to the benefit of the
one person in society who receives this last unit. If there are some
people who receive a higher benefit from the last unit than others, we
don’t have Pareto efficiency. Hence, the marginal benefit of a private
good must be the same across all people for the allocation to be
efficient. Hindriks and Myles explain this thoroughly if you want to read further on the Samuelson condition versus competitive equilibrium (see Chapter 8 in particular).

Edit: There is no benefit that producers get here. Think of the MRT as the opportunity cost of producing an extra unit of a good and of the MRS as the marginal benefit of another unit of a good. The marginal cost of a unit of public good is one unit of private good (i.e. the MRT). Therefore the rule says that an efficient allocation is achieved when the total marginal benefit of another unit of the public good, which is the sum of the individual benefits, is equal to the marginal cost of another unit. If the good were private then the marginal benefit of an extra unit can only be attained by a single individual, so there is no sum.


(Microeconomics) How to classify Public Goods and Common Resources in Externalities? In Microeconomics there are 4 types of externalities - Positive Consumption Externality, Positive Production Externality, Negative Consumption Externality, and  Negative Production Externality. And along with externalities you learn about common resources and public goods which are classified according to their excludability and rivalry.

I believe that common resources go in the category of negative externality since they tend to be over-produced, and that public goods go in the category of positive externality since they tend to be under-produced.

What I'm confused about is whether common resourses are classified as Negative Consumption Externality or Negative Production Externality. The two have different characteristics that are represented by their different graphs, where the Negative Production Externality has two Cost Curves (Private Marginal Cost and Social Marginal Cost) and one Utility curve, while Negative Consumption Externality has two Utility Curves (Private Marginal Utility and Social Marginal Utility) and one Cost curve. I'm confused as to which negative externality common resources belongs to.

The same goes for public goods. Are public goods Positive Consumption Externality, or Positive Production Externality? Like the explanation written previously, the two have different features and graphs as well depending on whether it is consumption, or production externality.

So to sum up I guess I have two questions which are a bit similar:

Are Common Resourses classified as Negative Consumption Externality, or Negative Production Externality?
Are Public Goods classified as Positive Consumption Externality, or Positive Production Externality?

I would sincerely appreciate it if someone can provide me with a detailed explanation regarding each question. Thanks! :)

First let's clarify some terminology: I assume that with "common resources" you mean goods that are non-excludable but rivalrous. The term that applies in those cases is open access. In common resources there are shared property rights so some excludability (this is one of the mistakes in Hardin's tragedy of the commons).

The answer to your two related questions then is: it depends on the type of good. Public goods and externalities are related but distinct problems.

Take a classic public good: national defense. That is not a problem where production or consumption have unintended effects and hence there are no externalities. It is also a very clear example of a public good: one cannot be excluded from national defense and one person's protection does not diminish someone else's protection.

That is not to say that the two never go together. Air pollution is a classic example of a production consumption externality (the companies do not intentionally pollute the air, but reduce consumer health nonetheless) and often a public bad: if one lives in the area one cannot be excluded and breathing the polluted air by one doesn't reduce the overall pollution level.

The classic Gordon-Schaefer fishing model is both an open access resource problem and a production-production externality: the catch of an additional fish by one producer raises the costs of other producers.


Lexicographic preference relation cannot be represented by a utility function I am stuck on the following exercise, related to preference relations and von-Neumann-Morgenstern utility function.

A farmer wants to dig a well in a square field $[0,1000]\times[0,1000]$. The preferences of the farmer on the possible locations are lexicographic, i.e:

If $x_1<x_2$ then $(x_1,y_1)\prec(x_2,y_2)$ for all $y_1,y_2$.
If $x_1=x_2=x$, then $(x,y_1)\prec(x,y_2)$ iff $y_1 < y_2$.

Initially, assume that the well location must have integer coordinates.
Is there a preference relation on lotteries, that satisfies the von-Neumann-Morgenstern axioms, and extends the lexicographic preference relation? If so, what is a linear utility function that represents this relation?

I think the answer is yes, and a possible linear utility function is: $u(x,y)=100000x + y$.

Now, assume that the well location can have real coordinates. Prove that there is no linear utility function that represents the preference relation on lotteries.
Which one of von-Neumann-Morgenstern axioms is violated by the preference relation on lotteries?

Here I am stuck. I don't understand why the utility function I suggested above doesn't work? And what axiom is violated here?

We can say more generally that lexical preferences are not representable using a continuous utility function. Lexical preferences are not continuous. Note the definition of a continuous preference relation.

The preference relation $\succeq$ is continuous if for any sequences of consumption bundles $(x_{i})_{i \in \mathbb{N}}$ and $(y_{i})_{i \in \mathbb{N}}$ with $x_{i} \to x$, $y_{i} \to y$, and $x_{i} \succeq y_{i}$ for each $i \in \mathbb{N}$, then $x \succeq y$. That is, continuity preserves the relation at the limit point.

Consider $(x_{i})_{i \in \mathbb{N}}$ defined by $x_{i} = (\frac{1}{2^{i}}, 0)$ and $(y_{i})_{i \in \mathbb{N}}$ defined by $y_{i} = (0, 1)$. Clearly, $x_{i} \succeq y_{i}$ for each $i \in \mathbb{N}$. However, $x_{i} \to (0, 0)$ while $y_{i} \to (0, 1)$. So this preference relation is not preserved at the limit point.

Even more generally, no utility function represents the lexical preference relation. I prove for $\mathbb{R}_{+}^{2}$, but this argument extends to $\mathbb{R}_{+}^{n}$ by projecting into $\mathbb{R}_{+}^{2}$.

Proof: Suppose to the contrary that some utility function $u : \mathbb{R}_{+}^{2} \to \mathbb{R}$ represents $\succ_{lex}$. We thus have $u(x, 1) > u(x, 0)$, as $(x, 1) \succ (x, 0)$. We construct the interval $I(x) = [u(x, 0), u(x, 1)]$. Now for any two distinct $x, y \in \mathbb{R}_{+}$, $I(x) \cap I(y) = \emptyset$ as we have either $x > y$ or $y > x$ (so WLOG, we have $(x, 0) \succ (y, 1)$).

Define $\mathbb{I} = \{ I(x) : x \in \mathbb{R}_{+} \}$, and let $\phi : \mathbb{R}_{+} \to \mathbb{I}$ be given by $\phi(x) = I(x)$. Observe that $\phi$ is an injection, as each $I(x), I(y)$ are disjoint for distinct $x, y$.

Note that $\mathbb{Q}$ is dense in $\mathbb{R}$. So there exists a rational number in each interval. Define $\tau : \mathbb{I} \to \mathbb{Q}$ such that $\tau(I(x))$ returns a rational number contained in $I(x)$. So $\tau$ is an injection. We have $\tau$ composed with $\phi$ an injection, which implies $|\mathbb{R}_{+}| \leq |\mathbb{Q}_{+}|$, a contradiction. QED.


Economic anthropology: Have there been cultures / societies without a market? I was having a discussion about whether the "market" or "markets" always and inherently emerge as long as human societies exist.
Given the definition of market as the interplay of supply and demand and given that in our reality there is always a demand to sustain one's own life and at the same time there are limited resources, it is tempting to suggest that markets are an inherent feature of human societies.
Thus, the question: Have there been cultures or societes for which there is an agreement that they did not produce a market?
In this context I was reading about the terms reciprocity and redistribution. Are these mechanisms distinguished from the formal definition of a market? I.e., could one say that a group of people that exercised reciprocity observed a society without a market?

Wherever a society exists, there is repeated exchange. And wherever there is repeated exchange, there are also markets.
(Of course, markets can vary in the degree of their development/functioning. One market may be better developed/functioning than another in that it has more frequent transactions, better enforced property rights and contracts, greater honesty/trust, lower transaction costs, greater competition, better flows of information, etc.)
Exchange should be understood broadly: It can be one tangible object for another or even a piece of information for another. It can also be gift exchange. (A common layperson error is to think that only exchanges involving money count as exchanges.)
Gift exchange includes instances where a hunter shares his big find with everyone in the group, with the tacit understanding that he has access to the group's other resources. It could also be where one group assists a second group, with the tacit understanding that the second group will provide assistance should the need arise.
Rough exchange ratios can be implicitly established and sustained through tradition: If lazy John is always skiving, then members of the community will notice and eventually sanction him (through e.g. ostracism). So, if John is consistently taking from the rest of the community $X$ amount of meat every month, then he must also contribute $Y$ in fruit or other goods and services monthly to the rest of the community, in order to avoid sanction. Over time, rough exchange ratios (and these may be a wide band) may form and be embedded in tradition. And if conditions change so that meat becomes more abundant/scarce relative to fruit, then the exchange ratios may change accordingly.
(There is also the evolutionary argument that these exchange ratios will tend to be "correct" in the sense that they correspond to actual scarcity: Groups that maintain the correct exchange ratios thrive, while those that don't die out. Groups that develop strong social norms to punish free-riders and reward appropriate levels of reciprocity thrive, while those that don't die out.)

If two isolated individuals each living on his own deserted island visits each other and engages in an isolated exchange once or twice in their lifetimes, then one could possibly argue that no market exists. But then one should also argue that these two individuals do not form a society (under any reasonable definition of the term "society").


Augmented Gravity Model I am currently using the gravity model for my dissertation on migration flows. Do gravity models need to be augmented by dummy variables only or can other explanatory variables (such as the unemployment rate in the destination/ origin country) be included please?

All your feedback is greatly appreciated.

The gravity model of migration, like the gravity model of international trade, is a theoretical model. When you try to estimate the model, there isn't a predetermined cap to what kind of variables or how many of them you can include in your regressions as controls in the reduced form econometric specification. The controls aren't just limited to dummies. Those estimated coefficients are often interesting in their own right.

As Bunea (2012) says (emphasis mine):

  Since Lowry (1966, pp. 1-118), the basic gravity model has been
  extended to the following form:

  $$
\begin{align*}
M_{ij} = &\beta_0 \times log(g) + \beta_1 \times log(P_i) + \beta_2 \times log(P_j) + \beta_3 \times log(X_i) + \\
&\beta_4 \times log(X_j) + \beta_5 \times log(D_{ij}) + \varepsilon_{ij}
\end{align*}
$$

  where $X_i$ is a vector of explanatory variables describing different features of the origin (i.e. push factors) and $X_j$ is a vector of explanatory variables describing features of the destination (i.e. pull factors). Push factors are those characteristics of the origin place that encourage (discourage) out-migration (in-migration), such as low incomes, high unemployment, high prices, in general few opportunities for development.

Whether some of these explanatory variables should be included in your regressions warrants a separate question and is dependent on the research context.


Can the function of the Central bank become automated? How plausible is the idea of the central banks function becoming automated. Central Banks seem to function as a feedback depending on the flow of money, inflation and fractional reserve. If the plausibility of automatizing the central bank, what kind of electronic data will it need - will it look something like blockchain technology where there is a ledger for each transaction?

It is conceivable that one of the functions of a Central Bank, of being a regulator of money supply, could ultimately be automated, if artificial intelligence were to advance enough. However, that role is much more complicated than merely feedbacks on flow of money, inflation and fractional reserve. So for the foreseeable future, that's a role filled by a panel of human experts.

And no, it will not be a distributed ledger,  because that's a hopelessly inefficient way to record Central Bank transactions. The Central Bank maintains its own ledgers, and will continue to do so. It is not the case that each transaction creates a new ledger. Instead, each transaction creates corresponding entries in two ledgers - that is the essence of double-entry book-keeping.


Using the Federal Reserve's Implicit Price Deflator to Convert to 2020 Dollars I have a list of government expenditures from 2011 to 2018 as shown below:
2011    2012    2013... 2018
$30m    $35m    $42m    $120m

How can I use the Federal Reserve's Implicit Price Deflator, to convert these figures into 2020 dollars? I'm not sure how to do this despite having read a lot online.
The index year for this table is 2012 equals 100 USD. I understand that this chart shows that 100 USD in 2012 is now worth approximately 112 USD, but I'm not sure how to use it if my index year changes etc.
The use case is I'm updating a report on short-run and long-run marginal costs associated with various activities and want to put them into 2020 dollars so that they are comparable.

If you want to change the base year of a deflator just divide all values by the value in base year and multiply by 100. For example, if you have deflator with base year 2012:
           2011  2012 2013 ... 2020
Def_2012:    87   100  101      112

Then deflator with base 2020 will be given as:
               2011             2012           2013     ...     2020
Def_2020:    87/112*100   100/112*100      101/112*100      112/112*100

This way you you rescale the implicit deflator to 2020. Afterwards you can just apply the deflator as you would normally do.
Edit:
In order to convert all values to 2020 you can use the following formula:
$$X_{2020} = \frac{X_{t}}{\frac{\text{deflator}_{2020}}{100}}$$
where $X$ is the nominal variable you want to adjust and  subscripts indicate the years with $t$ being an arbitrary year.


What does one-sided polynomial on lag operator $L$ mean? In some time series texts, there are some talks about one-sided polynomial on lag operator $L$. I tried looking up what this means, but I cannot find one.

So what does one-sided polynomial on lag operator, $a(L)$, mean?

The terms "one-sided"  and "two-sided" lag-polynomials, are used when the text considers the option to specify an equation with both "lags and leads", i.e a relation where both past but also future values co-vary with current value. When the lag-operator is "one-sided" it contains only lags in the one direction. When it is two-sided, it contains "lags" in both directions.
In some cases, scholars have used the term "forward" operator to verbally contrast it with the "lag" operator, but my impression is that it is not so widely used.

One may wonder "so if I read "one-sided lag-polynomial", does it mean that I will consider only past values, or only future values?". The answer is that in almost all cases the invertibility condition holds, so in principle and in theory, it makes no difference. In other words, in principle the term "one-sided lag polynomial" may mean either

$$\phi(L) = 1+\phi_1L+ \phi_2L^2+...$$

or

$$\phi(L) = 1+\phi_1L^{-1}+ \phi_2L^{-2}+...$$

where $L^{-1}x_t = x_{t+1}$.

A specification with a two-sided lag polynomial may appear to violate a basic causality postulate (the arrow of time), but we should remember that estimated relationships are relationships of association, not causation.

So when the concern and goal is purely forecasting, "anything goes" -or at least this is what many Time Series Theorists and Practioners essentially say.


Can Linear Supply-Demand Equilibria Be Understood as a Feedback-Control Process? I'm not an economist, but rather an applied mathematician working in control theory. I've recently been watching Berkeley's intro to econ course in a personal project to better understand economics and finance. The first few lectures are on linear supply-demand curves and how they create price equilibria.

To wit, suppose $\sigma(p)$ and $\delta(p)$ and the quantity supplied and demanded at price $p$ resp. Then the assumption of linearity implies

$$ \sigma(p) = S_\sigma p+\sigma_0,\ \ \delta(p) = -S_\delta p+\delta_0$$

(this is my notation). The numbers $S_{\sigma,\delta}$, $\delta_0,\sigma_0$ are presumed positive and correspond to the slope (what I believe is called "elasticity" in econ jargon) and the intercepts, which I've interpreted as the amount supplied and demanded at $p = 0$ (e.g. how much people are willing to supply or would demand if the good were free). Clearly we assume $p \geq 0$.

The equilibrium is met when supply and demand are equal--$\delta = \sigma$--and give the equilibrium price
$$
p_e = \frac{\delta_0-\sigma_0}{S_\delta+S_\sigma}.
$$

This much is trivial to a real economist.

Now to the interesting part:

Through the lecturer explaining how an out of equilibrium price eventually comes to equilibirium, I recognized the basic idea of feedback at work: the change in price is forced by the signed difference between supply and demand. I propose this is modeled by the equation

$$
\dot{p} = k(\delta-\sigma),
$$

or exactly a feedback control law ($P$-controller) with feedback gain $k$. Furthermore, since we assume the supply and demand curves are linear,   the entire equation is now a linear differential equation

$$
\dot{p} = k(\delta_0-\sigma_0)-k(S_\delta+S_\sigma)p.
$$

The equilibrium occurs when $\dot{p} = 0$ and is identical to that obtained using the graphical method.

But wait--there's more! Since we now have a differential equation, we can get a time-domain solution:
$$
p(t) = e^{-t/\tau}p_i+p_e(1-e^{-t/\tau})
$$
Here $p_i$ is the initial (out-of-equilibrium) price, $\tau = \frac{1}{k(S_\delta+S_\sigma)}$, and $p_e$ is the equilibrium price.

This implies that not only can we predict what equilibrium prices will be,  but we can also, subject to understanding $k$, predict the amount of time it will take to settle into the equilibrium. In fact, the form of this equation means that equilibrium is asmyptotic ($p_e = p(t= \infty)$) and percentages--63%, 86%, 95%, etc. for $t/\tau = 1,2,3,$ etc.--of equilibrium will be obtained. $t/\tau = 6$ is a good approximation of actual equilibrium (99.8%), so we gain the ability to state how long it will take for a price initially at $p_i$ to settle to $p_e$ (call this $t_e$, the "equilibirum time")

$$
t_e \approx \frac{6}{k(S_\delta+S_\sigma)}.
$$

Thoughts on $k$

So what is $k$ really? The short answer is I don't know. Mathematically it is a feedback gain, but that's not really so helpful.

It has the units of price/(quantity$\cdot$time), so I have pulled it into two factors $k = f_t/S_X$. I suppose $f_t$ to be the frequency at which transactions are made of the good. It makes sense that a less often traded good will reach equilibrium more slowly than a more often traded good. $S_X$ has the units of $S_{\delta,\sigma}$ but I have no good interpretation.  I've considered that we might have $S_X \approx S_\delta + S_\sigma$ to produce $\tau \approx 1/f_t$ but haven't got much farther than this.

The Bottom Line

I think I've found an interesting interpretation of supply-demand dynamics in terms of control theoretic principles, namely proportional feedback. Unfortunately I am so unknowledgeable in econ that I'm unable to know whether or not this is a fruitful line of inquiry which I should pursue, or whether I have just reconstructed a longstanding theory.

Can anyone help provide some context?

I think there might be an issue with this as currently constructed:

$\delta(p)= \infty$, $\forall P \in \mathbb{R_+}$ and $P_e = \infty =\dot{p},$ $\forall P \in \mathbb{R_+}$ This is because $\delta_0 = \infty $. Further, $\sigma(p) = S_\sigma p+\sigma_0= S_\sigma p$ since $ \sigma_0 = 0$.

In words - there should be infinite demand for any normal good whenever P is 0 and there should be zero supply of any good whenever P is 0.

That said, your intuition here is solid but I do think you're working on reinventing the tatonement wheel. Tatonement, in a nutshell, adjusts prices up when there is excess demand and prices down when there is excess supply.

If you are able to do this after watching a few lectures online, then I say keep going. You could make meaningful contributions once you have a better understanding of the field.

edit: If you'd like to message me privately, I would be more than happy to pass along my graduate school notes/materials to help guide your studies. At minimum, I recommend you followup your current study with some more advanced micro theory courses.


What is the relationship between population growth and economic growth? Population Growth is only one of several factors of influence on economic growth but it is not strictly necessary for the economy to grow.  I have also heard of the term called the population bonus - which describes how a change in the demographic breakdown in the working-age population can impact economic growth.

I want to understand what is the relationship between population growth and economic growth and how can population bonus potentially increase growth in GDP.   Can anyone help explain the importance?

This is actually quite complex issue. Taken narrowly answer to your question:

  what is the relationship between population growth and economic growth

Is that simply the relationship is positive. This is because in the question you ask only about growth of GDP per se. Growth ultimately depends on growth of production and labor is an important input in the most of production. One of the most widely used production functions is the Cobb-Douglas production function which has following form $Y = AK^{\alpha}L^{1-\alpha}$ and it is trivial to see that output depends positively on labor.

However, there are some problems with looking just at growth of GDP. Even though GDP will grow with more population you could in principle end up in a Malthusian trap. Yes GDP increases with more population but that is because we have more people and hence that does not mean that this increase in growth actually makes peoples lives better. This is why economists care not just about growth overall but rather per capita growth (that is economic growth per person).

When it comes to the relationship between per capita growth and population literature gets more murky. The most widely used growth model is the Solow-Swan model and in it increase in population has no effect on per capita growth. In Solow-Swan model all growth is exogenously given by growth of technology. Most of professional macroeconomist believe in Solow-Swan model because there is quite a lot of empirical evidence in its support but in the last 30 years endogenous growth theory is challenging the Solow-Swan model.

In endogenous growth theory population growth can actually lead to also per capita growth. This is because in such models growth depends not just on exogenously given (predetermined) growth of technology but can be affected by choices society makes. One of those choices is also to devote more labor towards R&D and knowledge development, hence in such model actually population growth can lead to sustained economic growth if that extra population is educated and channeled towards R&D/knowledge creation. This being said an important caveat is that although endogenous growth theory got more popular in recent years and even Paul Romer got few years back Nobel Prize in Economics for his contribution to the theory, it is still not widely accepted because its extremely difficult to test. So take that as you will.

  how can population bonus potentially increase growth in GDP

Population bonus is more widely known as demographic dividend. It referrers to situation where country's demographic structure improves in a sense that there are more working age people relative to either young or elderly. For example economy where you have 60 retirees 20 infants and 20 working age population will have (ceteris paribus) harder time providing for its people than economy where you have 10 retirees 15 infants and 75 working age population. This is because for most part all goods and services are produced by working age people (save some exceptions like child actors/paper delivery etc) and hence non-working age people do not really directly contribute to the economy.  Especially in those countries where state offers generous social benefits to non-working age people it is important to have enough working age population from which state can draw resources through taxes.

Thus having population bonus/demographic dividend can certainly increase GDP, if its sustained it can also lead to economic growth but it is worth noting that it is very difficult sustain growth this way. There is plenty of evidence that as people get richer their fertility declines and demography soon becomes unfavorable like in Europe or Japan. Hence, it is very unlikely demographic dividend can deliver long term sustainable growth but in can definitely help raise GDP while its happening and in principle can be sustained for few generations but not indefinitely.


Extensive Form Representation Question: Represent the extensive form of the game:

Suppose, Nature chooses either Up or Down with probabilities 0.5 and 0.5, respectively.

Then Player 1 chooses either Right or Left and does not observe nature's choice.

Finally, Player 2 chooses either G or H. Unlike Player 1, Player 2 observes the choice of Player 1 prior to making his choice. Player 2 also observes nature's choice if Player 1 has chosen Right but does not observe nature's choice if Player 1 has chosen Left.

My take:

Is this correct? I am not really sure about the instructions in bold.

Imagine you're Player 2. What different "scenarios" could you find yourself in?

If Player 1 chooses Right, then Player 2 could observe if Nature played Up or Down. So you have 2 information sets there.
If Player 1 chooses Left, then Player 2 couldn't differentiate between (Up, Left) and (Down, Left). How many information sets should there be here?

So in total, you should have how many information sets? You currently have 4. Which one should you "combine"?


How did the Hunt brothers' cornering the silver market drop gold prices in 1981? Muro's answer at Money SE discusses the attempt by brothers Nelson Bunker Hunt, William Herbert Hunt and Lamar Hunt to corner the silver mark. But Muro didn't explain the causation. I can't infer why this cornering caused gold prices to drop in 1981.

Since nobody is answering this, I will point out that the answer is half there.

The cornering attempt directly affected the silver price, but gold went along for the ride. People treated gold/silver as an asset class - “precious metals” - and they traded them on a relative basis (the gold/silver ratio).
The rules were changed, and the Hunts were forced to liquidate. This directly drove down silver futures.
Gold went along for the ride. People who were long silver and gold got hit my margin calls, and they had to liquidate both. The gold/silver ratio would rise if gold did not move. Finally, people unaware of the details of the silver market would just view this as the end of the run for “precious metals.”
Beyond that, it’s just opinions.


In the Solow growth model, does labor grow like technology at the steady state? The Solow growth model defines

$$k_t \equiv \frac{K_t}{A_t L_t}$$

Let our production function be Cobb Douglas

$$Y_t = K_t^\alpha (A_t L_t)^{1-\alpha}$$

We have

$$A_t = A_0 e^{gt}$$
$$L_t = L_0 e^{nt}$$

We find the steady state is

$$k^* = \left(\frac{s}{n+ g + \delta}\right)^{\frac{1}{1-\alpha}}$$

At the steady state, we notice

$$\frac{K_t}{L_t} = A_t k_t = A_0 e^{gt}  \left(\frac{s}{n+ g + \delta}\right)^{\frac{1}{1-\alpha}}$$

My Question

Does $$\frac{K_t}{A_t} = L_t k_t = L_0 e^{nt}  \left(\frac{s}{n+ g + \delta}\right)^{\frac{1}{1-\alpha}}$$

also grow at the steady state? In several lecture notes I found online, no one discusses this so I wanted to clarify.

Follow Up

But if $\frac{K_t}{L_t}$ and $\frac{K_t}{A_t}$ both grow this way, then

$$K_t = A_0 e^{gt} L_0 e^{nt}  \left(\frac{s}{n+ g + \delta}\right)^{\frac{1}{1-\alpha}}$$

So does this mean $K_t$ is a function of these parameters (e.g. $A_0$) or is this not a valid economic expression?

In deterministic growth models of closed economies with exogenous constant growth drivers (here, population and labor efficiency), the steady-state of the economy is in "growth rates", or under the newer terminology, it is a "balanced growth path".

The growth rate of the economy can only be the sum of the growth rates of these exogenous growth drivers. So in the balanced growth path, aggregate magnitudes grow at $n+g$, per capita magnitudes grow at $g$ and magnitudes per efficiency unit of labor are constant.

One can see this directly through the production function by considering

$$Y_t = K_t^\alpha (A_t L_t)^{1-\alpha} \implies \ln Y = \alpha \ln K_t + (1-\alpha)\cdot [\ln A_t + \ln L_t]$$

$$\implies \frac {\text{d}\ln Y}{\text{d}t} = \frac {\dot Y}{Y} = \alpha \frac {\dot K}{K} + (1-\alpha)\cdot \left[\frac {\dot A}{A} + \frac {\dot L}{L}\right]$$

$$\implies \frac {\text{d}\ln Y}{\text{d}t} = \frac {\dot Y}{Y} = \alpha \frac {\dot K}{K} + (1-\alpha)\cdot (g+n)$$

To have a balanced growth path, we must have equal and constant growth rates (so ratios like $Y/K$ etc remain constant), therefore

$$\frac {\dot Y}{Y} = \frac {\dot K}{K} = \gamma$$

$$\implies \gamma = \alpha \gamma + (1-\alpha)\cdot (g+n)$$

$$\implies \gamma = g+n$$

Note the importance of constant returns to scale, in order to obtain this result.


How do EMH proponents explain Black Monday (1987)? On Black Monday (19 Oct 1987), the Dow Jones Industrial Average fell by 22.6%. How do proponents of the Efficient Markets Hypothesis (EMH) explain this event?
Do they actually claim that the intrinsic value of the DJIA fell by 22.6% that day? Or do they have some other explanation?

Different proponents of efficient market hypothesis  (EMH) will have different explanations because EMH does not say which fundamentals/variables should drive prices. EMH just states that prices reflect avaiable information but it does not say which avaiable information is important and which isn't.
There are actually several versions of efficient market hypothesis  and answer might be slightly different for each of them (in details but not much overall). However, generally speaking efficient market hypothesis is not theory about why market prices fall or go up.
Efficient market hypothesis is a hypothesis about informational efficiency of market. As explained in Miskin & Eakins, Financial Markets and Institutions 8th ed. pp 158 :

Efficient market hypothesis (also referred to as the theory of efficient capital markets) ... states that prices of securities in financial markets fully reflect all avaiable information.

authors further explain:

The efficient market hypothesis views expectations as equal to optimal forecasts using all avaiable information. What exactly does this mean? An optimal forecast is the best guess of the future using all avaiable information. This does not mean that the forecast is perfectly accurate, but only that it is the best possible given the avaiable information.

Consequently, EMH does not say why prices go up or fall. It just says they contain all avaiable information and also refers expectations of future prices and again just says that these expectations will be the best forecasts given all publicly avaiable information. It does not even say these forecasts will be correct as they are just forecasts.
In order to say why prices increase or decrease you would have to combine EMH with some asset pricing theories that actually try to explain stock price behavior. For example capital asset pricing model (CAPM) could be used to explain what drives returns (which are changes in prices) and CAPM can be modeled together with assuming EHM but it can also stand separately. Furthermore, CAPM is just one example of a model that tries to explain asset prices, there are many more and in principle all can be combined with EHM.
For example, Stout (1997) uses EMH together with CAPM to show that in such model stocks might be sometimes undervalued (when there is large amount of uncertainity). So Stout who is proponent of EMH (at least to the extent that he uses it in his research) could say this might be because market undervalued stocks due to uncertainty.  The point here is that in order to know what proponents of efficient market hypothesis think about what caused Black Friday you need to enquire about what asset pricing model they are using. Different proponents of EMH will give you different explanations because being proponent of EMH does not mean you are necessarily 'married' to single asset pricing model. For example, if someone argues stock prices are given by present value of their dividend (something that could hold whether EMH holds or does not hold) would say that Black Friday was caused by expected net present values of dividends dropping and EMH would just state that those expectations were formed by all publicly avaiable information.


Could a carbon tax be a substitute for VAT? VAT is a tax on consumption. However, while in some instances it taxes negative externalities (e.g. fossil energy consumption) in many other cases it taxes services without externalities or potentially positive externalities (e.g. advisory services on how to reduce carbon emissions).

If consumption is being taxed arguably it would make more sense to tax consumption that actually causes negative externalities. Carbon taxes would be an obvious candidate (though arguably not the only one). The price of a product in a value chain would increase as carbon is added to it, just as with VAT, only that the tax is on "carbon added" rather on "value added".

Presumably a carbon tax would also be easier to administer than a VAT since carbon is taxed only where it enters the economy?

No, taxing externalities is not a good substitute for VAT for several reasons.

Pigovian taxes (taxes on externalities) have to be set up in an optimal way to actually reduce externalities. However, it is astronomically improbable that the same level of tax that mitigates the negative externality is exactly the same level that government has to raise for optimal provision of public goods.
It can be shown that Pigovian taxes can be imposed without deadweight loss on economy (which almost all taxes have) but such result hinges on a fact that they are made revenue neutral - hence to use Pigovian taxes in most efficient manner you should used them in a way that does not raise any additional revenue.
By discouraging the activity that creates negative externality the tax basically by substitution effect erodes its own tax base. Every tax does this to some extent - VAT for example encourage substitution of savings for consumption or substitution away from market goods to non market goods but since VAT is very broad this should reduce tax base much less than just selective tax on goods that create negative externality.

Consequently, Pigovian taxes are ill suited for  raising government revenue. Government should still pursue Pigovian taxes everywhere where there are externalities as correcting the externality is always optimal from social perspective. However, no modern government can realistically raise the kind of money it needs to run itself without VAT - in fact it is no accident that modern governments expanded precisely when modern VAT and income taxes were developed, if you check most government budgets VAT and income tax are always the biggest sections on revenue side of budget.


Does the break even analysis contradict the law of supply? The law of supply shows a positive relationship between price and quantity
However, a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even.
Does that mean that a firm producing at break even level is an exception to the law or that the analysis contradicts the law of supply ? ( please correct me if there are any inconsistencies in the logic)

The short answer is no.

The law of supply only states that increases in price result in higher quantities supplied, ceteris paribus. Break even is the point where Revenue = Costs.

Your confusion might come from this sentence:

  a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even.

If the price lowers, the firm will reduce its quantity supplied. That's it. The specific quantity depends on the variable and fixed costs.
It's not mentioned that the company has to break even - think of a monopolist for example.


Help with weierstrass’ theorem Question: Use the Weierstrass Theorem to show that a solution exists to the expenditure minimization problem of subsection 2.3.2, as long as the utility function II is continuous on $\mathbb{R}$ and the price vector p satisfies p » O. What if one of these conditions fails?

$X(\bar u)$ = $\{x\in$ $\mathbb{R}$$^n$$_+$| $u(x)$ $\geq$ $\bar u$

The objective is to solve:

Minimize $p\cdot x$ Subject to $x\in X(\bar u)$.

I do not understand how to actually apply the weierstrass theorem here. Can anyone solve this exercise?

This is the version of the theorem I will use:

Weierstrass theorem: Every continuous function defined over a compact set attains a minimum.

The function $p\mathbin{\cdot} x$ is continuous. The problem is that $X(\bar{u})$ is not compact. So, an additional trick is needed. Since we are working with a subset of $\mathbb{R}^n$ compact means the same as closed and bounded (Heine-Borel theorem). The continuity of $u$ is used to guarantee that $X(\bar{u})$ is closed. But it could be unbounded.

This is where $p\gg 0$ becomes useful. Take an arbitrary point $x_0 \in X(\bar{u})$ and consider the set $Y$ defined by  $$Y = \{\,y\in X(\bar{u}) \mid p\cdot y\leq p\cdot x_0\,   \}.$$
Since $x_0\in Y$, we know that $Y$ is nonempty. Using the fact that $p\gg 0$, you can show that $Y$ is bounded (I'll let you figure out the details). Since $Y$ is the intersection of two closed sets, it is also closed. Hence, it is compact. Therefore, by Weierstrass theorem, $p\cdot x$ has a minimum in $Y$.

I will also let you figure out the last step on your own, which is to show that the minimum of $p\cdot x$ in $Y$ is also a minimum in $X(\bar{u})$.


Nonparametric estimation: Interaction of two continuous variables The probability of observing a binary outcome y is a function of two variables, y = f(x1, x2).

Both x1 and x2 are continuous, and I have no prior on the functional form of f. I have shallow knowledge of kernel density estimation, but have only seen that done in the case of a univariate variable. How should I proceed here / is there some basic literature on this?

(I misunderstood the question at the beginning)
The model is a binary non parametric regression.
I don't know of a general model, I know of a semiparametric model by Klein and Spady
http://www.ssc.wisc.edu/~bhansen/718/NonParametrics7.pdf
The model is as follows let $x´=[x_1´ \quad x_2´]$
Now $P(y=1|x)=E[y=1|x]=g(x'\beta)=f(x)$.
The details in the link.
Another possibility is to estimate the joint distribution (that has both discrete and continuous variables). Let g(y,x) be the joint pdf, then estimate $g(y|x)=\frac{g(y,x)}{g(x)}=f(x)$. The estimation of a joint of this type needs a product kernel that you can check
Nonparametric estimation of regression functions with both categorical and continuous data
Journal of Econometrics, Volume 119, Issue 1, March 2004, Pages 99–130
Jeff Racine, Qi Li

For $y$ continuous the following answer works.
First of all is your data deterministic or random.
If deterministic you may wish to do interpolation. Splines are your best option.
If you think your dataset has noise or you have unobserved variables then you can do a non-parametric regression, I think this is what you want.
The first step is to actually make assumptions about the structure of the randomness.
The usual assumption is that
$y=f(x_1,x_2)+\epsilon$ and you let $x´=[x_1´ \quad x_2´]$ be orthogonal -indepdent- of the unobserved variable $x\perp\epsilon$. Then you can replace $f$ by some high degree polynomial on $x$ and estimate this using Non linear least squares, technically this is called series or polynomial estimators (NLS is used when you know f, but you can argue that a high degree polynomial approximates the true f).
Another way is using a ¨kernel¨ approach.
In fact  you can directly assume that f is the conditional expectation.
$E[y|x]=f(x_1,x_2)$. In that case you can estimate it using the joint distribution recall the definition of conditional expectation
$E[y|x]=\int yf(y|x)dy$, with $f(y|x)=\frac{f(y,x)}{f(x)}$. Replacing this elements by its kernel estimators you get the Nadaraya-Watson estimator, you can google it.
A third option is to use local linear regression. A more advanced option is to use Sieves estimators that use function basis to estimate the model.
If f is not the conditional expectation then you have to use non parametric regression with instrumental variables, this is more complicated there is a paper by Renault (and others) on Econometrica about this.
In summary it depends on the randomness structure that you are assuming.
The book to go if you want to learn non-parametric regression is Li & Racine http://press.princeton.edu/titles/8355.html
There is also a nice implementation of these econometric procedures in the open source statistical language "R" under the name (package) "np".


Who is the first person/paper to introduce "mixed strategy"? Who is the first person/paper to introduce "mixed strategy"? The PNAS by Nash used this notion without citing anyone.
Does the earlier book: Von Neumann, J., and Morgenstern, O., The Theory of Games and Economic Behavior
include the introduction of mix strategy?

Although I don't have the book here, I am quite certain that mixed strategies are already present in the book from 1944. I believe that I have read somewhere the claim that "mixed strategies" where actually introduced in an earlier version of the book (published in German in 1928, says Wikipedia). However, what I have read mostly is the claim that Emile Borel (yes, THE Borel) first wrote down this idea in a note in 1921.
I guess here is a man with enough authority to state that:

E. Borel was the first author to evolve the concept of a strategy, pure as well as mixed, although he did not go beyond the case of the symmetric two-person game.

says von Neumann here: "Communication on the Borel Notes", J. von Neumann and M. Fréchet, Econometrica 1953.


Are there Recent Studies of Price Elasticity of Demand for Housing in England? To address what is widely perceived as under-supply of housing in England resulting in high costs of housing, the UK government has set a target of 300,000 new homes annually (the target relates to England only because housing policy in Wales, Scotland and Northern Ireland is the responsibility of their devolved administrations).  I am interested in assessing the effect that achieving that target over a period of years would have on the cost of housing (for both buyers and renters).
Are there any recent (say last 10 years) academic studies of the own-price elasticity of demand for housing in England (or for the whole of the UK)?

I found a working paper  by Auterson (2014) which estimates that the price elasticity of demand for housing in UK was $-0.92$, but it is an auxiliary result as the paper's main intent was to forecast the housing prices in UK not to estimate elasticity - might be still useful though.


Structural VAR and Granger Causality Is it possible to Structural VAR (vector autoregression) model to imply Granger Causality?
In other words, if X and Y are determined at the same time, is it possible that X Granger-causes Y?
Thanks!

Every structural VAR (SVAR) model, e.g.
$$
B_0 y_t = B_1 y_{t-1} + u_t
$$
has an equivalent reduced form (VAR), e.g.
\begin{aligned}
y_t &= B_0^{-1} B_1 y_{t-1} + B_0^{-1} u_t \\
    &= A_1 y_{t-1} + \varepsilon_t.
\end{aligned}
where $A_1 := B_0^{-1} B_1$ and $\varepsilon_t := B_0^{-1} u_t$.

The reduced form can be used directly for testing Granger causality. In the example, one would test whether certain off-diagonal elements in $A_1$ are equal to zero. E.g. if $y_t$ is bivariate, then testing $H_0\colon a_{12}=0$ would tell us whether $y_2$ does not Granger-cause $y_1$. The structural form might even imply Granger non-causality if it implies that certain off-diagonal elements are zero.

However, Granger non-causality does not preclude finding an equivalent representation of the reduced form that has a nondiagonal $B_0$ matrix. In fact, you would just multiply any reduced form (whether Granger-causal or not) by any nondiagonal matrix, and that matrix would then be your $B_0$. Hence, $B_0$ being nondiagonal is not informative of presence or absence of Granger causality. (This is a sort of tautology, I think.)


R in econ departments? According to my personal observation the majority of (prominent) economists prefer use Stata for their statistical analysis and Matlab for other mathematical work. SAS and Excel are also used (especially in finance).

In my opinion R is a much better software for data cleaning, manipulation and analysis than Stata (not to mention that Stata costs) it is also seems on par with Matlab in what it does best. But I guess (as a graduate student) it will not make cooperation very smooth using a different statistical program than the rest. Use Stata like everyone else or suffer?

Thus a student who is an 'R expert' should, if having to choose between two equal departments, choose the one that uses R. But does such a department exist? A department where at least few researchers use R?

In my university the choice of
program is considered generally irrelevant. We focus on results, and it is up to each student to determine which program is best suited for the task and user preference.

You will find that using one language translates very well to another. With resources like stackoverflow, I would not be too concerned about which.

I would very carefully consider the statement "two equal universities". My experience has suggested to me such a thing does not exist, granting personal ambitions, career goals, and topic preference. I think programming choices are perhaps only a proxy for technical rigor and budget, little more.  Given you are only accepted to a finite number of schools, take time to evaluate this criteria closely.


How to change the observation for the first lag in an AR(1) model? I run a simple AR(1) model in my analysis using ols:

ar.ols(df$y, order.max = 1))

However, I work with generations as my unit of analysis. Therefore, the first lag of y would be the observation of y at time t-30. How can I specify this in the AR(1) model in R?

The ar.ols() function in R defaults to order $10log_{10} (N)$.  This implies that, to catch the 30th time lag, you need at least 1000 observations. That seems like a lot, but bear in mind that you are then running OLS on 30 features (lags t-1 to t-30). All else equal, it is not clear why 30 is the magic number for you here, and you can't just drop the more recent lags as they may have the dominant effect.

Without knowing more about your study, it's hard to recommend further.  Intuitively, though, I would not expect to be using annualized data to perform generational analysis unless I were running OLS (not AR) and explicitly constructing a lagged column vector to use as an additional feature alongside contemporaneous features/controls.


How to find price elasticity in the following demand system? We assume that the logarithmic function of $p_i$ equals the coefficients of the demand equation $w_i$. I have the following demand system:

$$ w_{a}=-0.03-0.01 \ \ nk +0.02 \ lcons $$
$$ w_{b}=-0.26-0.004 \ nk +0.08 \ lcons $$
$$ w_{c}= \ \ \ 0.96+0.03 \ \ nk -0.14 \ lcons $$
$$ w_{d}= \ \ \ 0.30+0.001 \ nk -0.05 \ lcons $$
$$ w_{e}= \ \ \ 0.07-0.005 \ nk +0.04 \ lcons $$
$$ w_{f}=-0.05-0.01 \ \ nk +0.04 \ lcons $$

Variable $w_i$ denotes the proportion of good $i$ of the total consumption (cons) of a family, and $nk$ (number of kids) is a dummy variable. $lcons$ is the logarithm of the total consumption. Plus, I know the income of the families.

I will appreciate your help with this situation.

The price elasticity is the derivative of the demand for each good over price. Demand is lcons * (wi) for each wi.


Is there a relationship between stock market hedge fund leveraging and the repo market? Lately there have been many articles attempting to link repo-market borrowing with equity market increases. 1

I can see a link if the hedge fund is buying treasuries or mbs and financing it by using the treasuries or mbs as collateral in the repo market. For example, I put up \$1B in equity and buy 30 year treasuries. I then use those treasuries as collateral in the repo market to raise more funds to buy treasuries (less any haircut). Rinse and repeat to lever up the \$1B in equity to whatever.

However, if the hedge fund invests in equities (the stock market) with that same \$1B how does the repo market enable hedging? I can't use the stock I bought as collateral in the repo market right? What am I missing?

"I can't use the stock I bought as collateral in the repo market right?"

In principle, you can.  Repos can and do happen between non-government actors, with any AAA-rated paper being preferred, but a slice of the repo market does deal in riskier instruments like corporate bonds and equities.   But that piece of the repo market likely isn't big enough to impact the money supply, and it isn't what's being alleged in the article you linked.

The article specifically addresses repo transactions between the Fed and financial institutions.  In these, the Fed buys Treasuries from banks under an agreement to resell them at a higher price, usually (but not necessarily) the next day.  The spread is the repo rate.  The intended effect is to increase short-term liquidity, by inducing banks to reduce prime rates and lend more freely.  This lending is supposed to support real economic activity that is otherwise not happening due to various frictions in the economy - it operates on the presumption that output is below potential, and that this is primarily because of credit constraints based in sentiment rather than fundamentals.

The issue is that if the market fundamentals have shifted so that the economy is in fact not producing below its potential (or not as far below as is believed), but is rather transitioning to a new and lower steady state, then there is no appetite for the liquidity being injected in this way.  So while banks are enticed to sell off their Treasury holdings because of aggressively low repo rates, because fundamentals in the economy are weak, they are reluctant to increase loans to economic actors.  Being bound to repurchase those treasuries in the short term, they pile the money into financial markets with the expectation of realizing their desired gains in this way instead.

Being collectively large enough to move share prices, when all of the banks are doing this together it produces a false signal: normally, markets rise because the underlying economic activity represented by stocks is rising.  Yet in this mechanism, markets are rising even as the underlying economic activity is stagnant or even declining.

This is what analysts mean when they talk of "decoupling".


Does Medicaid or Medicare represent a higher risk to the Federal Government Solvency and Debt going forward? I'm trying to understand not only which program is expected to be larger in terms of costs, but which one has the least capacity to adjust (reduce benefits if revenues for the program fall).

Yes, both presents higher fiscal risks up to the future, as they imply in a higher Federal deficit. The hability to adjust depends, unfortunately, not only on the program itself, but in a political risk as the congress could refuse to reduce the revenue to these programs as this would be not appealing for voters. At least USA does not have something like "free healthcare for everyone" written in it's constituition like countries such as Brazil have.
The broader the extent of the program itself determines it's costs.


Speculation: Why is it referenced and how do we account for it? I've been to a couple of presentations recently on the housing market in Toronto and I've heard the term "speculation" thrown around by economists/academics when discussing factors that are contributing to the rise in housing prices.

It seems to be just something to pin the blame of forecasting error on.

I am yet to see any clear measures of speculation but always see it mentioned again and again.

It is problematic to use as a word.  Some of the better authors in the last one hundred years have defined it, but there is no consistent definition.  I provide an operational definition for a narrow purpose in a paper as those possessing risk-loving preferences.  That definition, of course, would exclude many traders.  I use it with reference to implications for prudential regulation, but I didn't need to measure it for what I was doing as their presence or absence didn't impact the empirical component.

The problem of talking about "speculators," aside from the definition problem, is that it is not a behavior.  Indeed, it's not meaningful.  Imagine seeing a newspaper article about a plane crash, and "gravity" was blamed.  Excluding planes flying into mountains, how useful would it be to discuss the effects of gravity on planes instead of mechanical failure causing gravity to suddenly become more meaningful?

I can think of possible ways to mathematically segregate speculative behavior if you can clearly state what you mean by the word, but I would really want to ground that in a model so it would very clear what the predictions might be.  However, I can also imagine definitions where central banks could fall into that categorization in some cases so rigor will be both difficult and require a clear statement that it is an operational definition and not a canonical definition.

In the housing market, I doubt it could be readily measured because the measures in the housing market are so easily manipulated and there are plenty of incentives to manipulate.  That may not be true in the Canadian system, but in the American system, particularly with REITs, there is just a ton of ways to play games.  I think you would want a tamer market, such as equities, to build test runs of a methodology.


What happens to a company whose share price goes down to very little over dividend loss? This is a bit of an edge case scenario.

Let's say you have a company that is very profitable (and pays very high dividends off the profit) but has little/no growth potential.

The company decides to slash the dividends significantly (to the point that the total yield is super low, lower than most other equity) in order to pay significantly higher wages/benefits to their headcount.

Presumably, the share price of the company would plummet drastically, as nobody would want to own the shares (zero dividend + zero growth means you're better off investing in a savings account, at least outside of negative interest rate countries).

What happens to that company's finances longer term?

Their valuation just shrunk to almost nothing, but they aren't bankrupt as they still have the same cash flow, same same business, same outstanding debt and same profits as before - they merely redirected the profits from dividends to headcount costs.

(let's ignore real-world implications such that the company may start experiencing growth out of better publicity, employees using extra earnings to purchase their products ala Henry Ford-UAW parable, etc...).

What happens is completely dependent on the owners. They're the ones whose income has been taken anyway. They may be the only ones with the legal power to do anything (depending on the jurisdiction: there may be some countries where the State can intervene in such matters).

In some jurisdictions, the directors have a legal obligation to maximise returns to the shareholders over the long term; in other cases, they may have a contractual obligation to do so. And in most cases, that will be the expectation of the shareholders, anyway. They can intervene at an Annual General Meeting or an Extraordinary General Meeting: they can pass motions to change future dividend policy; or replace the directors - they could, of course just become the board of directors themselves. And if there's a legal or contractual obligation, then there may be recourse available through legal action too, to seek compensation for the directors' past transgressions.

If they decide to do nothing, then nothing in particular happens to the company's finances.


Do welfare payments for children incentivise having children? A common argument against safety net social welfare for people with children is that it can incentivise people without better options to have children. This could be considered a problem if it means people who aren't fit to be parents continue having children.

For example in New Zealand the the Domestic Purposes Benefit (a benefit paid to people with children, if child support from the other parent does not exceed the base rate) pays about the same rate as a minimum wage full time job.

An argument could be made that the economically rational decision for an uneducated person is to have children, where they can't be fired, aren't subject to starting times or drug tests etc.

Is there any evidence for what the effect of welfare payments for children has?

In particular I'm interested in any situation where the entitlement has either increased or decreased, and what the effects on birthrate or people applying for the benefit has been after that.

For example, "Assessing the Impact of the Maternity Capital Policy in Russia" (2014):

  Starting in 2007, the federal government has pursued an ambitious pro-natalist policy. Women who give birth to at least two children are entitled to "maternity capital" assistance ($11,000). In this paper we estimate a structural dynamic programming model of fertility and labor force participation in order to evaluate the effectiveness of the policy. We find that the program increased long-run fertility by about 0.15 children per woman.

Of course, the impact varies by country and incentive. See references and citations of this paper for more research.


How do I find the probability mass function of an individual observation for a multinomial logit model? I know what the probability mass function of a multinomial logit model is.  However, I do not know what the probability mass  function of an individual observation i, f(yi|xi), is for a multinomial logit model.  I would really appreciate if someone could explain how to get the answer and knows the correct answer.

A useful way to view the model is as follows. For each observation $i$, there are $K$ different possible observations (choices). So, there is a random variable $\varepsilon_{i,k}$ for each possible choice. We observe which choice is made and the values of the explanatory variables $\boldsymbol X_i$. Then, for a given distribution of the random variables $\varepsilon_{i,k}$ (given parameter values of the Type-1 EV), we know the conditional probability of picking choice $k$ conditional on $\boldsymbol X_i$.

This is explained in the following part of the relevant Wikipedia article:

  It is also possible to formulate multinomial logistic regression as a
  latent variable model, following the two-way latent variable model
  described for binary logistic regression.  This formulation is common
  in the theory of discrete choice models, and makes it easier to
  compare multinomial logistic regression to the related multinomial
  probit model, as well as to extend it to more complex models.

  Imagine that, for each data point $i$ and possible outcome $k$,
  there is a continuous latent variable
  $Y_{i,k}^*$ (i.e. an unobserved random
  variable) that is distributed as follows:

  $$\begin{align} Y_{i,1}^{\ast} &= \boldsymbol\beta_1 \cdot
\mathbf{X}_i + \varepsilon_1 \, \\ Y_{i,2}^{\ast} &=
\boldsymbol\beta_2 \cdot \mathbf{X}_i + \varepsilon_2 \, \\ \cdots &
\\ Y_{i,K}^{\ast} &= \boldsymbol\beta_K \cdot \mathbf{X}_i +
\varepsilon_K \, \\ \end{align} $$

  where $\varepsilon_k \sim \operatorname{EV}_1(0,1),$ i.e. a
  standard type-1 extreme value distribution.

  This latent variable can be thought of as the utility associated
  with data point $i$ choosing outcome $k$, where there is some
  randomness in the actual amount of utility obtained, which accounts
  for other unmodeled factors that go into the choice.  The value of the
  actual variable $Y_i$ is then determined in a non-random
  fashion from these latent variables (i.e. the randomness has been
  moved from the observed outcomes into the latent variables), where
  outcome $k$ is chosen if and only if the associated utility (the
  value of $Y_{i,k}^{\ast}$) is greater than the utilities of
  all the other choices, i.e. if the utility associated with outcome
  $k$ is the maximum of all the utilities. Since the latent variables
  are continuous, the probability of two having
  exactly the same value is 0, so we ignore the scenario. That is:

   \begin{align} \Pr(Y_i = 1) &= \Pr(Y_{i,1}^{\ast} >
 Y_{i,2}^{\ast} \text{ and } Y_{i,1}^{\ast} > Y_{i,3}^{\ast}\text{ and
 } \cdots \text{ and } Y_{i,1}^{\ast} > Y_{i,K}^{\ast}) \\ \Pr(Y_i = 2)
 &= \Pr(Y_{i,2}^{\ast} > Y_{i,1}^{\ast} \text{ and } Y_{i,2}^{\ast} >
 Y_{i,3}^{\ast}\text{ and } \cdots \text{ and } Y_{i,2}^{\ast} >
 Y_{i,K}^{\ast}) \\ \cdots & \\ \Pr(Y_i = K) &= \Pr(Y_{i,K}^{\ast} >
 Y_{i,1}^{\ast} \text{ and } Y_{i,K}^{\ast} > Y_{i,2}^{\ast}\text{ and
 } \cdots \text{ and } Y_{i,K}^{\ast} > Y_{i,K-1}^{\ast}) \\
 \end{align}

  Or equivalently:

   \begin{align} \Pr(Y_i = 1) &=
 \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,1}^{\ast})
 \\ \Pr(Y_i = 2) &=
 \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,2}^{\ast})
 \\ \cdots & \\ \Pr(Y_i = K) &=
 \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,K}^{\ast})
 \\ \end{align}

Edit:
Sometimes you'll see notation like this:
$$
Pr( i \rightarrow k) = \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,k}^{*}),
$$
where $i \rightarrow k$ means that individual $i$ chooses choice $k$, is fairly concise once the appropriate notation is defined.
If you're looking for a useful way to put this into a likelihood function, you might see something like this:
$$
P_i = \prod_{k=1}^K Pr( i \rightarrow k)^{\mathbb 1_{i \rightarrow k}},
$$
where $\mathbb 1_{i \rightarrow k}$ is an indicator function that is equal to 1 when $i$ chooses choice $k$ and zero otherwise and $P_i$ is the PMF of the observation associated with individual $i$.

Edit 2:

For an explicit formula, see the same Wikipedia article. Check out, for instance, the section with the following:

  As a result, it is conventional to set $C = -\boldsymbol\beta_K$ (or alternatively, one of the other coefficient vectors).  Essentially, we set the constant so that one of
  the vectors becomes 0, and all of the other vectors get transformed
  into the difference between those vectors and the vector we chose.
  This is equivalent to "pivoting" around one of the $K$ choices, and
  examining how much better or worse all of the other $K-1$ choices
  are, relative to the choice are pivoting around.  Mathematically, we
  transform the coefficients as follows:

   \begin{align} \boldsymbol\beta'_1 &= \boldsymbol\beta_1 -
 \boldsymbol\beta_K \\ \cdots & \cdots \\ \boldsymbol\beta'_{K-1} &=
 \boldsymbol\beta_{K-1} - \boldsymbol\beta_K \\ \boldsymbol\beta'_K &=
 0 \end{align}

  This leads to the following equations:

   \begin{align} \Pr(Y_i=1) &= \frac{e^{\boldsymbol\beta'_1 \cdot
 \mathbf{X}_i}}{1 + \sum_{k=1}^{K-1} e^{\boldsymbol\beta'_k \cdot
 \mathbf{X}_i}} \, \\ \cdots & \cdots \\ \Pr(Y_i=K-1) &=
 \frac{e^{\boldsymbol\beta'_{K-1} \cdot \mathbf{X}_i}}{1 +
 \sum_{k=1}^{K-1} e^{\boldsymbol\beta'_k \cdot \mathbf{X}_i}} \, \\
 \Pr(Y_i=K) &= \frac{1}{1 + \sum_{k=1}^{K-1} e^{\boldsymbol\beta'_k
 \cdot \mathbf{X}_i}} \, \\ \end{align}