Characterising a set of outcomes containing the collection of pure strategy Nash equilibria Consider a game with $N$ players, each indexed by $i=1,...,N$. Every player $i$ has to choose a $J\times 1$ vector of actions $a_i\equiv (a_{i,1},...,a_{i,J})$ where each $a_{i,j}$ can be zero or one. The payoff of each player $i$ is $u_i(a_i, a_{-i})$, where $a_{-i}$ denotes the actions of the other players.
A pure strategy Nash equilibrium (PSNE) of the game is $a^*\equiv (a_1^*,...,a_N^*)$ solving
$$
(1) \quad a_i^*\in argmax_{a_i\in \{0,1\}^J} u_i(a_i, a^*_{-i}) \quad \forall i=1,...,N
$$
Note that if $a^*$ is a PSNE, then
$$
(2) \quad \text{If $a^*_{i,j}$=0, then } u_i(a^*_i, a^*_{-i})\geq u_i(a^*_i+\{a_{i,j}=1\}, a^*_{-i})\\
 \quad \quad \text{If $a^*_{i,j}$=1, then } u_i(a^*_i, a^*_{-i})\geq u_i(a^*_i+\{a_{i,j}=0\}, a^*_{-i})\\
\forall i=1,...,N \quad \forall j=1,...,J
$$
where $a^*_i+\{a_{i,j}=1\}$ denotes $a^*$ where $a^*_{i,j}=0$ is replaced by one; $a^*_i+\{a_{i,j}=0\}$ denotes $a^*$ where $a^*_{i,j}=1$ is replaced by zero.
Claim: Assume that ties are zero probability events. Then, an action profile $a$ satisfies (2) if and only if
$$
(3) \quad a_{i,j}=1\{u_i(a_i+\{a_{i,j}=1\}, a_{-i})- u_i(a_i+\{a_{i,j}=0\}, a_{-i})\geq 0\}\\
\forall i=1,...,N \quad \forall j=1,...,J
$$
Question: Let $A$ be the set of PSNE. Let $B$ be the set of $a$ satisfying (3). Is $A\subseteq B$? If $B$ is empty, then $A$ is empty?

Condition 3 can be written in the following way: $a^\ast \in B$ iff for all $i$ and all $j$
$$
a^\ast_{i,j} = 1 \iff u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i})
$$
As there are no ties, this can be written as:
$$
a^\ast_{i,j} = 1 \to u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i}),\\
a^\ast_{i,j} = 0 \to u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i})
$$
If $a^\ast$ is a Nash equilibrium (i.e. $\in A$), then  for all $i$ and all $a_i$:
$$
u_i(a_i^\ast, a_{-i}^\ast) \ge u_i(a_i, a_{-i}^\ast)
$$

Is $A \subseteq B$?

Yes. Let $a^\ast \in A$. We want to show that $a^\ast \in B$. To show that $a^\ast \in B$, we only need to look at deviations where $a_{i,j}^\ast$ changes for a particular $j$. Towards this end, define $a_i = (a_{i,\ell}|\ell \le J)$ such that:
$$
a_{i,\ell} = \left\{\begin{array}{ll} a^\ast_{i,l} &\text{ if } \ell \ne j\\ 1- a_{i,j}^\ast &\text{ if } \ell = j \end{array}\right.
$$
So $a_i$ differs from $a_i^\ast$ only because $a_{i,j}$ switched value (from 1 to 0 or vice versa). To show that $a^\ast \in B$, we show the two conditions.

If $a_{i,j}^\ast = 1$ , we have:

$$
u_i(a_i^\ast, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 1\}, a_{-i}^\ast),\\
\ge u_i(a_i, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 0\}, a_{-i}^\ast).
$$
The inequality follows from the definition of a Nash equilibrium. The equalities follow from how we defined $a_i$.

If $a^\ast_{i,j} = 0$, we have:

$$
u_i(a_i^\ast, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 0\}, a_{-i}^\ast),\\
\ge u_i(a_i, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 1\}, a_{-i}^\ast).
$$
This shows that $a^\ast \in B$.

If $B$ is empty, then $A$ is empty?

This follows from $A \subseteq B$. If $B = \emptyset$ and $A$ is a subset of $B$ then immediately $A = \emptyset$.
Notice that the inclusion can be is strict, i.e. there are cases where $A \ne B$. To see this, consider the case where $J = 2$ and the payoff table is the following.

(00)
(01)
(10)
(11)

(00)
4,4
1,1
2,2
4,0

(01)
2,2
2,3
1,4
1,0

(10)
1,4
3,2
4,3
2,0

(11)
0,4
4,0
0,2
3,3

We have that $(11,11)$ is in $B$ but not in $A$ as it is not a Nash equilibrium. The only Nash equilibrium is $(00,00)$.


Which utility function yields a constant price elasticity of demand function? How do I know which utility function I can use to find an isoelastic demand function, e.g., $x(p)=Ap^a$?
And similarly, which cost function can I use to find an isoelastic supply function?
Does it work through trial and error, or is there a particular method?

By Roy's Identity we have that Marshallian (uncompensated) demand for good $x_i$ is

$$x_i^M = \frac {\partial U^*/\partial p_i}{\partial U^*/\partial B} \tag{1}$$

where $U^*$ is optimized utility over goods vector $\mathbf x = (x_1,...,x_i,...x_n)$ and $B$, which is the available budget, and given the price vector $\mathbf p = (p_1,...,p_i,...,p_n)$. To obtain a constant elasticity demand function we require

$$\eta \equiv \frac {\partial x_i^M}{\partial p_i}\cdot \frac{p_i} {x_i^M}=const. \tag{2}$$

Using $(1)$ we have that

$$\frac {\partial x_i^M}{\partial p_i} = \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{\big[\partial U^*/\partial B\big]^2} \tag{3}$$

Inserting $(3)$ and $(1)$ into $(2)$ we have

$$\eta \equiv \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{\big[\partial U^*/\partial B\big]^2}\cdot \frac{p_i} {\frac {\partial U^*/\partial p_i}{\partial U^*/\partial B}}$$

$$= \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{(\partial U^*/\partial B)\cdot (\partial U^*/\partial p_i)} \cdot p_i$$

$$\implies \eta = p_i\left(\frac {\partial^2 U^*/\partial p^2_i}{ \partial U^*/\partial p_i}- \frac{\partial^2 U^*/\partial B^2}{\partial U^*/\partial B}\right) \tag{4}$$

For this expression to be constant over the whole range of $p_i$ we need that

$$\left(\frac {\partial^2 U^*/\partial p^2_i}{ \partial U^*/\partial p_i}- \frac{\partial^2 U^*/\partial B^2}{\partial U^*/\partial B}\right) = \frac {C}{p_i} \tag{5}$$

for some constant $C$, which becomes the constant value of the price elasticity of demand.

That's the general condition that must be satisfied.

You could check whether the generalized Cobb-Douglas standard Utility function specification

$$U(\mathbf x) = \prod_{i=1}^n x_i^{a_i}$$

satisfies the condition, perhaps under some restrictions.

THE CASE OF QUASI-LINEAR UTILITY FUNCTION

The OP ponders the case of a quasi-linear utility function, so let's solve this forward. We have

$$\max_{x,m} [cx^{\theta} +m],;\;\;\; s.t. \;\;\;p_xx + m = B,\;\;\; 0<\theta <1,\;\; c>0$$

where $m$ is residual income for all other goods. The Lagrangean is

$$\Lambda = cx^{\theta} +m + \lambda[B-p_xx - m]$$
and first order conditions are

$$c\theta x^{\theta-1}=\lambda p_x,\\\;\;\; \lambda =1$$

So

$$x^* = \left(\frac {c\theta}{p_x}\right)^{1/(1-\theta)},\;\;\; m^* = B-p_xx^*$$

So the indirect utility function is

$$U^* = c\left(\frac {c\theta}{p_x}\right)^{\theta/(1-\theta)} + B-p_x\left(\frac {c\theta}{p_x}\right)^{1/(1-\theta)}$$

$$= \left[c(c\theta)^{\theta/(1-\theta)} - (c\theta)^{1/(1-\theta)}\right] \cdot \frac {1}{p_x^{\theta/(1-\theta)}} + B$$

One can easily verify that this indirect utility function satisfies the required condition $(5)$ for isoelastic demand, and also see how the preference parameters map to the demand parameters.


Why did large denomiations exist in the early 20th century? In the early 1900s, you could buy a newspaper for 5 cents, a pound of apples for 10 cents and the average salary was about \$40 a month.

Nowadays, an issue of the NYT costs \$2.50 , a pound of apples can cost \$1.15 per pound and the median monthly income is about \$2,200.

Today, it makes sense to have a \$100 bill, we can spend it all on a meal at a restaurant and you would need a few of them to pay rent or make a car payment. But in 1900, I can't imagine anything that could be bought with a $100 bill. Perhaps rent? Or a worker's wages?

Compared to the average salary in 1900, a \$100 bill would be the comparable to a $5,500 bill. I can't imagine anything costing that much money today that would justify producing such a highly valued denomination.

My question is: why did we produce \$100 bills in the early 1900s, when there was probably no practical use for them? And also, did people carry $20 bills around in their wallets or were these considered big money?

There were plenty of relatively large purchases that were regular enough to buy with large amounts of cash. If you wanted to buy a Model T in 1909 you'd need \$825-\$1000. A typical house could easily cost a few thousand dollars in an age when mortgage finance was much less common. So there were plenty of expensive things to need significant cash for, at least occasionally.

But why cash? The most important part of the answer is bounced checks. If you need to pay someone a lot of money by check you have problem. If Abby gives Betty a check and Abby walks off with the goods and the check bounces, Betty will have a problem. If Abby pays Betty but doesn't give Abby the goods until the check clears, Abby may have a problem if Betty never shows up with the goods after the check clears. You can do escrow, but that's expensive and complicated. Cash allows instantaneous settlement.

Other options existing in this era have other problems. Cashier's and certified checks, as well as money orders may serve in some cases, but don't easily allow negotiation because they denomination is fixed at the time of issuance. Wire transfers have existed since the age of the telegraph but like today remain an expensive technology for everyday use.

Hundred dollar bills are the (update: second) most common denomination today, which may surprise you:

Source: New estimates of U.S. currency abroad, the domestic money supply and the unreported Economy

This is mostly because the dollar is used internationally for business transactions.


Can one use the standard deviation of a variable as a regressor? I am wondering if you could use the standard deviation of a variable as a regressor in an econometrics model? Consider the following hypothetical model:

$$y_{it} = \alpha_0 + \alpha_{1}T_{it} + \alpha_{2}\sigma_{it}^T + u_{it}$$

Where $y$ is the some outcome variable (e.g. crop yields) in country $i$ at time $t$, $T$ is temperature, and $\sigma^T$ is the standard deviation of temperature. Could one interpret $\alpha_2$ in such a way that would indicate how increased volatility in temperature would impact crop yields?

i.e. testing if countries with greater temperature volatility $\implies$ more severe weather $\implies$ negatively impacts yields

What comes to mind is the use of volatility indices as independent variables, and I wonder if this would lead to a similar interpretation.

By including both regressors, you are effectively trying to find the partial effect of either the temperature level, or variance, on crop yield. That is, the effect of the one, after holding the other constant. Does that approach make sense? Do you need to know the effect of a rise in temperature, for a constant variance in temperature? Or, do you need to know the effect of a change in the deviation in temperature, for a given level in temperature?
The multivariate approach finds those partial effects. The coefficients can be biased if the two variables are correlated with each other—you can test multicollinearity by regressing the one regressor on the other (or, for higher numbers of regressors, one would look at the Variance Inflation Factor).


Can the Certainty Equivalent be negative? I am questioning if the CE of a lottery can be negative? For me it doesn't make much sense by definition.
I encountered this problem on the following exercise:
Imagine a case where we have a lottery(like the EuroMillions) with a price ticket of €2.5. By construction we win €13m with a probability of 1/139,838,160 and we loose with probability (1- 1/139,838,160).
Imagine that the utility function is $U(x)=-e^{(-x)}$.
I compute the CE as you can find in the picture. Is this wrong? How can I interpret it?

For the lottery with the prize of €13m, and let $n=139,838,160$:
$$U(x) = \left(\frac{1}{n} \cdot e^{-(13,000,000-2.5)}\right) + \left(\frac{n-1}{n} \cdot e^{-(-2.5)}\right) = -12.182$$
Therefore
\begin{align*}
U(CE)=-e^{-CE}&=-12.182\\
\ln\left(e^{-CE}\right)&=\ln(12.182)\\
CE&=-2.49999
\end{align*}

I hope it is clear.
Thanks a lot!

If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5.
Your exercise basically asks you to calculate what difference winning the lottery with a small probability makes. Given this utility function, not much.


Does the US Bureau of Economic Analysis use the generic growth formula to calculate percent change of GDP? My question is so basic, I'm embarrassed to ask it. (I study computational theory, not economics.) I'm looking at quarterly GDP numbers and growth rates. I've tried to arrive at the BEA's CPGDPAI numbers using the generic formula:

GDP(t)/GDP(t-1) = (1 + r)^t - t-1

Which fails miserably. I'm grateful for some light on this.

Update:

So, it's the data, not the math. I was pulling the nominal GDP data off FRED. After plugging in the the GDPC1 data, it was all right as rain. A rookie mistake, but, well, I'm a rookie. Thanks for the answers below.

Normally growth is calculated either as:

$$\frac{y_t-y_{t-1}}{y_t}$$

Or approximated by:

$$\ln(y_t)-\ln(y_{t-1})$$

Within year you can either continue using the  above definition or use year on year growth which compares every month quarter with the same quarter previous year.

If you are looking for Fred specifically here is the list of formulas:

https://fredhelp.stlouisfed.org/fred/data/understanding-the-data/formulas-calculate-growth-rates/

The list overlaps with what I written above and also the list shows other changes (which you might be interested to look into)


What's the economic argument (if any) for taxing dividends as ordinary income? My (limited) understanding is that dividends and share repurchases are economically very similar: while share repurchases are slightly less direct, both transactions are effectively ways for publicly traded companies to transfer wealth to their shareholders.
From what I can tell, the only real difference between these two transactions is the way that the tax code treats them: (non-qualified) dividends are taxed as ordinary income, while the gains resulting from share repurchases are taxed as capital gains. Companies often prefer to make share repurchases because they are effectively taxed at lower rates.
It seems to me that there's a likely inefficiency whenever effectively equivalent transactions are taxed at different rates. Is there some important difference between dividends and share repurchases that I'm missing that justifies the tax code's differential treatment?

Dividends have to be taxed as income, as otherwise people can incorporate, have their wages taxed as revenue for the corporation, and then pay themselves a dividend. (Canada adjusts taxes on dividends in an attempt to make these two approaches equivalent for taxes purposes.)
Capital gains are at a lower rate for reasons that are debatable. However, in order for a gain to qualify as a capital gain, the principle is that there has to be a possibility of a loss. (To what extent that principle is achieved can be debated, since there is an incentive to exploit this lower tax rate.) In the case of share buybacks, even though the firm is purchasing shares, there is no guarantee that the sellers are selling at a profit.

I am a consultant in Canada, and one of the first things to decide is whether to form a corporation or not. The logic varies upon the tax code in each jurisdiction. The following article discusses the situation in the United States: link to small business advice website.


Sequential price setting SPNE Firms A and B are in an oligopoly. They both face the linear market demand curve $X=A-\alpha P$, where $X$ is total market demand, and $P$ is price. Assume constant marginal costs $C_{A}$ and $C_{B}$, where $C_{B}<C_{A}$.

What will be the SPNE outcome if firm A sets price first? Is the following correct?

So long as firm A sets $P>C_{B}$, then it will get 0 profits (as firm B will profitably undercut and take all market demand). Therefore, it is indifferent between any $P>C_{B}$. If, however, A sets $P \leq C_{B}$, then firm B will not be able to profitably undercut, and so market demand will be split between the two firms, at some $P \leq C_{B}$. But then firm A will be making negative profits, and so can profitably deivate to setting some P>$C_{B}$.

Is this analysis correct? Or can firm A set some $p\leq C_{B}$ in an SPNE?

Your logic is correct. You need one more step. Using the same logic you can show that, in any SPNE, firm $A$ must set a price greater than the price that maximizes $B$'s profits. Firm $B$ then chooses such price.


Is the Occupy Wall Street famous 1% stable? Occupy Wall Street has a famous slogan "we are the 99%", refering to the "fact" that 1% of the people in the US take nearly the quarter of the national income.  However, this does not seem to consider that many people with extreme incomes have very variables revenues and have to borrow a lot from the private sector to finance their source of revenue, and that this revenue is unstable.  Are there any studies/data confirming that the 1% that has nearly 25% of the national income is in fact nearly composed of the same people year after year?  Any data that considers "permanent income" instead of "annual income"?

I've not see a strict analysis of the top one percent but I have seen an analysis of the top 400 tax payers, the very highest income taxpayers:

  Who are the rich? It depends who you ask and when. Since 1992, the IRS
  has tracked the top 400 earners in terms of adjusted gross income—the
  so called fortunate 400. ... Most importantly, this IRS report
  demonstrates there is a lot of income mobility at the top. Of all the
  filers who have made the list since 1992, 73 percent were on the list
  just once. Virtually no one remains on the list for all 18 years, but
  for privacy concerns the IRS did not report exactly how many did, if
  any. In last year’s report, just 4 people remained on the list for all
  17 years. This suggests that most top earners do not have a portfolio
  of big investments that can be cashed in year after year, but rather
  one big asset, such as a family farm or business or stock, the sale of
  which triggers a capital gain.

The Fortunate 400 By William McBride

A slightly more targeted answer:

  Considering those whose information was found in both years,
  approximately half  of taxpayers in the lowest and highest income
  quintiles remained in the same quintiles 20 years later. Nearly
  one-fourth of those in the bottom quintile moved up one quintile,
  while 4.7 percent moved to the top quintile. About one-fourth of those
  in the top 1 percent were also in the top 1 percent 20 years later,
  but nearly 70 percent remained in the top income decile. The overall
  results suggest that, while there is considerable persistence among
  observed taxpayers, there is also meaningful movement even within this
  narrow age cohort. Some taxpayers start from the bottom and move to
  the top and vice versa.

NEW PERSPECTIVES ON INCOME MOBILITY AND INEQUALITY
Gerald Auten, Geoffrey Gee, and Nicholas Turner (2013)


Distribution of US after-tax money income by household size I'm looking for the US household distribution of after-tax after-transfers income by household size, at a finer grain than quintiles. CPS has total money income (after-transfer) in ~$2,500 increments, but  I can't find an after-tax version. Do the full census, BLS, or think tanks estimate this?

You can find this information here. They provide an overview of their results, together with a link to the data and its description. The data is also available through LIS (second item).

Interestingly, this does not only include the US, but also many other OECD countries, in a homogeneous methodology (great for comparisons). Additionally, this is a longitudinal study of each country. In total, they have data for 27 developed countries between 1978 and 2013.

The data is very comprehensive, and it is at the decile level. It includes:

Entire population, decile cut-offs, equivalised household income
Working age population, decile cut-offs, equivalised household income
Entire population, decile means, equivalised household income
Working age population, decile means, equivalised household income
Entire population, decile cut-offs, per capita household income
Working age population, decile cut-offs, per capita household income
Deflator and PPP


Why is it good to create jobs? I was reading this article on planned obsolescence, which states

  On a macroeconomic scale, the rapid turnover of goods powers growth and creates reams of jobs

In context, it's implied that this is good - it is good to have people working to design, market, make, and distribute goods to replace other goods that break after a short lifespan.

Naively, it seems very bad to me, compared to the option of having goods that last a long time. Those people working on making replacement goods are now not available to work at other jobs or use that time on other pursuits. If the goods lasted longer, we could spend far fewer resources, as a society, supplying everyone with goods. Then we would have many more resources (raw materials in the goods, human labor making the goods, time consumers spend comparing and purchasing new goods, etc.) available to solve other problems or take on other projects.

Yet, I almost always hear people refer to a project or industry hiring lots of people to work on it as an innately good thing, unlike all other forms of resource expenditure. (I never expect to hear that planned obsolescence has an upside because, "look at all the copper it's using!" or "it really keeps our power plants busy providing electricity for the factories!".)

Why is "jobs" treated so differently as a resource, and should I change my naive approach to thinking about them?

In general, it is not:

  Despite the focus on ‘jobism’, jobs are a cost of production and
  consumption, not a benefit, and should be minimized

Mark J. Perry @ AEI

  Public discourse tends to be carried out in terms of jobs, as if a great objective was to create jobs. Now that’s not our objective at
  all. There’s no problem about creating jobs. We can create any number
  of jobs in having people dig holes and fill them up again. Do we want
  jobs like that? No. Jobs are a price and we have to work to live.
  Whereas if you listen to the terminology you would think that we live
  to work. Now some of us do. There are workaholics just like there are
  alcoholics and some of us do live to work. But in the main, what we
  want is not jobs, but productive jobs. We want jobs that will be able
  to produce the goods and services that we consume at a minimum
  expenditure of effort. In a way, the appropriate national objective is
  to have the fewest possible jobs. That is to say, the least amount of
  work for the greatest amount of products.

Milton Friedman in a 1980 lecture

  Under most circumstances, wages paid to people employed on a project
  should not be counted as benefits, although this practice is
  disturbingly widespread. As mentioned above, the opportunity cost of
  labor used on a project is the output that that labor would have
  generated elsewhere, either in paid work or in household production.
  Including wages paid to labor on a project as a benefit implies that
  that labor would have produced nothing otherwise and that the agency
  making the wage payments does not have standing.

Zerbe and Bellas (page 117-118):

I say in general, because there are some evidence that "jobs are good for you", and if increasing jobs diminishes the number of long term unemployed, that might be considered a benefit, even though it diminishes the quantity of leisure and home production they consume.


Aside from buying IPOs, what exactly is the point of shareholders? The more I look at the shareholder system in modern capitalism, the more it starts to appear as nothing more then an otherwise meaningless system that allows rich people to profit off others work while providing nothing and holding no liabilities.

Of course, I will explain why I think this, then I want someone to explain to me why and how I'm wrong. What exactly am I missing?

A company sells shares in order to raise money from the IPO, however:

  Why couldn't you just use bonds and other forms of debt instead?
  Why should shares give shareholders rights beyond receiving dividends? Couldn't they justify buying the security on that alone?
  Why should the company act for their shareholder's profit? They bought an IPO (or, most of the time, didn't) and are receiving dividends? Why should they arbitrarily be given more?
  Given that major shareholders are often vary rich; doesn't this system just facilitate unhealthy levels of income inequality given how the workers are treated as liabilities to be minimised?
  What's the point of money making via stock buy-backs if smaller companies who could use the money to better compete (and thus help the market) are more rarely in a position to do this?
  Don't hostile takeovers using this system do nothing more then create frustration and monopolies?

Again, I get the point of the system, but how does having all this gunk help? Again, it just looks to me like a simple idea that was pumped full of crap that conveniently helps no one except power interests.

But as I said; am I wrong? And if so, how?

The shareholder system is a way of distributing the ownership of the company - each shareholder owns a certain part of the company. Shareholders are the owners of the company, in several company legal forms, such as an LLC. They exist before and after an IPO (Initial public offer), meaning that both public (traded) and private companies may have shareholders.

The shareholders can fire and hire the Management team (CEO/CFO/ etc), which answers your questions 2 and 3. They usually signal the Management team which actions they expect the company to take to maximize the company value, reflected by the share value. If the shareholders have a short-term gain objective, this might harm the long-term sustainability of the company - a case where the management team and the shareholders might have different perspectives.

Raising funds through the sale of shares is attractive for companies because it is interest free, and might give you access to much more funds than you would ever be able to get from bonds - depending on how a company communicates its future potential. However, this comes with a loss of ownership. There is only so much unsecured debt one can sell - and debt is always paid back with interest.


What would the consequences of an economy-wide 500EUR/tonne carbon tax be? I recently calculated that Finnish car owners effectively pay 500 EUR per every tonne of carbon dioxide emitted from the tailpipe of a car. This payment is caused by three taxes that are in practice proportional to the emissions of a car:

Tax when buying a vehicle (it's practically proportional to the stated emissions)
Yearly tax when owning a vehicle (it's also practically proportional to the stated emissions)
Tax when refueling a vehicle (it's directly proportional to the real emissions)

A comment stated that it would be interesting to speculate what the consequences of economy-wide 500 EUR / tonne carbon tax would be. (Sorry @gerrit, I stole your question.)

So, what would the effects of such a tax be?

My understanding is that it would cause a massive transition to either nuclear or renewables. Power would be created by either nuclear power, or wind power and solar power, complementing each other. The very-short-term variability of wind power (wind can suddenly stop) would be balanced by massive lithium-ion batteries. The long-term variability of wind power and the variability of solar power would most likely be balanced by power-to-gas technology, where the gas can either be synthetic methane or synthetic hydrogen. If hydrogen, it would be stored to underground reservoirs. If methane, the carbon dioxide required to create it and emitted by burning it would not be a waste but rather something to be recycled, also stored to underground reservoirs separately from the methane.

The price of electricity would become extremely variable: sometimes, it's free (when there's lots of wind and/or sunshine), at other times it's so expensive that it makes sense to construct grid energy storage based on power-to-gas.

Fossil natural gas would no longer be treated as a clean fuel. Instead, synthetic methane created from water electrolysis and Sabatier reaction would be used.

Steel production would shift to using hydrogen (created by electrolysis of water) instead of using carbon. In areas that require heating, heat would be created from seawater or ground source, using massive heat pumps.

New cars would most likely be either based on electricity (passenger cars) and hydrogen (long-haul vehicles). The electric cars could be used to balance the short-term variability of wind power, using vehicle-to-grid (so the separate grid-connected batteries might not be needed). Old cars could be still used, as evidenced by the massive Finnish car fleet running mostly on gasoline and diesel, even though there is an effective 500 EUR per tonne carbon tax.

Aviation would use biofuels, the production of which would not be enough to displace all current oil uses, but would be enough to displace all current aviation oil use.

Forest owners would let the forest grow to a very high density, and when forest is chopped eventually after ending its growth, there would be lots of sawlogs that can be used to construct buildings in a manner that retains the carbon. The forest-based carbon sequestration could offset the emissions of cement production, which would still be necessary for a functioning world economy.

Thus, I believe it would not be a major disaster to world economy. In fact, it might be a good thing: it would cause a rapid mitigation of dangerous climate change.

It could be a disaster, however, to nations where the entire economy is built on export of fossil fuels.

Is my understanding correct? Or, would some of the technologies I listed be so expensive that it could still be a worldwide disaster?

My sense is that €500 / t CO2 is on the low side. It would change behaviour but it would not substantially alter the economy.

Here are some sample calculations how the tax would affect prices in the transport, food and electricity. This is of course before producers and consumers have adapted their behaviour.

Plane trip London-Paris return 0.244 t CO2 = tax €122 (244% of ~€50 lowest available fare)
Eurostar London-Paris return 0.022 t CO2 = tax €11 (18% of €66 lowest available price)
(Source1, Source2)

Carbon intensity of food consumption in Germany (Source)
1.9763 t x 1.2 (20% carbon intensity of last mile as per report) = 2.372 t
2.372 t / year x €500 = tax €1,186 / year
The proposed tax would add 27% to the average consumer spending on food in Germany of €4.320 / year (Source)

Carbon intensity of electricity generation in EU: 300 g CO2eq / kWh = 0.0003 t / kWh (Source)
Tax / kWh @ €500 t CO2: 0.0003 x €500 = €0.15 / kWh

Average price of domestic electricity in EU: €0.22, of which €0.09 taxes (Source)
The €500 / t CO2 tax would increase the electricity price in Finland (€0.17 incl taxes) to about the level in Denmark (€0.30 incl taxes): €0.17+ €0.15 = €0.32.

In sum, while the tax would likely change behaviour the proposed level appears to be in the range of other existing taxes for many products (VAT in EU is 17-27%). Arguably credits would have to be given to some households for them to be able to effort this new tax on basic necessities (these are typically VAT exempt today). That said, certain carbon-intense sectors that are not currently taxed at all would experience an important impact. Aviation is an obvious one. It would be important to identify them and their potential substitutes to determine the qualitative impact that the proposed tax would have.


How to find interest rates with fractional reserve banking? As I understand it, interest rates are set by supply and demand, like any price. Money can be thought of like any other finite commodity. Lenders are willing to supply a certain amount of money at each interest rate, and borrowers are willing to borrow a certain amount at each interest rate. The rate can be calculated by setting supply equal to demand.

However, with fractional reserve banking, this picture seems to fall apart. While the demand curve can still be found, the supply of money is not so easy to quantify. If the reserve requirement were low enough, banks could loan out an arbitrarily large finite amount of money. The only limit to the money supply is the reserve requirement. It doesn't make as much sense to say here that the interest rate is set by supply and demand.

I see a two possibilities:
1) There is a supply curve that depends on the reserve requirement. This means that the interest rate would be a function of the reserve requirement.

2) There is no supply curve. As far as I know, no one sets interest rates (the Fed included, since they only influence rates via open market operations: if they didn't do anything there would still be an interest rate). How can interest rates be calculated in this case?

This question is sufficiently confused about basic concepts that I'm reluctant to go through the effort of clarifying everything enough to provide a meaningful answer, but here is a quick stab...

First, you need to understand that interest rates are not set by the supply of cash provided by banks and the demand from borrowers. They're set in money markets, which is the other side of the credit intermediation that banks perform. Banks lend to borrowers and borrow in money markets (if they didn't borrow, then it wouldn't be fractional reserve banking) to fund a significant portion of their lending.

Second, reserve requirements are far from the only constraint on bank money creation, and I frankly cannot think of a single bank for which they're the binding constraint. Banks are more likely to be constrained by risk-weighted capital requirements, the liquidity coverage ratio, or stress test scenarios than by reserve requirements.

So to answer your question as to how interest rates are determined under fractional reserve banking: bank demand for funds is determined by their set of profitable lending opportunities under a set of constraints including capital requirements (and they face some competition for these funds from firms that have direct access to capital markets and which issue short-term liabilities), while the supply of funds is determined by the amount of uncommitted cash held by firms (and to a much lesser extent, households). These intersect at some point, which determines the interest rate. The Fed affects the risk-free rate by engaging in transactions which swap safe, usually interest-bearing collateral for cash, affecting the relative supply of each.


Difference between Unit Banking and Direct Banking As the title says, I want to understand the differences between Unit Banking Systems and Direct Banking Systems.

Here are the few questions that'll help me understand:

Direct banks provide remote services online. Does Unit bank do so?
It's said that Unit banks do not have branches in other areas. Now how big or small is this area? A city? District? State? Can it have more than one one branches in the same area?
Unit banks have at least one branch from where the it is operated, where people go and deposit their money physically. But direct bank has no such branch. All the work is done on line (Of course they have an office though.) Is it right?

I assume they're both independent units. So kindly explain these and if there are other points besides these, please let me know.

Direct banking is done online. Unit banking is just normal banks. Instead of working with some franchise direct banking allows you to go straight to the computers with the ones and zeroes and do banking. Whereas unit banking just offers you physical branches as well.

You already defined all this in your post. There is no question to answer here.


Making sure the p-values of my OLS estimates are correct I have learned the basics of the Classical Linear Regression Model and also various diagnostic tests to check if the assumptions of the CLRM are met, such as homoskedadticity, absence of near perfect multicollinearity, normality of residuals etc.
I wanted to know if it is possible to have an exhaustive set of things I need to test to be reasonably confident about the p-values I get, both overall and for each specific regressor's coefficient (in a multiple regression with cross sectional data).
For example, what are the things that a researcher would test to make sure that the p-values are technically correct (or at least acceptable) and publishable in an academic journal (not talking about the usefulness of the topic or whether there is causality etc, just the statistical inference, if that is acceptable or not)? Different textbooks/ lectures I have seen have put differing amounts of emphasis on different things. I wanted to know what would a professional researcher do?

The answer to your question:

I wanted to know if it is possible to have an exhaustive set of things I need to test to be reasonably confident about the p-values I get, both overall and for each specific regressor's coefficient

is "no", but for a different reason than is stated in the comments.
To start this, let me ground you in what a p-value is and what it is not.  First, it is not a probability statement.  It is a conditional frequency statement.  It is conditional on a utility function and a model of the world.  If you have a bad model of the world, then your p-values are suspect.
Frequentist tests, as opposed to Bayesian tests, attempt to show that some null hypothesis is false.  It does so by assuming that the null is true.  That is not a trivial statement for two reasons.  First, the null hypothesis acts as information.  You are conditioning all the observations on that assumptions, as if true.  If you would change your null, you would change the inference on your assumptions.  Second, there is no way to distinguish a chance effect from a false null.
A p-value is a statement that if you would repeat your experiment, possibly an infinite number of times, then the frequency of seeing the result you saw or a result that is more extreme is no more common than the stated p-value if the null model is the true model.
So, let us look at the things you need to get right for a p-value to imply its abstract meaning.
First, your utility function needs to be correct.  Least squares models assume that losses are quadratic, that is to say $$U(\hat{\theta},\theta)=-c(\hat{\theta}-\theta)^2.$$  If you change your model, then you change your estimator, which implies that your choice of p-values changes.
For example, if you would assume that $U(\hat{\theta},\theta)=-c|\hat{\theta}-\theta|$, then you would end up with median estimators and would use things such as Theil's or Quantile regression.  The results are more robust but less powerful.
Second, you need to get your specification correct.  P-values assume the null model is true, if you have a misspecified model, then your p-values will be suspect, although not automatically so far from reality as to discard them.
Third, you need to match the correct assumptions to the real null model.  This is the danger in testing for heteroskedasticity.  If you have heteroskedasticity, but it fails to reject the null, then you have a bad set of assumptions.  If you do not have heteroskedasticity, but it rejects the null, then you have a bad set of assumptions.
The actual fix for problems like that is to approach the assumptions logically.  That is why there is no single book or reference set for this problem.
Diamond prices are heteroskedastic in size but in a weird way.  Simple logic would tell you that they are probably heteroskedastic with respect to size.
Very small diamonds have only two purposes, industrial uses such as drilling, and as additions to jewelry.  As jewelry, their price is very high.  As an industrial material, the price is very low.
Most medium-sized diamonds are either close to their desired size for jewelry or need cut into very small pieces for industrial use.  However, the bulk of their use will be to make medium-sized diamonds cut to specification.  It will be easier to estimate their value as is.
Very large diamonds might remain highly valued, very large diamonds.  Think of the Hope Diamond as an example.  Others may make hundreds of small diamonds for industrial use.
The result is that you get a bowtie version of heteroskedasticity.  Small diamonds and very large diamonds have enormous price variability.  Medium diamonds are quite predictable as to the final cut and value and so are priced in a much narrower range.
You should never test for heteroskedasticity with diamonds because there is a way to think this through.  Testing should be reserved for things that you have no subject matter knowledge regarding.
Most p-values are robust because most Frequentist estimators are minimax estimators, that is to say, the estimator minimizes the maximum amount of risk that you must take under a specified utility function.
As a result, p-values tend to be conservative in their use of information and any resulting inference.
The reason that there is no one list is that if you can assert something about the population as true, such as diamond prices being logically heteroskedastic in size, then you can condition your model with that knowledge.
As a result, if you were to test diamond prices and they tested as homoskedastic, then you could ignore the test result as irrelevant.
There isn't a list.  Instead, you need to determine what researchers in a narrow domain do.
Let me give you one more example to see why that matters.
If a brick-and-mortar retailer needs to alter its revenue, it can issue coupons. It has the ability to control its revenue and its sales volume.
Electrical utilities, generally, cannot control their volume or price.  Even if they could mail coupons to use additional electricity, how would anyone monitor this sale?  Electric utilities control their costs, not their revenue.
That discussion alone tells you that the accounting information that management sees as important for the marginal decision is different.  If you give them a common specification, then your model will be misspecified.  Your p-values are off.  You omitted a variable, industry type.
Finding correct p-values are about more than overcoming technical obstacles, such as multicollinearity and heteroskedasticity, it is about understanding your domain of research.
Fortunately, p-values are usually robust.  If you set $\alpha=.05$ and $p<.01$ then you are probably safe.  On the other hand, if $p<.04999$ and your work is a bit shaky, then do not feel confident even if you followed the technical rules.
There is no one book, no simple list.  There are just domains where researchers discover unexpected things and correct for them.


In the Ramsey model, how does it make sense to say that capital equals wealth when wealth can be negative and capital cannot? In the Ramsey model, $K_t$ denotes capital, and $A_t$ denotes the assets/wealth of the households.

However, each household is identical. Since the economy is closed, the assets of each household equal their share of the capital. So, $A_t = K_t/H$, where $H$ is the number of households.

However, in theory, $A_t$ can be negative. Obviously a household can accumulate debt at some point in their lifetime. But $K_t/H$ can never be negative, since $K_t$ itself, total capital in the economy, cannot be negative.

So I do not understand how it makes sense in the Ramsey model to talk about "wealth" and "borrowing" and "debt" since $A_t$ is forced to be positive?

No matter in Ramsey model or the basic structure of Solow-Swan model, as long as is in the closed economy. The asset equals to the capital in the equilibrium.

$A_t$ is the total asset owned by the households, or by the representative household. In the market equilibrium, all the borrowing and lending must cancel within the economy. Therefore, the only asset in positive net supply is capital, which means $A_t = K_t $. The asset equals the capital stock quantity.

Therefore, the asset per person $a_t$ = the capital stock per person $k_t$. It seems like you confuse $A_t$ with $a_t $.


Why did coronavirus cause a food shortage? The argument I have seen is that since restaurants closed, demand for food is down.  I understand how demand being suddenly lower could disrupt supply.  What I don't understand is why demand is down.  I mean, people still need to eat just as much as they did before the coronavirus.  What is going on here?

This question is too open-ended, as it does not even specify a country. There are a number of issues that have shown up. These comments are based on my reading of news stories, and arguably not authoritative.

Logistics have been disrupted, putting shipping, planting, and  harvesting at risk. This is a worry for countries that are dependent upon food imports.
Some countries have put into place export restrictions (as a results of the previous concern).
In the developed countries, a significant portion of eating is at restaurants. The supply chain for restaurants is different than grocery stores. There are two issues: people eat different foods at home than restaurants (The example I heard was that most fish is eaten at restaurants in the United States, so there is a surplus of fish.) People have more time, and are baking more now than even during the seasonal peak at Christmas. In addition to the supply chains being different, the sizing and labelling is different (e.g. nutritional information must be displayed on grocery store food, and so a waiver had to be given to allow food without such labels to be sold). Flour is sold in 80 pound bags to restaurants, and households cannot work with such bags. (At my store in Canada, flour is being sold in small generic plastic bags.)


When treating a relative, normalized utility function as a pmf, what is the interpretation of Shannon entropy or Shannon information? Suppose $\Omega$ is a set of mutually exclusive outcomes of a discrete random variable and $f$ is a utility function where $0 < f(\omega) \leq 1$, $\sum_\Omega f(\omega) = 1$, etc.

When $f$ is uniformly distributed over $\Omega$ and $f$ is a probability mass function, the Shannon entropy $H(\Omega) = \sum_{\Omega}f(\omega)log\frac{1}{f(\omega)}$ is maximized ($=log|\Omega|)$, and when one element in $\Omega$ has all of $f$'s mass, the Shannon entropy is minimized ($0$, in fact). This corresponds to intuitions about surprisal (or uncertainty reduction) and outcomes and uncertainty (or expected surprisal) and random variables:

When $f$ is uniformly distributed, uncertainty is maximized, and the more outcomes there are for mass to be uniformly distributed over, the more uncertain we are.
When $f$ has all its mass concentrated in one outcome, we have no uncertainty.
When we assign an outcome a probability of $1$, we gain no information (are "unsurprised") when we actually observe it.
When we assign an outcome a probability closer and closer to $0$, the observation of it actually occurring becomes more and more informative ("surprising").

(This all says nothing about the much more concrete -- but less epistemic -- coding interpretation of Shannon information/entropy, of course.)

However, when $f$ has the interpretation of a utility function, is there a sensical interpretation of $log\frac{1}{f(\omega)}$ or $\sum f(\omega)log\frac{1}{f(\omega)}$?
It seems to me that there might be:

if $f$ as a PMF represents a uniform distribution over $\Omega$, then $f$ as a utility function corresponds to indifference over the outcomes that could not be greater*
a utility function where one outcome has all of the utility and the rest have none (as skewed of a utility as there could be) corresponds to very strong relative preferences -- a lack of indifference.

Is there a reference expanding on this? Have I missed something about the limitations on comparing probability mass functions and normalized, relative utilities over discrete random variables?

*I am aware of indifference curves and do not see how they might be relevant to my question for a variety of reasons, starting with my focus on a categorical sample space and with the fact that I am not interested in 'indifference' per se, but rather how to interpret utilities as probabilities and how to interpret functionals on probabilities when the (discrete) 'probability distribution' in question actually or (additionally) has the interpretation of a utility function.

After the exchange with the OP in my other answer, let's work a bit with his approach.

We have a discrete random variable $X$ with finite support, $X = \{x_1,...,x_k\}$, and probability mass function (PMF), $\Pr(X=x_i)=p_i, i=1,...,k$

The values in the support of $X$ are also inputs in a real-valued  cardinal utility function, $u(x_i) > 0\; \forall i$. We then consider the normalized utility function

$$w(X): w(x_i) = \frac {u(x_i)}{\sum_{i=1}^ku(x_i)},\;\;i=1,...,k \tag{1}$$

and we are told that

$$w(x_i) = p_i \tag{2}$$

Note that we do not just make the observation that a normalized non-negative discrete function of finite domain, satisfies the properties of a probability mass function in general -we specifically assume that $w(x_i)$ has the functional form of the PMF of the random variable whose values $w(x_i)$ takes as inputs.

Since $w(x_i)$ is a measurable function of a random variable, it, too, is a random variable. So we can meaningfully consider things like its expected value. Using the Law of the Unconscious Statistician we have

$$E[w(X)] = \sum_{i=1}^kp_iw(x_i) = \sum_{i=1}^kp_i^2 \tag{3}$$

This is a convex function, and if we try to extremize it over the $p_i$'s under the constraint $\sum_{i=1}^kp_i=1$ we easily obtain

$$\text{argmin} E[w(X)] = \mathbf p^*: p_1=p_2=...=p_k=1/k \tag {4}$$

and we have obtained a general result:

  The normalized utility function as defined above has minimum expected
  value iff the distribution of $X$ is Uniform.

Obviously in such a case $w(X)$ will be a constant function, a degenerate random variable with $E[w(X)]=1/k$ and zero variance.

Let's turn to Shannon's Entropy which is the focus of the OP. To be calculated, Shannon's Entropy needs the probability mass function of the random variable... so we should find the PMF of the random variable $w(X)$...

But it is my impression that this is not what the OP has in mind. Rather, it views Shannon's Entropy as a metric that has some desirable algebraic properties and perhaps can measure compactly in a meaningful way something of interest.

This has been done before in Economics, specifically in Industrial Organization, were Indices of Market Concentration ("degree of competition/monopolistic structure of a market") have been constructed. I note two that look particularly relevant here.

A) The Herfindahl Index, has as its arguments the market shares of the $n$ companies operating in a market, $s_i$, so they sum to unity by construction. Its unscaled version is

$$H = \sum_{i=1}^n s_i^2$$

which is an expression that has the exact same structure with the expected value of $w(X)$ derived above.

B) The Entropy Index
$$R_e = -\sum_{i=1}^n s_i\ln s_i$$
which has the exact mathematical form with Shannon's Entropy.

Encaoua, D., & Jacquemin, A. (1980). Degree of monopoly, indices of concentration and threat of entry. International Economic Review, 87-105., provide an axiomatic derivation of "allowable" concentration indices, i.e. they define the properties that such an index must possess. Since their approach is abstract, I believe it may be useful to what the OP wishes to explore and attach meaning to.


Rationalizable strategies in a game Consider a game in which, simultaneously, player $1$ selects any real number $x$ and player $2$ selects any real number $y$. The payoffs are given by:

$u_1 (x, y) = 2x − x^2 + 2xy$

$u_2 (x, y) = 10y − 2xy − y^2.$

(a) Calculate each player’s best-response function as a function of the opposing player’s pure strategy.

(b) Find and report the Nash equilibria of the game.

(c) Determine the rationalizable strategy profiles for this game.

I got the first two part

$\partial u_1 / \partial X = 0 \implies 2 – 2X + 2Y = 0$ $\implies$ Best Response of Player $1 = 1 + Y$

$\partial u_2/ \partial Y = 0 \implies 10 – 2Y – 2X = 0$ $\implies$ Best Response of Player $2 = 5 – X$

Nash Equilibrium

$BR_2 = 5-(1+Y)$

so that

$Y^* = 2$ and $X^* = 3$.

The Nash equilibrium is $(3,2)$.

I am unsure abut rationalizable strategy. I think it is rationalizable strategy for both players are all real numbers because there is no single number that is strictly dominated by another number. For example, if player 2 chooses 10, 11 is the best strategy for player 1 and dominated other number. But, if player 2 chooses 11, 12 is the best strategy for player 1 and 12 dominated 11 in this case. This logic then applies to all real numbers. I am unsure if this logic is correct. Any help is greatly appreciated.

Your calculation gives you all Nash equilibria in pure strategies. The unique Nash equilibrium in pure strategies is indeed the only Nash equilibrium, but this requires an argument. A careful look at the payoff functions shows that it is irrelevant for player 1 whether player 2 plays the pure strategy $y$ or a mixed strategy with expectation $y$. A similar argument applies to the other side. But then the calculation gives us that every player has a unique best response in pure strategies to every mixed strategy of the other. Moreover, mixed best responses must randomize over pure best responses, so all best responses must be in pure strategies. In particular, every Nash equilibrium must be in pure strategies.

Now rationalizable strategies are the strategies that survive iterated elimination of never best responses. Every strategy is a best response here and does never get eliminated, so every pure strategy profile is rationalizable. If you define rationalizability in terms of mixed strategies, the argument above shows you that you still get only the pure strategies as rationalizable strategies. There are some subtleties in how one defines rationalizability with infinite action spaces, but they are all irrelevant here.


How to treat different frequencies of a time series data? I have annual GDP, annual CPI, monthly exchange rate, and monthly export data. I would like to go for estimation being monthly stock return as the dependent variable. What are the methods to treat the annual figures in to monthly and/or the monthly in to annual? Please, I need your recommendations or books to read?

CPI is a stock while GDP is a flow. Re-sampling of stocks to higher frequencies can be approximated with a number of choices, but probably the most common is linear interpolation. In some contexts filling forward and filling backward are common. But in principle you could fit any function you want through all the points you have and use the values from the function instead. Flows are a bit more complicated because annual flows are the sum of monthly flows. Dividing the annual flow into twelfths across the entire year is analogous to filling forward or filling backward (depending on what month of the year is the date of annual measurement). For example, if GDP is 144 and measured in December of 2018, assigning every month in 2018 a value of 12 is similar to filling backwards. But again, as long as you remember that the flows are supposed to add up across months to equal or approximate annual flows you can you a number of functional forms to approximate them.

This is a big topic. I recommend searching for papers related to "interpolated time series". Resampling and Subsampling for Financial Time Series by E Paparoditis, DN Politis might be a good, if advanced, place to start. Modeling and Forecasting Time Series Sampled at Different Frequencies by Casals, Jerez, and Sotoca is another useful reference. In comparing the effectiveness different methods, Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment by Lepot, Aubin, and Clemens provides a nice list of methods but doesn't focus on the stock versus flow distinction.

Deterministic methods (fill forward or back)
Nearest-neighbor
Polynomial interpolation
Distance weighting methods
Fourier transformation based methods
Regression models
Auto-regressive models
Machine learning based methods
Kernel methods
K-nearest neighbors
Box-Jenkins models
Kriging-based methods


Subgame Perfect Nash equilibrium: two stage game Charitable organizations are ranked according to the proportion of their total budget that is spend on programs(what directly goes to the recipients of their services) relative to money spent on administration(e.g salaries paid to its staff). Suppose there is a charitable organization which spends P on its programs and S on administration (salaries). So its total budget is B = S + P and a proportion P/B goes to programs. The manager of the charity cares about the charity's reputation (ranking) but also cares about his salary. So suppose he maximizes $Q =(P/B)^a$ $S^{1-a}$ ,$ 0 <a< 1 $.

The only donors to the charity are players 1 and 2, who care about the total amount of money that goes to the recipients of the charity's services. Donor k has a budget W and a utility function
$U_k$ = $c_k$P, where $c_k$ is the dollar amount spent on a private good. Let player k's donation to the charity be $D_k$, k = 1 ,2 .
The timing of actions is as follows: In stage 1 the donors choose $D_1$ and $D_2$ simultaneously, and in stage 2 , the charity then chooses S and P.

a) Find the sub-game perfect equilibrium of this game.

b) Someone clams that the donors give less in total to the charity as the efficiency level of the charity (from their standpoint) increases. Verify the validity of this claim in this simple model and give the intuition for your answer.

c)Now suppose that in addition to the charitable organization above, there is another charity which is identical to first charity except that its manager's objective function is $Q =(P/B)^b$ $S^{1-b}$ ,$ 0 <b< 1$. Call these charities a-type and b-type. Assume that a >b . Denote player k's donation to charity j as $D_k^j$, k=1 ,2 and j=a ,b. What is the  sub-game perfect Nash equilibrium of the game? (Hint: write down the donors' Kuhn-Tucker conditions).

What I did is that

For a) I start from the 2nd stage game and maximize the charity's utility function

max $(P/B)^a$ $S^{1-a}$

s.t B = S+ P

The unconstrained problem becomes:
max $(P/B)^a$ $(B-P)^{1-a}$

Taking FOC wrt P  we obtain P*= aB

Then going in state 1, I determine the Best response function.Donor 1 wants to maximize

max (W -D1)(D1 +D2)

The best response function for donor 1 is D1 = (W- D2)/2 .
Doing the same for donor 2, the besr response function is   D2 = (W- D1)/2
Solving the optimal choices are D1*=D2*= W/3 .
Going back to stage 1 and substituting , P*=2W/3 , B*=2W/3a and S*=2W(1-a)/3a
This is the Nash equilibrium.

For part b)

The d Q/ d(P/Q) = a $( P/ B)^{a-1}$ $S^{1-a}$ > 0

and $d ^2$Q/d $(P/Q)^2$ = a(a-1)$(P/B)^{a-2}$ $S^{1-a}$ < 0

But I stuck and I do not know how to interpret it.

For part c) the donors' Kuhn-Tucher conditions  are :

$D_1^a$ + $D_2^a$ $\geq$ $D_1^b$ + $D_2^b$

$D_1^a$ + $D_2^a$ $\geq 0 $

$D_1^b$ + $D_2^b$ $\geq 0 $

But if I plug them into the donor's maximization problem I do not end up to a result.

Any help will be appreciated.
Thank you

In part a, if $B = D_1 + D_2$, then the SGPE should be

$\left\lbrace D_1 = \frac{W}{3},\ D_2 = \frac{W}{3}, \left\lbrace  P = \alpha (D_1 + D_2), \ S = (1 - \alpha) (D_1 + D_2) \right\rbrace \right\rbrace$

Don't say $P = \alpha \frac{2W}{3}$. That's an action, and the second stage best respond should a strategy (function) to make the equilibrium subgame perfect.


Does the Independence Axiom Require Statistical Independence? First: Given this definition of the Independence Axiom,

If for all $P$, $P'$, $P''$ in the set of lotteries over outcome space $X$, when:
$P$ preferred to $P'$ $\implies$ $aP + (1-a)P''$ preferred to $aP' + (1-a)P''$ for all $a$ in $(0,1)$."

Can I do what I do here?
Then $P$ preferred to $P'$ $\implies$ $P$ preferred to $aP' + (1-a)P$ for all $a$ in $(0,1)$.
Second:
Does satisfaction of the Independence Axiom rely on the statistical independence of the lotteries involved? It doesn't seem to be mentioned anywhere I've looked.
Any direction that can be offered is greatly appreciated.

First Question:

Yes, you can obtain $P \succsim P' \Rightarrow P \succsim a P' + (1-a)P \forall a\in (0,1)$ from setting $P''=P$ in the axiom. Thus, your new condition is a special case of the independence axiom as stated.

Second Question:

Statistical independence of $P,P',P''$ is not assumed in the independence axiom. The notion of statistical independence of lotteries is not even meaningful since you can only choose one lottery. Only when you would choose multiple lotteries at the same time their dependence would matter since you may be able to reduce/increase risk by combining lotteries which are not independent.


How is the growth rate form of the exchange equation derived? Equation of exchange states:
$ MV=PY $
where,
       $ M $ is Quantity of Money
       $ V $ is Velocity of Money
       $ P $ is Price Level
       $ Y $ is Real GDP
Could someone please share how the above common form of equation is converted to the growth rate form:
$ \%\Delta M + \%\Delta V = \%\Delta P + \%\Delta Y $

Edit: Sorry the latex doesn't seem to be working. Unable to figure that out.

Taking logs of the equation $MV=PY$ gives: $ln(M) + ln(V) = ln(P) + ln(Y)$.

Next, differentiating with respect to time yields $\frac{dln(M)}{dt} + \frac{dln(V)}{dt} = \frac{dln(P)}{dt} + \frac{dln(Y)}{dt}$.

Or, using $\frac{dln(X)}{dt} = \frac{\dot{X}}{X}$, with $\dot{X} = \frac{dX}{dt}$, the derivative of a variable $X$ with respect to time, you can immediately derive $\frac{\dot{M}}{M} + \frac{\dot{V}}{V} = \frac{\dot{P}}{P} + \frac{\dot{Y}}{Y}$, which equals $\%\Delta M + \%\Delta V = \%\Delta P + \%\Delta Y$.


What are the household holdings of financial products in China? What are the household holdings of financial products in China?
It seems to be generally accepted that real estate is the most popular savings product for households in China and perhaps it is also true in the wealthy countries for median households. I am specifically interested in financial products; not direct ownership in real estate; not life insurance; not small business participation or sole proprietorship.
Specifically what would the pie chart look like for the aggregate of all households if the categories were... certificates of deposit for fixed rates, CDs for floating rates, active equity mutual funds, "passive" equity mutual funds, equity ETFs, bond mutual funds, bond ETFs, China's so called "wealth management products" sold by banks and non-bank institutions, local government bonds, China bonds, deferred annuities, equity in widely held corporations ("stocks"). What are the categories of "wealth management products" in the way of short duration, long duration, fixed income, with deposit insurance, without deposit insurance, corporate issuer, government issuer, equity. If these are not the appropriate categories then feel free to make the categories more appropriate.

I do not think there is some dataset that has such an detailed breakdown that you want. However, one of the most widely used datasets on financial assets held by households are the various national surveys including the Survey of Chinese Consumer Finance and Investor Education (SCCFIE), China Household Finance Survey (CHFS), or Chinese Survey of Consumer Finance (CSCF). These datasets might not be directly freely avaiable but if you message the authors in the studies cited below they should provide them for replication purposes unless they happen to be confidential.
All three of the above are widely used in academic literature (just few examples include: Liao, Huang, & Yao (2010), Xie & Jin,(2015) or Chu, Wang, Xiao, & Zhang, (2017)).
These surveys are typically highly detailed but unfortunately won't have everything you cite you would like to see. For example, Liao, Huang, & Yao (2010) based on SCCFIE showed that the financial and non-financial assets were in 2008 distributed in China thusly:


Disagreement in Strategic Bargaining Construct a pair of startegies for the ultimatum game ($T=1$ bargaining game), that constitutes a Nash Equilibrium and together support the outcome that there is no agreement reached by the two players and the payoffs are zero to each. Show that the disagreement outcome can be supported by a Nash Equilibrium regardless of the number of bargaining periods.

I understand that in a bargaining game if $T=1$  then any division of surplus $x^*$ belonging to $[0,1]$, $(v_1, v_2)=(x^*,1-x^*)$ can be supported as a Nash Equilibrium.

In the question given above, I'm supposed to find a strategy such that there is no agreement reached and both players end up getting $0$ each. I have some trouble thinking about such strategies and whether it would constitute a Nash Equilibrium. One pair I could think of is: Player $1$'s strategy is to offer, say, $(x^*,1-x^*)=(0.6,0.4)$ and Player $2$'s strategy is Accept any offer $(1-x^*)$, $>$ or $=$ $0.5$. Hence we see that Player $2$ will reject the strategy. However, I cannot understand how a Nash Equilibrium startegy can be constructed (since the above stated startegy is clearly not optimal).

Another strategy could be, Player $1$ proposes $(0,1)$, and Player 2's strategy is Accept only strictly positive offers, i.e., reject $0$. However, this strategy cannot be Nash Equilibrium because the first mover has no Best Response.

Please help me understand the question. Hints for the next part would also be appreciated. Thank You!

Reference: Game Theory: An Introduction, Steven Tadelis

Consider:

Proposer offers $0$
Receiver always rejects the offer regardless of the amount

You should be able to argue that this is a pair of mutually best responding strategies for $T=1$. The $T>1$ cases follow a similar logic.


In a trade imbalance does the importing country's currency become more valuable? Suppose that country A buys freely from country B, but B uses steep tarriffs to block its citizens from buying from A. In that case country A's currency will pile up in B.

Other things being equal, will this cause currency A to become more and more valuable relative to currency B?

No. In fact the opposite happens. The value of a currency (or the exchange rate) is determined by supply and demand of the currency on the currency market. If country A imports more, the supply of country A's currency will increase more, therefore losing its value.

Suppose B's goods are priced in B's currency. Then, the supply of A's currency increases in your example, because it needs to sell its own currency to purchase more of B's currency to be able to buy B's goods. Therefore more of A's currency comes on the market, while more of B's is demanded. If priced in A's currency, the exporters in B will still want to have B's currency as they need it because they live in B. Then they will want to exchange the amounts of A currency obtained for B currency.

Hence, in both cases on the markets for currency, demand for B's currency increases, while supply for A's currency increases, therefore reducing the value of A's currency.

The currency would not (necessarily) "stockpile" however, as import tariffs in B do not influence the exchange of currency.


What is Tinbergen's "all other income when paid out" measuring? In When it all began by Dhaene and Barten (1989), the authors discuss in detail what they claim was the first ever macroeconometric model, Tinbergen's 1936 model of the Netherlands economy.

Some of their summaries of Tinbergen's variables are a bit vague though.

What, for example, is $E$, described as "All other income, when paid out"?

Very obscure indeed. If you see his paper in English (published in 1959), you can see the following (page 43):

By checking other variables, you can confirm that the variable letters are equivalent in this and your reference. Then, $E$ stands for "all non-wage income, taken at the moment received, instead of take when earned".

This is a standard differentiation between the time an income has been accrued ($Z$) and when it has actually been received ($E$).

You can see more about these definitions later on in the document. In page 49, you find an explanation on how these are calculated (19 is $Z$, 20 is $E$; see table in page 47-48):

It seems the author is assuming that income is received one further year after is is accrued. However, the text seem to me rather obscure. He mentions a "pamphlet regarding this working method which will be published later." (p.42). You might want to trace that one down.


Marginal rate of technical substitution for multi-dimension inputs? Suppose I have a vector x = ($x_1,x_2,x_3,x_4,x_5$) of inputs. How it is the MRTS($x_1,x_2,x_3,x_4,x_5$) ?

And which is the relation with the marginal products of inputs 1-5 ?

\begin{equation}
MRTS_{x_{i};x_{j}}=\frac{\partial f(X)}{\partial x_{i}}/\frac{\partial f(X)}{\partial x_{j}} \qquad \forall i\neq j
\end{equation}


Why does firing employees appear to increase the share price of a company? The last case is GM in the USA. I fail to understand why does this happen. I understand that laying off jobs means higher dividend for the share holder and thus higher stock prices but in the long run is this a viable strategy, doesn't this mean lower income (and thus dividend) for the next quarter or year? Being that not only the employees are being fired but also that the factories are being closed it doesn't seem to me as a case of automation taking over the job. Can someone explain this to me? (I know that GM is allegedly cutting down the production of models non selling well, I am asking in general and not the specific case)

Is firing employees to raise stock prices a good strategy for a
  company?

For the range of scenarios whereby more productivity can be obtained from employees (units of workforce), it certainly is.

By opting for layoffs, a company seeks to get rid of redundant capacity and its associated costs, such as insurance, retirement pensions, or vacations. Since labor is the production factor which can be adjusted in the short-term, the benefit from cutting labor costs is reflected in the company's financial statements quicker than when the company envisions to sell its facilities, real estate, and equipment. Market agents are mindful of that difference, whence they foresee that the company's numbers are about (or are likelier) to improve.

That "strategy" (or, said more properly, tactic) starts to fail where the remaining workforce burns out and/or gets demoralized. These conditions are evidenced in accidents; production failures; product recalls; costs of litigation for unfair labor practices or faulty products; sickness; sudden resignations that cause bottlenecks; and when excessive layoffs render the company unable to meet the demand. It also fails when managers make their firing decisions influenced by personal motives (politics: favoritism, animosity) instead of business-minded criteria.


What would happen if total welfare were evenly distributed? Suppose that there is a market with a large number of consumers and producers. Assume that the government knows the reservation prices of  all people in the market. Suppose that the government uses this information to tax all the consumer and producer surplus created in the market (by making everyone pay or receive the reservation price). The government distributes the total welfare equally to everyone.

What effect would this have on the economy compared to the situation without government intervention?

Firstly, let us fix the mode of taxation as lump-sum taxes. This means that after the redistribution we will still have an efficient allocation assuming there are no externalities.

Next, think about what happens to each individuals' income. Individuals with a high demand elasticity for the good will in general be taxed less than individuals with a low demand elasticity. If the demand elasticity is unrelated to the original income level, this means that this mode of taxation does not necessarily increase equality in income across individuals.

This may not be bad, because the redistribution may still be welfare enhancing if the marginal utility of income of high elasticity individuals is higher than the marginal utility of income of low elasticity individuals. However, we know that marginal utilities are unrelated to demand, since one can always apply any monotone transformation to the utility function and obtain the same demand functions. Thus, both equality of income and welfare effects are unclear.

Finally, we may consider what happens to demand in that particular market. Again, the effects are unclear. This time, it is the presence of income effects which obfuscate the picture. An individual with a positive income transfer may consume more or less of the good, depending on whether it is an inferior good or not.

In this answer, we have thus seen that without making stronger assumptions on preferences, we cannot say much about the effects of such a redistribution even if we assume lump-sum redistribution. If we go into second-best analysis, things may get even more complicated.


Finance vs. financial economics (terminology) What is the difference between finance and financial economics?
E.g. we have tags for both* and I wonder when to use which, but not only.
*And one more on financial markets – see this thread on Meta Economics SE.

Virtually same question was already addressed at quantitative finance stack by user Matthew Gunn.  The answer there can be summed as:

Financial economics is what economics calls finance. Finance is what finance calls finance.
Less flippantly though, there's a long debate on whether finance is a subfield of economics, ... Prof. Milton Friedman famously opposed awarding Markowitz a PhD in economics from the University of Chicago for his portfolio theory because while Markowitz's work was brilliant, Friedman didn't consider it to be economics. ... Myron Scholes also express views that finance is to some extent distinct from economics.

By many economists finance is not considered to be part of economics but rather to be a separate discipline. For example, according to Journal of Economic Literature (JEL) classification (which is authoritative source for classification in economics), there is no sub-field of finance in economics. There is only subfield of financial economics. This classification has a following guideline:

Guideline: Covers studies about issues related to various sub-fields in financial economics: general financial markets (both domestic and international) dealing with securities (stocks, bonds, and commodity and other futures); financial institutions and services; and corporate finance and governance.

This is something that probably most people who are in finance would recognize as subject matter of their field.
However, I would argue there is a subtle difference. People who study finance often do that in more practical way  with larger emphasis placed on analysis from the investors/financial firm point of view and applying concepts to some particular cases. While financial economics focuses much more on generalization and studying financial markets from much wider perspective than would be necessary for practical operation of financial firms. I will use a crude analogy but I think it is at least partially correct, many economists would consider economics to have similar relationship with finance as physics and engineering or biology and medicine.  This is admittedly a crude analogy as I think when it comes to finance/financial economics the distinction is more subtle and less obvious but that is the best analogy I can think of.


I'm studying the effect of a law on a sector: how do I select the control group? I'm planning on using a regression discontinuity design (RDD) to estimate the impact of a sector-specific law on the firms in that sector. I need to create a control group and I have read somewhere in the literature that you should use "a similar sector". When it comes down the selecting the similar sector, I don't know where to start. Does anyone have any suggestion (readings and such)?
The sectors which would be (separately) analysed are: telecommunication, education, construction, banking, finance.

Ideally, but not always feasible, the first option would be to select a similar region where the law is not in place and compare them (e.g. different country / State / city)

If it is not possible to compare with another region, then chose a similar industry. I would try to justify it well though, and comparability would be harder to prove. For example, if you want to study subsidised education, you might want to compare it with subsidized healthcare - the argument could be they're both services, and provide basic needs at some point, with standardised outcomes expected.


consumer and producer surplus So, I am trying to evaluate the consumer and producer surplus. In my notes it is written that the new consumer surplus (defined by the change of the graph from pre-subsidy to post-subsidy) is G + A + D + E - which I do understand. But now, the new producer surplus is defined as the area H + D + A + B. This, one the other hand, doesn't maky any sense to me. Shouldn't the producer surplus be the area H + the "tunnel" that is not marked below H?

TL;DR version: "the tunnel" and D+A+B have exactly the same area.

You are right to say that post-subsidy producer surplus is equal to the blue area in the following figure:

However, it turns out that The Tunnel (i.e. the dark blue area) is exactly equal in size to D+A+B.

Intuitively, there are two ways to think of a unit subsidy:

Paying the seller a subsidy of $S=p^s-p^d$ per unit sold has an effect equivalent to reducing his marginal cost by $S$. The seller's marginal cost curve shifts down in the same way as the supply curve in your figure. From the seller's point of view, its as if he is paid price $p^d$ for each unit he sells but had a low marginal cost. When we calculate producer surplus (by looking at the area between the marginal cost curve and the price) this creates some new producer surplus equal to 'the tunnel' (which represents the seller's lower perceived cost). This way of thinking about things gives us the blue area above.
The subsidy does not reduce the seller's cost [so the seller's perceived marginal cost curve coincides with S(pre-subsidy)], but the subsidy is paid to 'top-up' the price received by the seller from $p^d$ to $p^s$. So from the seller's point of view it is as if he is being paid price $p^s$ with a high marginal cost. When we calculate producer surplus we look at the green area in the following figure:

Since these are two ways of thinking about the same subsidy, they must yield the same value for producer surplus. But you should be able to see by comparing the blue and green areas that this necessarily implies that the tunnel has the same area as D+A+B.

You can check this is true in an example by drawing the curves $Q_S=0.5+p$ and $Q_D=2-p$ and then applying a subsidy of $0.5$ to produce a new supply curve $Q_S'=p$. It looks like this:

Geometrically calculating the areas, you should find that 'the tunnel' and A+D+B both have area 3/8.


What does it mean for a firm to have a 0.95% dividend? Here's an Apple's dividend:

What does this 0.95% mean? Is it 0.95% of share capital? If not, what does it mean?

As of the market close on 2/14/2020, Apple's stock price was \$324.95 per share. The forward looking dividend (based on announced dividend policy over the next year) is \$3.08 per share. The forward looking dividend yield is therefore $\frac{3.08}{324.95} = 0.00948$ or 0.95%. This means that for every share, the firm is planning to pay out \$3.08 over the next year, so the effective yield is that payout over the current stock price. Essentially, every \$100 invested in Apple will throw off 95 cents of cash over the next year.

Source: Yahoo Finance


How could Pablo Escobar earn so much money? I read it's about 30 billion. But I read there were 6 million users in 1985. Let's say he gets 10 percent of the sales price, it means each person paid 50000 dollars for cocaine. It sounds way too much. Any calculations how they reached this amount?

Not sure an economist can answer this particular question about Escobar's wealth and not only because all the given estimates are uncertain. A more interesting question for an economist (or at least for me) would be: What is Escobar's (a drug firm) mark-up? Or what is the economic size of the international market for cocaine?

The most serious source of information I was able to find is the 2012 World Drug Report of the United Nations Office on Drugs and Crime (UNDOC). UNODC estimates suggest that

The total retail market for cocaine amounts to some US $85 billion in 2009 (p.70).
Columbian production proxied in terms of hectares is around 89% (based on 2009's estimates (68,000+73,000)/158,800 - Table 12).

So, if you assume that all Columbian production is produced by Escobar's cartel his total benefit is about USD $75 billion for the year 2009.

Edit: However, we cannot assume that Escobar's cartel got 89% of the retail value (thanks to @Alecos Papadopoulos for this point). The USD 75 billion is the equivalent of the "box office" for a movie - but the producers of the movie (i.e. Escobar's cartel) do not get all the box office.

Then, the question is what is the proportion retained by the Escobar's cartel? I have no clue! However, Levitt and Vetakesh QJE paper document that "earnings within the (drug) gang are enormously skewed (..) with high-level gang members earning far more than their legitimate market alternative." So, by extrapolating from this result and assuming that the drug market is vertically integrated and controlled by the Escobar's cartel, we may speculate that the latter (and Escobar himself) may get a big part of the box office.

In this report, you may also find more reliable estimates about

The total number of cocaine users: 16 million (Table 7)
and the estimates per region.
Purity-adjusted prices per gram in those regions

So if you find some data on consumer's average consumption and Escobar's production you may get a better estimate of his wealth.


Effect of default risk on the interest rate of bonds So I want to calculate the effect of default risk on the required interest of bonds (not on the price of bonds as that is normalized to one).

I thought of using the consumption capital asset model and calculating the following.

$$ U = u(C_1) + \beta (1-p)E_tu(C_2) + \beta p u(C_2) $$

Where the following applies:

$$ p = no \; default ,\; (1-p) = default $$

and the following budget constraints apply:

$$C_1 = Y_1 - T_1 - S_1$$
$$C_2 = Y_2 - T_2 + (1+r)S_1$$

I got to the following result using Lagrange, such that the consumer maximizes utility by deciding on how much to invest:

$$1 = \frac{\beta E_t[U'(C_2)p(1+r)]}{(u'(C_2)}$$

However, I want to get it ultimately in an equation that uses covariances but I do not know how.

To make sense of your question, your $U$ should be
$$
U = u(C_1) + \beta (1-p)E_1 [ u(C_2) ] + \beta p E_1[u(Y_2 - T_2)].
$$
(This definition of $U$ assumes that the default event is independent of other sources of uncertainty in the economy.)

If the bond defaults, saving $S_1$ from the first period is wiped out.
If there is uncertainty in period-2 production/endowment/transfer/etc, then period-2 consumption is still random in the event of default.

Assuming that agent only chooses consumption $(C_1)$ (therefore $C_2$ in the event of no default), then real interest rate $r$ is determined by
$$
1 = E_1[\beta \frac{u'(C_2)}{u'(C_1)} (1-p) (1 + r)].
$$
This is just an extension of the standard Euler equation for interest rate
$$
1 = E_1[\beta \frac{u'(C_2)}{u'(C_1)} (1 + r)]
$$
when $p = 0$.

If equilibrium consumption remains the same (e.g. equals exogenous endowment), then higher probability of default $p$ leads to higher interest rate $r$, as you would expect. High probability of default incentivizes agent to consume today instead of saving for tomorrow. To keep him from borrowing and clear the bond market, equilibrium interest rate must be higher.
Equivalently, higher default risk means agent demands higher liquidity premium on bonds.

If $p$ and $r$ are small, $(1-p)(1+r) \approx e^{r-p}$, which implies that $r$ moves one-to-one with respect to $p$.


Is global economic collapse inevitable? The global economy depends on growth, but growth can never be indefinite in a finite world with limited space and resources.  Moreover, growth always tends to decline over the long term due to the law of diminishing returns, as we choose opportunities which give the best returns first, leaving those which give lower returns for later.

So does this mean economic collapse is inevitable?  Could the global economy survive indefinitely with zero growth, while borrowing more and more money to avoid decline?

Under what scenario would the economy be indefinitely sustainable, without a huge collapse?

The global economy depends on growth, but growth can never be
  indefinite in a finite world with limited space and resources.

It is only partially correct to say that the global economy depends on growth, because an economy per se does not depend on it. But it certainly isn't wrong to say so, because a large part of the world's economic activity is funneled into investments. Without growth, there would be few investments and new enterprising going on.

Growth has certainly been one of the primary political goals of the 20th century and continues to be so, as the main technique for progressing the human race and bringing order to the global society. In our new century, it has much to do with continuous improvement of quality of life, and lifting millions (billions) of people out of poverty.

Furthermore, since the growth paradigm is so central to the way we order the economy, much of economic activity depends on it. We count on growth to make our pensions grow so we can afford to retire. Society counts on growth when investing in new expensive infrastructure, so that huge projects will pay for themselves over the long term. The financial system depends on growth of businesses in return for lending out their capital. And so forth.

  Moreover, growth always tends to decline over the long term due to the
  law of diminishing returns, as we choose opportunities which give the
  best returns first, leaving those which give lower returns for later.

Diminishing returns is used to refer to a point at which the level of profits or benefits gained is less than the amount of money or energy invested. This means that after a certain point, it is no longer profitable to invest more to extract more value out of a proposition, and it may be necessary to divert resources to other parts of the economy. For instance, in the early days of the oil industry, there were huge wells of oil to drill and a lot of oil was wasted due to early technologies. Since there was plenty of oil, the spill didn't matter so much. As oil became a scarcer resource and drilling became more difficult, new technologies were developed to counter diminishing returns. With modern efficient technology, we are producing more oil today than we ever have before, even though the oil has demonstrated diminishing returns.

Your assumption is correct in the sense that: the closer we come to the production possibilities frontier which is the current technological limit, we will see a decline in the growth rate. However, because growth is compounded, we have really seen an exponential growth of world GDP the whole time.

To answer your question: with current technology, we would undoubtedly run out of finite resources at one point. That doesn't automatically mean that there would be an 'economic collapse' – instead we would see ever more diminishing returns and a decrease in activity within that particular sector, while more profitable opportunities would emerge. Society would have to change and find a new way of living. There wouldn't be a sudden "energy catastrophe", but a slow and gradual decline over many decennia. It is plausible that we would see lower birth rates, not because people would die, but because it becomes more expensive to have many children while maintaining a certain standard of living.

With that said, as technology progresses, we become more apt at extracting resources efficiently. The big challenge of this new century is to become independent from fossil fuels and base our economy on renewable energy or other sources. There are also other potential shortages, as some rare-earth minerals have supply limits with practical implications for our way of life. Hopefully, there will be a replacement or recycling of these, and/or ways to locate and extract them more efficiently.

The challenges humanity face are certainly great, and no-one knows the solutions yet. But the hope is that we will be able to cope with the change via new technologies and/or changes to society. That doesn't mean that an economic collapse is inevitable, but it does mean that economic, and probably societal, reorganisation is inevitable.

Regarding your other questions, I hope someone else can elaborate on them in a more technical sense:

  Could the global economy survive indefinitely with zero growth, while borrowing more
  and more money to avoid decline?

  Under what scenario would the economy be indefinitely sustainable,
  without a huge collapse?


What is the Economic Rationale for Discounted Early-Bird Parking? Most urban parking structures/lots in the US seem offer so-called "early bird" discounts for parking, where a substantially discounted flat rate applies to customers entering before a certain cutoff time (typically 9 or 10 AM) and exiting before late evening (typically 6 or 7 PM).

The practice is nearly universal, so there must be some economic advantage to it (unless the various players are locked into some kind of weird Nash equilibrium).

My best guess is that this is a kind of price discrimination targeting people commuting in to work, but price discrimination usually soaks business customers in order to offer discounts to discretionary customers, and this situation is the opposite, so I'm not sure that's it.

It could be that it's especially valuable to kick out a large number of vehicles by early evening so that a "second crop" of customers can be brought in. But evening rates (at least when no special events are being held nearby) are also typically discounted.

It could also relate to minimizing turnover costs and traffic congestion. But turnover costs for a parking lot (especially an automated one) seem to be a tiny fraction of the overall cost.

And it could be the result of a price war between suppliers. Knowledge of a particularly cheap nearby lot would tend to spread quickly among coworkers.

Is there a definitive slam-dunk reason these discounts exist?

EDIT: Another guess is that commuters are high-knowledge customers and shoppers are low-knowledge customers and so it makes sense that the latter would tend to get charged more.

There seems to be some research on price discounting. For example, this paper studies two forms of dicounting, last-minute discounting and early-bird discounting, in settings with monopoly, duopoly and competitive markets. They propose a model and run some experiments. Some excerpts from this paper:

  There might be many reasons for such a pricing policy [Early-Bird Discounting]. Early bookings for example reduce uncertainty for
  airlines, who might be willing to forgo some profit in order to reduce this
  uncertainty. Introductory pricing in markets with network externalities
  might be another reason. Here we want to investigate intertemporal price
  discrimination between different customer groups, who enter the market
  at diferent points in time. One might think of holiday makers versus
  business travellers in the market for airline tickets. Holiday makers
  usually plan ahead and enter the market for a particular flight quite early,
  while business travellers often cannot plan well in advance. Additionally,
  the reservation price of business travellers is much higher than that of
  holiday makers. A monopolist airline will try to persuade as many low-
  valuation customers as possible to book early, such that it can charge
  higher prices later when the high-valuation customers have entered the
  market.

This is also interesting:

  When customers with lower valuations for the good start
  searching earlier for cheap prices than consumers with higher valuations
  then it is possible to sell to these customers at early-bird discount prices,
  while the high valuation customers who enter later can be charged higher
  prices. The main objective of the monopolist is to get the low-valuation
  customers out of the market before the high-valuation customers arrive.
  The low-valuation customers accept early, as they anticipate that the
  prices will go up over time.

They conclude that:

  In the early-bird scenario competition even seems to enhance the ability
  to price discriminate initially, while with increasing experience this advantage is eroded. [...] The monopolists had much less pricing
  power than predicted by standard theory, while the duopolists - also
  contrary to the prediction - were able to retain at least some pricing
  power. The most striking advantage of competition is the increase in total welfare.

Perhaps not entirely pertinent to the case of parking, this entry in The Economist claims that early-bird tickets are useful in events like concerts to build up a minimum mass of goers:

  T in the Park, a large pop shindig held in Scotland, pioneered the practice of offering discounted “early bird” tickets to build buzz.


How is Regression different from Econometrics? I am an undergraduate student and am planning to take a course in Regression Analysis next semester whose description is

"The analysis of continuous response data. The focus is on linear and multiple regression with theoretical and practical training in statistical modeling. A hands‐on, applied course where students will become proficient using computer software to analyze data drawn from a variety of fields, and will learn what assumptions underlie the models, how to test whether the data meet the assumptions, and what can be done when the assumptions are not met. "

I am signing up for it because the university I attend currently does not offer a course in Econometrics and Regression is the closest thing to Econometrics that is offered. I want to understand Econometrics at the level of Stock and Watson's Introduction to Econometrics and/or Jeffrey Woolridge's Introductory Econometrics textbook.

Can someone please explain what the differences might be between the Regression course mentioned above and a course that would be taught out of the Econometrics textbooks mentioned above? If in a hypothetical world, I could take a course next semester supplementing the Regression course to patch up my understanding of Econometrics, what book/ source would you recommend that I read? Any advice is greatly appreciated!

Aside: I want to get acquainted with the world of economic research and eventually pursue a doctoral degree in Economics.

To put it simply regression modeling is a technique, a tool, used by econometricians. Regression deals with dependence amongst variables within a model. But it cannot always imply causation.

As economists, we are biased towards establishing causal relationships. So, this explains our focus on endogeneity issues and strategies of identification to establish causality.

In other words, as Ragnar Frisch (one of the founders of the Econometric Society) would say

  Econometrics by no means the same as economic statistics

See also the answers to What is an econometrician? where the key idea is that economic theory is crucial to econometrics and understand measurement. Neither theory nor measurement on its own is sufficient to further our understanding of economic phenomena. We need both and measurement without theory is unlikely to provide a satisfactory explanation of the way economic forces interact with each other.

In summary, regression is super useful but not sufficient to make causal inference and to understand economic phenomena.

Stock and Watson's Introduction to Econometrics and Jeffrey Woolridge's Introductory Econometrics are both excellent textbooks.