Ad-driven skills gap? In a Business Insider piece I read that:

Michael Madowitz, an economist at the Center for American Progress, says he is "unusually unsympathetic to the evergreen skills gap critique."
"First the Econ 101 'pay more for more if you want more' cuts against this well in a period of little wage growth," Madowitz said. "Second, there's fascinating/depressing empirical research showing a 'skills gap' occurs because employers add and remove qualifications for the same job postings depending on the labor market, ensuring there is always a skills gap."

(Emphasis mine.)
Unfortunately, the article has no further details on the latter research/issue to back up this statement that at least some of this "skill gap" is just employer advertisements vagaries, presumably regardless of what the job actually requires. And I'm curious if this is remotely quantifiable, i.e. how much of the "skills gap" in some sector (under study) turned out to be of this ad-based kind.

I don't buy the explanation of this being vagaries of advertisements, regardless of the job.

Instead, I see it as employers showing rational and predictable flexibility in response to a changing supply of labour.

At times of plentiful supply, they can reduce recruitment costs by being far more selective at the first stage - filtering by essential requirements in the job ad. Finding good staff can be an onerous process. All the filtering stages after writing the ad are labour-intensive, and the labour is often line-managers and upwards, so not cheap. If I know I can get a decent number of interview candidates even with very stringent essential requirements, I'll do that, and that will reduce the number of applications that I have to process. This reduces my costs without significantly impairing the outcome.

At times of restricted supply, it's more cost-effective to advertise for lower essential requirements. That way, you cast a wider net, get more applicants, and are more likely to find a suitable candidate, without overwhelming the staff who have to read through the applications and process them.

As an occasional recruiter, I've done all this myself.


Exchange Rate Determination Most introductory international economics textbooks use the Uncovered Interest Rate Parity condition for deriving bilateral exchange rates. In other words, it is solely interest rate differential brought about by monetary policy that cause exchange rates to fluctuate, as investors demand/supply more attractive currency, which then adjusts exchange rate till the return in both currencies are equalized.

The question is- how do trade in goods affect this phenomenon? For instance, say China increases money supply, effectively decreasing interest rate, and this causes flight away from China towards American dollars, causing the American dollar to appreciate. China then benefits by improving its Current Account position. However, as the Chinese exchange rate has depreciated, Chinese goods are now cheaper, so more people will demand Chinese goods. Would this not have an offsetting effect on the depcreciation as people now demand the Yuan more?

1st of all, regading your remark

  ... , it is solely interest rate differential brought about by monetary
  policy that cause exchange rates to fluctuate...

That's just because it these models the internet case is isolated (like the joke of economists needed to change a light bulb)

As to your question, its always a combination between interests/ exchange rates and trade:

Per your example:
When China lowers rates, effectively increasing money supply, the Yuan decreases and make exports more profitable
Once exports occur, dollars come back to China, and converted to Yuans, making the Yuan increase back.
To keep exports more profitable to China, a constant increase of money, lower interests rates, are needed. This, in turn, can create local price surges. Essentially the local economy subsidizes exports

What China does though, is not exactly that,  even if similar
China holds its foreign exchange rates low, and its interest higher. That sounds a bit contradicting, but its possible, if China's government absorbs dollars.
So on one hand, Chinese exports are cheap, and on the other hand, the Yuan is a strong local currency. Furthermore, due to yhe e, export accelerated growth, wages and wealth has increased...


How is the interest on fractional reserve money creation paid? In fractional reserve banking commercial banks create money when they make loans. When these loans are paid back the account is zeroed, the created money disappears, but the bank is still entitled interest. Where does the money to pay the interest come from? Does the central bank necessarily have to inflate the currency to pay it? Does money in circulation generally cover it?

Repayment of interest does require an expansion of the money supply, but not in a way that is inflationary.

Consider first the way that commercial banks make loans. The whole purpose of loans is to borrow against future output— so if you imagine a firm that wants to purchase a machine that will allow it to produce more stuff— and thus have greater future income— the firm may go to a commercial bank and ask to take out a loan. The commercial bank will price that loan at a premium to the risk-free interest rate that reflects the risk of the loan, so that in expectation (taking into account that some loans will be repaid only in part or not at all), banks making riskier loans will be repaid at a slightly higher rate than ones making safer loans.

So firms will be offered an interest rate, and if that rate is less than their expected return on their investment, they'll make the investment. Notably, in this case, they're only borrowing money and investing because their future output will be higher. So when all of this works out (that is, when there isn't a credit bubble in which people are borrowing against future output that won't materialize), future output $Y$ (i.e., GDP) will be higher by an amount that is  the amount of additional money $M$ required to repay the interest on the loans.

$$ \% \Delta Y \geq \% \Delta M $$

Now if you recall the relationship between the money stock $M$, output $Y$,  the price level $P$, and money velocity $V$:

$$ PY=MV \rightarrow P=\frac{MV}{Y} $$

It's easy to see that if $V$ is constant, then:

$$ \% \Delta P = \frac{\% \Delta M}{\% \Delta Y}$$

Which, when combined with the observation that $ \% \Delta Y \geq \% \Delta M $, implies that an expansion of the money supply that is merely sufficient to allow for repayment of interest would actually be deflationary. This should make perfect sense: in expectation, output must increase by more than the amount captured by banks in the form of interest, otherwise firms in the real economy would never bother investing in fixed capital. So the money supply will in all likelihood increase by an amount strictly greater than the level required to repay interest, yet without any increase in inflation.

Loans to consumers are similar, so I won't give them a full treatment. Consumers are borrowing against their own future output, and foreclosures are generally priced in to the interest rate on a loan.

You'll note that I left out the case where in the aggregate, future output is less than that required to repay outstanding loans. This is intentional, as it's worth treating as a separate question.


Preference relation value representation without uncertainity Let $\succeq $ be a weak order in $\mathbb{R}^n$. If

$ x\geq y \Rightarrow x\succeq y$
for any $x\succ y\succ z$ there exists a unique $\lambda \in (0,1)$ such that $y\sim \lambda x+(1-\lambda)z$,

then there exists a value representation of $\succeq$.

Could anyone give me an example where the Value representation theorem(with out uncertainity) in the $n$ dimensional Euclidean space can be applied and a counter example that cannot be applied?

First, consider the preference relation $\succeq$ defined by
\begin{equation*}
x \succeq y \text{ if } x_1 + \cdots + x_n \geq y_1+\cdots+y_n
\end{equation*}

This preference relation satisfies your assumptions. Indeed, suppose that $x \succ y \succ z$. You can check that $y \sim \lambda x + (1-\lambda) z$ for a unique $\lambda$ defined by
\begin{equation*}
\lambda = \dfrac{y_1+\cdots+y_n-(z_1+\cdots+z_n)}{x_1+\cdots+x_n-(z_1+\cdots+z_n)}
\end{equation*}

And these preferences obviously admit the utility representation $u(x)=x_1+\cdots+x_n$.

Now, consider the lexicographic preference relation $\succeq$ defined (I take $n=2$ for simplicity) by
\begin{equation*}
x \succ y \text{ if } (x_1>y_1) \text{ or } (x_1=y_1,x_2>y_2)
\end{equation*}
Lexicographic preferences have two important properties:

$x \sim y$ if and only if $x=y$
they have no utility representation

If $x=(1,1)$, $y=(0,1)$ and $z=(0,0)$, we have $x \succ y \succ z$. In addition, any $\lambda>0$ is such that $\lambda x + (1-\lambda) z \succ y$. Thus assumption 2. is not valid, and the conclusion of the theorem (existence of a utility representation) is not valid either.


Productivity and wages in a task-based model In a task-based model like the one developed in Acemoglu&Autor 2011 with two types of worker, can the low type have a higher wage with respect to the high type?

More specifically: if the productivity of the "high type" is greater for each task in the economy, can we say that, regardless of the aggregate supply of labor, the equilibrium relationship is $w_h>w_l$?

Thank you in advance for any help!

I guess you are referring to a situation in which abundand supply of high-skilled workers decrease productivity such that individual $w_h < w_l$
Considering Equation (5) in Acemoglu & Autor, $H$ and $L$ capture supply of both groups of workers. If $\frac{H}{L}$ becomes sufficiently high, I don't see why the relation between both wages should not become lower than 1.


Why obligation acceleration triggers Credit-Default Swaps (CDS)? ISDA Credit Event definitions state that CDS can be triggered in an event of Obligation Acceleration. What does that mean? If for example, a country pays a bond 1 year prior to maturity are CDS triggered?

There is an example of its use at definedterm.com from a Cemex 2006 Note saying

  "Obligation Acceleration" means one or more Obligations in an aggregate amount of not less that the Default Requirement have become due and payable before they would otherwise have been due and payable as a result of, or on the basis of, the occurrence of a default, event of default or other similar condition or event (however described), other than a failure to make any required payment, in respect of a Reference Entity under one or more Obligations.

What this seems to mean (IANAL so this is not legal advice) is that a default or similar event (perhaps of a coupon) may lead to some or all of the amount borrowed becoming repayable earlier than originally planned, i.e. accelerating the obligation


Optimal Pricing with Advertising Below are three different demand curves (i) - (iii), which depend on advertising (A).
(i) Q(P,A) = A $\times$ ($\alpha$ - $\beta$P), where $\alpha$, $\beta$ > 0
(ii) Q(P, A) = $\alpha$ + A - $\beta$P
(iii) Q(P,A) = $\alpha$ - $\beta$(B - A)P
For each of the cases, what happens to the optimal price as advertising increases? How does your answer depend on the behavior of the cost function C(Q)?

Assuming constant marginal cost of $c$ and that the firm is a monopoly:
$$\max (\pi(\text{A}, \text{P})) = (\text{P} - c)\text{Q} - \text{A}$$
WRT (i),
$$\max (\pi(\text{A}, \text{P})) = \text{A}\alpha\text{P} - \text{A}\beta\text{P}^2 - \text{A}\alpha c - \text{A}\beta\text{P}c - \text{A}$$
For that, we calculate
$$\frac{d\pi}{d\text{P}} = \text{A}\alpha - 2\text{A}\beta\text{P} - \text{A}\beta c = 0$$
and
$$\frac{d\pi}{d\text{A}} = \alpha\text{P} - \beta\text{P}^2 - \alpha c - \beta \text{P}c - 1 = 0$$
We get to
$$\frac{e\text{A}}{-e\text{P}} = \frac{\alpha\text{P} - \beta\text{P}^2 - \alpha c - \beta\text{P}c - 1}{-\text{A}\alpha + 2\text{A}\beta\text{P} + \text{A}\beta c}$$
Any continued guidance would be greatly appreciated

Given how the problem is set up, I think they want you to take A as given when you choose price.  If that is the case, then you can find the comparative static $\partial P / \partial A$

We can think of p in the Foc as being a function of all the other parameters: $A\alpha-2A\beta P(A,\alpha,\beta,c)-A\beta c=0$

Now differentiate this wrt A: $\alpha-2\beta P(A,\alpha,\beta,c) -2A\beta (\partial P / \partial A)-\beta c=0$

Now you can solve for $\partial P / \partial A$


Can Eurozone countries issue their own bonds? Can Eurozone countries issue their own bonds, or is this a delegated competency?

If not, how do such countries raise funds for big uncosted spending such as infrastructure projects?

Issuing bond is indeed the competence of the individual states.

For example, you can buy 10 year bonds both from Italy and Germany, both of which are Eurozone countries:
https://www.investing.com/rates-bonds/de-10y-vs-it-10y


Profit maximization under uncertainity I have a seller say S and I have a buyer say B. Buyer’s willing to pay is equal to x which is private information. But Seller believe that it falls in the range [0,x1]. Seller’s belief distribution is F and cdf of F is f which is positive.
Take it or leave it price offer is p which is offered by seller. Buyer accept it iff p is less then and equal to x. How can I write seller’s profit maximization.

How can I construct maximization problem. I cannot imagine it. But after construction I can solve it. Please give me a hint to construct it.

What I did is

$$\Pi_s=(1-F(x))q.r-pq$$
where r is income q is quantity.

Let $G$ be the cumulative distribution function for the buyers' willingness to pay. If there is no cost, expected profit is simply expected revenue. The revenue is quantity times price. If the unit of the good is actually sold, this is just the price. Now the probability that the good is sold at price $p$ is the probability that the willingness to pay is not smaller than $p$, which is $\big(1-G(p)\big)$. So in total, expected profit is simply

$$\big(1-G(p)\big)p.$$

Two things are implicit in the problem already: First, the willingness to pay already includes all relevant aspects of the buyer's income. Moreover, unless we have the willingness to pay for each amount of the commodity, it is likely that the expected quantitty is a unit quantity of $1$.


Diminishing Nature of Utility Function From Convexity of Preference $\succsim$? Is diminishing nature of utility function in consumer theory stemming from the assumption of convexity on the preference relation $\succsim$?

My understanding is:

  $\succsim$ is convex. Then, $u(.)$ is quasiconcave. By the definition of quasiconcavity, the indifference sets are convex. Also, by another definition of quasiconcavity, the cross-section of the utility function versus a commodity shows a concave function indicating the diminishing nature of utility in any good $x$.

Diminishing marginal utility is a cardinal property. In other words: it is not invariant to arbitrary monotonic transformations. This indicates that it cannot be associated with any restriction on preferences (i.e., over a binary relation over consumption), since restrictions on preferences are by nature ordinal.

To see this, consider preferences defined over $\mathbb{R}_{++}$, perhaps thought of as monetary outcomes. If we take $x \succeq y$ if and only if $x \geq y$, we have a convex preference. Notice that both $U(x) = ln(x)$ and $U(x) = e^x$ represent our preference. Both are quasi-concave functions, although only $ln(x)$ is also concave.

If you are willing to entertain additional functional restrictions on $U$, then you can find axiomatic restrictions on $\succeq$ guaranteeing the concavity of $U$ over $\mathbb R$. The obvious example is the expected utility model, which assumes that $U$ is linear over the set of simple lotteries over $\mathbb R$, i.e., takes the form:
$$U(p) = \sum_{supp(p)} p_i u(x_i)$$
where $p$ is a lottery over $\mathbb R$. In this case, the risk aversion axiom---that the certainty equivalent of $p$ is less than the expected value of $p$---will ensure the concavity of $u$.

This is possible only because we have taken a stand on the cardinal normalization of $U$. In other words, any EU representation of $\succeq$ will have a concave $u$ but there still exist other convex non-expected utility representations.


Is exogeneity guaranteed for a lottery-generated instrumental variable? If using $z$ as an instrument for $x$, to study the effect $x$ has on $y$ and given that $z$ was indeed generated through a lottery, is it definite that exogeneity of $z$ will hold?
I have looked at this as two-stage OLS and I believe that the fact $z$ is lottery-generated can satisfy the exogeneity condition, i.e. $\text{corr}(z_i, \epsilon_i) = 0$. Is that true?

The simple answer is no.
Randomness does not imply exogeneity.
In my opinion, the following example from Deaton: "Randomization in the tropics, and the search for the elusive keys to economic development" is quite illuminating.
Assume that you want to estimate the effect of having a railway-station on poverty. We have data on the poverty rate $P_c$ and the presence of a station $R_c \in \{0,1\}$ for a large number of cities $c$. Assume that the true data generating process is given by:
$$
P_c = \alpha + \beta_c R_c + \varepsilon_c,
$$
Notice here that the slope $\beta_c$ is city specific, so the effect of a railroad is heterogeneous across cities.
The coefficient $\beta_c$ cannot be estimated, but you might wonder whether it is possible to estimate the mean of $\beta_c$ over all cities that decide to build a railroad.  $\overline{\beta} = \mathbb{E}(\beta_c|R_c = 1)$. Then:
$$
P_c = \alpha + \overline{\beta} R_c + (\varepsilon_c + (\beta_c - \overline{\beta})R_c) = \alpha + \overline{\beta} R_c + \upsilon_c,
$$
where $\upsilon_c = \varepsilon_c + (\beta_c - \overline{\beta})R_c$ is the new error. Although $\mathbb{E}(\upsilon_c) = 0$, you might be worried that:
$$
\mathbb{E}(R_c \upsilon_c) \ne 0.
$$
Indeed, we have:
$$
\mathbb{E}(R_c \upsilon_c) = \mathbb{E}(\varepsilon_c R_c = 1) + \mathbb{E}(\beta_c - \overline{\beta}|R_c = 1) \Pr(R_c = 1) = \mathbb{E}(\varepsilon_c R_c).
$$
This will only be zero if the decision to build a station is uncorrelated to anything that might otherwise influence the poverty rate in the city. As this is probably not satisfied, we need an instrument.
A random instrument
Assume that the government randomly allocates cities to development areas, captured by the instrument $Z_c$ (equal to 1 if city $c$ is a development area and zero otherwise).
If designations to development areas motivate the building of railways, we have that there is positive correlation between $Z_c$ and $R_c$. Also as $Z_c$ is random, we can assume that $\mathbb{E}(Z_c \varepsilon_c) = 0$. However:
$$
\mathbb{E}(Z_c \upsilon_c) = \mathbb{E}(\varepsilon_c Z_c) + \mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1)\Pr(R_c = 1, Z_c = 1).
$$
The first term on the right hand side is zero by the randomness of $Z_c$. As such, $Z_c$ will be uncorrelated to $\upsilon_c$ iff:
$$
\mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 1, R_c = 1) - \mathbb{E}(\beta_c|R_c = 1) = 0.
$$
This will be the case if the mean effect of building a station over all cities that are a development area equals the mean effect over all cities that build the railroad (whether they are a development area or not).
The last condition can be rewritten as:
$$
\begin{align*}
&\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|R_c = 1),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_C = 1, R_c = 1) \Pr(Z_c = 1) + \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0|R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)\Pr(Z_c = 0|R_c = 1),\\
\to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0, R_c = 1).
\end{align*}
$$
This will be the case if either:

$\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)$ which means that the mean effect among all cities who are a development area and decided to build a railroad should be equal to the mean effect among those who build a railroad and are not a development area. So if the behaviour (i.e. the decision to build a station) changes due to being designated as a development area, this condition will not be satisfied.
Or $Pr(Z_c = 0, R_c = 1) = 0$ which means that no city that is not a development area builds a station.

In short: randomness of the instrument makes sure that $\mathbb{E}(\varepsilon_c Z_c) = 0$ but does not guarantee that $\mathbb{E}(\upsilon_c Z_c) = 0$.


Vickrey Auction question I'm looking at this webpage here:
https://gilkalai.wordpress.com/2013/05/18/test-your-intuition-21-auctions/

I've generated the R code below to simulate the value for each Bidder 1000's times. I'm trying to figure out the optimal allocation strategy as well as the expected revenue.

Assuming a Vickrey Auction, it seems that the sample space is as follows:

Event         Probability     Revenue
3x300           .1^3            300
2x300, 1x100    .1^2 * .9       300
1x300, 2x100    .1 * .9^2       100
2x300           .9^3            100

This gives me an expected revenue of $105.6. This is inline with the simulation below.

However, I'm confused by what's exactly meant by "optimal allocation" strategy. Any suggestions?

generateBid <- function(x){
  epsilonValue <- runif(1, min = -1 , max =1)
  isNostalgic <- rbinom(1, size = 1, prob = .1)
  finalValue <- epsilonValue + isNostalgic*300 +(1-isNostalgic)*100
  return(finalValue)
}

cameraBid <- data.frame(bidderA = sapply(1:1000, generateBid), bidderB = sapply(1:1000, generateBid), bidderC = sapply(1:1000, generateBid))

VickreyAuctionProfit<- apply(cameraBid, 1, function(x){
  indexSecond <- rank(x)
  return(x[which(indexSecond ==2)])

})

mean(VickreyAuctionProfit)

Also, looking at the mean, it seems that the expected revenue of the auction should be less than 115?

Optimal allocation refers to whom the seller awards the item, in order to maximize revenue. In a Vickrey auction, it is a Nash equilibrium for everyone to bid their valuations. This is stronger than a Bayesian Nash equilibrium, which is the solution concept in Bayesian games (which include auctions). The Bayesian Nash equilibrium is the solution concept in which every player seeks to maximize their expected utilities, subject to their respective beliefs about the world. In the Vickrey auction, agents' beliefs about the world do not really matter.

The goal of the Vickrey auction is to illicit truthful responses from the agents. That is, they bid their precise valuations. The first-price auction is revenue-equivalent to the Vickrey auction; however, agents do not truthfully bid their valuations.

In particular, if we have a symmetric $n$-player auction where players' valuations are drawn from a probability distribution $F([0, \omega])$ with density $f$, we have the expected revenue for the seller as:

$$n \cdot \int_{0}^{\omega} (n-1) \cdot y \cdot f(y) \cdot F^{n-2}(y) \cdot (1-F(y)) dy$$

Combinatorially, we select the highest two bidders in $n(n-1)$ ways, since order matters. Suppose the second highest bid is $y$. This occurs with probability $f(y)$. The remaining $n-2$ players bid less than $y$ with probability $F^{n-2}(y)$. The highest bidder has valuation greater than $y$ with probability $1-F(y)$. Each of these selections are independent; so by rule of product, we multiply. As this is an expected value, we include $y$ in each product. By rule of sum, we add up over each possible value of $y$ (hence, the integral).


What are some non-von-Neumann-Morgenstern preferences used in economics? Von-Neumann Morgenstern preferences are preferences over lotteries that can be represented as the expectation of a "deterministic" utility function over outcomes.

"non-von-Neumann-Morgenstern" preferences are therefore preferences that cannot be described this way, so that we cannot apply expected utility maximization.

I know of at least one non-von-Neumann Morgenstern preference relation, namely so called "EZ preferences".

What are some other non-von-Neumann-Morgenstern preferences that are actually used in economics (or some other field)?

The von Neumann-Morgenstern (vNM) utility function takes the form
\begin{equation}
U(p,x)=\sum_{i=1}^np_iu(x_i)
\end{equation}
where $x=(x_1,\dots,x_n)$ with $x_i$ being the (monetary) payoff associated with outcome $i$ and $p=(p_1,\dots,p_n)$ with $p_i$ being the probability that $i$ occurs.

In behavioral economics, generalizations of the vNM utility usually happen in either (or both) of the following two channels:

Non-linear probability weighting: instead of using $p_i$'s, which can be considered objective probabilities, a model may use decision weights $w_i$'s which map the objective probability distribution into a new distribution.
Reference dependent utility function: instead of $u(x_i)$, which depends only on the payoff in outcome $i$, a model may consider a utility function $u(x_i,r)$ that depends on both $x_i$ and some reference payoff $r$.

Modifications made through either of these will give rise to a non-expected utility function, which is supposed to improve the model's descriptive accuracy of people's decision under risk.

Example of 1: Rank-Dependent Utility

Suppose $x_1<x_2<\cdots<x_n$, and calculate decision weights ($w_i$'s) as follows:
\begin{equation}
w_i=\pi\left(\sum_{j=i}^np_j\right)-\pi\left(\sum_{j=i+1}^np_j\right),
\end{equation}
where $\pi(\cdot)$ is called a probability weighting function (PWF).
A common PWF is the Tversky-Kahneman PWF:
\begin{equation}
\pi(t)=\frac{s^\gamma}{(s^\gamma+(1-s)^\gamma)^{1/\gamma}}.
\end{equation}
Then the rank-dependent utility with T-K PWF is
\begin{equation}
RDU(p,x;\gamma)=\sum_{i=1}^nw_iu(x_i).
\end{equation}
Note that $w_i=p_i$ if $\gamma=1$, so that vNM EU is a special case of RDU.

Example of 2: Disappointment Aversion

\begin{equation}
DAU(p,x;\overline U)=\sum_{i=1}^np_i[u(x_i)-D(u(x_i)-\overline U)],
\end{equation}
where $D(\cdot)$ is a function that captures how disappointment ($u(x_i)<\overline U$) or elation ($u(x_i)>\overline U$) affects an individual's evaluation of a prospect. Here, the reference utility level $\overline U$ can be something like the certainty equivalent level of utility. EU can also be derived as a special case from DAU by requiring that $D(\cdot)$ be a constant function.

Example of 1+2: Prospect Theory

Prospect theory (PT) incorporates both non-linear probability weighting and reference dependence. Again, assume $x_1<\cdots<x_n$. PT distinguishes between gains ($x_i>r$) and losses ($x_i<r$). For gains, decision weights are given by
\begin{equation}
w_i=\pi^+(p_i+\cdots+p_n)-\pi^+(p_{i+1}+\cdots+p_n),
\end{equation}
while for losses the decision weights are
\begin{equation}
w_i=\pi^-(p_i+\cdots+p_n)-\pi^-(p_{i+1}+\cdots+p_n),
\end{equation}
where $\pi^+(\cdot)$ and $\pi^-(\cdot)$ are both PWFs. For instance, one can have both PWFs be the Tversky-Kahneman form, but with different parameters. In addition, the utility function also takes a reference dependent form:
\begin{equation}
u(x_i,r)=
\begin{cases}
(x_i-r)^\alpha &\text{if }x_i\ge r\\
-\lambda(r-x_i)^\beta &\text{if }x_i<r
\end{cases}
\end{equation}
where $\alpha,\beta\in(0,1)$ are parameters of risk aversion (differentiated between gains and losses) and $\lambda>0$ is the coefficient of loss aversion.
Thus, the non-expected utility function under PT is
\begin{equation}
V(p,x;\text{parameters})=\sum_{i=1}^nw_iu(x_i,r)
\end{equation}


Ultra simple DSGE to implement I want to create a DSGE model in Python. I want to build it myself to help me understand it (not just use a library)
Can someone direct me to a very simple model which would be straightforward to understand, and quick to implement?
(I hear there are some 3- or 4-equation models, but I'm wondering if there's a quick summary of the actual equations so I can get straight to coding)
Thanks!

As mentioned by Michael quantecon has good resources. Here is an example of very simple DSGE model from quantecon.
I am not sure if you can get to as low as 3 equations but here is very simple example of real business cycle DSGE by Chad Fulton. I won't copy all the code from the site, you can just follow the link which has full code, but it is based on the following set of equations and it is quite simple
$$ \psi c_t = (1−α)z_t(k_t/n_t)^α \text{ static FOC}$$
$$ 1/c_t = \beta E_t\left(\frac{1}{c_{t+1}} \left[αz_{t+1}(k_{t+1}n_{t+1})^{α−1}+(1−δ)\right]\right) \text{Euler Equation}$$
$$ y_t = z_tk^α_tn^{1−α}_t \text{ Production Function}$$
$$ y_t = c_t + i_t \text{ Resource Constraint} $$
$$k_t =(1−δ)k_t+i_t \text{ Capital Evolution}$$
$$1 =l_t+n_t  \text{ Labor-leisure Tradeoff}$$
$$\ln z_t=ρ \ln z_{t−1}+ε_t\text{ Technology Shocks}$$
Where; $y_t$ is output, $c_t$ consumption, $i_t$ investment, $n_t$ labor, $l_t$ leisure, $k_t$ capital stock, $z_t$ technology and rest are parameters. $β$ is discount rate, $ψ$ marginal disutility of labor, $δ$ depreciation, $α$ capital share of output, $ρ$ technology shock.
You could probably simplify it a bit further but not much more.


Stability analysis and dimension of a dynamic control system I have an optimal control problem where I have two control and one state variable.

$$max\int_{0}^{\infty}\left(u\left(c\right)-P_{M}M\right)e^{-\rho t}dt\tag{1}$$

where $P_{M}$ is the unit price of carbon abatement activity ($CO_2$ abatement at atmosphere) and $M$ is the control variable, which is the level of abatement that we choose optimally.

The state variable is

$$\dot{S}=R\left(S\right)-c+\eta\left(M\right)\tag{2}$$

where $S$ is the environmental quality, $c$ is consumption, $R(S)$ is the regeneration of environment (like the production function in models with capital accumulation) and $\eta (M)$ is the carbon abatement function, which is supposed to be increasing and concave function. As it is logical, this function contributes to the increase of environmental quality.

I write the Hamiltonian ;

$$\mathcal{H}=u\left(c\right)-P_{M}M+\lambda\left[R\left(S\right)-c+\eta\left(M\right)\right]$$

The FOC are ;

$$u_{c}=\lambda\tag{3}$$

$$P_{M}=\lambda\eta_{M}\left(M\right)\tag{4}$$

$$\dot{\lambda}=\rho\lambda-\lambda\left(R_{S}\left(S\right)\right)\tag{5}$$

So, in this model, I have 2 control variables, consumption $c$ and abatement $M$ and a state variable $S$.

I have a doubt about writing the Jacobian Matrix, If I did not have the second control variable $M$, I would write a 2 dimension differential system of $\dot{c}$ and $\dot{S}$ but as I have a second control variable $M$, I am not sure if I can always describe the whole dynamics of the economy by two differential equations.

When I try to see the dynamics of the control variable $M$, with differentiating equation $(4)$ according to time, I have ;

$$\frac{\dot{\lambda}}{\lambda}+\frac{\eta_{MM}}{\eta_{M}}\dot{M}=0\tag{6}$$

According to the equation $(6)$ the dynamics of carbon abatement $M$ is represented by the dynamics of costate variable.

In this case, can I represent all dynamics describing this economy by only two differential equations which are $\dot{c}$ and $\dot{S}$ ?

Thanks in advance for suggestions and hints.

Differentiating $(3)$ with respect to time we get

$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$

Inserting into $(6)$ we obtain

$$\dot M = -\frac {\eta_M}{\eta_{MM}}\frac {u_{cc}}{u_c} \dot c$$

So the fixed point of $M$ will happen under the same conditions that the fixed point of $c$ will. Also, the optimal time-evolution of $M$ is a scaled linear value of the optimal evolution of $c$. So it appears all aspects are captured if you "ignore" the $M$ variable.

A bit more formally, since your two decision variables are linearly dependent it means that the $3 \times 3$ Jacobian of the system will be singular at the fixed point. You will get a double eigenvalue and a single one, and you should be able to show what stability properties hold (or what is required for the desired stability notion to hold), in such a situation. Due to the existence of the double root, this requires a bit of a different mathematical approach.


Price Derivation in a Production Function Here is a composite goods production function:

And here is the price ratio of Ys and Yg, derived from their marginal products:

Then the author normalized the price of final goods Y to 1 and somehow got the following relationship:

How is the last equation derived?

Thank you!

Rewrite the first equation as
$$Y^{\frac{\rho-1}{\rho}} = Y_s^{\frac{\rho-1}{\rho}} + Y_g^{\frac{\rho-1}{\rho}}. $$

Divide both sides by $Y_s^{\frac{\rho-1}{\rho}}$ to obtain

$$\left(\frac{Y}{Y_g}\right)^{\frac{\rho-1}{\rho}} = 1 + \left(\frac{Y_s}{Y_g}\right)^{\frac{\rho-1}{\rho}}.$$

Substitute the price-ratio and production-ratio relationships as in your second equation and make some algebraic changes, and you will get your last equation.  Note: I interpreted ``the price of $Y$ normalized to $1$'' as $\frac{p_g}{p_Y} = p_g= \left(\frac{Y}{Y_g}\right)^{\frac{1}{\rho}}$.


Why is a company allowed to change the number of shares it’s divided into after its IPO? I read that even after an IPO, a company can change the number of shares that it’s divided into based on a board vote. What’s to prevent them from arbitrarily increasing the number of shares to raise capital, thereby diluting the value of the outstanding shares?
I understand that a company might refrain from releasing all of their stock into the market initially so they can sell it later to raise capital, but it seems strange that they would be allowed to raise money by decreasing the value of an asset that shareholders have already bought.

If it were impossible to raise equity capital once a corporation has gone public, it may be impossible for it to expand if lenders have reached the prudential limits for lending to the corporation. This is highly constraining, and could lead to the company’s bankruptcy.
Does raising capital lower share prices? If we make the assumption that the share price is always equal to book value (this obviously is often violated in the real world), when it sells new shares, it gets cash equal to the number of shares. The result is that book value and market capitalisation increase by the same amount. However, this is somewhat short-sighted way of looking at it. The corporation may only be able to survive in the long term via issuing new equity.


Why doesn't Costco simply raise prices and nix the membership fee? I know little about the field of economics. I realize this question may be too rudimentary for this forum but I am sure someone here can explain this.

My question concerns Costco, which is a wholesale retailer that sells groceries, as well as electronics, clothes, and home goods. To even enter the store, one must have a Costco membership.

My question is simple: What purpose does the Costco membership fee serve? More specifically, why doesn't Costco get rid of the the whole "paid membership" idea, open its doors to anyone and everyone, and raise its prices to make up for the lost revenue?

Costco basically offers two different memberships, the standard membership with an annual fee of \$55 and the Executive membership costing \$110. The primary benefit of the Executive membership is that members receive 2% cash back on every Costco purchase (which is in addition to whatever cash back your credit card is earning for you). Thus, it only makes sense to purchase the Executive membership if one's yearly spending at Costco exceeds \$2,750 per year.

This is Costco's defense:

  "Costco's membership fee provides a means of covering part of our operating costs and overheads thereby reducing our prices on the products we sell. This way the more members we have, the lower our prices and the more you buy, the more you save."

If Costco did not require a membership fee, Costco would generate more revenue from a significant increase in customers. Costco could raise prices slightly on every item to compensate for the loss.

If Costco is indeed making the smart decision economically by requiring a membership fee, then why don't more stores adopt this model?

Because there is not a single successful model of actually doing business. Costco's model is pretty text-book level, but not economics textbook level, but rather marketing textbook level:

Differentiate ("In Costco we do things differently").
Make the customer feel special (hey, here is "members only"!).
Induce loyalty or at least "loyalty" ("now that I paid the executive fee I might as well buy from Costco to get part of it back")

Whether it will indeed be successful or not, depends on time, place, culture it operates in, as well as a few boring economic facts like, whether Costco's prices are indeed lower, etc.

Economics analyzes firm behavior at a much more abstract level, "losing sight" of all these different business strategies because Economics is ultimately interested in market outcomes (and then economy-wide outcomes), not individual-firm outcomes.


Finding Mixed Nash Equilibria in a $3\times 3$ Game I am tasked with finding all of the Nash Equilibria (pure or mixed) in the of the following game:

$$\begin{array}
\\&L&C&R\\
T&2,2&2,3&1,2\\
M&0,3&3,2&1,1\\
B&3,1&0,2&0,3\end{array}$$

I have already deduced that there are no pure strategy equilibria here by using the "underline the best response" method. In order to find mixed strategies, I know that one must be indifferent to all of his own options given a probability distribution from the other player, i.e. the expected utility from any of his choices is the same given what his opponent might play.

To do this, I started by finding the expected utility of each of the pure strategies by assigning probabilities to each of their actions. For Player $1$, $T$, $M$, and $B$ got probabilities of $x$, $y$, and $1-x-y$, respectively; I did a similar thing for Player $2$, with $p$, $q$, and $1-p-q$. Then, I calculated each players' utility given that his opponent picked a pure strategy; for Player $1$:

$$U_1((x,y,1-x-y),L)=3-x-3y$$
$$U_1((x,y,1-x-y), C)=2x+3y$$
$$U_1((x,y,1-x-y), R)=x+y$$

and for Player $2$:

$$U_2(T, (p,q,1-p-q))=q+2$$
$$U_2(M, (p,q,1-p-q))=2p+q+1$$
$$U_2(B, (p,q,1-p-q))=3-2p-q$$

Using these, I hoped to be able to see whether or not any of the pure strategies was strictly dominated given some other mixed strategy (e.g. $M$ is strictly dominated by some combination of $t$ and $B$). However, I'm not confident that this is the best way of actually doing this problem. Any suggestions as to where to go/what method to employ would be sincerely appreciated. Cheers.

Assume the pure strategy $s_i$ is strictly dominated by $s_i'$. This means that $s_i$ is not a best response to any of the opponent's pure strategies. For whatever strategy the other player is playing, the strategy $s_i'$ would yield a larger payoff. Even if $s_i'$ is a mixed strategy, for any strategy the opponent is playing you can select a pure component of $s_i'$ with a larger payoff than $s_i$ as the payoff of $s_i'$ is the average of the payoffs of its pure components.

A reversal of this is that if a pure strategy is a best response to any pure strategy of the opponent, it cannot be strictly dominated.

From the "underline the best response" method you will see that all the strategies of all the players are best response to at least one strategy of the opponent, hence none of them are strictly dominated.

Unfortunately, this still does not tell you much about the mixed equilibria. With different supports it is possible to have more than one.

An example:

In the game
$$
\begin{array}
\\&L&C&R\\
T&1,1&0,0&0,0\\
M&0,0&1,1&0,0\\
B&0,0&0,0&1,1
\end{array}
$$
the strategy profile where all strategies are mixed with probability $1/3$ is a mixed equilibrium, but the strategy profile where the Row player mixes $T$, $M$ while the Column player mixes $L$ and $C$ with probability $1/2$ is also a mixed equilibrium. (There are two other mixed equilibria and three pure ones.)

So there is no trivial method to find all the mixed equilibria, one usually has to use a trick or work through the different cases for all the supports.

Wilson's oddness theorem tells you that there are an odd number of equilibria, except in special "degenerate" cases.


How is the defintion of the mean preserving spread (MPS) not too general? The mean preserving spread is defined as follows:

Consider two lotteries g and h. Let $x_g$ und $x_h$ denote the corresponding random variables. Then h is a mean preserving spread (MPS) of g, if:
$x_h = x_g + z$ for some random variable z with $E[z|x_g] = 0\ \forall\ x_g$.
(This definition is supposed to be complete and exhaustive and I have found it in multiple sources)

An example for an MPS is the following:

Lottery g is replaced with a new lottery (lets call it h) by adding some noise, which makes the payoffs more extreme but preserves the mean.

My problem is: As far as I understand it, g (the old lottery) is also a mean preserving spread of h (the new lottery). At least the definition from the beginning allows for this: If I set z in a manner that the first outcome becomes larger and the second one smaller, I can still have $E[z|x_g] = 0\ \forall\ x_g$, but the total risk is now smaller. I have basically spread the lottery negatively. Just that MPS is now an entirely useless concept, since it does not allow for judgements with regards to the value of utility for an individual with concave utility functions (or any other shape for that matter.

I realized my mistake whilst writing this, so I will just share the answer as well.

Essentially my mistake was, that I assumed this aspect of the definition ($E[z|x_g] = 0\ \forall\ x_g$) to mean that the expected value of z has to be 0 given, that we are in the lottery that is given by the random variable $x_g$. This reading was however wrong. It actually means, that the expected value of z has to be 0, given that we are at a particular value of $x_g$. This has to be individually true for all values of $x_g$.

Thus, using this definition correctly a negative spreading is not possible anymore, the MPS of the original lottery is therefore always at least as risky as the original lottery. A risk averse decision maker will thus never want to choose an MPS.


How to Calculate the productivity multiplier? Given a Cobb Douglas

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

$ K_{t+1} = sY_t + (1-\delta) K_t$

How do we get the multiplier on productivity to be equal to $ \frac{1}{1-\alpha}$?
I understand that if productivity increases, output increases, thus we get more capital and thereby more output and so on. But I can't reach this multiplier.

My Attempt:

If have x increase in A,
then
$Y_t=(1+x)A (K_t^\alpha L_t^{1-\alpha})  $

That is an x increase in Y.

$ K_{t+1} = s(1+x)A (K_t^\alpha L_t^{1-\alpha})  + (1-\delta) K_t$

$Y_{t+1}=(1+x)A * K_{t+1}^\alpha *L_{t+1}^{1-\alpha} $
$Y_{t+1}=  (1+x)A * (s(1+x)A (K_t^\alpha L_t^{1-\alpha})  + (1-\delta) K_t)^\alpha  * L_{t+1}^{1-\alpha} $

I think the increase here should be $x * \alpha $ but I can't see it.
so that for a unit increase in productivity i.e x=1;
 $\Delta Y  =  1 + \alpha + \alpha^2 +... =\frac{1}{1-\alpha} $

Also in the steady state we  have,
 $ Y = A^\frac{1}{1-\alpha} * \frac{s}{\delta}^\frac{\alpha}{1-\alpha} * L$
taking logs we have that percentage change in A increases Y by $\frac{1}{1-\alpha}$.

Let $y=Y/L$ and $k=K/L$ be the per-worker levels of output and capital. Observe that $y=Ak^\alpha$.

Steady state is given by: $$k^*=sy^*+(1-\delta)k^*,$$ or $$k^*=sA(k^*)^\alpha+(1-\delta)k^*.$$

Doing the algebra: $$k^*=\left(\frac{sA}{\delta}\right)^{\frac{1}{1-\alpha}}.$$

And: $$y^*=A\left(\frac{sA}{\delta}\right)^{\frac{\alpha}{1-\alpha}}=A^{\frac{1}{1-\alpha}}\left(\frac{s}{\delta}\right)^{\frac{\alpha}{1-\alpha}}.$$

I am not sure what is meant by the "multiplier on productivity". I would interpret this term to be the answer to the question, "Given a small unit change in $A$, what is the resultant change in $y^*$?" That is, the following expression: $$\frac{\partial y^*}{\partial A}=\frac{1}{1-\alpha}A^{\frac{\alpha}{1-\alpha}}\left(\frac{s}{\delta}\right)^{\frac{\alpha}{1-\alpha}}.$$

However, clearly, this does not correspond to your desired answer. So I suspect what is really meant is the elasticity of $y^*$ with respect to $A$: $$\frac{\partial y^*}{\partial A}\div \frac{y^*}{A}=\frac{1}{1-\alpha}.$$


Predicting Shifts in Supply and Demand for Loanable Funds I know the answer is A.  but I need an explanation on why. I thought government savings would decrease since they have less tax revenue, shifting savings left and interest rates up. I could see how public savings could also increase due to this new taxation cut but consumers would spend part of the tax break, resulting in less of an increase in savings compared to decrease in government savings. What am I missing here? Thanks.

The best way to think about this in terms of an intro course is to break down “total savings” into “private savings” plus “public savings.”

Private savings: Y-T-C

Public savings: T-G

If you sum these, taxes drop out, so changes in the level of taxation do not affect the supply of loanable funds.

However, this tax change specifically affects the taxation of savings. It’s a decrease in the tax on interest income, which should spur households to save more and consume less. Thus, the result of the change is an increase in private savings.

Not to confuse the issue too much, as I don’t know how you’re being taught, but I will note that the notion that government “savings” contribute to loanable funds is disputed, as it involves an implicit assumption that government spending does not result in productive investments.

From a practical standpoint, it’s often better to think of government borrowing as contributing to the demand for loanable funds instead, though many intro courses do not teach it this way.

Most governments (outside of typically oil-producing nations that have sovereign wealth funds) are net debtors, so even if they run a surplus, that shows up primarily as a decrease in government bonds outstanding (i.e., lower demand for funds).

If you think about it this way, then you can let changes in the level of taxation be neutral with respect to the supply of loanable funds, but you’ll still see “crowding out” effects as a result of deficits, as the increased demand for funds results in a higher equilibrium interest rate.


Is a self-sustaining "loop" of economy a thing? Let's have a hypothetical situation with 3 companies. Company A produces a machine part to certain extreme specifications. Their only client for this product is company B. Company B uses these to build gearing systems for very accurate control of a tool. Company B sells these exclusively to Company C. C makes these into manufacturing machines, which are bought by A because they are the only tool that is capable of building the part for B to the required specification.

This is a very simple example, but I think anyone can see how this can be extended to include multiple companies (perhaps even competing ones) where the only reason they exist is because they keep eachother going.

Is there a term for this in economy? Does this happen in reality?

This is not a "sustainable loop" -you neglect the concept of value-added, created by the use of additional factors of production in each company.

Company $A$ has revenues $p_AQ_A$, with which it must cover all its expenses (including returns to capital).

These are only part of the costs of company $B$, because it also uses additional capital and labor to build the gearing system. So the revenues of company $B$, that have to cover expenses in order for $B$ to keep going, and given that it has purchased $Q_A$ units from $A$, is $p_BQ_B > p_AQ_A$.

These in turn, represent only part of the costs of company $C$. With the same reasoning, we end up with $p_CQ_C > p_BQ_B >p_AQ_A$. Not that the number of machines build by $C$ are the number of machines that $A$ demands in order to produce $Q_A$.

Under the described loop, company $A$ must now pay $p_CQ_C$ in order to do its job (plus any other costs for the labor and any additional capital it may be needed). But since $p_CQ_C > p_AQ_A$, this will mean that company $A$ will have losses, even before paying for the other factors of production it needs to hire. Hence, the loop is not sustainable.

A possible (misguided) objection ¨

  "But $p_CQ_C$ represents an investment for company $A$, it is not
  an operating expense"

Essentially this objection implicitly assumes that with $Q_C$ machines in hand, company $A$ will be able to produce more than $Q_A$ product units. But if this is the case, then after the first purchase, company $A$ won't demand more machines from $C$, hence $C$ won't demand additional gearing systems from $B$, hence $B$ won't demand any additional product units from $A$, and so any remaining productive capacity of the $Q_C$ machines will not be used.
If this objection does not assume the existence of this additional capacity, then the $Q_C$ machines will fully depreciate in order to produce $Q_A$ product units -and then, there may be initially classified as "investment", but at the end of the production cycle of $A$ they will be recorded as "depreciation charge", i.e. a bona fide running expense (which is what they are, since they fully depreciate).

I would suggest to study Input-Output models of an economy.


Present value heuristics problem No idea if this is the right stackexchange for this,  feel free to point me elsewhere!

I'm teaching business calculus and one of the problems the students have is to figure out the present value of a continuous income stream $\$$$A$ per year earning $r$ interest rate.  The integral comes to $$P.V=A\int_0^te^{-rz}dz=\frac A {-r}(e^{-rt}-1)$$
I've noticed as a heuristic that for relatively short values of t ($t<10$) and small values of r ($r<.04$) (the parameters the learning software uses),   the present value comes to be just a bit less than $A\cdot t$. (usually within about 10 percent)  Does anyone know an economics rational why this is the case,  and would it hold for larger values of $r$ or $t$?

Intuition

Well, when $r$ is zero, you are just getting value $A$ per time unit, so total present value will be $A \cdot t$.

The approximation

You are asking about the approximation
$$
\frac A {-r}(e^{-rt}-1) \approx A \cdot t.
$$
This is equivalent to
$$
\frac 1{rt}(1-e^{-rt}) \approx 1,
$$
which is true for small $rt$ values. You can see this by taking the limit of the left hand side and using L'Hospital's rule.
$$
\lim_{rt \to 0} \frac{\frac{\text{d} (1 - e^{-rt})}{\text{d} rt}}{\frac{\text{d} rt}{\text{d} rt}}
= \lim_{rt \to 0} \frac{e^{-rt}}{1} = 1.
$$

The approximation in another form

The approximation
$$
1-e^{-h} \approx h
$$
seems to be a variant of the more well known approximation
$$
\ln(1+h) \approx h
$$
for small values of $h$.

How good is this approximation

To get a sense of the error term for large values using your approximation, you can use Taylor-approximation for the function
$$
f(x) = \frac{1-e^{-x}}{x}
$$
starting from some very small $x$ value.


Change in Consumer Surplus with Two Price Changes: Is it Path Dependent? The diagrams below showing demand for goods X1 and X2 are adapted from Johansson P-O (1991) An Introduction to Modern Welfare Economics (p 43).  Demand for each good is assumed to be a function of its own price and the price of the other good.  At the initial price P2i of X2, demand for X1 is D1i, and at the final price P2f of X2, demand for X1 is D1f.  Similarly, demand for X2 increases from D2i to D2f when the price of X1 falls from P1i to P1f.  (Thus the goods seem to be complements, but that’s incidental).

Question: What is the change in total (Marshallian) consumer surplus resulting from the combination of the two price changes.

Johansson argues that it depends on the sequence of the price changes.  Suppose that the price of X1 changes first.  Considering the change in price of X1 with the price of X2 remaining at P2i, the relevant demand for X1 is D1i and we can infer that consumer surplus increases by area A.  Considering the change in price of X2, however, by assumption the price of X1 has already fallen to P1f, so the relevant demand is D2f and consumer surplus increases by area B + C.  In total, therefore consumer surplus increases by A + B + C.  But if the price of X2 changes first, then parallel reasoning leads to the conclusion that total consumer surplus increases by A + B + D.

It seems to me that this should instead be analysed as follows:

The change in total consumer surplus is the difference between a)
the initial total consumer surplus and b) the final total consumer
surplus.
The sequence of the price changes has no bearing on either (a) or (b).
(a) is E + F and (b) is (A + D + E + G) + (B + C + F + H).
Hence the change in total consumer surplus is an increase equal to (b) minus (a) or A + D + G + B + C + H.

Am I missing something?

Your reasoning is correct (i.e. the book is wrong). First, let's follow the book's logic a little more carefully step by step, beginning with the case where p1 changes first:

a fall in p1 leads to a gain of A in market 1 and of H in market 2 (because D2 shifts).
a subsequent fall in p2 leads to a gain of B + C in market 2 (we are now using the new demand curve in market 2); moreover, it causes D1 to shift so the consumer gains G + D.
thus, the total gain is A + H +G + D + B + C.

If p2 changes first:

a fall in p2 leads to a gain of B in market 2 and of G in market 1 (because D1 shifts).
a subsequent fall in p1 leads to a gain of A + D in market 1 (we are now using the new demand curve in market 1); moreover, it causes D2 to shift so the consumer gains H + C.
thus, the total gain is A + H +G + D + B + C.

We see that the order doesn't matter.

Another quick way to verify that the order doesn't matter is to note that these demand curves are coming from a consumer who chooses the optimal demand vector for given prices. If the prices are the same then the solution to the maximisation problem will also be the same.


Game Theory: Continuity in equilibrium profits? Consider 2 agents $A_i$. $A1$ moves before agent $A2$. Each of their utility functions is continuous in each agents' decision $0<s_i\in \mathbb{R}$ and a parameter $x$. Additionally, each agent's utility function has a unique interior maximum in its own decision $s_i$.
The agents' have lexicographic prefeences (primarily needed so that the other agents can anticipate strategies): $A_1$'s maximize its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for $A_2$. $A_2$'s maximizes its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for $A_1$.
Statement one: This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.
Statement two: The agent's utility profiles are continuous in $x$
(Remember, each of their utilities was continuous in $x$.) My reasoning for statement two is that an incremental change in x implies that the ne NE will be very close to one of the NE before the change in $x$. And as these NE had the same utility profile, the change in $x$ will only have an incremental effect on the utility profiles.
Do you agree with the two statements and do you perhaps have a more solid argumentation (partcularly for the second).

Let me first try to answer Statement 2:

The agent's utility profiles are continuous in x

Let's try to flesh out the problem. Let $S_1$ and $S_2$ be compact, convex, strategy sets of the players (say subsets of $\mathbb{R}^n$) (for simplicity) . Let $u_1(s_1, s_2, x)$ and $u_2(s_1, s_2,x)$ be the continuous utility functions of the two players.
Let's solve for the SPNE. At stage 2, player 2 maximizes her utility given the action $s_1$ of player 1. This gives:
$$
v_2(s_1, x) = \max_{s_2 \in S_2} u_2(s_1, s_2, x)\\
s_2(s_1, x) = \arg\max_{s_2 \in S_2} u_2(s_1, s_2, x).
$$

The problem is well defined due to the theorem of the maximum.
The function $v_2(s_1, x)$ is continuous (Berge's maximisation theorem).
The best response $s_2(s_1, x)$ is only guaranteed to be an upper hemi-continuous, non-empty correspondence.
If $s_2(s_1, x)$ is single valued, then it is continuous. However single valuedness will depend on the functional form of $u_2(s_1, s_2, x)$ (e.g. strict quasi-concavity) and therefore does not follow from the assumptions so far.

Let $\tilde s_2(s_1, x)$ be a selection of $s_2(s_1, x)$. In stage 1, Player 1's optimisation problem is:
$$
v_1^\ast(x) = \max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).\\
s_1^\ast(x) = \arg\max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).
$$

If the selection $\tilde s_2(s_1, x)$ of Player 2 is the optimal choice from $s_2(s_1, x)$ for Player 1, then the objective function appears to be upper semi-continuous, (but not, in general, continuous). Due to upper semi-continuity of the objective function, the maximisation problem is well defined.
Although $v_1^\ast$ will be (I think) upper semi-continuous, it is not necessarily continuous. Anyway, the best response correspondence $s_1^\ast(x)$ does not need to be continuous.
If $s_2(s_1, x)$ was single valued, then it is continuous, which means that $v_1^\ast(x)$ is also continuous. If, in addition $s_1^\ast(x)$ is also single valued, then it will also be continuous.

Let $\tilde s_1(x)$ be a selection from the best response $s_1^\ast(x)$ from player 1. Then, we can define:
$$
v_2^\ast(x) = v_2(\tilde s_1(x), x),\\
s_2^\ast(x) = \tilde s_2(\tilde s_1(x), x).
$$

In general, neither $v_2^\ast(x)$ nor $s_2^\ast(x)$  need to be continuous.
If both $s_1^\ast(x)$ and $s_2(s_1, x)$ are single valued, then $v_2^\ast(x)$ and $s_2^\ast(x)$ will both be continuous.

It thus appears that Statement 2 holds if both $s_2(s_1, x)$ and $s_1^\ast(x)$ are single valued.

Single valuedness of $s_2(s_1, x)$ holds if (for example) $u_2(s_1, s_2, x)$ is strictly quasi-concave in $s_2$
In turn, single valuedness of $s_1^\ast(x)$ holds if $u_1(s_1, \tilde s_2(s_1, x), s_1)$ is strictly quasi-concave in $s_1$. Notice that strict quasi-concavity of $u_1(s_1, s_2(s_1, x), x)$ is a strong assumption as it puts shape restrictions on the best response $s_2(s_1, x)$ (and not only on the function $u_1$ itself).

I hope this answers Statement 2.
For statement 1:

This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.

As shown above, the existence of a SPNE can be shown using only continuity of $u_1$ and $u_2$ and compactness of the strategy sets, and under the assumption that $\tilde s_2(s_1, x)$ selects the best option for Player 1.
Different SPNE will in general not give the same utility. Assume that at stage 2, Player 2 is indifferent between more than one optimal strategy, so $s_2(s_1, x)$ is multivalued (for example, assume that $u_2(s_1, s_2, x)$ is the constant function, so Player 2 is indifferent between every strategy in $S_2$). If different selections $\tilde s_2(s_1, x)$ from $s_2(s_1, x)$, provide different utilities for player 1, then it is possible that $v_1^\ast$ takes different values depending on the choice of player 2 in stage 2 of the game. So different SPNE may give different utilities.


Why can the job offer have a negative relationship between salary and quantity of work? Excuse me the following question arose, is it possible that the job offer has a negative relationship between salary and the amount of work?

Sure it is! Look up Borjas, Labor Economics, in the Labor Supply chapter.
In the following graph you notice that the supply curve is upwards sloping until wage $W = 20$, but for any greater level of compensation, the worker does not need to work as hard anymore, and starts reducing their hours of labor.

For an example, you might want to imagine a young music performer. If their earnings from concerts are, say, $ W = \\\$800 $ a month, they might have to gig a lot. As their music becomes known, their earnings from concerts might exceed $ W=\\\$8000 $, at which point, they may decide to work less, thus, reducing their number of concerts.


Is the money market equilibrium interest rate the same as the IS-LM equilibrium rate? If IS-LM equilibrium interest rate is, let's say, 5%, does it also mean that the money demand and supply equilibrium interest rate would also be 5%?

Yes. The LM curve is a combination of all such points where money market is in equilibrium. You can see this from the picture below I took from Blanchard et al. Macroeconomics: a European perspective.

The IS-LM equilibrium occurs when IS-LM curve intersect as seen on picture below (again taken from Blanchard et al.)

No matter where IS-LM points intersect there will be a direct mapping from any point along LM curve (as shown in first picture) directly onto the money market. Hence any equilibrium IS-LM interest rate will be also money market equilibrium interest rate (as correctly pointed out by Giskard in comments these are the same interest rate). Hence, in the standard textbook exposition of the model these will be always same.


Utility function and homogenous of degree zero I've a utility function which is given by

($x_i$-$b_i$)$^{c_i}$ $\sqrt{x_2}$

. What values of b and c can I input to ensure Homogenous of degree zero in prices and wealth? I think c will be positive. However, I have never done these types of questions before. What are the steps that will guide me to solve questions like this?

Consider the utility maximisation problem:
$$
\max_{x_1, x_2} u(x_1, x_2) \text{ s.t. } p_1 x_1 + p_2 x_2 \le m.
$$
If we multiply all prices and income by the same number $t > 0$, we obtain the problem:
$$
\max_{x_1, x_2} u(x_1, x_2) \text{ s.t.} t p_1 x_1 + t p_2 x_2 \le t m
$$
However if we divide the left and right hand side of the constraint by $t$ we get the original problem back.
In other words, the two problems are identical. This also means that the two problems will also give exactly the same solution.
Summarizing, all demand functions are homogeneous of degree zero whatever the utility function because multiplying all prices and income by the same positive number does not change the problem: everything is $t$-times more expensive but you also have $t$-times more income.


What is the tolerable level for multicollinearity? and possible remedial measures for it? My data has issues of multicollinearity. Upon estimating, I found out that my data have high levels of VIF In however I was confused as to what level of VIF can I possibly ignore the issue of multicollinearity.

As I search the web, some says that it should be less than 2.5, others 10 or 14.

My VIF results is this:

varA: 18.74
varB: 11.14
varC: 10.03
varD: 5.99
varE: 2.86

What step should I probably do to address the issue? I have fixed sample size and I can't log transform my model since one of my variables ranges from a negative to positive number.

I cannot post this as a comment but researchers such as Farrar and Glauber (1976) have shown that if VIF ≥ 10, this indicates the presence of multicollinearity between explanatory variables. There is no black and white threshold for this within the literature, and it highly depends on your sources and on the person reviewing your work. Moreover, you have not made any mention of the estimator you are using, but as you may know, multicollinearity makes the OLS estimator is less precise. That being said, have you tried standardizing by subtracting the mean? More information on your model and or your own knowledge on this would be useful to get help, in my view.


Evidence that open source production processes increase efficiency and/or consumer surplus? Is there peer reviewed evidence that open source production processes increase efficiency and/or consumer surplus? It seems that the first theorem of welfare economics requires complete markets which requires that all actors have perfect information. Yet, producers zealously seek to keep their own production processes secret rather than public.

For example, Soda Co. has an unpatented secret recipe at time $t_0$ that sells in the market. Are there economists who argue that if Soda Co. must patent and publicly reveal the secret recipe at time $t_x$, efficiency and/or consumer surplus would increase in comparison to current economy?

If not, are there good economic reasons why economic regulation does not require all suppliers to publicly reveal secret recipes, i.e., all information about the production process that would allow other producers to compete in production of the identical product (whether that be Soda Co.'s management techniques, scheduling practices, shipping arrangements, leasing terms, model number of vats, speed of conveyor belts, actual secret recipe for a beverage, etc.)? That is to say, are there good economic reasons that secrets anywhere in the entire production process are not simply barriers to entry?

Maybe Drink Co. can produce and sell Soda Co.'s recipe for 0.01 less than Soda Co. after they clone Soda Co.'s land, labor, capital, entrepreneurship, and recipe due to one tweak? Maybe consumers will only pay 0.10 less after they possess perfect information on how Soda Co. produces the beverage?

There is a large economic literature on intellectual property rights. However, the issue seems far from settled on what even the optimal duration for patents are. Note that open source is even a step further than a 0 day patent duration. A strong case for your view would probably be found in Boldrin/Levine:
http://levine.sscnet.ucla.edu/general/intellectual/againstnew.htm

Here are several further starting points in the literature:

Scotchmer, Suzanne. "Standing on the shoulders of giants: cumulative research and the patent law." The Journal of Economic Perspectives (1991): 29-41.

Besen, Stanley M., and Leo J. Raskind. "An introduction to the law and economics of intellectual property." The Journal of Economic Perspectives (1991): 3-27.

Posner, Richard A. "Intellectual property: The law and economics approach." Journal of Economic Perspectives (2005): 57-73.


Why Need a Numeraire Good in Dynamic Analysis Often in macro pure exchange economy models (sequential trading), authors make the initial period good price $P_0=1$ asserting this set-up is to make the consumption at period $0$ the numeraire.

But I don't see any detailed explanation or justification for doing this.

Just to add some background story:
The usual set-up is that agents are endowed with some units of goods, say oranges. So the unit of their consumptions is oranges. $P_t$ often the price, at $t=0$, of an orange consumption promised for delivery in period $t$. Why make $P_0=1$?

Is there any reason not to make $P_0 = 1$?
In microeconomics (which your problem seems to be about, rather than macro) consumer's demand is homogeneous of degree zero in prices & income. That is if all parameters measured in money change by the same ratio (say they are all multiplied by 10) consumer decisions are unchanged. The root of this is that what usually matters is the price ratio between two goods. But
$$
\frac{p_1}{p_2} = \frac{r \cdot p_1}{r \cdot p_2}
$$
for all $r > 0$, so any proportional price change does not change the price ratio between two goods.
As a result of this, we are usually left with a degree of freedom when solving these models. To do away with it and simplify equations it is best to just arbitrarily select a good and appoint it numeraire.
In macroeconomics you have similar issues if you assume the neutrality of money.


Am i correct here? #greenTaxes #priceMechanism #inefficientTax #DeadweightLoss Now accoring to this article: https://www.telegraph.co.uk/news/2016/07/30/britain-banishes-plastic-bags-as-5p-tax-sees-usage-plummet-by-6/?fbclid=IwAR3avBXNeV1n3Emb8weHygZnCJdKo6xLIPWJB0J0gyHhXXh6FQf8MJKrznc

just by putting a 5p tax on plastic bags. The total number used plastic bags has fallen from approx 9 billion down to 1 billion.

Now based on that i wanted to see why this tax led to a distortion of the price mechanism, creating socially inefficient tax rates that also resulted in deadweight losses as seen in the uploaded figure. Is this correct done/understood?

Yes you are correct, provided of course that plastic bags have no external effects. Then that triangle is the deadweight loss.

However, green taxes are typically implemented for a reason and if plastic bags have an external effect the supply curve is not equal to the social marginal costs. The idea of the taxes is to bring the private marginal costs in line with the social marginal costs. If plastic bags do have an external effect than that triangle is not deadweight loss but part of a cost that society bears.


How to compute the Marshallian demand for this specific utility function $$u(x) = x_1^\alpha \cdot (x_2+x_3)^{1-\alpha}, \text{ with } \alpha \in(0,1)$$

I tried to set up the Lagrangian and it turns out $\lambda$ have no solution unless $p_1 = p_2$.

$$L = x_1^\alpha \cdot (x_2+x_3)^{1-\alpha} -\lambda(x_1 p_1 + x_2 p_2 + x_3 p_3 -1)$$

$$\frac{\partial L}{\partial x_2} = (1-\alpha)x_1^\alpha(x_2+x_3)^{-\alpha} -\lambda p_2 = 0$$

$$\frac{\partial L}{\partial x_3} = (1-\alpha)x_1^\alpha(x_2+x_3)^{-\alpha} -\lambda p_3 = 0$$

I noticed this function is similar to C-D utility function. So I'm wondering if I can combine $x_{3}$ and $x_{2}$ as one good and then apply C-S utility function. But this method seems unreliable to me.

Any one can help?

It means that the level surfaces of $u$ don't touch the surface $x_1p_1+x_2p_2+x_3p_3=1$ unless $p_2=p_3$. But, since $x_i\ge0$, it follows that the admissible domain of $x_i$ is compact and the maximum is still achieved somewhere. Namely, in some boundary point of the  admissible domain $-$ for $x_2=0$ or $x_3=0$.


A working model of Supply and Demand I've just started to read an excellent textbook "Macroeconomics" by Gregory Mankiw. In the beginning he explains how economists think:

  Young children learn much about the world around them by playing with toy
  versions of real objects. For instance, they often put together models of cars, trains, or planes. These models are far from realistic, but the model-builder learns a lot from them nonetheless. The model illustrates the essence of the real object it is designed to resemble. (In addition, for many children, building models is fun.)

  Economists also use models to understand the world, but an economist’s
  model is more likely to be made of symbols and equations than plastic and glue.

Then he explains what endogenous and exogenous variables are and

  To make these ideas more concrete, let’s review the most celebrated of all
  economic models—the model of supply and demand

The above is obviously a toy model of Supply and Demand. But since nobody will start building a new car having only a toy model of it, I suppose that there must be some empirical research showing how Supply and Demand curves behave in real life and how equilibrium is reached which are used by economists for decision-making etc.

Is there any article or probably textbook which contain an example of working model (i.e. an opposite of the toy model) of Supply and Demand?  I'd like to grasp the difference between toy and working model of Supply and Demand.

Industrial Organization is the field concerned with estimating supply and demand relationships in individual product markets. You can see for example Einav and Levin for a description of what empirical IO people are up to. The specific models are usually much more complicated than what you learn in Mankiw, tailored to specific markets, and very strong assumptions are required to estimate them. For recent IO papers, you can see for example the NBER working paper series.


why the currency crisis must occur at time t0 when the pegged rate equals the shadow exchange rate Explain why the currency crisis must occur at time t0 when the pegged rate equals the shadow exchange rate and not before or after t0.

Suppose the crisis were to occur at a date before and after t0 and examine the incentives of an individual investor. Is it in his or her interest to also sell the currency on that date?

I assume you are referring to first generation currency crisis models, a la Krugman.

Before t0: Here there is of course no benefit to attacking and causing a currency crisis (by definition). That is, before the pegged currency is overpriced (i.e. it is not overpriced) there is no profit from attacking/short selling the currency as a drop in the value of the currency is not expected and therefore short selling is not profitable. So before it is profitable to cause a crisis there can be no crisis.

After t0: It would be profitable to cause a crisis and short sell a currency as long as it has not happened yet. However perfectly rational agents would react immediately, why wait? Like with investing in stocks, once you find a mispriced stock, you can profit anytime, but why wouldn't you profit immediately? Or lets say you find 100USD on the ground. You can profit from picking it up anytime after you find it, but why wait and risk that it could be too late later?

Hence it happens exactly at t0 as there is no reason to wait. An investor in currency would try to be faster than others to get arbitrage profits so others would try to be even faster than him and this goes on as much as possible. Its not possible to attack before (as its not possible to pick up 100USD found on the ground before its found there) so it happens as soon as it is possible, i.e. in t0.

In a second generation crisis a la Obstfeld it would take a coordination of several investors attacking at once to cause the crisis, so you don't have the effect of everyone trying to be faster than the other as they must "work together" to be successful and thus the time of the attack/crisis is not so straightforward to predict.


Two quasilinear utility functions Suppose there are two agents $\{A,B\}$ with the following utility functions: $U_A(x_A,y_A)=x_A+f(y_A)$ and $U_B(x_B,y_B)=x_B+g(y_B)$

$f$ and $g$ are strictly increasing and strictly concave. How do I prove that in the set of pareto efficient allocations $y_A=y_B$ ?

The statement is not true.
Let $x_A + x_B = 1$, $y_A + y_B = 1$.
Let $U_A(x_A,y_A) = x_A + \ln(y_A)$, $U_B(x_B,y_B) = x_B + \ln(y_B + 1)$.
$f \neq g$ are both strictly increasing & concave. For all $z \in [0,1]$ the distributions
$$
(x_A,y_A) = (z,1), \ (x_B,y_B) = (1-z,0).
$$
are Pareto-efficient. This follows from $MRS_A(x_A,y_A) \geq MRS_B(x_B,y_B)$.
 In these cases $y_A \neq y_B$.


Perfect Complements - Walrasian Equilibrium For a homework , I struggled to solve the following question but couldn't go further:

endowment of person 1 = (30,0)

endowment of person 2 = (0,20)

utility functions are such that:
\begin{eqnarray*}
    U (a_1,b_1) & = & \min(a_1,b_1) \\
\\
    U (a_2,b_2) & = & \min(4a_2,b_2).
\end{eqnarray*}
What I am doing is setting a1 equal to b1 and 4a2 equal to b2. After that I'm writing these;
$$
p_1 a_1 + p_2 b_1 = 30p_1 \mbox{ and } p_1a_2 + p_24a_2 = 20p_2
$$
Finally, by looking at feasibility condition, I'm writing down:
\begin{eqnarray*}
    \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\
\\
    \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20.
\end{eqnarray*}

Here, if I do some calculations, they result in $p_1 = p_2$ and accordingly $a_1=b_1=15, a_2=4$ and $b_2=16.$ But then the problem is that there is excess demand for good2 and there isn't a WE.

Alternatively, I am considering to have $p_2 = 0$ so that there exist a Walrasian Equilibrium.

BUT, I'm stuck at this point and lack the correct intuition for solving next steps. Or, I might be in a completely incorrect way. Please, explain me what will be done with this excess demand?

Thank you in advance.

If the prices $p_1$ and $p_2$ are positive than as you pointed out the equations
\begin{eqnarray*}
    \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\
\\
    \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20.
\end{eqnarray*}
hold. This is troublesome, because subtracting the first equation from the second yields
\begin{eqnarray*}
\frac{3 \cdot 20p_2}{p_1+4p_2} & = & -10
\end{eqnarray*}
which has no solution for positive prices. However if a good has zero price, than the demand equations are different as no one minds buying superflous goods as long as they are free. For $p_2 = 0$ you would have
\begin{eqnarray*}
    \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\
\\
    \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & \leq & 20.
\end{eqnarray*}
This is because $\frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2}$ is a minimum that the buyers need but they do not mind having more as good 2 is free. There is also a formulation of equilibrium, the 'value-equilibrium' which is that for good $i$
\begin{eqnarray*}
p_i \cdot D_i(p) = p_i \cdot S_i(p) \mbox{ and } D_i(p) \leq S_i(p)
\end{eqnarray*}
meaning that there can be oversupply of zero price goods.
(Think about air. Would you buy air from someone?)

Unfortunately this equation system would still imply
\begin{eqnarray*}
\frac{3 \cdot 20p_2}{p_1+4p_2} & \leq & -10
\end{eqnarray*}
which has no solution in non-negative prices. On the other hand, for $p_1 = 0$ you would have
\begin{eqnarray*}
    \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & \leq & 30 \\
\\
    \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20.
\end{eqnarray*}
which implies
\begin{eqnarray*}
\frac{3 \cdot 20p_2}{p_1+4p_2} & \geq & -10.
\end{eqnarray*}
As $p_1 = 0$ any positive $p_2$ is a solution for this.


Using the Slutsky equation Suppose we have utility:

$$U(x,y)=x^{0.5}y^{0.5}$$

Then Marshallian Demand for good $x$ is:

$$x(p_{x},p_{y},I)=\frac{0.5I}{p_{x}}$$

And Hicksian Demand for good $x$ is:

$$x^{c}(p_{x},p_{y},U)=p_{x}^{-0.5}p_{y}^{0.5}U$$

If $p_{x}=1$, $p_{y}=4$, $I=8$ and $U=2$:

Marshallian Demand for $x$ is 4 and Hicksian Demand for $x$ is 4.

Supposing that the price of good $x$ rises to $p_{x}=4$, how can the Slutsky equation show us the change due to substitution effect and income effect?

Substitution effect: $\frac{\partial x^c}{\partial p_{x}}=-0.25p_{x}^{-2}I$ according to the Slutsky equation. If we plug in $p_{x}=4$, the result is -0.125. How is that meaningful in any way? The Hicksian Demand should drop from 4 to 2 after this price increase but I don't see how the Slutsky equation shows us that.

In your context, Slutsky Equation says, after $p_x$ increases from 1 to 4, the following is true:

\begin{align*}
\text{total demand change in $x$} & \\
= \text{demand change in $x$, keeping $U$ fixed at 2} & \\
+ \text{demand change in $x$, accounting for the income change if we do not fix $U$ at 2.} &
\end{align*}

In the example, the substitution and income effects are of the same sign, so I use absolute values for the demand changes in the equation above. Thus the plus sign rather than the minus sign in Slutsky Equation.

The total demand decrease in $x$ is $3$, that is, $$x_0 - x_1 = 4 - 1 = 3.$$

The demand change in $x$, keeping $U$ fixed at 2 is: $$x^{c}_0 - x^{c}_1 = 4 - 2 = 2.$$

To fix $U$ at 2 under the new price $p_x = 4$, the consumer needs the new income to be $I_1 = 16$ instead of 8. With \$16, the consumer buys 2 units for both $x$ and $y$. If we do not fix $U$ at 2 when $p_x$ increases, which means we allow income to decrease from 16 to 8. The demand reduction in $x$ induced by this \$8 income decrease is: $$0.5 \times 16 \div 4 - 0.5 \times 8 \div 4 = 2 - 1 = 1.$$

Therefore we have $3 = 2 + 1$, as Slutsky Equation tells us.


Interpretation: Elasticitity of Substitution I have this production function:

$$P(x_1,x_2)=x_1+x_1*x_2$$

I am trying to find the elasticity of substitution, and I found this:

$$\sigma = -\frac{d \ln (\frac{x_2}{x_1})}{d \ln(\frac{x_1}{1+x_2})}$$

Then I have these conditions:

$x_1 >0$ and $x_2=0$

When putting this numbers in, I have ln (0) which is not possible, what is the interpretation of this elasticity of substitution?

Will the elasticity be zero or infinity?

The ratio of marginal products $\frac{\partial P}{\partial x_1}/\frac{\partial P}{\partial x_2}$ (the marginal rate of tech. substitution) is equal to $1/x_1+r$ where $r\equiv x_2/x_1$.

The derivative of the MRS wrt to $r$ ($\frac{d MRS}{d r}$) is equal to $1-\frac{g}{MRS-r}$, where $g\equiv \frac{dln(x_1(r))}{dr}$.

Note how $\frac{dln(MRS)}{dr}$ which is equivalent to $\frac{d MRS/d r}{MRS}$ is equal to $\frac{MRS-r-g}{MRS(MRS-r)}$. Also note how $\frac{dln(r)}{dr}=1/r$.

Rearranging the last two relations in order to acquire expressions for $dln(MRS)$ and $dln(r)$ and substituting into the definition of the elasticity of substitution yields $-\frac{MRS(MRS-r)}{(MRS-g-r)r}$.

Finally, note how $r=0$ when $x_2=0$. Also, $MRS=1/x_1$ when $x_2=0$. The behavior of σ depends of the value of $g\equiv \frac{dln(x_1(r))}{dr}$ at $x_2=0$. If the value of $g$ is defined and is finite and constant then $σ$ approaches $- \infty$.

Intuitively, the behavior of $g$ near $x_2=0$ depends on the shape of the isocurves. $g$ denotes the (proportionate) rate of change in $x_1$ as a result of an infinitesimal change in $r$. The following graph shows that for values of  $x_2$ near zero (notice the values on the vertical axis), the isocurves are really steep.

Very informally, in this setting, (really) small changes in $r$-which is equal to the ratio of $x_2/x_1$ (which is equivalent to the tangent of angle $θ$)- have almost no effect on $x_1$. For $Δθ$ approaching zero $Δx_1$ is also small and probably close to zero too. Equivalently and very loosely speaking, in this region $g$ should be close to zero too.


Explaining Stagflation High demand for goods drives up prices, and also encourages firms to hire more; and likewise high employment raises demand.
Then why stagflation?
According to above explanation,  it should not exist.

For a long time many people thought that stagnation was an impossible economic experience to occur as Keynes economic theory of the Philipps curve which portrayed macroeconomic policy as a trade-off between unemployment and inflation.   However during the 1960s in UK and in 1970s in the US stagnation occurred but was due to external situations which caused prices to increase. U.S. and other industrialized countries entered a period of slow economic growth occurring simultaneously with high rates of inflation.

https://www.investopedia.com/articles/economics/08/1970-stagflation.asp

It appears, which Neo-Keynesians are believing, that stagflation is caused by cost-push inflation (when some force or condition increases the cost of production). So external factors such as a shortage of natural resources or an act of war are the main contributors to this situation.


What does a "sell-off" mean if every stock transaction is balanced? If every stock transaction is "balanced" (1 sell = 1 buy), then what does a "sell-off" actually mean and how does it move the price of a stock?

Ex. If millions of people sell a stock, there's necessarily an equal millions of buys of that stock.

The Cambridge English dictionary has a definition here: link to definition

I will paraphrase their definition: a sale of equity shares that causes its price to fall.

As the question notes, that appears to make little sense, if for every buyer, there is a seller. If stock prices were negotiated by individual verbal contracts, the implication was that the seller was more morivated to sell than the buyer to buy, and so the price had to be adjusted lower.

However, the common usage of “sell off” in the financial press is different: it is a synonym for a stock price (or stock index) experiencing a price fall on a day versus the previous day. The implication is that all the sellers were motivated than buyers, and so the price fell, like in the verbal contract.

However, if the observed price on the exchange is lower, we only know two things:

For every buyer, there is a seller.
The price of the transactions on the current day is lower than the previous day.

There’s any number of ways the price can end up lower, and short of questioning everyone involved in trading that day, we have no idea what really happened.

I will now give an example how intuition about supply and demand can be misleading.

Imagine that Acme Inc. traded yesterday at \$10. The next day, market makers set the price around that price, with a bid-offer spread. There is no trading.

Then imagine that at 10 am, Acme announces a successful lawsuit against it by the plaintiff Wile E. Coyote. The market makers are shocked, and drop the mid of the bid/offer prices to \$5, and still there is no trading.

Meanwhile, retail stock traders who did not know about the lawsuit saw the new price, and rushed to buy. 100 customers buy 100 shares each for the market makers, who are happy to sell out of inventory.

From the persepctive of an outside observer, we saw 100 retail traders rush to buy shares - and the price fell by half.


Alternative measure for market capitalization, company "value" I wonder what the term "value" of a company, currency, etc could mean. The most obvious, to me, is market capitalization (related: Clarifying the meaning of market capitalization)

Is there also something like measuring how much money has been spent for the shares that are currently circulating.

E.g. there is an IPO for 1000 shares each at 1\$, all of them are sold. Then the price rises to 2\$ and 500 of the shares get traded for this value. Then the classical market cap would be 2000\$, and the measure I wonder about would be 1500\$.

Does this have a name?

This is the reverse of the accounting procedure discussed in my answer to this question: link to question

What you want is the average cost basis for all holders of company.

There is no practical way that this could be calculated for a public firm. Nobody is required to report that information, and not everyone uses that accounting method for equities for taxes.

It would also be highly biased against firms where the founders hold a significant portion of shares. In an IPO, only a portion of the total number of shares are sold, sometimes only 10% of the firm. (Your example implies that 100% of the shares are sold in the IPO, which is misleading.) it is entirely possible for a founder of a company to retain control of a large percentage of the shares for decades. This method would assign almost no value to that portion of the company, which makes the valuation given meaningless for analysis.


Why isn't the Law of Demand true for Marshallian demand? My professor said, in practice, Marshallian demand follows the law of demand (ie that increase in price decreases demand). But he said in theory, the Marshallian case is ambiguous and it does not follow the law of demand.

MY QUESTION:

Why isn't the Law of Demand true for Marshallian demand?

Marshallian demand is also called uncompensated demand. That is a price changes and you see what changes in what the household buys. This consists of two effects.

The good itself is more expensive
The household is poorer because they can no longer afford their old consumption bundle.

The effect of (1) is always to make you consume less of the good. The effect of (2) is what is ambiguous. An inferior good is one where as you get richer (poorer) you consume less (more) of it. For an inferior good forces (1) and (2) oppose. You consume more of the good because you are poorer and less because it is more expensive. A good (check out Giffen goods) could be so powerfully inferior that the effect of (2) is bigger than the effect of (1). No one seems to be able to find evidence that they exist, but they certainly are possible. Often potatoes in Ireland are used as an example, though they were not.


Can setting bank's reserve requirement ratio in 100% simulate gold standard? Statement:

  If you want a gold standard, it's easier to set bank's reserve requirement to 100%.

Context:
This was a comment in a video telling how government actions influence monetary inflation.

Question: Is the statement true?

P.S.: Please, give a source to the answer.

Examine this statement:

  If you want a gold standard, it's easier to set bank's reserve requirement to 100%.

The only sense in which this is true is that it eliminates fractional reserve lending.

However, a key defining characteristic of the gold standard is that government-issued money can be converted into gold upon demand. This creates a real constraint upon government fiscal policy. Having a 100% reserve requirement does not imply such a constraint on government policy, the currency can be a fiat currency.


What's the role of money multiplier in the banking system? On the internet , there are lots of people claming that the banking system has an immense power because it can multiply money obtained by deposits ( 'make money out of thin air' ) and then lend this money just with the provision of the reserve ratio , according to the theory of money multiplier.

They say that asking a real collateral for money without any previous existence is a fraud. It gives a disproportionate power to banks.

However, they seem not to consider that when the banks multiply money they gain a liability towards the depositor , and every time they multiply money is always a new liability due to deposits , so in the end the only profit for the banks ,say in a mortgage , should be the charged interest and not the entire mortgage (if the creation hadn't a liability it would be true) as they claim.

Are my considerations good? Can someone underline other relevant factors? Is this theory a total nonsense?

At present, this question resembles three questions.

Are people on the internet saying incorrect things about bank money creation?
Is money creation by banks fraudulent?
Is money creation by banks limited by the fact that deposits are bank liabilities?

These are questions that mainly do not fit the desired format of this website, so I will handle them very quickly. If there are follow up questions, it might be best to create a new narrow technical question.

I would argue that bank creation of money has led to a lot of incorrect statements about it on the internet. Unless a particular statement is to be analysed, there is nothing else to say.
The belief that bank lending is fraudulent money creation show up in some popular Austrian economics writing. The answer requires delving into the definition of “fraud.” It is safe to day that no legal system accepts that view, as that would imply that a significant portion of the financial system would lose legal backing.
Yes, the fact that bank deposits are liabilities limits bank money creation. It is not the same thing as creating counterfeit banknotes, which are not liabilities.


Effects of shocking budget or indifference curves on optimum point? I am supposed to determine if shocking either the income or preferences would certainly change the optimum solution in case of a corner solution?

I'm not sure what you mean by "shocking the preferences". But it's very easy to find an example in which the optimal solution is still at the corner after a (small) positive income shock.

Suppose $u(x,y)=\sqrt x+y$, $p_x=p_y=1$, and income is $m=0.1$. Performing utility maximization subject to budget constraint, we get
$$
x^*=m,\qquad y^*=0.
$$
Now give income a positive shock: $m'=m+\delta$. As long as $\delta\le0.15$, the optimal solution is still going to be a corner solution: spend all the income on  $x$ and zero on $y$.


How much leverage has the Federal Reserve taken on today (Jan 2016)? http://nymag.com/daily/intelligencer/2015/12/big-short-genius-says-another-crisis-is-coming.html?mid=twitter-share-di

  What makes you most nervous about the future? Debt. The idea that
  growth will remedy our debts is so addictive for politicians, but the
  citizens end up paying the price. The public sector has really stepped
  up as a consumer of debt. The Federal Reserve’s balance sheet is
  leveraged 77:1. Like I said, the absurdity, it just befuddles me.

How did Michael Burry arrive at the 77:1 figure? Is this close to reality? How could the Fed be leveraged when they can simply print money to pay down debt in the first place?

Page 4 of this document has the balance sheet as of November 2015: http://www.federalreserve.gov/monetarypolicy/files/quarterly_balance_sheet_developments_report_201511.pdf

Their assets are 4489 liabilities are 4431. Thus Equity is 58, L/E = 76 (approx).

It would not matter even if Fed had infinite leverage ratio. However, as far as I know congress has agreed to keep Fed solvent (so the ratio can't go negative). Although, the institution is not even solvent, as some of it's assets are marked mark to model, thus having much lower fair value than presented in the balance sheet.

The balance sheet reflects created money (growth in base money) and not debt. Created money generates inflation, but significant amount of bank credit was destroyed which also constituted money people transacted with. Thus the newly created Fed money has just replaced the bank credit and there has been no significant inflation, as the M2 money supply has only raised slightly. Some of the currency has also been exported overseas as USA continues to have a balance of payment / trade deficits.

Based on the quote, it seems the author has confused two issues. If you want to investigate the debt and liabilities of the government, you need to look at the government debt and it's other liabilities (not merely the Fed). He is correct that the debt is at a quite high historical level (only during world war 2 was it higher), while other liabilities (such as social security and medicare liabilities) I believe are at an unprecedented historical level.


What is the rate of new house construction in Auckland, New Zealand? What is the rate of new house construction in Auckland, New Zealand, where the housing market seems to have gone to the roof? Assuming that the current high prices of Auckland houses are a result of less supply compared to demand (and not a bubble), is the number of houses to number of buyers going to stabilize anytime soon?

Also, if so many new houses are warranted in Auckland, is it also having high activity in the construction industry?

If you're really serious about researching this data, I'd suggest contacting the Real Estate Institution of New Zealand: https://www.reinz.co.nz/reinz/public/reinz-statistics/reinz-statistics_home.cfm

As long as there are reasons to move to Auckland (more work etc.) and the supplied housing is not high enough to match the demand, the prices will continue to rise. This effect is visible in many major cities where the supply of housing is (for some reason like goverment rules or slow building) inelastic.


How is the Corona $2 trilion stimulus package going to be financed? Perhaps this is a simple question but as a non-US citizen I don't understand how the US government is going to finance the $2 trillion stimulus package to combat the economic damage due to the Corona pandemic. Is the government going to somehow delegate this to the FED and if yes, how?

I believe that the government is going to encourage the financing of the $2 trillion dollar stimulus to the FED  and this will take place through the FED purchasing US treasuries.

It is also in the best interest of the FED to do so in order to foster the stability, integrity, and efficiency of the nation's economy (Federal Reserve Bank mission).

This is actually taking place now as The Federal Reserve Bank of New York will purchase $37 billion in Treasury bonds to boost liquidity in the crucial government debt market, announced on the 13th.  (https://thehill.com/policy/finance/487427-ny-fed-to-accelerate-bond-purchases-to-keep-treasury-market-flowing)

The New York Fed will also offer $1 trillion in short-term loans to banks and brokers in exchange for Treasury bonds. The Fed will eventually be repaid with interest on the loans it offered to banks and brokers in exchange for returning the bonds it purchased.
The government and the FED are going to be working together to help jump start the economy, according to the article.

During the Great recession the US government used the American Recovery and Reinvestment Act to help jump start the economy while the Central Bank used quantitative easing (QE), zero interest rates, buying US treasuries, and other tools to as well.

On a side note:

A stimulus package can either be in the form of a monetary stimulus or a fiscal stimulus. One of monetary stimulus tools involves cutting interest rates to stimulate the economy. When interest rates are cut, there is more incentive for people to borrow as the cost of borrowing is reduced.

Which is what the Fed is doing now.

An increase in borrowing means there’ll be more money in circulation, less incentive to save, and more incentive to spend. Lowering interest rates could also weaken the exchange rate of a country, thereby leading to a boost in exports. When exports are increased, more money enters the economy, encouraging spending and stirring up the economy.

Fiscal stimulus refers to increasing government consumption or transfers or lowering taxes. Effectively this means increasing the rate of growth of public debt, assuming that the stimulus will cause sufficient economic growth to fill that gap partially or completely.

When a government opts for a fiscal stimulus, it cuts taxes or increases its spending in a bid to revive the economy. When taxes down, people have more disposable income . An increase in disposable income results in more spending to boost economic growth. When the government increases its spending, it injects more money into the economy, which hopefully decreases the unemployment rate, increases consumer spending, with the hopes to counter the impact of a recession.

A potential problem of fiscal stimulus is that to increase public spending the government has to increase its borrowing. There is a chance/rick that people may actually choose to save the excess disposable income instead of spending it, which could render the stimulus package ineffective.

American Recovery and Reinvestment Act (ARRA),February 2009, is an example of  economic stimulus package. It instilled the confidence needed to boost economic growth. It also aimed to restore trust in the financial services industry.

ARRA cut taxes by 288 billion dollars. It spent 224 billion in extended unemployment benefits, education, and health care.   While also creating jobs using $275 billion in federal contracts, grants, and loans.