How to describe a utility function in words? Suppose I have a utility function of Cobb-Douglas form

$$U(x, y) =x^{0.2}*y^{0.8}$$

I want to describe it in words. I would say like:

  The utility of consumer is captured by number of good x and number of good y. The
  indifference curve is downward sloping and convex, which means the
  marginal rate of substitution is diminishing as x increases. The
  consumer prefer y to x since y gives a higher utility level.

Is this correct and conprehensive?
Any help will be appreciated.

The best way to go about it is to examine the marginal rate of substitution.

Take
$$
x^\alpha y^\beta = C
$$
and use the implicit function theorem to derive the slope of the indifference curve:
$$
x^\alpha y(x)^\beta = C
$$
$$
\alpha x^{\alpha-1} y(x)^{\beta} + x^\alpha \beta y(x)^{\beta-1} y'(x) = 0
$$
$$
MRS(x) = y'(x) = -\dfrac{\alpha y(x)}{\beta x} = -\dfrac{y}{4x}.
$$
This gives the rate at which the consumer is willing to give up $x$ for $y$ and stay on the same indifference curve. In words, "If the consumer currently has the bundle $(3,4)$, he is willing to give up $4/12$ of a unit of the $x$ good for a bit of the $y$ good."

You might worry that the MRS does not full capture the Cobb-Douglas utility function, but you can recover Cobb-Douglas preferences by "solving" the differential equation $y'(x) = -\alpha y(x)/\beta x$, since it can be rewritten
$$
\dfrac{y'(x)}{y(x)} = - \dfrac{\alpha}{\beta x}
$$
and the left-hand side is the derivative of $\log(y(x))$ and the right-hand side is the derivative of $\log(x)$, implying that
$$
\log(y(x)) = - \dfrac{\alpha}{\beta}\log(x) + v
$$
so that
$$
\log(y(x)^\beta x^\alpha) = v
$$
and
$$
y^\beta x^\alpha = e^v = u(x,y)
$$
so that the MRS fully characterizes the Cobb-Douglas utility function.


Is marginal cost different from extra cost? I have been reading this resource which talks about why derivatives can be used to find the marginal cost.

https://www.dummies.com/education/math/calculus/how-to-determine-marginal-cost-marginal-revenue-and-marginal-profit-in-economics/

I am slightly confused about a statement referring to marginal cost and extra cost. They write "If you look very closely at the right side of the above figure, you can see that the extra cost goes up to the curve, but that the marginal cost goes up a tiny amount more to the tangent line, and thus the marginal cost is a bit more than the extra cost".

Are they stating that the marginal cost is different from the extra cost? Isn't that incorrect as the marginal cost is the extra cost from producing one more unit of a good? Or is marginal cost separate from the extra cost?

is it saying that marginal cost is different from extra cost?

In this graph they use marginal cost to describe a sort of linear approximation of the change in the cost function $C(x)$ at some point $x_0$, that is
$$
MC_{x_0}(x) = (x-x_0) \cdot \frac{\partial C(x_0)}{\partial x}.
$$
This is indeed not the usual way the term marginal cost is used.


Perfect substitutes and Lagrange How does one solve utility maximization of perfect substitutes using Lagrangian function?
Consider the problem
$$\max_{x,y} ax +by  $$
subject to the constraint that
$$px + qy \leq I$$
where $a,b,p,q,I>0$.
Note problem statement are subject to the standard implicit assumptions that consumption cannot be negative.

Your Lagrangian would be
$$L = (ax+by)+\lambda (I−p_x x−p_y y) +\mu_x(x−0)+\mu_y(y-0),$$
where the final two terms represent the restriction that $x,y\geq0$.
You then arrive at conditions
$$\frac{\partial L}{\partial x}= a -\lambda p_x +\mu_x=0$$
$$\frac{\partial L}{\partial y}= b -\lambda p_y +\mu_y=0$$
$$I=p_x x+p_y y$$
and complementary slackness conditions
$$\mu_x x =0 , \quad \mu_y y =0.$$
You then consider many different cases: $(\mu_x>0,x=0,\mu_y=0,y>0),(\mu_x=0,x>0,\mu_y=0,y>0), (\mu_x=0,x=0,\mu_y=0,y>0), ....$
and for each of them you look at your conditions:
Suppose $(\mu_x>0,x=0,\mu_y=0,y>0)$. Then,
$$a -\lambda p_x +\mu_x=0 \Rightarrow a -\lambda p_x <0$$
$$b -\lambda p_y +\mu_y=b -\lambda p_y=0$$
$$\Rightarrow a -\frac{b}{p_y} p_x <0 \iff \frac{a}{p_x} <\frac{b}{p_y}$$
This condition can be checked as $a,b,p_x,p_y$ are given. If it does not hold, $x=0,y>0$ cannot be the solution. If it does hold, it means that the utility per dollar is higher for good $y$ and all money should be spend on $y$ (which $x=0,y>0$ and the budget constraint imply).
You have to check all the cases, but you will arrive at a solution of the above form.


Difference-in-differences Matching I have a difference-in-differences analysis. My treatment is at the county level. Each county received the treatment at a certain point in time.

I want to proceed with a propensity score matching as an extension to get DiD-matching results . I know everything about PSM, however, I am not quite sure how to get the STATA command in order to:

Generate Pre- and Post-treatment dummies since I don't have a
treatment at one point in time; i.e. DD is staggered.
Use the DD with pscore

You can use mark. For example, if the treatment was done in the year 2000, and your time variable is called time, the syntax is

mark treatment if time >= 2000 & time <.

where treatment is the dummy that is generated.


Demand curve from logit model Is it possible to derive a demand curve from a discrete choice multinomial model. For instance, a logit model would give the probability of purchasing each good. But can I translate that into a demand curve.

We cannot obtain "demand" in the usual sense, because demand is a random variable. The "best" we can do is first, to obtain the Conditional Expectation of individual demand (conditional on the variables that determine the probability of whether the consumer will demand/buy).
Let a situation where consumers decide to buy or not to buy a single (for simplicity) item of a good $A$. We can model this underlying utility framework by

$$u_i(A) = \alpha p_i+\mathbf x_i'\beta +e_i$$

with $i$ denoting the consumer, $p_i$ is the good's price, faced by consumer $i$, $\mathbf x_i$ containing various other variables pertinent to the case, and $e_i$ representing a "random" preference shock, assumed, say, to follow the standard logistic distribution, conditional on the $x$'s.
The consumer will demand/buy, $q^d_i=1$, if (by convention) $u_i(A)>0$, so we can model the conditional probability of buying by

$$E(q_i^d =1\mid p, \mathbf x_i) = \Lambda(\alpha p_i+\mathbf x_i'\beta)$$

Then the conditional expectation of Market demand is

$$Q_d = \sum_{i=1}^nq^d_i = \sum_{i=1}^n\Lambda(\alpha p_i+\mathbf x_i'\beta)$$

This is still a random variable, and with unknown parameters. To obtain something one dimensional, given a sample of size $n$ on $\mathbf x_i$ and on relevant transactions (and so on $p_i$), one can estimate $\hat \alpha, \; \hat \beta $, say by maximum likelihood. Then we can obtain some concept of "average market demand curve" by using the sample means of the $x$'s as

$$\hat Q_d(p\,; \mathbf {\bar x}) = \sum_{i=1}^n\Lambda(\hat \alpha p+ \mathbf {\bar x'}\hat \beta) = n\cdot \Lambda(\hat \alpha p+ \mathbf {\bar x'}\hat \beta)$$

where here the price $p$ is not indexed anymore, and we vary it to obtain a curve.

As a toy numerical example, assume there are $100$ consumers, that ${\bar x'}\hat \beta =1$, and  $\hat \alpha = -2$. Then $\hat Q_d(p\,; \mathbf {\bar x_i})$ will look like

MULTINOMIAL LOGIT

In a multinomial logit where we examine the choice to buy "this or this or this" (mutually exclusive choices $k=1,..,K$), we end up with (after the usual normalization)

$$\hat E(q^d_{ik} \mid p_{ik}, \mathbf x_{ik}) = \frac {\exp\{\hat \alpha p_{ik} + \mathbf x'_{ik}\hat \beta\}}{1+\sum_{k=2}^K\exp\{\hat \alpha p_{ik} + \mathbf x'_{ik}\hat \beta\}}$$

Then we can obtain $K$ demands (each conditional on the rest) using (say, for good $1$)

$$\hat Q^{(1)}_d(p_1) = \frac {n\cdot \exp\{\hat \alpha p_1 + \mathbf {\bar x}_{1}'\hat \beta\}}{1 + \sum_{k=2}^{K}\exp\{\hat \alpha \bar p_{k} + \mathbf {\bar x}_k'\hat \beta\}} $$

Note that here we have to average the prices of the other goods also.


How to derive utility function from indirect utility in this exercise Derive the consumer’s direct utility function if his indirect utility function has the form

$$v(\boldsymbol{\mathbf{p}},y)=yp_1^\alpha p_2^\beta$$

where $\beta ,\alpha$ are negative.

Use Roy's identity to find demand functions:
$$\displaystyle x_1=-\frac{\frac{\partial v}{\partial p_1}}{\frac{\partial v}{\partial y}} =-\frac{\alpha yp_1^{\alpha-1} p_2^\beta}{p_1^\alpha p_2^\beta}=-\frac{\alpha y }{p_1} $$

$$\displaystyle x_2=-\frac{\frac{\partial v}{\partial p_2}}{\frac{\partial v}{\partial y}} =-\frac{\beta yp_1^{\alpha} p_2^{\beta-1}}{p_1^\alpha p_2^\beta}=-\frac{\beta y }{p_2} $$

Since $p_1x_1 + p_2x_2 = y$, from the above demand expressions we get $-\alpha -\beta = 1$.
Demand equations above can also be rewritten as:

$$x_1^{-\alpha}  = (-\alpha y)^{-\alpha}p_1^\alpha$$
$$x_2^{-\beta}  = (-\beta y)^{-\beta}p_2^\beta$$

Multiplying them we obtain the following:
$$x_1^{-\alpha}x_2^{-\beta} = (-\alpha y)^{-\alpha}p_1^\alpha (-\beta y)^{-\beta}p_1^\beta = (-\alpha)^{-\alpha}(-\beta)^{-\beta}yp_1^\alpha p_2^\beta $$

Comparing it with the indirect utility function, we get the direct utility function as:

$$u(x_1, x_2) = \frac{x_1^{-\alpha}x_2^{-\beta}}{(-\alpha)^{-\alpha}(-\beta)^{-\beta}}$$


How to calculate the after tax WACC So i have this question:

Assume the following data for U&P Company: Debt (D) = $100 million; Equity (E) =$300 million; rD = 6%; rE = 12%; and TC = 30%. Calculate the after-tax weighted average cost of capital (WACC):

I know that the formula is indeed

After tax WACC=(1-TC)rD(D/V) + rE(E/V).

If i correctly replace all the numbers i get that the after tax wacc is 6%. For example, in order to get D/V i do 100/130 since V=E+D=130.

However on the answer sheet it states that :

After tax WACC=  (1/4)(1 - 0.3)(6) + (3/4)(12) = 10.05%.

I do not understand this answer; where is the (1/4) and the (3/4) coming from ? Shouldn't D/V=100/130 and E/V=30/130?

Thanks for the help!

$V=E+D=300+100=400$

So

$\frac{E}{V}=\frac{3}{4}$ and

$\frac{D}{V}=\frac{1}{4}$

You probably overlooked one zero and thought equity was 30, but in your question you yourself state that $E=300$.


Calculating optimal level of negative externality I am trying to solve the following question(s):
Let $h \geq 0$ represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility function and attaches a utility of $\phi(h) = -2h^2$ to the externality. The firm's profit function is $\pi(h)=120-2(h-10)^2 $.

Compute the pareto optimal level of $h$, $h^o$.

Compute the firm's optimal level of $h$, $h^f$.

Suppose the consumer rather than the firm can determine the specific externality levelh with which the firm operates. Compute the optimal level of $h$, $h^c$, from the consumer's perspective.

My approach so far:

To find $h^o$ I maximize the aggregate surplus/welfare of h. $$\max_{h \geq0} \pi(h) + \phi(h)$$
$$ W(h)=120-2(h-10)^2 - 2h^2 = 0 $$
$$ \frac{dW}{dh}= -4(h-10)-4h = -8h + 40 = 0 $$
$$ 8h=40 \rightarrow  h^o=5$$

To find $h^f$
$$FOC\ of\ \pi(h): \frac{d\pi}{dh} =-4(h-10) = -4h+40 = 0 \rightarrow h^f=10 $$

To find $h^c$
$$ FOC\ of\ \phi(h): \frac{d\phi(h)}{dh} = -4h = 0 \rightarrow h^c=0 $$

Is this correct?

Almost correct. Setting $W(h)=0$ is wrong (but inconsequential for the solutions). Checking the SOC for completeness should be included, but is somewhat obvious here. Correctness of 3. holds only under the assumption that the consumer owns no shares of the firm, which seems to hold in this exercise.


They measure different things.  In particular the Gini index measures inequality and is strongly affected by the number of individuals with almost nothing, while the Herfindahl index measures the diversity of the shares (for example the choice in a market) and is almost unaffected by those with almost no share

If $n$ people had equal shares then the Gini index would be $0$, not changing as $n$ increased since inequality would not change, while the Herfindahl index would be $\frac1n$, reducing as $n$ increased to reflect greater diversity

As an an illustration, the following $36$ values give roughly the same results as your two examples

V2 <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
        0.05, 0.19, 0.19, 0.19, 0.19, 0.19)
Gini(V2)
# [1] 0.8527778
Herfindahl(V2)
# [1] 0.183

But the zeros have no impact on the market shares and the Herfindahl index: having somebody extra with no share is as irrelevant as having nobody extra, and somebody with a tiny share is almost as irrelevant.  Removing those zeros would lead to

V3 <- c(0.05, 0.19, 0.19, 0.19, 0.19, 0.19)
Gini(V3)
# [1] 0.1166667
Herfindahl(V3)
# [1] 0.183

with the Gini index being much smaller (there is less inequality when those with nothing are excluded from consideration) but the Herfindahl index staying the same, corresponding to somewhere between $5$ and $6$ equal shares

You could looking at the Lorenz curves for your examples with something from the package like

plot(Lc(V0), col="blue")
lines(Lc(V1), col="red")

or something homemade like

plot(c(0,1), c(0,1), type="l")
points((0:length(V0))/length(V0),cumsum(c(0,sort(V0)))/sum(V0),
    type="b", col="blue")
points((0:length(V1))/length(V1),cumsum(c(0,sort(V1)))/sum(V1),
    type="b", col="red")

gives a picture like this with V0 in blue and V1 in red

So you can see that there is a greater area between the diagonal and the red curve (V1) than between the diagonal and the blue curve (V0), explaing why V1 gives a higher Gini index.  This is largely due to V1 having more extremely low values than V0 does

But on the right hand side, the top two values are greater for blue (the cumulative sums of the others are lower), so shares are more concentrated for V0 leading to a higher Herfindahl index


Does GDP typically refer to Real GDP or Nominal GDP? I previously believed GDP typically refers to nominal and real GDP is specified explicitly. However, in data sources like this, does it refer to Real GDP growth or Nominal GDP growth? Real GDP growth seems to be the more useful figure which makes me reconsider what GDP refers to by default.
Edit: Apologies I originally posted the wrong link. The link is now updated to the world bank data source.

The dataset states "Annual percentage growth rate of GDP per capita based on constant local currency" so it real GDP growth in this case.
GDP can refer to both nominal and real GDP, there is no preferred meaning. You have to read what is written in text/dataset.


"omission bias" vs "action bias" -- how can they be reconciled? I'm having trouble reconciling these two biases. They seem to be the exact opposite of each other. Yet humans are allegedly pre-disposed to both.

How can they be reconciled?

Omission Bias:

  The omission bias is an alleged type of cognitive bias. It is the tendency to judge harmful actions as worse, or less moral than equally harmful omissions (inactions) because actions are more obvious than inactions.

Action Bias (academic source: Bar-Eli et al., 2007)

  Simply put this means that just about everyone, when faced with ambiguous situations, especially those circumstances associated with risk, gets the feeling that they need to take some action regardless of whether this is a good idea or not.

This is a great question, you're not the only one confused. We just completed a working paper on this very topic: "What is action, what is inaction? A review of action-inaction biases and recommendations for term use and typology" and we cover these two biases and many other action-inaction biases.

See page 5:

  Building on the action-effect is the omission bias. The omission bias
  has been defined in many different ways, and is sometimes regarded to
  be the same as the action-effect. We view the two biases as different,
  with the omission bias extending the action-effect to focus on the
  choices made between action and inaction or about moral attributions
  to action and inaction, rather than the experience of negative
  emotions or regret as a consequence of action-inaction. In its
  broadest definition, the omission bias is the phenomenon that people
  prefer omissions to commission when there is the possibility of a
  negative outcome (Spranca, Minsk, & Baron, 1991; Ritov & Baron, 1990).
  Using this definition, if we consider the classic action-effect
  scenario, the two investors facing a risky situation and presented
  with the options of taking action and switching versus not taking
  action and sticking to the initial investment would prefer inaction to
  action. Linking with the action-effect, the bias could partly be
  explained as a strategy to minimize anticipated-regret (Anderson,
  2003; Wong & Kwong, 2007; Zhang & Fishbach, 2005), possibly further
  heightened by related biases such as loss aversion (Tversky &
  Kahneman, 1991) and risk aversion (Kahneman & Tversky, 1979; Tversky &
  Kahneman, 1992).

Briefly, the "action effect" is the tendency to regret actions more than inaction (negative outcomes) leading to a preference for omissions. In Bar-Eli's paper the action-effect is reversed to an inaction-effect (Zeelenberg etal. 2002) due to normality, the expectations for goal keepers to act. Therefore, the reconciliation is by using normality from norm theory (Kahneman & Miller, 1986).

Hope that helps, we'd appreciate feedback if things aren't clear.


What metrics would indicate a house bubble rather than genuine market values? There are concerns that Auckland, New Zealand, is currently experiencing a housing bubble.

Auckland is one of the top 10 cities in the world on a housing unaffordability index.

The question is - how as economists can we tell whether housing prices reflect a bubble or genuine market value?

I would be looking at average wages, crime rates, happiness, and other social measures of the city (ie. it's reasonable that people would be willing to spend more on a house in a utopian city). A good answer should comprehensively list these and explain the relevance of these metrics.

The first metric to look at is house-price-to-rent ratios. Rental prices capture the value of the housing (and housing-linked) services provided by a property, including things like how safe a neighborhood is, how good the schools are, et cetera. In contrast, house prices are the capitalized value of the future stream of housing services PLUS the price of an asset (mostly land; structure value beyond the housing services it provides is negligible). As a result, a higher ratio of house prices to rents means that, controlling for the value of housing services provided, we observe people paying more for housing assets (i.e., land).

This doesn't mean that the asset is necessarily overvalued, but it at least moves you in the right direction by separating out the asset price from the value of housing services. You can then start thinking about the supply/demand fundamentals that apply to housing assets, and what you'd have to assume to support a particular valuation of the asset (things like, as Jamzy noted, whether growth in price-to-rent ratios is linked with increased home purchase borrowing).


Securities Prices in Financial Markets Hello in the moment I am working with the book "Introduction to the Economics and Mathematics of Financial Markets" by Jakˇsa Cvitani ́c and Fernando Zapatero, which can be accessed at the link.

However, at the moment I am struggling with the third chapter "Models and Securities Prices in Financial Markets". Right in the beginning at page 56 there is an introduction of a formula, which I do not understand completely.

X(1) should display the worth of all the assets (securities,stocks, bonds etc.). Therefore B=Bonds, S=Securities or Stocks. The numbers 0 and 1 illustrate the time. So 1 for the end and 0 for the beginning.

But what is the this sign "δ". Actually it is a delta isn't it? And it actually illustrates a changing process. However, here it is described as follows "Denote by δi , i = 1, . . . , N the number of security i shares that Taf holds between time zero and time one. We denote by δ0 the amount invested in the bank account at time zero, “the number of shares” of the bank account." (page 55, at the same book). My problem is that I really do not understand the meaning of this sign. Also the Sentence, which is illustrated in the picture talks about a vector. I completely do not understand the word vector in this relationship.

I hope I can get some insight to this topic through your help. Overall it should be an introduction to the two periods binomial models.

The total wealth (as is pointed out in the book) is given as :

$$
X(1)=\delta_{0}B(1)+\delta_{1}S_{1}(1)+...\delta_{N}S_{N}(1)
$$
The wealth at period one is just the sum of the securities (including amount invested in the bank) times their number. In other words, the total wealth is the amount of money this individual has invested in the different assets. It has nothing to do with a "changing process." Asssume that the individual does not buy any more assets during period 1 and the assets promise a return of $(1+r_{i})$ where $i$ denotes the asset in question. In period 2, the individuals wealth is then
$$
X(2)=\delta_{0}B(1)(1+r_{0})+\delta_{1}S_{1}(1)(1+r_{1})+...\delta_{N}S_{N}(1)(1+r_{N})
$$


What is the difference between a Start-Up and a small company which is not a Start-Up? What exactly makes a small and youg company a "Start-Up", among other small and young companies which are not necessarly start-ups?

Probably, there's no clear-cut definition, however, there's an article of Paul Graham called Startup=Growth which represents the opinion many agree upon: startup is a company that plans to grow fast (... and scale). A quote:

  A startup is a company designed to grow fast.
      Being newly founded does not in itself make a company a startup. Nor is it necessary for a startup to work on technology, or take
  venture funding, or have some sort of "exit." The only essential thing
  is growth. The only essential thing is growth. Everything else we associate with startups follows from growth.

That's why a new barbershop isn't a startup since

  A barbershop doesn't scale.

This definition probably still leaves some space for interpretation, but not that much.


Why do cities with rich areas still contain poor areas? A cynic would attribute the disparity (and the failures to gentrify) to indifference or inattention, (or worse) disdaining neglect or ignoring, by the politicians and rich of the poor in the poor areas.

I exemplify with cities whose disparities I have witnessed, followed by links to the maps overlaying location with income, but I exclude downtown areas which always appear rich:

New York: (Lower) Manhattan vs. The Bronx and Brooklyn (e.g. Bedford-Stuyvesant, Bushwick)
<Google search: 1, 2>

London (Boroughs): Hammersmith & Fulham, Kensington & Chelsea, Westminster
vs. Hackney, Tower Hamlets, and the southern boroughs like Lewisham, Croydon.
<Google search: 3, 4, 5>

Toronto: Midtown (along Yonge St. subway line, like Lawrence Park) vs. Jane & Finch, Rexdale
<Google search: 6>

Vancouver: West & North Vancouver, West Side, South Vancouver vs. Downtown East Side, Surrey
<Google search: the Vancouver Sun has many maps.>

Regarding NYC, it has mostly to do with public housing and rent stabilization.

Here's a map of public housing in NYC. Notice the large public housing developments in lower Manhattan, The Bronx, and Bushwick.

About half of all apartments in NYC are rent stabilized, and these apartments tend to be priced at a fraction of the market value and inhabited by tenants who never move (because then they'd lose their precious deals). Thus rents in NYC have a bimodal distribution.


Why does the Fed control interest rates? The Fed more or less controls interest rates using open market operations, but why?

Why are interests rate not controlled by supply and demand?

The fed controls interest rates via supply and demand.

The fed directly controls only one rate, the rate at which banks can borrow from the government.

Lowering this rate permits banks to lower the rates they charge while making the same profit. They don't have to do so, but supply and demand says that those banks who do lower their rate can take business away from those which don't, so the market pressures them to more-or-less follow the fed rate's shifts.


Time costs and the St. Petersburg paradox In the St. Petersburg paradox, we end up with the problem that a rational agent should be willing to play the game for any wager, if we look at expected income or utility of expected income. The standard "solution" to this is to instead look at expected utility of income, where the marginal utility of income is decreasing.

Intuitively, decreasing marginal utility (for instance, where utility is logarithmic) of income makes sense, since one could argue that the utility of income comes from purchasing power, and standard indifference curves are convex.

But I've seen people (in a statistics department) point out that playing the game also has a time cost in addition to the monetary wager. This suggests potential other trade-offs that could affect decision making here, such as the fact that this time could be spent on more enjoyable pursuits, or in working and making money, even if the decreasing marginal utility is still the most convincing explanation for people's behavior to me.

How might be time-costs factored into the St. Petersburg problem, in terms of limiting the player's wager, and what literature might provide a good coverage of this?

Consider the version of the paradox from Wikipedia:

  A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, 8 dollars if a head appears on the first two tosses and a tail on the third, 16 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2k dollars, where k equals number of tosses. What would be a fair price to pay the casino for entering the game?

If there is a finite fixed per-period time cost, $c$, of playing the game then the paradox would not be resolved. The payoff from the lottery would then be
$$-c+\sum_{t=1}^\infty \left[\frac{1}{2^t}\left(2^t-c\right)\right]=\sum_{t=1}^\infty \frac{1}{2^t}2^t-\sum_{t=0}^\infty \frac{c}{2^t}=-2c+1+1+1+\ldots=\infty$$

Thus, as in the original paradox, the value of the lottery is infinite.

An alternative might be to suppose that the marginal cost of time spent is not constant (e.g. it is increasing). However, it is hard to see how one would justify an increasing cost of time without invoking some idea of the dis-utility of time (but the whole purpose of this exercise is to get away from invoking a utility function).

Two solutions that are related (in the sense that they involve a time dimension) and that might be more appealing are:

(1) discounting. Even if your utility of wealth is linear, you might put a low weight on future payments because you have to forego some return during the period between now and when the payment occurs. If the future is discounted at rate $\delta<1$ then the lottery would have value
$$\sum_{t=1}^\infty\delta^{t-1}\frac{1}{2^t}2^t=\frac{1}{1-\delta}.$$
Thus, the value of the lottery is finite for any $\delta<1$ and the lottery is more valuable to more patient people (i.e. people with larger $\delta$).

(2) finite lifespan. This is formally equivalent to discounting, but conceptually different. Now the idea is that the player is not going to live forever so there is a chance that they 'die' before having the opportunity to collect the lottery's payoff. In particular, suppose that the player dies at the end of each period with probability $\lambda$ and that the payoff is zero if s/he dies before the end of the lottery. The value of the lottery is therefore
$$\sum_{t=1}^\infty\lambda^{t-1}\frac{1}{2^t}2^t=\frac{1}{1-\lambda}.$$
Thus, players who expect to live a long time have a higher willingness to pay for the lottery, but the value of the lottery is finite for anyone who does not expect to live forever.


Calculating price elasticity by asking a "this" or "that" question Let's say 2 goods exist: A for 10 and B for 20 (B is more premium in this example)

If I decrease the price of both goods by 10% and ask:

"Do you prefer A for 9 or B for 18 "

What sort of insights (elasticities, etc.) would I be able to get from this? I figure the usual arc elasticity is not possible for this, since the units are discrete.

Is it also possible to get other insights if I lower the price of the premium product more than the cheap product? i.e.:

"Do you prefer A for 9 or B for 17?"

Or even lower the price of the cheap product more than the premium, i.e,

"Do you prefer A for 8 or B for 18?"

I'm thinking about any insights that I can get from asking these sorts of questions, but the traditional equations of elasticity don't seem to apply.

The elasticity of demand measures how changes in prices affect changes in quantities demanded So if you have the total quantity demanded at the original prices and reduced them by, say, $10\%$, then observe the total quantity demanded of both goods, you can calculate elasticity by simply using the arc formula,  or the midpoint formula, etc. In fact, by just varying one good at a time, or reducing one good more than the other, you could also estimate cross-price elasticities.

In your example, however, it sounds like you have only information on a single consumer that buys either one good or the other. So you only know if they were buying at the original prices and which good and if they start buying after prices decrease, or if they decide to change the good they buy.

This exercise cannot really measure any elasticity because there is no sensitivity to measure. Best case scenario, if the person was not buying a good and after the decrease in price starts buying it, it can give you a lower bound on their valuation for the good.

In contrast, if the question was changed to how many units of each good would the consumer buy? then you could estimate demand elasticity and cross-price elasticities with careful exercises of lowering each price at a time.


How did the derivatives market 'collapse' in 2008? I just watched the film 'The big short', and have been trying to think in context with the financial markets in 2008. I think the film does a great job of explaining how sub-prime mortgages came to be issued and sold by opportunists to investors who were blinded to their junk value.

One key thing that I feel though is brushed over though is what the market for derivatives on the mortgage industry looked like at that time, and how the eventual defaults on the mortgages caused this to 'collapse'. They claim that the derivatives market had grown to 20 times the size of the market itself. According to the film,

  if... the mortgage bonds were the kerosene soaked rags, then the 'synthetic CDOs' were the atomic bomb with the drunk president holding his finger over the button

but what did these derivatives look like? I understand how someone may invest in a mortgage, which is money lent that will have to be written off in case of default. But how did the actual defaults magnify in the event to hit investors twenty fold? What were people investing in on top of the mortgages themselves? And why did these objects all necessarily become junk when the mortgages themselves did?

The question is quite complex and an answer should be far beyond looking only at the structure of the derivatives, but i try my best.

The initial situation:

Between 1998 and 2006, the price of the typical American house increased by 124%
Global saving glut due to the dotcom-bubble in the early 2000s, seeking institutions for alternative investments, cutting out financial risks
The Community Reinvestment Act which helped low- and moderate-income Americans to get mortgage loans

These points led to a situation, where in fact everyone could get a mortgage to buy a house, not only for using them as own-property, but as a financial asset. Also people with "no income, no job, no assets", leading to the term Ninja-Loan, were granted loans and mortgages. Times were great, as long as house prices increased. The US financial markets were flood by money from investors, wanting to become part of that upswing. How they took part, answers how the derivatives look like, and how their structure led to the financial crisis:

CDOs and CDSs

Lets take a look an a classic CDO:

CDOs pay investors in a prescribed sequence, based on the cash flow the CDO collects from the pool of mortgage loans. There are several tranches, which catch the cash flow of interest and principal payments in sequence based on seniority.

As CDOs became popular, some investment banks repackaged selected tranches into yet another iteration, known as CDO-squared, a complex and intransparent structure of financial derivative.

A CDS (credit default swap) is a financial agreement that the seller of the CDS will compensate the buyer in the event of a loan default. It can be considered as an insurance for a creditor of the referenced loan.

Synthetic CDOs

Synthetic CDOs are CDOs, which do not only bundle the mortgage loans in the picture above, but also other assets and derivatives. Up to 2008 it was popular to construct CDOs, bundling single CDSs, rather then mortgage loans. This is the structure of a synthetic CDO, created in times before 2008.

  How did defaults of mortgages hit investors twenty fold?

Synthetic CDOs described above can be considered as a bet on the performance of CDSs, rather than the performance on a real mortgage security. In fact, the derivatives were detached from a single mortgage due to repeated securization in CDOs and there use as an underlying for CDSs. They became solid derivatives offering high returns.

As stated in the New York Times (1), from 2005 through 2007, at least $108 billion synthetic CDOs were issued, but the actual volume can be much higher because synthetic CDO trades were unregulated.

Debate and further topics

Synthetic CDOs are well known for their destructive character during the financial crisis. They were easy to construct, barely supervised or regulated and offered high returns. As the former Citigroup CEO said, referring to the leveraged lending practice: "As long as the music is playing, you’ve got to get up and dance. We're still dancing." As long as there was a market for these derivatives, they were constructed. Current research address the missing incentives for each, buyer and seller of CDOs, CDSs and synthetic CDOs during the time before the financial crisis: As long as the risks of defaulting mortgages could be transfered to other parties, the credit-worthiness of a borrower does not matter any more. Furthermore, speculatives without any real insurable interest could invest in these derivatives, which increased the demand for these derivatives additionally.

For further information to the structure of the derivatives mentioned above see here or (2). On the role of rating agencies see (3).

(1): The New York Times, Banks Bundled Bad Debt, Bet Against It and Won,  by Gretchen Morgenson and Louise Story, Published: December 23, 2009

(2) Joseph R. Mason, Josh Rosner: Where Did the Risk Go? How Misapplied Bond Ratings Cause Mortgage Backed Securities and Collateralized Debt Obligation Market Disruptions,  May 14, 2007

(3): Markus K. Brunnermeier, DECIPHERING THE LIQUIDITY AND CREDIT CRUNCH 2007-08, NBER Working Paper No. 14612.


Interpreting the q-intercept of the demand curve Given a demand curve for a particular commodity, which I'll interpret as a function $p(q)$, the $p$-intercept can be interpreted as the choke price, i.e. the lowest price at which quantity demanded is 0.
Assuming the demand curve eventually intercepts the $q$-axis, the $q$-intercept can be interpreted, by definition, as the quantity demanded by consumers if the commodity were free. In other words, any production level greater than this quantity will necessarily result in excess due to the fact that consumers won't want more of the commodity even at a price of 0. Is there some term for this theoretical quantity? Something analogous (but opposed) to the notion of the choke price?
I'm a mathematician, and not an economist, so my economics vocabulary is fairly limited. I was reading about the notion of "market saturation" which seems related but not quite what I'm looking for. Thanks in advance!

There isn't usually a name for this, but we could borrow from the theory of the individual consumer the term satiation point.


on economic growth and Second Welfare theorem While I was studying eighth edition of Mankiw's Macroeconomics, in chapter 9 on Economic Growth(pg. 245), the author mentions :

"Some economists have proposed increasing the incentive to save by replacing the current system of income taxation with a system of consumption taxation."

Here, the book was discussing on the topic of government policies that affect rate of saving and spur growth. My question is that doesn't this finding contradicts with Second Welfare Theorem in Microeconomics, according to which, the taxation should be on endowments and not on consumption goods so that we can have Pareto Efficient allocation(refer Varian's Intermediate Microeconomics eighth edition Ch. 31, pg 605)? Or am I wrong in superimposing the micro concept over macro?

The two statement say two completely different things. So, no, there is no contradiction.

The second welfare theorem essentially says that a system of transfers that results in an efficient allocation can be supported by a competitive outcome. The statement about replacing income taxes with a consumption tax will increase peoples' incentives to save. These are two completely different statements.

Your mistake is that you suppose that the second welfare theorem prescribes a system of taxation to achieve the hypothetical transfers. It doesn't. The second welfare theorem just supposes that these transfers can be made---nothing more. Importantly, the alleged social planner who makes the transfers has complete information about the agents in the model. When information about the agents is hidden from the social planner (information such as the skills and true abilities of each individual), such transfers become difficult and costly. That is, they create distortions. This was a point that was famously made by James Mirrlees in his seminal paper "An Exploration in the Theory of Optimum Income Taxation" (1971). The point of this paper is that there is a trade-off between equity and efficiency. Once we consider the fact that transfers are costly and create distortions, the question is, how do we create a tax system that minimizes the amount of distortions and still is able to make desired transfers of wealth. This serves as a motivation for why people might study the incentives created by an income tax or consumption tax, for example.


Perfectly competitive firms. Economies of scale My textbook says the following:

"Perfectly competitive markets only achieve productive efficiency if you assume that there are no economies of scale in the industry."

Why is this the case? And by "in the industry" does it mean external economies of scale?

Put simply, a competitive market is one with a larger number of smaller firms — but economies of scale are best exploited by a smaller number of larger firms.

This doesn’t mean that a competitive market can’t take advantage of economies of scale, just that such a market is only guaranteed to reach productive efficiency if the size of the firm isn’t a factor.


The value of money in a model economy. Has this been addressed? Consider the following question:

"Given a simplified economy which produce and trade some products A,B,C etc... with N agents and a total money supply of M. What will be the price of A, B, C as measured in units of that money?"

It seems to me that this is one of the most fundamental questions in economics and yet I am struggling to find any papers that address it. I guess there may be some agent based models out there which address it but I don't know what keywords or phrases to use in a search. Any ideas?

What you are asking about is "General Equilibrium with Money", and there is literature on the matter.

To get you started, check out

Duffie, D. (1990). Money in general equilibrium theory. in Friedman and Hahn (eds), Handbook of monetary economics, vol. 1, 81-100.

The chapter starts with the sentence

  This chapter provides conditions for the existence of general monetary equilibrium in a simple setting, and discusses the role of
  money in general equilibrium theory.


Sequential Bargaining Consider a sequential game with two players who use the following procedure to share two desirable, identical and indivisible objects. Player 1 proposes an allocation in stage 1 which player 2 either accepts or rejects. In the event of rejection, neither player receives either of the objects and in the event of acceptance the objects are allocated according to the proposal of player 1.

The subgame perfect Nash equilibrium of the game is where player 1 allocates both the objects to himself and player 2 receives nothing. Is there any Nash equilibrium of this game other this?

What is the difference for player 2 between accepting nothing or rejecting? None. Therefore, we may say that from the point of view of player 2, he is indifferent between accepting nothing or rejecting.

What is done, as in Rubinstein's (1982) Bargaining Game, is to assume that in indifference, player 2 accepts receiving nothing, which also has more implications when the game is extended to an extra round where players change positions in case of player 2 rejecting the proposal on the first round, or when is included an outside option.

EDIT:

Following the comment from denesp, I will complete my answer.

The previous reasoning is applied mainly to SPE. If we want to see if there is any other equilibrium, we can start by representing the game in normal form, such that, we can identify two equilibria.

Player A offers 2, and Player B accepts
Player A offers 0, and Player B accepts.

However, the last one is not SPE because precisely in Rubinstein's model (and in other bargaining models), it is assumed that if player B is faced with two options that have the same payoff (when involves accepting, and the other rejecting), then he accepts.


Does there exist a mathematical formalization of Hayek's ideas about information and the price system? Hayek's idea, roughly, is that individuals in aggregate have distributed knowledge about their own utility functions and about specific supply-conditions, that no single individual can oversee, and that markets can aggregate this information efficiently, while planning bureau's cannot.

Does there exist a mathematical formalization of Hayek's ideas about this?

Eric Maskin has a paper called “Friedrich von Hayek and Mechanism Design”, which tries to connect some of Hayek’s ideas to the (formal) theory of mechanism design. This paper, and the references therein, might interest you.


Why isn't there an "ideal value" for a given currency? If a currency can be worth too little (e.g. needing \$10,000,000,000 to buy a loaf of bread) and worth too much (e.g. being able to buy a loaf of bread for $0.00001), why isn't there an "ideal value" (a point, range, or a shifting set of priorities based on market conditions) that a currency could be fixed or drawn toward?

Many everyday transactions are still done in cash, requiring mental calculations from the transacting parties. Too large numbers and too small numbers create transaction costs that, at least for the extremes, cannot be ignored.  So conceivably, some "middle" point could be considered "ideal" as regards the efficiency of money as medium of transactions.

What forces may prevent such an "ideal value" for a currency from being established?

Because money is really just arbitrary numbers, and those numbers only really have any meaning in relation to the other numbers in the set. For example:

Average salary is \$50,000 and a entry-level car costs \$10,000. If the price drops by \$5,000, it's well known that sales will go up, ceteris paribus. But what we increased all prices and all wages by a factor of 100,000? Would that change people's behavior in an otherwise identical system? E.G. now the average salary is \$5,000,000,000 and the average car costs \$1,000,000,000 and the price drops 500,000,000, then what happens?

Well there is a fancy economic name for the theory that covers this (which I forget at the moment), but the answer is No, nothing would change. It's quite obvious that humans are very insensitive to absolute price and very, very sensitive to relative price. (There have been several economy-wide currency revaluations in other countries that serve as examples of this.) Adding or taking away a couple zeros at the end of the numbers doesn't change the really make a difference (provided you apply the change uniformly to all numbers everywhere).

The two major problems that hyperinflation a la Weimar Republic cause are 1.) governments are slow moving beasts and don't react quickly enough to start printing 1 billion dollar notes, etc. so you end up having to use a wheelbarrow full of cash for purchases (which in today's economy isn't as big of a concern since almost all money is just electrons in a computer somewhere,) and 2.) changing prices and adjusting to changing prices and adjusting your forward looking predictions about prices are all not costless actions. It takes time and energy to do any one of those three things. Because of that fact, it imposes a very real dead weight loss on the economy as a whole when prices increase or decrease very rapidly. The faster things move, the more costly it is to adjust and the more often you have to do so, so the DWL becomes exponentially greater than a more slowly-paced situation.


Perfect Competition and costs of production question The table below shows the marginal cost of production for a firm that is in an industry that is perfect competition. The market price of the product is $300. Use this information to answer the questions below.
$$quantity \quad marginal \ cost$$
$$1 \quad 270$$
$$2 \quad 210$$
$$3 \quad 180$$
$$4 \quad 240$$
$$5 \quad 285$$
$$6 \quad 315$$
$$7 \quad 345$$
$$8 \quad 375$$

Suppose the firm’s total fixed costs are $200.
a. What quantity does the firm produce in the short run to maximize profits?
b. What are the firm’s profits in the short run?
c. Does the firm stay in business in the long run? Why?

Welcome to Economics SE. We ask that when asking about homework questions that you show your work so we can see what you need help with. The site is not meant to be a free homework solving machine.

Remember that a firm's profit is revenue minus costs, where costs can be fixed (200) or variable (manipulation of the marginal costs).

$$\Pi(q) = pq - c_f - c(q)$$

where $p$ is price, $c_f$ is fixed costs, and $c(q)$ is the cost with respect to $q$. We don't necessarily need to expression the variable costs as a function in this case.

So using the information we have, we know that for producing one unit of the good, our profits are:

$$\Pi(1) = 300 \cdot 1 - 200 - 270 = -170$$

For two goods, we have

$$\Pi(2) = 300 \cdot 2 - 200 - (270+210) = -80$$

And so on...

So you do this calculation for all the quantities and you see which one maximizes profits. That will also allow you to answer part b. As for part c, you have to see if there is a way for profits to be above 0 for some quantity produced.


What amount of share buy-backs are legitimate? My understanding of a shareholder buy-backs is the definition given by Investopedia:

  A stock buyback, also known as a "share repurchase", is a company's
  buying back its shares from the marketplace. You can think of a
  buyback as a company investing in itself, or using its cash to buy its
  own shares. The idea is simple: because a company can't act as its own
  shareholder, repurchased shares are absorbed by the company, and the
  number of outstanding shares on the market is reduced.

In "The Entrepreneurial State: Debunking Public vs. Private Sector Myths" by Dr. Mazzucato claims that shareholder buy-backs are a sign of stagnant economy that prioritizes stock prices and executive pay over things benefiting society:

  We are witnessing an increased financialization of the business
  sector, with many companies spending more on share buybacks—to boost
  their stock prices, stock options, and executive pay—than on areas
  like human capital and R&D. As the work of Lazonick (2014) has shown,
  in the last decade, nearly $4 trillion has been spent on share
  buybacks by Fortune 500 companies.

I've reviewed the book if you're looking for more context to this quote.

Furthermore, Robert Reich believes they should be made illegal, as they were before 1982:

Stock buybacks are artificial efforts to interfere in the so-called “free market” to prop up stock prices. Because they create an artificial demand, they force stock prices above their natural level. With fewer shares in circulation, each remaining share is worth more.

  Buybacks don’t create more or better jobs. Money spent on buybacks
  isn’t invested in new equipment, or research and development, or
  factories, or wages. It doesn’t build a company. Buybacks don’t grow
  the American economy.

  So why are buybacks so popular with Corporate CEOs?

  Because a bigger and bigger portion of CEO pay has been in stocks and
  stock options, rather than cash. So when share prices go up,
  executives reap a bonanza. The value of their pay from previous years
  also rises – in what amounts to a retroactive (and off the books) pay
  increase on top of their already outrageous compensation.

Is there some amount of shareholder buy-backs that could be seen as the sign of a healthy economy? Wherein "healthy" means still benefiting society via redistribution of wealth to those participating in the labour that created the profits or increase of collective human knowledge. Alternatively, what regulation has been proposed to correct short-sighted buy-backs?

If the company has excess cash or excess liquidity that it cannot effectively deploy in capital investments, and it has reason to believe that the expected value of returns from purchasing shares will be higher than alternatives (such as buying equities or financial instruments), this would be legitimate, where "legitimate" can be defined as an increase in the long-term value for anyone holding shares in the company.

As for determining how often this is done "legitimately", as opposed to being done manipulatively in order for inside traders to profit from reducing the supply of shares and thus illegitimately driving up prices, that's a different question.

The question of whether it is "legitimate" in terms of share buybacks giving the executive board control out of proportion to shares held, this should be considered separately from the question of whether excess cash legitimizes the buyback from a long-term profit maximization perspective.

As for the question of whether excess cash that cannot be deployed effectively is a sign of a "healthy economy" or not, this need not be determined by the observation of excess cash of a specific corporation considering share buybacks. For example, one company may have excess cash that it cannot deploy effectively in a situation where it expects to earn higher returns from a share buyback than investing in alternative financial assets or securities, without this necessarily applying on an economy-wide basis (which would contradict a variety of theoretical assumptions about financial markets always efficiently reallocating capital to its most profitable use ... but that could happen.)

Relevant regulatory approaches to "illegitimate" share buybacks could fall under shareholder rights (other shareholders would be poorly represented by executives deciding on share buybacks to temporarily inflate share prices for their personal profit), or possibly in some cases insider trading. The regulatory framework, and its application, varies across jurisdictions.


Applying ensemble modelling to VAR models It is ok to apply ensemble modeling to VAR models? I mean, using several specifications of the VAR model instead of just one specification. So, for example, if you want to check that $x_{t-1},...,x_{t-p}$ granger causes $y_t$, you compare results across the different specifications. If you have 5 specifications and 4 of them say yes, and 1 says no, then we conclude that $x_{t-1},...,x_{t-p}$ granger cause $y_t$. If 4 of the specifications say no, and 1 says yes, we conclude that $x_{t-1},...,x_{t-p}$ do not granger cause $y_t$. This idea is from machine learning. Not sure if there is a paper out there that has used something similar.

Emsemble learning is beneficial in forecasting where statistical adequacy of any model in the ensemble is of limited importance. However, when models are used for inference, statistical adequacy is quite important. The correctnes of the assumed null distribution of the test statistic depends (to a larger or smaller degree) on the statistical adequacy of the model. Violations of some modelling assumptions may lead to distortions of the critical values and thus the test size, making inference unrealiable. This is a common problem known from the model averaging literature. Only statistically adequate models may be averaged when the goal is inference (rather than forecasting), as otherwise the inference is unreliable. My explanation is a bit simplistic, but I hope it conveys the main idea.


Question about the relationship between Weak Axiom and Slutsky Matrix We know that if a differentiable Walrasian demand function $x(p,w)$ satisfies Walras' law ($p^Tx=w$), homogeneity of degree zero ($x(\alpha p,\alpha w)=x(p,w)$), and the weak axiom of revealed preference, then at any $(p,w)$, the Slutsky matrix
\begin{equation}
S(p,w)=D_px(p,w)+D_wx(p,w)x(p,w)^T
\end{equation}
is negative semidefinite.

My question is: if $S(p,w)$ is negative semidefinite, then what can we say about the demand function $x(p,w)$? Can we conclude that $x(p,w)$ satisfies weak axiom?

It is almost true.

There are examples of demand that have a negative definite Slutsky matrix but fails the Weak Axiom.

However, if we ask that
$$v \cdot S(p,w) v <0 $$
whenever $v \not = \alpha p$ for any scalar $\alpha$ (i.e. $S$ is negative definite for all vectors except those proportional to price), then the Weak Axiom holds.


Uniform bounds on rate of merging for Bayesian learners Update. Cross posted at Cross Validated.

In a well-known paper, Blackwell & Dubins (1962) show that the posterior probabilities of two Bayesian agents, whose priors agree on events of measure $0$, will become arbitrarily close to each other under an increasing stream of information.

Mathematically, the result is as follows. Let $(\Omega, \mathcal{F}, \{\mathcal{F}_n\}, Q)$ be a filtered probability space with $\mathcal{F}_n \uparrow \mathcal{F}$. Let $P$ be a probability on $(\Omega, \mathcal{F})$ with $Q \ll P$. Then,
$$d(P^n, Q^n): = \sup_{A \in \mathcal{F}}|P(A \mid \mathcal{F}_n) - Q(A \mid \mathcal{F}_n)| \to 0 \text{ a.s. $Q$ as } n \to \infty.$$
We say that $P$ and $Q$ strongly merge.

In a more recent and also very influential paper, Kalai & Lehrer (1994) introduce the notion of weak merging. The definition is as above, except the $\sup$ is taken over finite horizon events; tail events are ignored:
$$w(P^n, Q^n) : = \sup_{A \in \mathcal{F}_{n+1}}|P(A \mid \mathcal{F}_n) - Q(A \mid \mathcal{F}_n)| \to 0 \text{ a.s. $Q$ as } n \to \infty.$$

For weak merging it is possible to find uniform bounds on the rate of convergence (Fudenberg & Levine, 1992; Sorin, 1999). I am wondering if there are any results in this direction for strong merging.

This paper by Acemoglu, Chernozhukov and Yildiz (2016) and the references therein may be of interest.

The results they derive are in a much more limited environment, but I think they still gesture toward the direction you're looking. Otherwise, their literature review should also prove useful.


Are tariffs ever levied on services and does MFN apply? I am interested in tariffs on services.

Are tariffs ever levied on services and do most-favoured nation rules apply as they do with goods?

GATS doesn't apply tariffs, but

  The GATS comes in two parts; firstly a framework of rules, principles and concepts which underlie
  obligations regarding measures affecting international trade in services; and, secondly, the specific
  negotiated commitments listed in countries' schedules for service sectors. The GATS covers four
  modes of supply: (1) cross-border supply; (2) supply through the movement of consumers to the
  location of the supplier; (3) supply though the establishment in a country of the commercial presence
  of legal entities from another country; and (4) the supply through natural persons of one country in the
  territory of another.

  The key articles in the GATS regarding the scheduling of commitments are Article XVI (on market
  access) and Article XVII (on national treatment). Market access as such is not defined in the GATS.
  Rather the approach of the GATS is to list six categories of measure which are prohibited unless
  specified in a countryís schedule (for each of the four modes delivery). The categories cover the
  following: (a) limitations on the number of service suppliers: (b) limitations on the value of service
  transactions or assets; (c) limitations on the number of service operations or on the quantity of service
  output: (d) limitations on the number of natural persons who may be employed; (e) limitations on the
  type legal entity through which a service is supplied; and (f) limitations on the permissible size of the
  participation of foreign capital either in terms of a maximum percentage limit on foreign shareholdings
  or in terms of the total value of an individual entityís or of aggregate foreign investment. National
  treatment is defined as treatment no less favourable than that accorded to like domestic services and
  service suppliers. Under Article XVII measures entailing deviations from national treatment are also to
  be included in a country's schedule.

Basically limitations (a)-(f) allow various forms of quotas for services from other countries.

As for MFN, GATS has its own version/definition of it. Basically:

  Article II of the GATS constitutes a general obligation which is, in principle, applicable across the board by all Members to all services sectors. Article II:1 of GATS states:

    "With respect to any measure covered by this Agreement, each Member shall accord immediately and unconditionally to services and service suppliers of any other Member treatment no less favourable than that it accords to like services and service suppliers of any other country".

And as with similar WTO agreements, there were exceptions allowed

  Article II:2 states:

    "A Member may maintain a measure inconsistent with paragraph 1 provided that such a measure is listed in, and meets the conditions of, the Annex on Article II Exemptions". The liberalizing content of the GATS depends to a large extent on the nature of these specific commitments.

  The annex makes it clear that no new exemptions can be granted, at least by this route: any future requests to give non-MFN treatment can only be met through the WTO waiver procedures. [...]

  While Article II applies across the board, Article XVI on market access and Article XVII on national treatment only apply to sectors included by a Member in its schedule of specific commitments.

  [... T]he "Scheduling of Initial Commitments in Trade in Services: Explanatory Note" states that "where an MFN exemption has been granted for a measure, a Member is free to deviate from its Article II obligations, but not from its Article XVI and XVII commitments. Therefore, in such cases, a Member may accord treatment more favourable than the minimum standard to some Members, as long as all Members receive at least the that minimum standard of market access and national treatment appearing in its schedule."

  It follows that an MFN exemption would give a Member who had made no commitments in a sector considerable freedom to discriminate. However, an MFN exemption which coexists with fully liberal market access and national treatment commitments would have little meaning - unless a Member wished to provide more favourable treatment to certain foreign suppliers than it does to its own.


How to write budget constraint if the agent sell the goods that buy Della’s utility is $U(G,R)$

G is the consumption of gasoline per week

R is the the consumption of other goods

Just help me to write down budget constraint.

I think that

$$R*1+(G-20)*10+20*1=1000+10*X$$

where

X is the amount of the gasoline that she sell on the black market.

Agent is a buyer of gasoline in the black market if his consumption of gasoline exceeds 20 i.e. $G > 20$, and he buys $(G-20)$ units from the black market in this case. So, his budget constraint is $R + 20 + 10(G-20)=100$

Likewise, agent is a seller of gasoline in the black market if his consumption of gasoline is below 20 i.e. $G \leq 20$. In this case, he sells $(20-G)$ units in the black market. So, his budget constraint is $R + 20=100 + 10(20-G)$

To conclude, his budget constraint (in all cases) is $R + 20 + 10(G-20)=100$.


Why does nation's total debt matter and not NIIP? Most people in the media talk about how the total debt level of a nation relative to its GDP is a measure of how "screwed" the country is. For example, see this: http://www.zerohedge.com/news/goldmans-sigma-x-hints-who-next-contagion-target

It seems to imply that UK is in bad shape but nobody seems to pay attention to it?
Japan also not looking good.

However, look at the Net International Investment Position (NIIP) of these countries:
UK: about 0% of GDP (equal amount of debt and assets)
Japan: >50% of GDP (Japan holds more assets than debt)

"A country's external debt includes both its government debt and private debt, and similarly its public and privately held (by its legal residents) external assets are also taken into account when calculating its NIIP"

If Japan needed to pay back its debt, couldn't it just sell some of the assets it holds and repay that way?

Is there something I'm missing here? Which is the right way to look at things?

NIIP includes (and in fact is predominantly) personal assets.

The government of Japan could not sell the assets of its citizens to satisfy its sovereign debt.


Are food stamps considered near cash? I was wondering if certain government coupons like food stamps can be considered a form of near cash. I recall my professor saying something like near transfer, but I am unsure if the terms "near transfer" and "near cash" are interchangeable.

It think it depends on:

the liquidity and value of government coupons: if you could sell them very easily for cash and with little discount, then there is little difference.
the accessibility/familiarity of the expenditure they are to represent: if they are food coupons to be exchanged in any major supermarket, then beneficiaries could simply using them for their monthly shopping. If they are to be used only in a super-healthy store, not so easily accessible, and valid only to buy children's food, maybe some of the coupons will not be used, or will be resold at a high discount. In that case, they are less like cash, since it is not very rational to destroy cash, or sell cash for less cash.

The two points are surely connected.


Sustainable growth in the long run Suppose that, in addition to physical capital and labour, an economy requires a fixed factor and a non-renewable resource to produce the final good. Prove that an economy with these input requirements may still be able to sustain growth in the long run.

What I think is that

Using the concept of resource intensity, we can examine how economic growth, population growth, and the growth of non-renewable resource use are related. Define y as GDP per capita, L as the size of the population, I as resource intensity, and R as resource consumption. The definition of resource intensity is:

$$I=R/yL$$
$$R=IyL$$

Now we can rewrite this equation in terms of growth rates

$$\hat{R} = \hat {I}+ \hat {y} + \hat {L}$$ .

This equation says, for example, that if output per capita is growing at 1% per year, population is growing at 1% per year, and resource intensity is constant, then total resource use will grow at 2% per year.
The equation can also be turned around and used to illustrate how resource limitations affect economic growth. For example, assume that the quantity of a nonrenewable resource available for use is constant, as in the case of a renewable resource that is al- ready being exploited at the maximum sustainable yield. This assumption implies that Rn = 0. The previous equation can then be rewritten as:
$$ \hat {y} =-\hat {I}— \hat {L}$$ .

In this form, the equation says that for output per capita to have a positive growth rate, resource intensity must be falling faster than population is growing.

For other nonrenewable resources, the ratio of the ultimately recoverable quantity of the resource to current use is higher, but the same result holds: Eventually, if use continues as its current rate, the re- source will run out. Indeed, a moment’s thought makes it obvious that no rate of depletion of a nonrenewable resource is sustainable indefinitely. In the case of renewable resources, it is possible to sustain use indefinitely, but only at a limited level. If GDP is growing and resource use is already at the maximum sustainable yield, then resource intensity (the quantity of resource used per unit of output) must fall over time.

These considerations would seem to suggest that our current level of resource consumption may indeed not be sustainable. The question is: Does that mean that our level of income is not sustainable?
The answer to this question is a definitive no, for two reasons. First, even though the resources that we use now exist in fixed supply, there are frequently possibilities for substituting different resources. Second, although resource deple- tion is a drag on growth, other factors, most notably technological progress, may be sufficient to overcome this drag and allow income to keep rising. We address these issues in turn.

I think so. What do you think about this question? And what is fixed factor? I cannot include it.

Thank you

As I said in the comments, without additional details proving anything is either impossible or trivial. Impossible if I do not make any assumptions, trivial if I am allowed to make any assumption I like.

Let me give you a hint on how to solve this in the framework how it is usually taught.

The assumptions made are:

There are decreasing returns to scale to any single production factor
There are constant returns to scale to all production factors together
Technology and the labour force grow at an exogenous rate (which may be 0)
The production function is cobb-douglas form

Then, denoting national income at time $t$ as $Y_{t}$, capital as $K_{t}$, labour as $L_{t}$, technology as $A_{t}$, a fixed factor such as land as $X$ (which is independent of time) and a non-renewable resource extraction as $E_{t}$ we can formulate the production function as:

$Y_{t}=A_{t}K_{t}^{\alpha}X^{\beta}E_{t}^{\eta}L_{t}^{1-(\alpha+\beta+\eta)}$

We can determine the income per capita (or more precisely the income per labourer) by dividing both sides by $L_{t}$. Using small letters to denote a variable in per labourer terms we get:

$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$

Note that whereas $X$ is independent of time, $x$ is not because the labour force grows over time so the amount of land per labourer changes over time. You can now rewrite this equation in terms of growth rates to determine how it is (im)possible to continue to grow income per capita even in the presence of a non-renewable and fixed production factor.


What is an economic bubble? How do one know whether an appreciation in the price of an asset is a bubble or not? For example, during the 2007-2008 crisis, there was a housing bubble, which is only mainly agreed upon when the bubble actually starts to burst. Beforehand, however, the rise in housing price was not commonly accepted as a bubble, especially during the period when the housing price was still rising before the crisis.

So how do we know whether an appreciation is a bubble or not?

Any suggestions of articles about the issue are appreciated.

This a highly disputed question! There is a new and interesting paper by Òscar Jordà, Moritz Schularick and Alan M. Taylor about "Leveraged bubbles". They study bubbles in housing and equity markets in 17 countries over the past 140 years.


Where can I get national debt data? I'm trying to find out about the history of South Africa's government debt. However, I couldn't manage to find suitable literature or websites showing the history. It is also very difficult to find data on the national treasury website of South Africa.
Does anybody have any hints or literature tips?

Several international organisations provide open access to macroeconomic data such as national debt - usually the World Bank, economic development organizations such as the OECD and the African Development Bank, as well as public institutions in the USA (data.gov) and in Europe (as an example the ECB data warehouse). The national central banks should also provide the data you need, but its ease of access might vary.

The World Bank Open Data has the data you need, both in LCU (local currency) or in % of GDP, but only up to 2000 for the specific case of South Africa.

The IMF also collects data for several countries, so check if the page for South Africa has the data you are looking for.


Why would option chains' IV differ, for the same stock and expiration? I believe that the right-hand side refers to options sold before CHK split its stock on Apr 15 2020 1 for 200. 1. But why do the IV differ so much, e.g. at the strike price of 3?

Why hasn't this difference in IV been arbitraged?

With illiquid stocks, the B/A spreads for the options are Holland tunnel wide.  For some it's maybe 50 cents, for others, over $4.  With such a wide range of price, where is the actual market?  What's fair value on which to base an IV calculation?
Most likely, the broker is basing IV on the midpoint.  However, the midpoint isn't an accurate representation of price.  For example, the $8 put is 4.00/7.80 with a midpoint of 5.90.  Suppose I offer to sell at 6.80?  That changes the midpoint to 5.40 and therefore lowers the IV.  It's all meaningless.
As for arbitraging the difference, how do you arbitrage when there's a spread of anywhere from 50 cents to over 4 dollars?  It's impossible.


Models for online markets with reputation system The only relevant model I'm aware of is Liu Qingmin(2011 R.E.S). Is there any other decent models dealing with the mechanism of online markets under reputation system, and perhaps linking to regulation?

I would suggest you start by looking at

  C. Dellarocas. "The Digitization of Word-of-Mouth: Promise and
  Challenges of Online Reputation Systems". Management Science 49
  (10), October 2003, 1407-1424.

for a review of relevant literature and

  Friedman & Resnick "The Social Cost of Cheap Pseudonyms". Journal of
  Economcs and Management Strategy. 10 (2), 173–199.

and

  "Sustainable Reputations with Rating Systems," 2011. Journal of
  Economic Theory, 146, 2, 479-503

for some models of online reputation systems that comes to mind. If I think of any others, I will update this answer.


Shifts in demand curves When plotting a demand curve for a good, lets say carrots with price on the y-axis and quantity demanded on the x-axis. A shift any determinant of demand other than price of carrots results in a shift of the curve. Let us say that determinant in question is income. National income increases and as a result demand for carrots increases (as carrots are a normal good). The shift for an increased demand is referred to as a 'shift to right'. However is this not a vertical shift up. A vertical shift up of the demand curve means that for the same price people will demand more and for the same quantity people will pay a higher price. Is 'a vertical shift up' and 'a shift to the right' just different ways of describing the same movement or does a 'a shift to the right' achieve something different that I have not understood.

Consider a curve with a negative slope (such as the demand function for a normal good).
If you draw a second curve with a negative slope that does not intersect this one then it can either be in the "right-up" or the "left-down" direction from the first.
When they talk about shifts they sometimes just mean this general direction where the new curve will be. Sometimes (as with taxes) there really is a shift.
Let quantity be the vertical axis. If $D(p) = m - \frac{1}{p}$ then an increase in income will shift the demand curve up. But if $D(p) = \frac{m}{p}$ then an increase in income causes something that is not really a shift but a rescaling of the curve. You might still call this a shift up, as by moving up from the original demand curve you will reach the new one.


Appropriate estimator for FDI "gravity" model I am trying to model Foreign Direct Investment (FDI) flows and am facing several issues;

There is the well-known condition that under heteroskedasticity log-linearized OLS models are biased and inconsistent. Santos-Silva and Tenreyro (2006) suggest using Poisson pseudo-maximum-likelihood (PPML) to overcome this, and subsequently PPML, negative binomial and Poisson modeling became commonplace in the literature, but...

... FDI flows are often negative (divestment), which cannot be accommodated by count data models or PPML. Negative flows are economically meaningful and therefore cannot just be dropped without loss of consistency. An alternative is to use stocks rather than flows, but this does not seem to fit very well with the theory of gravity equations which seem to me to inherently deal with flow concepts.

Can anyone suggest an appropriate estimator? I have heard the suggestion of multinomial logit (where dep.var. is share of global FDI flows) but this would also seem to falter in the case of a negative flow.

An alternative is to use stocks rather than flows, but this does not
  seem to fit very well with the theory of gravity equations which seem
  to me to inherently deal with flow concepts.

Using stocks rather than flows and avoiding the disinvestment issue is a good alternative. Gravity equations have been shown to do a good job fitting stocks of FDI (see the Handbook of International Economics, Gravity Equations: Workhorse, Toolkit, and Cookbook, page 149).

This paper considers a model in which FDI takes the form of acquisitions. Using the discrete choice framework in a way that resembles Eaton and Kortum (2002), the authors develop a gravity equation for FDI which fits the data well. This paper extends the model to greenfield investment by imagining that instead of bidding for assets, each corporation selects the best "investment project" across all host countries.


maximize profit of competitive firm in short-run Consider a competitive firm with the short-run cost function $C(q) = 20 + 6q +5q^2$. The firm faces a market price of $p$ for its output.

(1) Find the firm's Profit Maximization problem.

(2) Find F.O.C and S.O.C.

I'm uncertain how to write the maximization problem up. As I understand competitive firm in short-run, we simple need to maximize profits minus cost by choosing the optimal quantity $q$.

So my guess is that the answer is:

$\max_{q} p-C(q)$.

Assuming that is correct, I am then uncertain how to make sense of the S.O.C.

Is the cost given the marginal or cumulative cost? If it's the cumulative cost, then the profit is  pq-C(q): your price is money per item, so you have to multiply by the number of items to find the total gross revenue. The problem is then to maximize the profit, i.e. find $\max_{q} pq-C(q)$.

If C(q) is the marginal cost, then the marginal profit is marginal revenue minus marginal cost, i.e. marginal profit = p-C(q).

The FOC is that the firm should keep selling the product until selling more would cause their marginal profit to go negative. For a continuous function to go from positive to negative, it has to go through zero. Thus, the FOC is that their marginal profit be zero.

While the marginal profit going from positive to negative means that it's going through zero, going through zero doesn't necessarily mean that you're going from positive to negative; you still have to check which direction it's going. So the SOC is that the marginal profit be decreasing.

FOC: marginal profit = 0
SOC: marginal profit decreasing

If the marginal profit is decreasing, then when it hits zero, it's going from positive to negative.


Existence of maximum utility with two bads I am working with a consumption set $X = R_+^2$ and preferences that are complete, transitive, continuous and strongly monotonically decreasing. The economy is characterized by the presence of two economic bads. I need to show that "there always exist a solution to the consumer's utility maximization problem".

My point is that if I had a continuous utility function over a compact budget set, using Weierstrass theorem I have the certainty of the function reaching its extrema. The problem here, if I am right, is that the budget set is no longer bounded and thus neither compact.

Are there other results I can use to show the statement?

I assume, by a strongly monotonically decreasing preference $\succsim$, you mean: for all $x,y\in\mathbb R^2_+$,
\begin{equation}
x\succ y \text{ only if $x\le y$ and $x\ne y$}.
\end{equation}
Then it is clear that $(0,0)$ is the most preferred bundle in $\mathbb R^2_+$, i.e. $(0,0)\succ x$ for all $x\in\mathbb R^2$ such that $x\ne (0,0)$.

Therefore, any utility function representing $\succsim$ must also satisfy $u(0,0)\ge u(x)$ for all $x\in\mathbb R^2_+$. Hence,
\begin{equation}
(0,0)=\arg\max_{x} u(x)
\end{equation}


Do perfect complements have to be normal goods? If so, why? Two goods $x,y$ are perfect complements if they have the utility function
$$U(x,y) = \min \lbrace ax,by \rbrace $$
$$a,b \in \Bbb{Q}^+$$
My professor said $x,y$ have to be normal goods but didn't explain why well enough that I could understand.

My Question:

Are perfect complements always normal goods? If so, why?

A good is normal if its demand is increasing in income. So let $p_x$ and $p_y$ be the price of the goods with quantities $x$ and $y$ and let $m$ be income.

Suppose $ax>by$. Then $\min\{ax,by\}=by$. By slightly reducing $x$ by and spending the saved money on $y$, one gets a better bundle. For an optimal bundle, this cannot be.

Similarly, it cannot be optimal that $by>ax$. So in the optimal consumption bundle, it must be the case that $ax=by$. It is also not that hard to see that the consumer will spend all her income. So rewrite the condition as
$$y=\frac{a}{b}x$$ and plug it into the budget equation $$p_x x+p_y y=m$$ to get
$$p_x x+ p_y\frac{a}{b}x=m=x\Big(p_x+p_y\frac{a}{b}\Big).$$
Therefore, we get the demand function given by $$x(p_x,p_y,m)=\frac{m}{p_x+p_y\frac{a}{b}},$$ which is clearly increasing in $m$. Similarly, one shows that the other good is normal too.

Pedantic remark: A differentiable function can be increasing at every point without the derivative being strictly positve everywhere. The function given by $x\mapsto x^3$ has derivative $0$ at $0$ but is everywhere increasing.


Numerical Question on a subsidized monopoly Suppose a government agency has a monopoly in the provision of internet connections.
The marginal cost of providing internet connections is $ \frac12$, whereas the inverse demand
function is given by: $p = 1 - q $. The official charge per connection is set at 0; thus,
the state provides a subsidy of $\frac12
$
per connection. However, the state can only provide budgetary support for the supply of $0.4$ units, which it raises through taxes on consumers.

Bureaucrats in charge of sanctioning internet connections are in a position
to ask for bribes, and consumers are willing to pay them in order to get connections.
Bureaucrats cannot, however, increase supply beyond $0.4$ units.

(a) Find the equilibrium bribe rate per connection and the social surplus.

(b) Now suppose the government agency is privatized and the market is deregulated.
However, due large fixed costs of entry relative to demand, the privatized company
continues to maintain its monopoly. Find the new equilibrium price, bribe rate
and social surplus, specifying whether privatization increases or reduces them.

(c) Suppose now a technological innovation becomes available to the privatized monopoly,
which reduces its marginal cost of providing an internet connection to $c$, $0 < c < \frac12$.
Find the range of values of $c$ for which privatization increases consumer surplus.

I understand that the consumers will be willing to trade away all the consumer surplus as bribe to receive the connection. But each consumer has a different willingness to pay as demonstrated by their demand curve or MB. How do I find a single equilibrium bribe rate for the problem? Should I just take the $Total \ Benefits \ = trapezoid \ area = 0.4\frac{1 + (1-0.4)}2 $. And divide this area with total connection which is $0.4$ and thus get the per connection average bribe.

Also total taxes = total subsidy, and also consumer surplus will be just traded away as bribe. So isn't the social surplus the same overall all the time?

You probably want to assume that every consumer pays the same bribe. Also, the social surplus depends on the quantity consumed, so it might (spoiler alert: it will) change when the internet is privatized, and when the marginal cost changes.


Proving that the utility is concave Consider a household which solves the following problem:
$$v(k,r,w)=\underset{c,l\in B{(k,r,ω)}}{\ {max}} \{u(c,l)\}$$
where $u : R_+^2 \rightarrow R$ is a strictly concave, twice continuously differentiable, strictly increasing function in its two arguments: consumption, $c$, and leisure, $l$. The constraints the household must obey in selecting $c, l$ are summarized by $B$:
$$B(k, r, w) = {\{c, l : 0 ≤ c ≤ rk + w(1 − l), 0 ≤ l ≤ 1\}}$$
Here, $k, r, w > 0$ are numbers over which the household has no control.
Prove that $v$ is concave in $k$ and that the derivative of $v$ with respect
to $k$ exists for ‘interior $k$’. Display a formula for the derivative of $v$.
What I was thinking for solving this is by following Benveniste & Scheinkman theorem on differentiability $ω : D → R$ defined on the neighborhood $D$ of $x_0$, i.e.
$D ⊂ X$ and $x_0 ∈ int(D)$ such that: $ω(x) ≤ v(x)$ and $ω(x_0) = v(x_0)$.
And $ω(.)$ is concave and differentiable, then $v(.)$ is differentiable at $x_0$ and $v'(x_0) =ω'(x_0)$.
I'm guessing we have to substitute c with $rk + w(1 − l)$ in the utility function, but I'm confused a bit with leisure. Because after that, I think we should just get the envelope theorem or not?

Sharing my answer out there, correct me if I'm wrong.
$u(c,l)=u(rk+w(1-l), l)$
$U$ is strictly concave and differentiable. Let $max$ u attained at $(c^*, l^*)$ i.e $(k^*, l^*)$.
Then $v(k^*)=u(k^*)$ and $u(k) ≤ v(k)$
Then by Theorem 4.10 (Benveniste & Scheinkman), $v$ is differentiable at $k^*$.
$V_k^*=U_k(rk^* + w(1-l^*, l*)
      = U_c(rk^* + w(1-l^*, l*)$


Inflation in items of daily use I was reading a document released by our Central Bank and I came across this line:

  Finally, inflation in the items of daily use was much lower as
  compared to other items. The contribution of essential items of daily
  use was only 1.1 percentage points in the overall inflation of 4.5 %
  in FY15.

What does this mean in very simple terms? How do I explain this to school going children, for example?

This means items of daily use did not experience price increases as much as items that are not of daily use.

Let's say half expenditures are on items of daily use. Then an increase of 2.2 percentage points will make these contribute 1.1 percentage points in total inflation. To make numbers match, the remaining 3.4 percentage points of total inflation must then have been contributed by a 6.8 percentage point price increase in items not of daily use.


Does creating more jobs actually help the economy as a whole? So I recently came across an interesting point on why NASA isn't a waste of money, and it basically stated how NASA supports thousands of jobs, creates community events, and uses a part of its budget as the salary of some of the smartest people on the planet. But is this a huge impact?

Whenever a large company is created, it creates hundreds maybe even thousands of decent paying jobs. But does this only benefit the people who are working in that company? Or does it benefit the community as a whole (in terms of economic success)?

Generally, whenever people have more income, it doesn't only benefit those people with more income. Imagine you somehow received \$100 that fell from the sky (or through NASA job). Would you save the entire \$100 and never spend it? More likely, you would spend some fraction of it... let's say you spend \$60. This money you spend will go to a shopkeeper, who then keeps a fraction and spend some of that \$60 on something else, and so on. So this extra money benefits other people too, not just the person receiving the money.

If you're interested, you can read more about this here.


What's the definition of social welfare in a procurement auction? In single item forward-auctions, the social welfare is defined as:
$$\sum_{i=1}^{n} v_i x_i$$
where $v_i$ is buyer $i$'s valuation for the item, and $x_i$ is a binary variable indicating whether or not $i$ gets the item.
What is definition of social welfare in a reverse auction(procurement auction)?

You need to add $x_0 v_0$ to your social welfare, where $v_0$ is the value the seller assigns to keeping the good and $x_0=1$ in that case. Then, the efficient (social-welfare-maximizing) allocation is that the highest-value buyer gets the good if her value is above $v_0$ - otherwise it is efficient that the seller keeps the good.
Similarly, in a procurement auction, we have one buyer who values the good to be procured at $v_0$ and $n$ sellers who can produce the good at private cost $c_i$. Social welfare is then
$$\sum_{i=1}^n x_i (v_0 - c_i),$$
and, hence, it is efficient to let the lowest-cost firm produce the good given its cost is below $v_0$ - otherwise it is efficient that the good is not produced.
For sufficiently high values $v_0$, you can also define it as $\sum_{i=1}^n x_i (- c_i).$


First Order Condition for Profit Maximization in Gambling Industry I am working on a model of optimal payout percentages in the gambling industry.

Because the nominal price of a \$1 ticket is always \$1, we use an effective price strategy where Q = \$1 in won prizes. If a game pays out 50%, the effective price is \$2, since that is what would need to be spent to win an expected \$1 in prizes. Pretty simple, right?

Well, I ran into this footnote in some research, and can't figure out how they got to the First Order Condition for Profit Maximization from the first equation:

"Let $C(Q)$ represent operating costs as a function of quantity units, where one quantity unit is defined as one dollar in expected value of prizes.

The lottery agency's net profits are given by

$$N = PQ - Q - C(Q)$$

where $P$ is the price charged for a quantity unit.

The first-order condition for profit maximization can be written

$$-E_{PQ} = P(1 - C')/[P(1 -C')- 1] $$

If marginal operating costs are $6$ percent of sales and the payout rate is $50$ percent, we have $P = 2$ and $C' = .12$, implying that the price elasticity of demand at maximum profit is $-2.3$.

For an increase in the payout rate to increase profits, $E_{PQ}$ must exceed $2.3$ in absolute value."

-[Citation] Clotfelter, Charles T, and Philip J Cook. "On the Economics of State Lotteries." Journal of Economic Perspectives: 105-19.

In the FOC equation, $-E_{PQ}$ is the effective price elasticity of demand. That normally would be found by taking the derivative of $P$ with respect to $Q$ in the first equation.

How did they end up where they did? There has to be something I'm missing.

I am having trouble understanding how that particular First Order Condition was reached- whether it was a result of some some derivative process on the Net Revenue equation, or if it is simply an external condition being applied.

Thanks!

The expression in question is in footnote $11$ of the referenced article. Reading the paper, we see that the decision variable here is "the payout rate", which is the reciprocal of $P$. So equivalently, we can solve the maximization problem with respect to $P$ (and not w.r.t. $Q$). More over, "price elasticity of demand" involves the derivative of $Q$ with respect to $P$, and not the other way around:

$$E_{PQ} = \frac {dQ/dP}{Q}P $$

and we expect it to be negative (higher price means lower payout rate leading to less demand for the quantity measure here, i.e. less "demand for prizes").

We can write the maximization problem as
$$\max_{P}N = \max_{P}\left[P\cdot Q(P) - Q(P) - C(Q(P))\right]$$

The  first-order condition is

$$\frac{\partial N}{\partial P} = Q + P\cdot Q'  - Q' - C'\cdot Q' = 0 \tag{1}$$

Multiply throughout by $P/Q$:

$$Q\frac PQ + P\cdot Q'\frac PQ   - Q'\frac PQ  - C'\cdot Q'\frac PQ  = 0$$

$$\Rightarrow P +P\cdot E_{PQ} - E_{PQ} - C'\cdot E_{PQ}=0$$

$$\Rightarrow -E_{PQ} = \frac {P}{P-1-C'} \tag{2}$$

This makes sense. Plugging in the values presented in the reference, we have

$$-E_{PQ} = \frac {2}{2-1-.12} = \frac {2}{0.88} \approx  2.27 $$

which is very close to the value resulting from the equation presented by the authors. I haven't been able, by whatever algebraic manipulations I tried, to replicate their formula, but eq $(2)$ is correct in any case. If a reconciliation comes up, I will update.


How does Brouwer's fixed point theorem relate to Walrasian equilibrium? I am trying to understand Walrasian equilibrium and its connection to fixed points, especially how we can apply Brouwer’s fixed point theorem to the notion of Walrasian equilibrium. I understand the implications of Brouwer’s fixed point theorem, and that conversion of prices to the unit simplex is an admissible transformation such that we know, by the theorem, there must be at least one fixed point. I understand Walrasian equilibrium as a price and consumption vector such that utilities of all households are maximized s.t. budget constraints and markets clear.

What I do not understand is the link between the two. I struggle with the intuition as to how fixed points map onto equilibrium prices.

Please excuse me if I am asking a stupid / obvious question. I cannot find an explanation of the intuition of the link anywhere I have looked.

Thanks in advance for your help,

R.

In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed point is, in fact, a Walrasian equilibrium.

Depending on the exact construction of the proof, sometimes the fixed point will only give you the equilibrium price vector. However, this will be done so that the implied demands will then clear the market.

In other proofs, like this one, the fixed point will give you both the equilibrium price and allocation.


von-Neumann-Morgenstern v. Bernoulli Utility Function A great deal of time is spent distinguishing the big $U$ (von-Neumann-Morgenstern)v. small $u$ (Bernoulli Utility Function). The v.NM function maps from the space of lotteries to real number as it represents the preference defined on the lottery space while the Bernoulli is defined over sure amounts of money.

Why is this distinction so important in the theory of expected utility? Also, what does this distinction enable us to achieve that is important in the expected utility theory? What is the most intuitive way to understand the expected utility theory is constructed in this way?

Bernoulli utility represents preference over monetary outcomes. In a way, this is no different from the typical utility functions defined over consumption bundles.

vNM utility, in contrast, represents preference over lotteries of monetary outcomes. Thus, the argument of vNM utility is an object related to, but categorically distinct from, the object that is an argument of Bernoulli utility.

For example, Bernoulli utility function allows us to compare the utilities from having $\$5$ to having $\$7$, while vNM utility allows us to compare utilities from the lottery $(0.2\otimes\$5,0.8\otimes\$7)$ --- having $\$5$ with $20\%$ and $\$7$ with $80\%$ --- to the lottery $(0.6\otimes\$5,0.4\otimes\$7)$ --- having $\$5$ with $60\%$ and $\$7$ with $40\%$.

In this sense, the distinction between Bernoulli and vNM utility functions are necessary (as they are applied to different objects) rather than important (since they both represent some kind of preference in the end), as @denesp says in his comment.

Moreover, if we only consider degenerate lotteries, i.e. the probabilities are either $0$ or $1$, then vNM and Bernoulli utilities coincide.
That is, the Bernoulli utility of having $\$5$ is exactly the same as the vNM utility of having $\$5$ with $100\%$. Therefore, distinguishing Bernoulli from vNM utility functions enables us to examine the effects of uncertainty apart from the mere quantity of "stuff" (be it goods or money).

Lastly, defining utility over money also allows us to study people's attitudes towards risk.


Why does deflation cause banks to increase their interest rates? I have read that in a deflationary spiral, lower prices cause banks to increase their interest rates. Is this true and what is the logic behind it?

I suspect there is a small mistake in your notes.

Deflation does not cause banks to increase their interest rates. However it is true that a deflationary spiral (or plain deflation for that matter) causes real interest rates to increase. These are the interest rates that matter for the economy anyway, which is why in macroeconomics we often refer to the real interest rate even if it's not explicitly stated so.

The nominal interest rate, set by banks in part, is the interest rate in terms of money. It determines how much money we have to pay back for credit.
The real interest rate  is the rate in terms of goods and services. It determines how many goods and services we have to pay back for credit.

The real interest rate is defined as:

$ (1+r) = (1 + i) / (1 + \pi) $, which can be approximated by:
$r = i - \pi$, where r is the real interest rate, i is the nominal interest rate set by banks and $\pi$ is the inflation rate.

When we have deflation we get a negative $\pi$, which, as can be seen from the equations above, increases the real interest rate r. The smaller $\pi$ is, the bigger r gets. When $\pi$ is negative, even if we set $i$ to its lowest possible value, which is 0, the real interest paid on it will be positive. This means the amount paid back in goods and services is larger than the amount borrowed and increasingly so the bigger the deflation (smaller $\pi$) is.


Economists studying the eldery and intergenerational transfers Does anyone knows who are the leading economists in the field of elderly and intergenerational transfers?

I'd like to know what are the current topics they are working on.

As far as intergenerational transfers are concerned, I would suggest you start with Wojciech Kopczuk:

Kopczuk, Wojciech. "Taxation of Intergenerational Transfers and Wealth." Handbook of Public Economics 5 (2013): 329.


If the consumer is not a price taker, does she still set marginal rate of substitution equal to the price ratio? Consider some consumer who consumes good $x$ and $y$. The price of $x$ is fixed to be $k$. The price of $y$ is $c(y)$. The consumer's marginal rate of substitution is given by $\frac{-x}{y}$. If the consumer optimises and is a price taker, I know she should set the absolute value of this equal to the price ratio. However, her quantity demanded affects the price ratio, as her demand influences the price of $y$. So, does it still hold that she would set

$$\frac{x}{y}=\frac{k}{c(y)}$$

Or does this result fail since her quantity demanded has an affect on the price of $y$?

We have the utility function  $U(x,y) = Ax^ay^a$ and we want to solve

$$\max U = Ax^ay^a,\;\;\; s.t. xp_x + yp_y(y) = I$$

The Lagrangean is

$$\Lambda = Ax^ay^a + \lambda [I-xp_x - yp_y(y)]$$

and the first order conditions are

$$\frac {aU}{x} = \lambda p_x$$
$$\frac {aU}{y} =\lambda p_y + \lambda y\frac{\partial p_y}{\partial y}$$

It follows that at the optimum

$$xp_x = yp_y + y^2\frac{\partial p_y}{\partial y}$$

or that

$$\frac{x}{y} = \frac{p_y}{p_x} + y \frac{\partial p_y/\partial y}{p_x}$$

But the previous expression is more interesting because it can be viewed as a quadratic polynomial in $y$, and be solved for its roots