Logistic (or probit) regression How does the use of a logistic (or probit) regression help address the endogeneity concerns: explanatory variable is correlated with error term?

Logit and probit regressions do not help with endogeneity at all. What they do is bound the predicted value (aka conditional expectation) of your response variable between 0 and 1. As such they are appropriate for performing regression analysis for binary outcomes like:

Whether someone wins or loses a sports game
Whether someone is going to be employed or unemployed

To understand this, consider the univariate regression model

$$y_i = \beta x_i + u_i$$

Say $y_i = 1$ if player $i$ wins a tennis match and $y_i$ = 0 if a player loses a tennis match, and you are trying to work out whether they will win based on the difference between their world ranking and that of their opponent, $x_i$.

If you perform ordinary linear regression you might get an estimate of $\beta$ which is, say, $0.05$. Perhaps most of your observations for $x_i$ (i.e. the difference in world rankings) are between $0$ and $20$, which means that your predicted value of $y_i$ (i.e. $\beta x_i$) is between $0$ and $1$. You can then easily say whether you are predicting a win or a loss for your player depending on whether that predicted value is greater than or less than $0.5$. However there may be certain situations where the difference in world rankings is $>20$, in which case you will get a prediction >1.

In one sense this is unproblematic because based on your model this player is clearly the forecast winner. However you may prefer not to have unbounded predicted values, in which case a probit or logit transformation or model is appropriate.

Taking probit as an example, you specify that the probability that a player wins, conditional on the difference in rankings is not $\beta x_i$ as before. Instead you state

$$p(y_i = 1 | x_i) = \Phi(\beta x_i)$$

where $\Phi$ is the cumulative distribution function (CDF) of the normal distribution. As you can see in the normal CDF diagram here, no matter how high or low the value of $\beta x_i$ is, the variable on the y-axis never exceeds $1$!

In addition, if the distribution of your error term $u_i$ is logistically distributed and you use logit, or normally distributed and you use probit, your estimation technique will be more efficient (i.e. you will have smaller standard errors) than if you performed the simple linear regression. This point is more complex though so I don't provide a proof.


Ambiguous definition of Substitution and Income effect? Consider the (schematic) indifference curve diagram of two good-quantities Q1, Q2:

According to Wikipedia we call the vector AB' the substitution effect and the vector B'B as the income effect.

I now wonder why I could not, instead of translating the tangent through B onto the lower indifference curve, translate the tangent through A to the higher indifference curve, thus denoting AA' as the income effect and A'B as the substitution effect.

Note that these two definitions do not coincide for (sufficiently) different shaped indifference curves.

First, you can make any definition you want as long as you use it consistently, but it's usually better to adopt the standard definition if you want to avoid confusion.   Similarly you could ask why we don't call a garage a dogpound and a dogpound a garage.  We perfectly well could, but it's better if we all use words the same way.

Second, assuming we agree to adopt the standard definition, your picture doesn't give enough information to determine a substitution and an income effect.  That depends on where you started and what changed, which you haven't told us.

IF you started at $A$ and the price dropped, then, according to the standard definition, the substitution effect is from $A$ to $B'$ and the income effect is from $B'$ to $B$.

If you started at $B$ and the price rose, then, according to the standard definition, the substitution effect is from $B$ to $A'$ and the income effect is from $A'$ to $A$.

Third, if you want to adopt your alternate definition, you will find that your definition and the standard definition agree arbitrarily closely for sufficiently small changes.  So if you imagine an infinitesimal price change, it doesn't matter which definition you adopt.  But for finite price changes, what matters is not that one definition is better than another, but that we find a definition we can all agree to use.


Why in maximization in OLG model with 1 generations we assume that both consumptions are positive? I have an OLG model with constant level of technology and population. The utility of agent who is born in period t is:

U() = c1,t + 1/1+pC2,t+1

My textbook asks me to write the maximization problem and FOC for the agent  who is born in period t with the assumption that agent can not refuse the consumption in any of the periods.

I know how to write maximization problem

U() = c1,t + 1/1+pC2,t+1 -  max c1,t ≥0 С2,t+1 ≥0
c1t + st = wt
c2,t+1 = (1 + r,t+1)st

The only thing which I do not get is why we need the assumtion that agent can not refuse the consumption in any of the periods. What does it give us?

I am sorry of the question is obvious, I am just studying OLG model on my own and do not get everything.

I think all the textbook is trying to say is that there can be no corner solutions to your maximization problem.

Not refusing consumption in any period, I think translates to consumption in each period > 0.


Monopsony theory of Single Payer One theory of how single payer/universal healthcare systems are less expensive than the US private insurance market is that each country functions as a monopsony: a market structure in which a single buyer substantially controls the market as the major purchaser of goods and services offered by many would-be sellers. For example, with drug pricing, a country can ultimately tell Pfizer, "We don't want to pay more than X for your new drug. If you don't agree, you can take your business elsewhere", and Pfizer is forced to agree because otherwise they lose an entire market. I have two questions on this logic:

Pfizer is a global company, they sell to many countries, how can any single small country be considered a "monopsony" on Pfizer's products? . In a global marketplace, with many payers, it would seem that none of those payers individually can claim to have a monopsony. And this is especially true for small payers, such as Sweden, Norway, Denmark, New Zealand, which are individually a very small part of the global customer base. And this brings me to the next question, the flipside:

Why don't US health insurers have the same "monopsony" power? Why can't Aetna, which is bigger than many countries, tell Pfizer the same thing: "We don't want to pay more than X for your new drug. If you don't agree, you can take your business elsewhere"? How is their "monopsony" any less than Sweden's, which is a much smaller market?

If a firm cannot cover costs and make profit then it has no profit motive to develop and sell drug XYZ. So the profit motive is a "cost plus" model. The price must cover costs plus profit whether there is a single buyer or multiple buyers of the output goods.
If drug XYZ makes a net profit in a domestic market, and if there is a marginal profit in other global markets, then prices can be set lower in some parts of the global market and there is still an incentive to provide drug XYZ for marginal profits there.
Bottom line is price floors must be set by the market and/or government purchases if long term investments are to be recovered from future cash flows. This looks like cost plus pricing of output goods. Otherwise a firm that cannot recover costs from cash flows goes bankrupt and the assets transfer to another firm with less long term debt and then the cost can come down.


Sen's property $\alpha$ holds when limited attention in choice? Consider the limited attention choice framework by Matejka and McKay (2015).

This framework can give rise to consideration sets, as roughly summarised below.

Consideration sets in the limited attention choice framework:
  Since attention is a scarce resource, the agents may be induced to contemplate only a subset of the available alternatives, ignoring all the others. More precisely, this framework assumes that the decision maker (hereafter, DM) knows the feasible set but is not fully aware of the value of each alternative in the feasible set. The DM is endowed with a prior on the value of each alternative in the feasible set and can "buy" an  information structure to update such a prior.  An information structure consists of a  signal that, once realised, allows the DM to update her prior and then choose the alternative from the feasible set maximising the expected utility given the refined information. On one hand,  the DM prefers a  signal that is as informative as possible on the value of each alternative  in order to minimise ex-post regret. On the other hand, limited attention constraints imply that a more informative  signal is more costly because it requires more cognitive effort to be processed. Such a trade-off can lead to equilibria featuring information structures such that some alternatives in the feasible set are never chosen by the DM for any realisation of the signal. Hence, these alternatives are not part of the DM's consideration set. Complementarily, the DM consideration set's consists of the alternatives which are chosen by the DM with positive probability  for at least some realisations of the signal.

Question: Consider the Sen's property $\alpha$ stating that
if an alternative $x$ is chosen from a set $T$, and $x$ is also an element of a subset $S$ of $T$, then $x$ must be chosen from $S$. Does this property hold in the the limited attention choice framework by Matejka and McKay (2015)? The authors have a discussion about IIA (independence of irrelevant alternatives) but I couldn't understand whether the Sen's property $\alpha$ holds or not.

My math is a bit rusty, but I believe you can show that, for a well-behaved (e.g., open and connected) feasible set $T$ and a consideration set $T'$, $x \in S \subseteq T' \subseteq T$, e.g. Sen's property $\alpha$ holds.

That doesn't seem like a very interesting question/result though, as it just follows directly from the definition of "well-behaved".  Maybe I'm missing something.


Empirical measurements of consumption and production elasticity of substitution? The consumption elasticity of substitution  (ES) is the ES between different consumption goods.

The production ES is the ES between factors of production.

My question is: are there any serious empirical studies of the numerical values of these elasticities, and if so, where can I find them?

The elasticity of substitution of production factors derived from empirical data depends on the data. In the case of simple production, you can estimate the production function (fit your data to the functional form of the production function) and from it derive the elasticity of substitution and other functional parameters.
The elasticity of substitution in the case of a production function (accounting for output) has it's roots in the classical consumption function (accounting for utility), as you mention.
Empirical studies are plenty and the numerical results vary depending on country, industry (aggregate level) or type of company (micro level). However, the robustness of the results may not be up to par, because aggregate (and otherwise), especially neoclassical, production functions are subject to numerous restrictions and assumptions.
For empirical studies you can start here Chirinko R., Fazzari S., Meyer A., 2004

In the case of utility (elasticity of substitution in consumption function), I don't know of any empirical studies. I'm not one to be asked this, but my first hunch is - additive utility is a REALLY strong assumption. Classical and austrian economists have argued that utility is subjective. So, numerical studies of elasticities of substitution in consumption would have to be verry biased.


Who was the inventor of Utility function? To my knowledge, the idea of representing weak-order with a function dates back to Cantor. So my questions are:

1) Was Cantor the first person to rigorize these kinds of representation?

2) Were there anyone in the ancient world who (though not rigorously) developed the idea of representing social choice or preference by a function?

Not to be too obvious about it, but couldn't you just say Jeremy Bentham? The calculus of choosing bundles that maximizes pleasure specified a very simple utility function in words.


What are Giffen Goods? What exactly are Giffen goods and are they of purely theoretical interest or has there been empirical evidence of their existence?

Consider the Slutsky equation,
$$
\frac{\partial x}{\partial p} = \frac{\partial x^c}{\partial p} -
\frac{\partial x}{\partial I} x.
$$

A giffen good is the case where the income effect $\frac{\partial x}{\partial I} x$ is negative and large (in magnitude) enough so
that $\frac{\partial x}{\partial p} > 0$.

From Wikipedia:

  There are three necessary preconditions for this situation to arise:
  (1) the good in question must be an inferior good,
  (2) there must be a lack of close substitute goods, and
  (3) the good must constitute a substantial percentage of the buyer's income, but not such a substantial percentage of the buyer's income that none of the associated normal goods are consumed.

(Sufficient when (1) also adds that the good is so inferior that the income effect is greater than the substitution effect.)

A Giffen good does not generate utility directly through its price. Contrast this to a Veblen good where there the user actually derives utility directly from the price.

See this link for some info about empirical evidence.


Visual Representation of mixed strategy profile using a simplex in $R^3$ I am looking for a reference to learn how to plot mixed strategy profile in a two player game with three pure strategies.

Looking forward to any advice or suggestion.

(Here is the question regarding simplex I posted in mathematics.stackexchange)

I'm not sure what you're asking is possible.

Start with the intuition from the case with 2 pure strategies. A player's mixed strategy is a 2-vector $\sigma_i(p)=(p,1-p)$, which can be visualized on a 1-dimensional simplex (or simply a line). Therefore, a profile of two mixed strategies can be represented as a point in a 2-dimentional plane, because this profile is controlled by two parameters (say $p$ for player 1 and $q$ for player 2) like the following

With 3 pure strategies, a mixed strategy is a 3-vector $\sigma_i(p_1,p_2)=(p_1,p_2,1-p_1-p_2)$. It is controlled by two parameters, and so we need a 2-dimensional simplex to represent one (mixed) strategy. For a profile of two mixed strategies, we need two 2-dimensional simplices, or planes. Naturally they should be perpendicular to each other just like the two 1-dimensional simplices are in the 1-dimensional case, although this is not a hard requirement. The difficult part is that we now have four controlling parameters, e.g. $p_1,p_2$ for player 1 and $q_1,q_2$ for player 2. So the strategy profile lives in a 4-dimensional space, which is impossible to visualize.

The following is a failed example. Player 1's mixed strategy $\sigma_1(\hat p_1,\hat p_2)$ lives in the blue simplex and Player 2's mixed strategy $\sigma_2(\hat q_1,\hat q_2)$ lives in the orange simplex. The two simplices share the vertical axis, but this is not necessary. As the two red dashed lines show, it's not always possible to find an "intersection" of the two strategies (the two lines have different heights).


Perfect complements indifference curve What would be the I.C of this function? $u(x,y)=\min(x,√y)$
I understand since it is a perfect complements case it should be "L" shaped but I wanted a more detailed graph.

For perfect complements like $u(x,y) = \min( g(x_1), h(x_2))$ the points at which kinks occur are such that $g(x_1)= h(x_2)$. In this case that would be $x= \sqrt(y)$ so in this case indifference curves would look like:

The picture above plots two indifference curves and the dashed line gives you all kinks of every indifference curve. Normally the kinks would lie on $45^°$ line because usually complements are given by functions where respective slopes of $x$ and $y$ are linear (e.g. $\min (\alpha x, \beta y ))$ but here you have case where you have $\sqrt(y)$, so the  $x$ has to be consumed with exponentially increasing quantities $y$.


How to extrapolate the data? I have bond yields for Indian G-Sec. For short term yields the proxy are (15-91 days) , (92-182 days) , (183-364 days). For long term yields the proxy are 5,10 & 15 years. Data ranges from 1996 - 2016.
But the yield values for 15 years bonds are unavailable from 1996 to 1999. So in order to have a complete data-set, i need to compute the 15 year bonds yield those 3 years on monthly basis.

Please check the below link for data file.
https://spaces.hightail.com/space/9wrYy

You are never going to be able to fill in the gaps exactly, so the best approach may depend on what you want to use the data for.

One reasonable approach might be to use an average offset from the 10-year rate, e.g. adding about $0.3$ to it.  If you did that, then you would get the red data on the chart below, which compares reasonably well with the actual blue data for the 15-year rate, though not perfectly.  So you might use the red line as a proxy for the earlier missing blue data.  This might not be such a good idea if the purpose of your analysis was precisely considering the changing gap between the 10-year and 15-year rates


How much physical currency does the Fed transfer to banks per year? I just wonder since money has to enter the system somehow. Currently the Fed will supply whatever banks ask to meet their capital needs. But how much physical currency does the Fed, treasury, and mint transfer to banks per year? And which banks?

The amount of currency in circulation (bills and coins) is published by the Federal Reserve. (Available at the FRED web site.) The annual change in this time series is the amount created.

This is the amount outstanding, it does not tell us how much currency is being destroyed and replaced. Also, there is seasonal demand for currency, and some cash is shipped back and forth.

The creation of currency is not really a "transfer" (gift). The banks need to buy the currency by exchanging reserves (settlement balance at the Fed) for the notes.

Finally, the Federal Reserve does not single out how much it deals with any particular banks by name under normal circumstances. (During some of the special operations undertaken after the Financial Crisis, the Fed published some information about their counterparties.) The most you would get would be on the basis of the category of bank.


Do points inside the PPF curve always mean productive inefficiency? If a firm produces combinations of goods along the PPF curve, it has achieved its productive efficiency. And when a firm reaches productive efficiency, it means that all factors of production have been fully utilized. And another way of measuring productive efficiency is to look at its average cost. If a firm has reached productive efficiency, it means that the firm is producing at the lowest average cost.
But what about the points that fall inside the PPF?
Let’s say A (15,10) is on the PPF curve, and B (15,5) lies beneath it. Assume that a firm is producing at point B, and the reason why it isn’t producing at a higher level is that there’s still some cash deposit in its business account that hasn’t been withdrawn yet.

Since cash deposit is NOT a factor of production, even the firm is producing at points beneath the PPF, it doesn’t necessarily imply that the firm has not utilized all factors of production. Where did I go run?

Well the problem here is in a set up. It is inconsistent to say that firm is producing below PPF not due to underutilization of factors of production but due to cash, since cash will not enter production function, or if you assume that cash is here used to buy factors of production it is inconsistent to say that firm fully utilizes all avaiable factors while having excess cash.
For example, take very simple production function $F=L^{1/2}$ and firm has access to 100 units of labor, with all the standard background assumptions. I used this simplistic case because here PPF is just one point instead of set of points, and in this case the PPF would just be point F=10. If firm would be using 100 units of labor it would be operating at PPF and if not it would be operating below PPF. However, you cant just arbitrary announce that this firm  is producing F=8, due to not using all cash if cash does not enter as factor of production. That is simply inconsistent with having production function $F=L^{1/2}$
Furthermore, let us assume that firm does not just have 100 units of labor but that it has 100 dollars and has to purchase labor on labor market for 1 dollar per labor. Well in this case if firm has still some excess cash, for example if it purchased only 70 units of labor, then it still does not uses all factors of production in most efficient manner because given its 100 dollars it still has 30 units of labor avaiable for hire that it could use. Consequently, the problem in the setting above is that it just arbitrary announces that firm is utilizing all factors of production as much as possible while not being at PPF. That is like asking, from math we know that unit circle is given by all points that satisfy equation $1=x^2+y^2$, but I assume point (100,10) lies on unit circle and it clearly does not satisfy the equation. It is just inconsistent reasoning.


Why is the Australian Government financing the stimulus via debt rather than the RBA? The Australian Government is engaging in quantitative easing for the first time. The RBA is creating money that is being used to buy bonds on the secondary market so that there is enough liquidity for banks and financial institutions to buy new bonds to fund the covid stimulus measures.

Why are financial institutions necessary as an intermediate step? Why not fund the measures directly from the RBA without introducing additional debt (and the associated interest)?

I'd also be curious to know if Modern Monetary Theory has a different answer to this question than conventional economics.

As noted in comments, the reason why the RBA acted the way it did most likely comes down to its legal framework. Before the 2020 crisis, very few countries allowed their central bank to directly fund the fiscal arm of the government, which I will call the Treasury. I had often seen Canada cited as the only developed country that allowed the central bank to buy government bonds at auction (I am unsure that no other example exists). I think such financing was allowed in the 1945-1970 period in more places.

I know very little about the legal framework the RBA operates under. I will instead just discuss the economic effects.

The Modern Monetary Theory (MMT) view is that the addition of banks into the loop does not matter. Having the Treasury issue bonds that are just bought back bu the central bank - which is owned by the Treasury - has exactly the same economic effect as the Treasury directly borrowing from the central bank.

If we look at the mathematical models used by other economic theories, they say the same thing, since the final balance sheets are the same. The usual description of this is that this is “money financed deficits.” However, the argument seems to be that the difference is that behaviour of the fiscal arm will change, and so there is a greater bias towards inflationary policies.


Constrained Optimization using Lagrangian method The stationary points that we derive by solving the first order conditions of the Lagrangian are those points global optimum points or local optimum points?

In general, FOCs produce local, not necessarily global, optima. Consider
\begin{equation}
\max_x\;f(x)=x \sin(x) \quad\text{s.t. $x\in[0,7]$}.
\end{equation}

As the plot below shows, $f(x)$ has three stationary points in the domain $[0,7]$: $x_1=0$, $x_2\approx 2.03$, and $x_3\approx4.91$. Among them, $x_2$ is the local maximum that would be picked up by the FOC. However, the global maximum happens at the boundary, where $x=7$. Since $f'(7)\ne0$, the global maximum would be missed if you rely only on the FOC.


Economic theory journals for a refinement theorem about utility function representation I would like to ask which are the (mathematical) economics journals that publish papers about economic theory and that focus mainly on the mathematical aspects of it. Let me be more precise. If I have a refinement of a representation theorem for preference relation axiomatization, which journals are most suitable to get the paper accepted?
Of course, I mean a significant generalization of the theorem and, of course, I am not thinking about Econometrica or Journal of Economic Theory since I understand my question is about a refinement of a theorem, and not a completely original economic issue. Nevertheless, I think my generalization fills an important gap in the subject and I would like to ask for a list of economics journals (maybe listed by ranking of importance) that may be the right target for my result. Thank you in advance!

For theory you have in order of prestige... (I know subjective)

Journal of Economic Theory
Theoretical Economics
AEJ-Micro (only micro)
Mathematics of Operations Research (OR related)
Games and economic behaviour (only game theory)
Economic theory
Journal of Mathematical Economics
Social choice and welfare (only social choice)
Theory and Decision
Mathematical social sciences
$\ldots$

There are of course other journals that publish theory, like:

International journal of Game theory
Economic Inquiry
Economics Theory Bulletin

Feel free to add to the list.


Time series in monetary policy I wanted to learn how time series analysis is used to study monetary policy/ money and banking data, such as how and which techniques are used to study which data, what kind of problems are studied with which techniques, how are effects of policies gauged etc.

What are some good books/ online courses or other resources for that?

For time series analysis a classical text is the time series analysis from Hamilton - although nowadays the text is slightly outdated it has an excellent and detailed overview of models that are still in use or serve as basis for more advance models.

The time series and panel data econometrics from Pesaran is also excellent - it’s also more contemporary but also deals with models not used in monetary economics.

If you have no background in econometrics the introduction to modern econometrics from Wooldridge is really good. Or also Verbeeks guide to modern macroeconomics has good treatment of time series models and in terms of difficulty it is somewhere between Wooldridge and Hamilton or Pesaran.

For a more practical handbook I would recommend international macroeconomics and finance from Nelson, and if you care mainly about forecasting and policy evaluation the kind usually done in central banks or ministries of finance economic forecasting and policy from Carnot, Koen and Tissot is quite good practical text


Can't Derive Expectation in Second Price Auction I'm working on a SPA where we solve:

$\max_{\beta \ge 0} Pr\{Winning\}[v - \mathbb{E}(b^{[2]} \ | \ b^{[2]} \le \beta)]$

We assume all beliefs on bids are independently and identically distributed on $[0,1]$ given by CDF $F(b)$.

$b^{[2]}$ denotes the second highest bid.

I've found $Pr\{Winning\} = Pr\{\beta \ge b_1, \ldots , b_n\} = F(b)^N$

In my lecture notes we are given:

$\mathbb{E}(b^{[2]} \ | \ b^{[2]} \le \beta) = \frac{1}{Pr(b^{[2]} \le beta)} \int_0^{\beta} bdF(b)^N$

I'm assuming this comes from $P(A|B) = \frac{P(A \cap B)}{P(B)} $, but can't quite figure it out. Especially the $F(b)^N$ in the integral.

There are $N+1$ bidders. So we only care about the second highest bid. That's the same as finding the highest bid amongst the remaining $N$ bidders. The expected second highest bid is given by:

$$b \cdot Pr[b] \cdot Pr[b \geq b_{-i}] = b \cdot f(b) \cdot F(b)^{N-1} = b \cdot dF(b)^{N}$$

As expected value is an average, we divide out by the probability that $b \leq \beta$, which is $F(\beta)$.

Thus, we have:

$$\mathbb{E}[b^{[2]} | b^{[2]} \leq \beta] = \dfrac{1}{F(\beta)}\int_{0}^{\beta} bdF(b)^{N}$$


Find pareto optimal allocations Consumer 1: $U_1(x_1,y_1)=x_1y_1$
Consumer 2:$ u_2(x_2,y_2)=min\{x_2y_2 , 4\} $
Initial endowments e1=(1,4) and e2=(4,1)
I want to find Pareto optimal allocations and show its edgeworth box
My solution
First I solve utility maximization
For consumer 1,
$MRS=P_x/P_y$ And budget constraint $x_1P_x + y_1P_y= P_x+4P_y$
$x_1^*=(P_x+4P_y)/2P_x$
$y_1^*=(P_x+4P_y)/2P_y$
For consumer 2
At optimum, $x_2y_2=4$
Budget constraint $x_2P_x+y_2P_y= 4P_x+P_y$
Then $x_2[(4P_x+P_y)/P_y -P_x/P_y x_2]=4$
Feasibility constraint
$$x_2=5-x_1$$
$$y_2=5-y_1$$
Then I solve together
$$(5-x_1)(5-y_1)=4$$
When I insert $x_1$ and $y_1$ I will obtain
$$(9P_x-4P_y)(6P_y-P_x)=16P_xP_y$$
From this point I cannot proceed the solution
How can I find the Pareto optimal allocations ?

Set of Pareto efficient allocations is given by the dashed line in the Edgeworth Box. It is the set of feasible allocations satisfying $y_1 = x_1$ and $x_1y_1 \geq 9$
.


How do I adjust for heterogeneous labor markets when calculating the wage premium? I have a dataset with mean and median wages for particular geographic areas around the United States. I am interested in calculating the wage premium between some areas. I can already do a simple version of that but I would like to adjust my model for heterogeneous labor markets because the dataset includes data split by industry group for each area. How would I go about doing that exactly?

Consider this example:

+------+-------------+--------------------+-------------------+
| Area | Industry    | Mean Annual Income | Number of Workers |
+------+-------------+--------------------+-------------------+
| IL   | Medical     | $100,000           | 50                |
+------+-------------+--------------------+-------------------+
| IL   | Natural Gas | $50,000            | 50                |
+------+-------------+--------------------+-------------------+
| NY   | Medical     | $150,000           | 60                |
+------+-------------+--------------------+-------------------+
| NY   | Natural Gas | $90,000            | 40                |
+------+-------------+--------------------+-------------------+

I could just calculate the wage premium in Natural Gas and Medical between IL and NY, then take a weighted average. But what should the weights be? The proportion of workers in each industry is different for the two areas here.

This looks very similar to textbook price index problems. Could I just calculate a chained price index?

It all depends on how you define the 'wage premium between different areas'.

The average NY resident makes $51,000 more per year than the average IL resident.

There are 110 medical workers and 90 natural gas workers in NY and IL combined, so you could try weighting the wage premium of medical workers by 110/200 and the wage premium of natural gas workers by 90/200 to give a combined wage premium of $45,500. This will tell you how much more or less a person chosen at random from the whole population will earn if they work in the same industry that they are currently working but in a different area.


Essential difference between Neo-Classical and Neo-Liberal economical philosophies From what I learned about Neo-Classical and Neo-Liberal economical philosophies in different sources such as documentary series or Wiki site articles I understand both are "neo" versions of Classical and Liberal economy, and that all four are branches of capitalism,

but how would you explain the essence of difference (if there is one), between Neo-Classical and Neo-Liberal?

To expand on my comment. Your question is not really about economics but about philosophy or political science. The reason for this is that the Economics is usually defined following the definition of Lionel Robbins as: "Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses."

Neo-liberalism is an ideology. It is a set of values or set of ends that based on the given values are worth pursuing. For example, from Neoliberal value of freedom one could derive the end goal of free speech. But economics does not examine either ends or values. Economics treats ends and values as given by an individual. Hence there is not really anything like Neo-liberal economics (although I seen the term used as a pejorative, like trickle down economics - but those are not really economic terms that have any meaning in economics - you generally wont find them in papers or textbooks or conferences) as that entails value judgments that can only be judged based on philosophy not science.

There is a Neoclassical school of economics. However, it has nothing to do with capitalism. Capitalism is a term that’s poorly defined and not used in economics. In most economics textbook you would not even find the word, outside some inserted exempts of newspaper articles in undergraduate books. Neoclassical economics is an umbrella term so its easier to classify theories. The term for example implies that models used will assume rational agents (as opposed to lets say behavioral school that relaxes this assumption). This being said the classification is used mostly by historians of economic thought as in present day its not uncommon for people to use different models and set of assumption to fit given purpose, so there are many economists who use neoclassical model in one work but behavioral model in another.

Also, Neoclassical economic models have no relationship to idea of capitalism (many neoclassical models suggest quite large role for government intervention in economy or redistribution given certain welfare function). For example, Jozef Stiglitz's work is Neoclassical, meaning almost all his papers use neoclassical models, but he does not like the idea of 'classic capitalism'. Another example would be Greg Mankiw, who mostly works with Neo-Keynesian models in his research, but personally he is supportive of the idea of 'classic capitalism' - his ideology could be described as quite libertarian or Neo-liberal.


Introduction of an asset tax in the AK model Let's start by analyzing the family's problem with the imposition of a tax. Assuming a CRRA utility function:
\begin{equation}
    U = \int_0^\infty e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] \operatorname{dt} \hspace{5mm} (1)
 \end{equation}
Agents maximize $(1)$ subject to the assets' restriction:
\begin{equation}
        \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a
\end{equation}
The Hamiltonian is given by
\begin{equation}
    \mathcal{H} = e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] + \lambda \cdot [ w - c + (1-\tau)r \cdot a - n \cdot a ]
\end{equation}
The maximum principle conditions are:
$$\begin{cases}
    \frac{\partial \mathcal{H}}{\partial c} = 0 \implies e^{-(\rho-n)t} \cdot c^{-\theta} = \lambda \\ \dot{a} = \frac{\partial \mathcal{H}}{\partial \lambda} \implies \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a  \\ \dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial a} \implies \dot{\lambda} = -\lambda \cdot [(1-\tau)r - n] \\ \lim_{t \to \infty}  [\lambda \cdot a] = 0
\end{cases}
$$
Is my reasoning correct so far? Or should I have included a transfer from the government to the agents somewhere? Furthermore, whether the government spending is productive or not would also change how we deal with the tax in our AK model, right?
UPDATE: Thinking a little bit more, I've reached the conclusion that the problem should be reestated as:
Maximize $(1)$ subject to assets' restriction
$$
\dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a + v
$$
where $v = \frac{V}{L} = \frac{\tau \cdot r \cdot a}{L}$ is a per capita lump-sum transfer from the government to the families. Therefore, the Hamiltonian is now given by:
\begin{equation}
    \mathcal{H} = e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] + \lambda \cdot [ w - c + (1-\tau)r \cdot a - n \cdot a + v]
\end{equation}
The maximum principle conditions are:
$$\begin{cases}
    \frac{\partial \mathcal{H}}{\partial c} = 0 \implies e^{-(\rho-n)t} \cdot c^{-\theta} = \lambda \\ \dot{a} = \frac{\partial \mathcal{H}}{\partial \lambda} \implies \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a + v \\ \dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial a} \implies \dot{\lambda} = -\lambda \cdot [(1-\tau)r - n] \\ \lim_{t \to \infty}  [\lambda \cdot a] = 0
\end{cases}
$$
Using the equations above, we can find that:
$$
\frac{\dot{c}}{c} = \frac{1}{\theta} \cdot [(1-\tau)r - \rho]
$$
Considering now a production function in per capita form given by
$$
y = Ak, A > 0
$$
we have that the firms profits are (note that I didn't include the tax in $r$ here, per the reasoning that since $r$ enters profits as an expense, in this case it's not taxable, because it's not an asset income as in the representative family's problem)
$$
\pi = Ak - \underbrace{w}_{=\frac{\partial y}{\partial L} = 0} - \delta \cdot k - r \cdot k
$$
which means the first-order condition of profit maximization will be given by
$$
\frac{\partial \pi}{\partial k} = 0 \implies A = f'(k) = \delta - r
$$
or
$$
r = A - \delta
$$
Therefore,
$$
\boxed{ \frac{\dot{c}}{c} = \frac{1}{\theta} \cdot [(1-\tau) \cdot (A-\delta) - \rho]}
$$
Now, in equilibrium, $a = k$. If we substitute this and $v = \tau \cdot r \cdot k$ in the assets's restriction, we have:
\begin{align*}
\dot{k} &= -c + (1-\tau) \cdot r \cdot k - n \cdot k + \tau \cdot r \cdot k \\ &= -c - (r+n) \cdot k
\end{align*}
and, therefore,
$$
\boxed{ \dot{k} = -c + (A-\delta-n) \cdot k}
$$
Is what I did correct?

I think your math is mostly correct but I have to admit that I am not used to AK models.
A short answer for your main question:  is it ok to introduce taxation in the model without including the government?
Answer: Yes, of course. There are overwhelming literature on dynamic or static theoretical modeling of the effect of tax policy without introducing government as an agent.
For a most famous example, see J.E. Stiglitz, Effects of wealth, income, and capital gains taxation on risk taking. Quarterly Journal of Economics, 83 (1969)
In this line of research, people almost never introduce government. You could search for the works by Arrow (1965), Cass (1972), Fishburn, Richter (1960), Tobin, Mossin (1968), Merton (1969), Russell and Smith (1970)
Also you could see the first paper published by Acemoglu.
But, as you might already explored, many recent papers include government as an agent. For example: Farhi, Emmanuel. "Capital taxation and ownership when markets are incomplete." Journal of Political Economy 118.5 (2010): 908-948.


Is $y$ always output per person in DSGE/RBC models? As title says, is $y$ output per person in DSGE/RBC model? If so, why did macroeconomics decide not using Solow/Cobb-Douglas style of using full labor to characterize economy?

Model reference: http://crei.cat/people/gali/pdf_files/monograph/slides-ch3.pdf

For Gali's model, would $y$ represent output per worker? Or the whole output of the economy represented by representative household?

Reference: why is labor $h_t$ often average working time percentage of non-sleeping time in RBC?

this for me seems to say that full maximum working time is assumed to be 1 with $h_t$ representing working time per labor unit. Thus it seems to be that $y$ is output per person..

As far as Gali is concerned he defines (on page 16 of his book) $N$ as "hours of work or employment", noting that "$N_t$ can be interpreted as the number of household members employed, assuming a large household and ignoring integer constraints".

Then on chapter 3 he talks about a continuum of firms $Y_t(i), \ \ i \in [0,1]$ producing the aggregate output:
$$
Y_t \equiv \left( \int_0^1 Y_t(i)^{1 - \frac{1}{\epsilon}}\mathrm{d}i\right)^\frac{\epsilon}{\epsilon -1}$$

But small-case $y_t$ refers to the output of the log-linearized equations. Somebody can use $y_t$ as the logarithm of the Gross National Product, so as I perceive it that, this is not output per person.


Transformation of random variables and second order stochastic dominance Suppose $X$ and $Y$ are two random variables where $X$ has SOSD (second order stochastic dominance) over $Y$. Let $g(\cdot)$ be a monotonic function and $X'  = g(X)$ and $Y' = g(Y)$.

Under what conditions of $g(\cdot)$ $X'$ has SOSD over $Y'$? I know that if $g$ is linear, SOSD property is reserved. Is there any sufficient and necessary conditions of $g$ that assures SOSD property?

As long as $g(\cdot)$ is Nondecreasing and Concave

Definition: For any lotteries $F$ and $G$, $F$ second-order stochastically dominates $G$ if and only if the decision maker weakly prefers $F$ to $G$ under every weakly increasing concave utility function $u$. Link


Returns to scale - Constant Function Suppose we have a production function $f(z)=2$.

I am asked to determine whether the function exhibits increasing, decreasing, constant or no returns to scale.

For $t>0$, $f(tz)=2$.

I'm not sure about the answer: should I say the function exhibits no returns to scale whatsoever or take different values for $t$ ($0<t<1 \implies$decreasing returns to scale, $t=1 \implies$ constant returns to scale, $t>1 \implies$ increasing returns to scale)?

You want to find a relation between $tF(z)$ and $F(tz)$ for all $t>1$ (or $0$ for CRS).

So since $2t=tF(z)>F(tz)=2$ for all $t>1$, we see decreasing returns to scale.


Is a resource allocation problem in mechanism design a direct or indirect mechanism? From L. Hurwicz's work and book "Designing Economic Mechanisms," I cannot figure out whether a resource allocation problem in mechanism design is a direct or indirect mechanism.

I think the answer is yes since a resource allocation mechanism can be described as a tuple of two components, i.e., $\langle M, g \rangle$, where $M = (M_1, \ldots, M_n)$ and $M_i$ defines the set of possible messages of agent $i$. By writing the agents' complete message space as

$$\mathcal{M} = M_1 \times M_2 \times \cdots \times M_n$$

we can define the outcome function $g$ as

$$g : \mathcal{M} \to \mathcal{O},$$

where $\mathcal{O}$ is the output space defined by

$$\mathcal{O} = \{(\theta_1, \ldots, \theta_n), (t_1, \ldots, t_n) \; | \; \theta_i \in \mathbb{R}_{ > 0}, \; t_i \in \mathbb{R} \}.$$

Then, the outcome function $g$ determines the outcome, namely $g(\mu)$ for any given message profile $\mu = (m_1, \ldots, m_n) \in \mathcal{M}$ and the payment function is defined as

$$t_i : \mathcal{M} \to \mathbb{R}$$

which determines the monetary payment made or received by agent $i \in \mathcal{I}$.

By definition, a direct mechanism is a mechanism that asks all agents for their types and then produces some outcome. Formally, it is a mechanism $\langle M,g\rangle$ in which $M_i$ is wlog equal to $i$'s type space, usually denoted by $\Theta_i$.

In your example, there is no such restriction. However, for any indirect mechanism there exists an equivalent direct mechanism. This result is called the revelation principle.


Are energy and capital complementary factors of production? Why energy and capital are complementary factors of production?

In the literature of environmental taxation, it's often argued that energy and capital are complementary factors of production. So for instance if price of energy goes up, so will the price of capital. I am struggling to understand the relationship.

You say "energy and capital are complementary factors of production. So for instance if price of energy goes up, so will the price of capital.". I think you might have confused yourself because this kind of positive correlation in prices is not necessarily a property of complements (indeed, often one would expect the reverse to be true).

It is more helpful to think of complementarity as "if the price of energy goes up then the demand for capital will go down" (in much the same way that people will buy less butter if bread becomes more expensive).

The reason for this is not far from that proposed by andersan. Capital is often invested in machines (e.g., computers, robots, trucks, farm equipment) that require energy to operate.

Consider a factory producing cars. It can assemble the car in various ways (a) by having a team of workers (labour) do the work, (b) installing robots on an assembly line, or (c) some combination of the two. If the price of electricity doubles then having a bunch of robots on the factory floor suddenly becomes more expensive because the energy needed to run them became more costly. Thus, the guy who was just indifferent between having a robot and a worker will now strictly prefer to hire a worker and the demand for capital decreases.

Even if you have a business (e.g. long-distance transportation) where you cannot easily substitute capital for labour, there will buy some entrepreneur who is indifferent between buying some capital goods (e.g. planes) to start a business and not. An increase in the cost of energy (e.g. fuel) will reduce the prospective profitability of his firm and thus leave him strictly preferring not to invest.


Why was 'structural' chosen to mean 'structural unemployment'? Source: Economics (3 ed, 2014) by N Gregory Mankiw, Mark P. Taylor

  structural unemployment [=] unemployment that results because the number of jobs available in some labour markets is
  insufficient to provide a job for everyone who wants one

Please help me dig deeper than the definition above, the meaning of which I already understand and so ask not about. Instead, I don't understand why structural was selected for the above.  Why not adjectivize this type of unemployment with another word?

This definition (below) doesn't answer my question; all unemployment is caused by structure of the economy. So what's so structural about structural unemployment?

  c. :  involved in or caused by structure[,] especially of the economy

Cyclical unemployment is named after the business cycle, as in the part of unemployment that varies with the business cycle. Structures are things that last a long time, so structural unemployment is the part of unemployment that is driven by features of the economy, like structures, that last a long time.

But as you intuit, surely the length, frequency, and other details of business cycles are driven by such structural features. That's almost certainly true, and major reason why structural change and good policy are thought by many to explain the great moderation in business cycles.


Arguing uniqueness of Bayes-Nash equilibrium in an auction setting In an auction setting with interdependent values, let $\theta_i$ denote the type of player $i$ and $m_i$ that player's message (a bid, essentially). I have calculated the best response function as:
$$m_i^* = \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j)$$
Obviously, then, truth-telling by all parties ($m_k = \theta_k$ for all $k$) is a symmetric BNE, regardless of $\gamma$. I want to also argue that it is the unique BNE. If $\gamma = 1$, it seems like there would be assymetric BNE if there are only two players in which one player's positive misreport ($m_i - \theta_i$) would equal the other player's negative misreport ($\theta_j - m_j$). If $\gamma<1$, it looks like the truth-telling BNE is unique, but how would I prove that?

Also, so far I have not used a common prior assumption. Do I need it here to argue the existence of the truth-telling BNE, or to argue it's uniqueness? If so, we can suppose $\theta_i$ is drawn i.i.d. uniformly from $[0,1]$.

Let $BR=(BR_1,BR_2,BR_3,...)$ denote the best response mapping. This gives a mapping given message vector $m$. (For any $BR_i$ only the messages of non $i$ players are used to derive the best response mapping.)

Let $m^* = (m_1^*,m_2^*,m_3^*,...)$ denote the symmetric equilibrium. As it is an equilibrium, for any $i$ you have $BR_i(m^*) = m_i^*$, so
$$
BR(m^*) = m^*.
$$
Suppose we have a non truth-telling equilibrium $m'$. As this is an equilibrium
$$
BR(m') = m'
$$
so
$$
\forall i\in N: m_i' = \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j').
$$
Rearranging this
$$
\forall i\in N: 0 = \theta_i - m_i' + \gamma \sum_{j \ne i} (\theta_j-m_j').
$$
Adding up all the equations yields
\begin{eqnarray*}
0 & = & \sum_i (\theta_i - m_i') + \gamma \sum_i \sum_{j \ne i} (\theta_j-m_j')
\\
0 & = & \sum_i (\theta_i - m_i') + \gamma \cdot (n-1) \cdot  \sum_i (\theta_i - m_i') \\
0 & = & (1 + \gamma \cdot (n-1)) \cdot \sum_i (\theta_i - m_i')  .
\end{eqnarray*}
There are two ways this can hold. Either $\sum_i (\theta_i - m_i') = 0$ or $\gamma = -\frac{1}{n-1}$.

Case 1. Suppose $\sum_i (\theta_i - m_i') = 0$. Then from
\begin{eqnarray*}
m_i' & = & \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j') \\
m_i' - \theta_i & = & \gamma \sum_{j \ne i} (\theta_j-m_j') \\
(m_i' - \theta_i) - \gamma \cdot (m_i' - \theta_i)  & = &  \gamma \cdot (\theta_i - m_i') + \gamma \sum_{j \ne i} (\theta_j-m_j') \\
(1 - \gamma) \cdot (m_i' - \theta_i) & = & \gamma \sum_{j} (\theta_j-m_j') = 0.
\end{eqnarray*}
In the truth-telling equilibrium this is true for all $i \in N$. If $\gamma = 1$ non truth-telling equilibria are also possible for any number of players, e.g. $n-1$ players reporting their type $+1$ and one player reporting her type $-n+1$. If $\gamma < 1$ and the equilibrium is non truth-telling then this is not possible.

Case 2.
Suppose $\gamma = -\frac{1}{n-1}$.   Let $x_i$ denote $\theta_i - m_i'$, so the equation system
$$
\forall i\in N: 0 = \theta_i - m_i' + \gamma \sum_{j \ne i} (\theta_j-m_j')
$$
becomes
$$
\forall i\in N: 0 = x_i + \gamma \sum_{j \ne i} x_j.
$$
For $\gamma = -\frac{1}{n-1}$ the matrix representation of this equation system is invertible so there is only a single solution, and $\forall i: x_i = 0$ is clearly a solution. This means that the only equilibrium is the truth-telling equilibrium.


Eric Reinert's conspiracy theory? After read Eric Reinert's "How Rich Countries Got Rich ... and Why Poor Countries Stay Poor", I've got the question:

Reinert theory is:

Rich West countries are investing to advanced branches at their home to get rich, and ...
Buy raw products from poor countries to send them back to stone ages.

If Reinert theory is true, why rich West is investing so much to advanced business branches on new East companies, like Foxconn?

I've read Eric Reinerts book and I do not think it is conspiracy theory because he explains also the economic background of the both symptoms you show.

One point he describes is how free trade discussion prefers the developed countries. Industrial production profits a lot more from scale effects than agricultural production. So he says, it is not fair to hinder countries to develop an industrial sector. Your argument about the so called tiger states has 'roots' in another development strategy of Japan for neighboring countries. The so called flying geese paradigma was one the reasons why it is now possible to buy high technology products from east asian countries. So it is more an example that strengthens Reinerts arguments. There was no leader goose for many other areas in the global south. Maybe china now plays this role. With the BRICS countries the situation today is more complex.  But the fact that the world is changing, can't be an argument against a theory which tries to be rooted in history.

That Eric Reinert points out still valid issues can you see in the discussion about the Economic Partnership Agreements.


Is it the most efficient to allow free transactions in financial markets? Most beginner level economics books include explanations why it is most effecient most of the time to allow free trade in the market of goods and services. (e.g. the theory of comparative advantage) However, few of them talks about similar things for financial markets (e.g. stock exchange, forex). The books will tell you how those financial instruments work, but they do not discuss whether it is the most efficient to allow free transactions in financial markets (just like the free trade in goods/services markets). Could anyone tell me if there are any theories on the topic?

What makes me curious is that, despite their huge role in our life, financial instruments are not classed as any factors of production, and seems to create no value economically. For example, fund managers are NOT enterpreneurs.

So should the government intervene to prevent speculation, or should they not intervene? Is there any theory on this?

As a part of the overall financial system, such instruments can increase market efficiency and facilitate exchanges.  Since all such instruments are essentially backed by government monopoly rights (at the most basic, the monopoly held by governments to create money) they would fall under "land" in terms of the classical factors of production.  While it may seem strange to count financial instruments as a form of "land" it makes sense if you understand "land" to really mean "any factor of production associated with monopoly / exclusive use rights."  As a form of land, the returns to financial instruments are thus economic rent and the root cause of speculation.  So yes, they should be regulated / taxed.

Note, however, that it is not true that they "create no value economically" -- monopoly rights exist for a reason, often specifically to increase overall market efficiency.  It makes more sense for government to grant rights-of-way to utilities and such, to prevent redundant over-building of utility lines, etc.  But the value so created is being created by government, not the private entities that often claim the rent as their own.  Hence they should be taxed at (nearly) 100% so that the rents go back to the community, rather than free-loading rentiers.

There is plenty of theory on all of this, though it is considered heterodox today.  Henry George (and his peers) wrote about it at length around the turn of the 19th-20th century, and economists such as Mason Gaffney continue to publish current works in this area today.  The New Physiocrats are another contemporary group publishing ideas along the same lines.


Value Function For Durable-Good Monopolist with General Distribution It is known that with a unit mass of consumers, each of whom has a value distributed between 0 and 1, one can think of the monopolist solving
\begin{equation}
\max_{p} \ p[1-F(p)]
\end{equation}
when marginal costs are 0 and where $F$ is the CDF of the consumers' valuations.
This yields the solution
\begin{equation}
p^*=\frac{1-F(p^*)}{f(p^*)}.
\end{equation}
$\textbf{Question}$: How would one write the value function (optimal profit function) of the monopolist in this case, given that the solution is implicitly (?) defined?

Can one just say that it is
\begin{equation}
\text{optimal profit function}=\frac{[1-F(p^*)]^2}{f(p^*)}
\end{equation}
or is this incorrect?

Many thanks.

You probably want to say that the optimal profit is $\pi=\frac{[1-F(p^*)]^2}{f(p^*)}$ where $F(x)$ is a given probability distribution, $f(x)$ is its density and $p^*$ satisfies $p^*=\frac{1-F(p^*)}{f(p^*)}$.

If you are dealing with a general $F$ function, you may want to state the conditions that guarantee that $p^*$ exists and it is unique. If $F$ is uniform or some other well-known distribution, however, you should probably find in closed form the value of $p^*$ and give an actual number as optimal profit.


Is income inequality decreasing? Mr. Yaron Brooks from the Ayn Rand Institute argues that global income inequality is decreasing. True/false?

Here is a fairly instructive figure showing a cross-country measure of income inequality:

It shows that inequality was increasing for most of the last 35 years, but has started to decrease again recently. (You can read more at the source for this figure: http://www.conferenceboard.ca/hcp/hot-topics/worldinequality.aspx)

Once data is weighted by population, we see a longer-term trend for decreasing inequality (due to rapid growth in China and India, which contain around a third of the world's people:

As @FooBar notes in a comment, a pattern of decreasing income inequality is consistent with economic theories of convergence.


Why isn't the Fed funds rate greater than interest rate on loans to the pubic First of all, fed funds rate is the rate banks borrow reserves' from each other. Interest rate on loans is the rate the bank charges to public for its loan. From one line of logic, we think of the rate bank charges to public should clearly be greater than the interest rate itself bear to borrow since banks are making a margin. And empirical data clearly support it.

However, another line of logic does not quite support it. We know the banks borrow reserve money in fed funds market, which is clearly better than the money it lends to the public in the following sense: one unit of fed funds can support 10 units of additional deposit if the reserve-deposit ratio is 10%. So the banks with this addtional 1 unit of reserve, can theoretically make 10 more units of loans and therefore creating 10 more units of deposit. So reserve is better money. Hence, theoreticaly, the fed funds rate should at leastbe higher than the rate banks make loans to the public, if not 10 folds.

My question being whether there is a logical fallacy in my understanding and if not, what kind of reality deviated it from theory.

I think you are missing the credit risk attached to any loan. You, as an individual borrower, are far more likely not to reimburse your loan than a bank. Thus the loan is far less risky.

Central bank money could only be qualified as "better" (even that does not mean much) because it is issued by the Central bank, deposits are issued by commercial bank, so you bear the risk that your bank fail, but it happens that in most country individual deposits are insured by some state agency. So I cannot really see why central bank money would be better than deposits.

Nowadays, the amount of reserves that a bank must hold does not actually play a big role in the economy (as opposed to what most of economic textbooks and economic professors are teaching in undergrad schools). The most important part is the interest rate that is paid on those reserve, which is fixed by the central bank and which act as an opportunity cost for the bank when it decides to lend to an individual rather than holding reserves.

However I see your point, but the fact is that this way of thinking is misleading. Banks create (in theory) as many loans as they want, because there are no physical constraints, if you want, that prevent them to do so. The only constraint is a regulatory constraint that states that in order to remain solvent (in order to be able to repay the depositors) banks should invest x% of the assets in reserves. Those reserves would be held on a central bank account in central bank money, and the return on those reserves will be the fed fund rate decided by the central bank (the same way a commercial bank decide how much interest it will pay on your savings account). So it is true that one more unit of reserves gives the right to issue more than one unit of loans but this does not mean that one money is better or worse than another. It's is just a matter of regulation.

It may be hard to get, but the point is really that central bank money is safer so cheaper than deposits, no matter what regulatory constraints apply to banks.

You may want to read this paper which may be a bit complicated but you should get the intuition. This two docs may be simpler (with nice pictures) : doc 1, doc 2.


Reason for Economic Slowdown in Western Nations Why do the U.S and other major Western Countries (e.g, Canada, Germany, in contrast to India, Australia, and China) have stagnated GDP growth? Comparing the GDP growth in the post-2000s from the 1990s and previous decades, the trendline seems to go down. Government debts have also increased by a lot among many countries. Is there a reason applicable to all of these nations that explains this?

The primary reason is that it is easier to catch up than it is to forge ahead.  India and China have lower GDPs per capita, so they can copy what other countries have done and gain.  But the United States, Canada, and Germany can't copy from India or China, as they are already higher.  Copying would make them lower.  And there's no one higher from whom they can copy.  They can copy a little from each other, but not enough to maintain high growth.

A good example is Japan.  Japan used to experience fast GDP growth.  Then it caught up to countries like the US and Germany.  Since then, it has stagnated.

Presumably this will also happen with China and India, but their high populations mean that it might not happen until their GDPs are bigger than those of the United States.

Note that China remains below the world average in GDP per capita.  Presumably it can grow until it is closer to the US or Germany.

Of course, it is also possible that there is something about the Chinese system that will limit it before then.  We can only speculate until their GDP stabilizes.  Some have speculated that China's growth is stabilizing now (it's been decreasing steadily since 2010), but we will need more time to make that determination.  At least ten years after it hits a steady rate, which it has yet to do.


Applying Economic Principles to Other Studies I was recently reading this article about some research that was done concerning the illegal killing of elephants.  There were two quotes that caught my attention:

  ...illegal killing [of the elephants] increased markedly after 2008 and was correlated strongly with the local black market ivory price and increased seizures of ivory destined for China...

and

  ...Illegal killing rates were strongly correlated with black market ivory prices in the Samburu ecosystem...

In the article, the correlation is treated as a previously unknown fact that was "discovered" by the research team and appears to be one of the major finds of the study.

However, it does not seem like the researchers can take credit for this "discovery."  It seems that an economist can reach the same conclusion by a much simpler method:

Increased seizures of ivory reduce the supply of ivory on the market.
Since the supply of ivory has decreased and the demand has remained constant, the price of ivory will increase.
The higher price will motivate individuals to enter the ivory market as suppliers (which, therefore, will lead to more elephants being killed).

Is there a particluar reason as to why the economic principles of supply and demand were not applied to this study in order to more rapidly reach this result?

This paper seems to be about practical policy. It seeks to understand what drives the killing of elephants. While what you write makes sense in theory, there might be some hidden factor that we do not perceive that has a much bigger effect (for example the actors of the market may not be rational) so looking at the data is always welcome.

I would also say that your economic reasoning is not the only possible one. I could also provide a line of reasoning reverses causality

More elephants are killed (this may be random, or it may be a result of unemployment in other sectors).
Because demand is unchanged, prices fall.
Since authorities always seize half of the ivory contraband, more contraband is seized.

If this is true then the seizures and the price change is a result, implying a different market dynamic. Unfortunately the paper only tested correlation, not causality, but this highlights that there can be more than one explanation to a phenomenon. In fact it is even possible to argue that the price of ivory and the number of elephants killed is negatively correlated:

More elephants are killed (this may be random, or it may be a result of unemployment in other sectors).
The authorities, who until now looked the other way, notice this and start seizing ivory, leaving almost none on the market.
The price of ivory rises.

Which of these explanations you find logical is somewhat subjective. Because there can more than one, a logical explanation may not be the right one. That is why econometric papers (could) provide useful insights.


The stochastic discount factor $M_t$ in a simple RBC model In these class notes, page 3, the author defines the stochastic discount factor as $M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$.

I'm trying to find the rationale behind it.

A cash flow at period t, is usually discounted by $\prod^t_{i=0} \frac{1}{1+r_i}$ (if I'm not mistaken). I was thinking of using equation (5) - in the notes - where we have that $\frac{1}{1+r_t}=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$.

However, since $\frac{E_{t-1}(u'(C_{t}))}{u'(C_{t})}\neq 1$, I'm not exactly sure how to proceed.

Also, just below equation (10), the author states that $E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$. How is this possible? Isn't the tower property exactly the inverse: $I_0\subseteq I_t\implies E(E(M_{t+1}|I_t)|I_0)=E(M_{t+1}|I_0)$ ?

Any help would be appreciated.

Somewhat inconsistent/careless use of indices. In

$$M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$$

it is assumed that we stand at period $0$ for any $t$ ahead we want to look at.

In

$$E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$$

it is assumed that we stand at any $t$ and we look one period ahead.

The two expressions don't give the same formulae for any given $t$, and there lies the problem/confusion.


Multiple Forgone Job Opportunities and Economic Profit I'm told that forgone job opportunities are implicit costs, to be included in calculating economic profit, but I can't seem to see how I could list and consider many different forgone opportunities. My question here is how do we do so?

For example, if I have a piece of land or big city real estate, there could be a great many different uses for it. If I considered each of these forgone opportunities an implicit cost, I would think the expression of my economic costs would be useless to me.

So I suspect that either I'm suffering from an oversimplified ECON101 explanation of implicit and opportunity costs or simply an incorrect one. Any references would be appreciated.

We don't use all possible opportunity costs. We only use the cost of forgoing the next-best-alternative.


Why do money markets fluctuate so much? Compared to commodity markets, money markets move a lot. My guess is that commodity markets are less elastic due to fixed production costs. But I don't know if that is the correct explanation.

The first reason for which volatility is higher is the liquidity of currencies. It has no storage or transportation costs. It can be used to purchase goods or traded again without additional costs. It doesn't age as fast as some commodities (agricultural products don't last forever). Therefore, the volume of trade is very high, and market participants can very easily enter or exit the market.

Forex is extremely suitable for speculation: money relates to a lot of political decisions, macroeconomic trends and many other driver that are easily and publicly available. Therefore it might be easier to small private traders to trade currencies than firms, which require deeper knowledge. It is indeed easier to read a standard newspaper than to read an earning call transcript.

Moreover, Forex market's design also favor volatility: currencies trade 24/24, not 9 to 5. That means more people have access to it at any point in time. Another critical point is that there is a gigantic quantity of derivatives based on currencies. They can be used both for speculative purposes (thus generating leverage and increase the potential upside of the bet), or to generate protection against currency movements.

All these elements contribute to make foreign exchange markets more volatile than commodities'.


Why is money in circulation a liability of the central bank? We know that the money today is fiat currency, that it is money because the government says it's so. So when new money is printed or loaned out to the commercial banks by buying treasury bonds, government does not technically "owe" any thing. The only thing that is stopping it from creating new money every time need arises is the possibility of inflation.

Given all of that why is still money in circulation listed as a liability of the central bank? Am I missing something here?

Balance sheets always balance, so assets equal liabilities.

Imagine a commercial bank goes to the central bank and wants cash. The central bank provides the cash, but asks for some of the commercial bank's loans (or government bonds) in return.

The central bank now has the loans (or government bonds) as assets and the cash as liabilities.

The cash is a liability, because if the commercial bank goes back to the central bank and gives back the cash, the central bank will have to give back the loans (or government bonds).

So while it's true that cash is not backed by gold, it is still backed by something. You can take your cash to the central bank, exchange it for government bonds, earn cash interest on your government bonds and then use this cash to pay your taxes. More generally cash is backed by the goods that you can purchase for it.


One-step Binomial model's Radon-Nikodym derivative In the one-step binomial model...

Question 1: What exactly is a '$\frac{d \mathbb Q}{d \mathbb P}$'?
I think it's $\frac{d \mathbb Q}{d \mathbb P} = \frac{q_u}{p_u}1_u + \frac{q_d}{p_d}1_d$, so it's some asset with payoffs $\frac{q_u}{p_u}$ and $\frac{q_d}{p_d}$, expected value of 1 and replicating portfolio $(x,y)$ of
$$x=\frac{1}{1+R}\frac{u(\frac{q_d}{p_d})-d(\frac{q_u}{p_u})}{u-d}$$
$$y=\frac{1}{S_0}\frac{\frac{q_u}{p_u}-\frac{q_d}{p_d}}{u-d}$$
This seems to be some replicating portfolio that is expected to payoff 1 at $t=1$.
Well, $(x,y)=(\frac{1}{1+R},0)$ seems to give the same payoff but with -100% lower risk

Question 2: What exactly is a/an '$X\frac{d \mathbb Q}{d \mathbb P}$'?
We could say that the price of X uses not
$$E[X] = X_up_u+X_dp_d$$
but rather
$$E^{\mathbb Q}[X] = E[X\frac{d \mathbb Q}{d \mathbb P}] = X_u\frac{q_u}{p_u}p_u + \frac{q_d}{p_d}p_d$$
I guess it's some asset that pays $X_u\frac{q_u}{p_u}$ or $X_d\frac{q_d}{p_d}$ then replicating portfolio is...then idk. Not sure we need one since we're using real world probabilities anyhoo

In economics, the Radon-Nikodym density $\frac{d \mathbb Q}{d \mathbb P}$ of the risk-neutral measure $\mathbb Q$ with respect to the physical measure $\mathbb P$ is the price of Arrow-Debreu securities. It is a price, not a claim.

In the binomial setting, there are two AD securities, $1_u$ and $1_d$. The former entitles the holder to 1 unit of numeraire if, and only if, state $u$ realizes.
The no-arbitrage price of $1_u$ is $\frac{q_u}{p_u}1_u$ (say $R = 1$). Similarly for $1_d$.

In general, the price of the AD portfolio $1_{\Omega'}$---which pays off 1 unit of numeraire if, and only if, state $\omega \in \Omega'$ realizes---is
$$
E^{\mathbb P}[ 1_{\Omega'} \frac{d \mathbb Q}{d \mathbb P}] = \mathbb Q (\Omega').
$$
So
$$
E^{\mathbb P}[\frac{d \mathbb Q}{d \mathbb P}] = 1
$$
is then the price of a bond/risk-free security (discount accordingly if $r \neq 1$).

Extending from $1_{\Omega'}$ to a general $X$ gives the usual risk-neutral pricing formula $E^{\mathbb Q}[X]$, as you have written out.


How to understand share value vs real worth of the company When read in the news articles for stock market price, I learned that the share value of a company changes in a year.  My question is does the share value really reflect if the company is doing good or not?

Example: In the recent aPIGEE article it says,  the company in a 52 week duration share value ranged from \$5.45 to \$16.34 .

When a stock price changes, it implies that the present value of future cash flows from the company to investors has changed.

A multitude of different things can change that number, and ascertaining why a stock price changed is often difficult or near impossible.

Simple example: one period cash flow, no uncertainty

Let $r^f$ be the risk free rate. Let $c$ be some cash flow one period ahead.

The present value of the cash flow is:

$$ p = \frac{1}{1+r^f} c$$ And that's what it would trade for in efficient market. Two key observations:

If the discount rate $\frac{1}{1 + r^f}$ were different, then the price would be different.
If the future cash flow $c$ were different, the price would also be different.

Conceptually, the price of a financial asset can be decomposed into some discount rate times the quantity of a future cash flow.

More complicated setup, one period cash flow, uncertainty

With the addition of uncertainty, you get a conceptually similar result: the price of a financial asset should be the inner product of a discount factor with the future cash flow. Let $\mathbf{s}$ be a random variable denoting the stochastic discount factor, and let $\mathbf{c}$ be a random variable denoting the future cash flow. The Law of One Price (LOOP) implies the existence of a stochastic discount factor $\mathbf{s}$ such that:

$$ p = E[\mathbf{s}\cdot\mathbf{c}] $$

Let $\mathcal{F}_t$ be the set of information available at time $t$. As new information becomes available, the conditional expectation will change:

$$ p_t = E\left[\left. \mathbf{s}\cdot \mathbf{c} \, \right| \, \mathcal{F}_t \right] $$

This looks fancy, but it's almost the same idea as the simple, one period cash flow case. The price of an asset today is the product of a discount factor and future cash flows. In an efficient market, changes in asset prices may be due to news about discount rates or news about cash flows!

If the share price triples, that's almost certainly due to positive news about future cash flows. Small changes though, or changes at the level of the overall S&P500 index are trickier.

Practical Implications

In the fall of 2008, the stock market collapsed! Asset prices fell. What did this mean though? The decline may have be due to declining expectations of future cash flows from the corporate sector. But it also may have been due to rising discount rates, that risky future cash flows were being assigned a lower price! In a strict sense, disentangling discount rate news from cash flow news is nearly impossible. In general though, when looking at an individual company, you're almost certainly seeing more variation due to cash flow news than discount rate news, but strictly speaking, there's no way to make a clean decomposition.

Sensitivity of the current price to future cash flow news...

A number of features can make the price very sensitive to news:

Leverage: If a company has a lot of debt and is near bankruptcy, the cash flows to equity may be massively different depending on whether the company survives or not.
Time horizon: How much is something like Uber worth? Will it take over global transportation with robotic, AI vehicles? Or will it sink under a morass of rising labor costs and government regulation? The big payoffs from something like Uber are in the distant future, and forecasts of the distant future may be volatile, sensitive to certain types of news.


Why don't governments ban businesses from passing tax costs onto their customers? So, after a bit of discussion in the comments of this question on the Politics SE, I figured I'd post it here:
One of the arguments against taxing large businesses is that they'll just pass the costs on to their customers, effectively raising inflation and harming the working class, or cut costs by making employees redundant.
Why don't countries pass laws imposing taxes and legally mandating that companies are forbidden from doing either of these things, and that it must be paid directly out of their profit and/or owners' equity? This might possibly take the form of the government mandating that all businesses over a certain size must issue X% of their shares to the government without granting the government voting rights, so that the government owns X% of all of their profits.
Are there any countries who have done anything like this? How did it work for them, if so?

I don't see how the government taking X% of a business's profits is stopping the business from increasing prices (or is this just the penalty in the enforcement clause?), so I am going ignore that part.

What you are describing is essentially setting prices, determining what prices a business can and cannot set. (If you merely prohibited passing on costs, the business would claim that they are not doing this, they merely happen to be raising prices because of other reasons.)
This usually means a planned economy rather then a market based one, as the most important function of the market is to set the prices. Allowing the market to do this results in all sorts of nice things: it balances supply and demand, provides incentives for innovation, etc.
To some extent even market based economies engage in price setting. Sometime governments set price ceilings on some products. The usual reasons behind this are

Fairness: at the lower price demand will outstrip supply (in fact the low price will likely diminish supply), but everyone will have a chance of buying some of this good, not just the wealthy who could afford the higher equilibrium prices. (See rent control.) The dynamic effects are assumed to be negative: the lower price will disincentivize future investment.
Non-competitive markets: if the market is not functioning properly, e.g., someone has a monopoly on the good, price setting may actually increase the quantity sold and have a positive effect on net welfare.
Anti-price gouging: if there is a short term demand spike and supply is strongly inelastic, prices may skyrocket in a given market. (Such was the case in some good markets at the start of the Covid-19 pandemic.) In this case, a price ceiling may transfer welfare from producers to consumers without significant welfare loss (if supply is really strongly inelastic). The term "anti-price gouging" is somewhat unfortunate as it is emotional with sensationalistic overtones.
There are also non-economic reasons: such a measure may be politically popular, even when there are downsides, as these are not immediately visible.

Recommended further reading:
REVIEW OF ECONOMIC THEORIES OF REGULATION


Relation between linear utility function and U=max{x,y} I'm studying general equilibrium theory, and in the study guide I came across a utility function of the type $U=\max\{x,y\}$, which I'm not that familiar with. I study mainly from two books: Intermediate microeconomics by Varian and Nicholson's Microeconomic Theory and couldn't find any details about the nature of this kind of utility function in either.

My teachers over the semesters have mentioned some details about it as a curious fact, so I know the choice of the consumer is driven by the relative price of the goods, in this case x and y, so he'll choose to consume the cheaper good, and if the relative prices are equal he'll be indifferent towards both. Also I know the indifference curves have the shape of an inverted L.

What is not clear to me is how is it related to the Perfect Substitutes utility function since both seem extremely similar to me, but the indifference curves are very different.

Is the utility function $U=\max\{x,y\}$ a more general case?
More importantly, when the relative prices are the same, how can we illustrate consumer's choice in that inverted L indifference curve, doesn't it turn linear?

The optimal choice set for a max function and a perfect substitutes function with equal relative prices share some solutions [i.e, boundary solutions], but in general, the indifference curves, and hence non-boundary solutions, are different.
Main Idea
For both a max(x1 x2) and perfect_sub(x1 x2) utility function, the point, say, m/p1 (or m/p2) would maximize utility. So both utility functions share boundary solutions. But think about the IC of m/p1 for a perf_subs consumer and think about the IC of m/p1 for a max consumer. You see the points that the two consumers think are 'just as good' as that m/p1 point are very different.
That is, the same boundary bundle might be a solution to both utility functions, but the other solutions will differ.
Logic
The reason is that U = perfect_subs is a (not-strictly) convex utility function, whereas U = max isn't. That is: the consumer is either indifferent between, or actually prefers, less extreme combinations to more extreme combinations for the former. As for the latter? Well, they just care about the max; they like extreme combinations (e.g, [C = (m/p1, 0)])
That's why for U = perfect_subs with relative prices the same, both A = (100,0) and B = (ß(x1), (1-ß)x2) have the same utility. That is,B lies on a straight line IC connecting (100, 0) to (0, 100). Whereas for U = max(x1 x2), B would not be on the same IC as A (unless ß is 0 or 1, in which case, we're back to talking about boundary solutions!)
The word, max is confusing here. Essentially, the perfect_subs (x1 x2) consumer wants to maximize total goods. The max(x1 x2) consumer just cares about maximizing the larger quantity in his {x1, x2} bundle.
Tl;DR
To be more concrete: If A = (100,0) and B = (75, 25), a perf_subs consumer is indifferent between A and B; a max consumer is not.


Monetary policy rate and fiscal deficit A colleague in work mentioned today that Chile is keeping a flat monetary policy rate given "there is a higher expected fiscal deficit".
If there is an expected fiscal deficit (i.e., spending more than what you earn), why would you keep the monetary policy rate low (i.e., lend money cheaper). To me it sounds a little contradictory, but wanted to hear other views.
Is there another reasoning why you would keep a low flat monetary policity rate if there is an expected increase in the fiscal deficit?

If a country is on a deficit it would want to keep rates low since it will most likely have to borrow money to pay for it's deficit.
It also has the possibility of debasing it's currency which in turn debases it's current/future debt.


Expected inflation in the real interest rate equation Real interest rate = Nominal rate - Expected inflation

In the above equation, in a quarterly data-set, which expected inflation shall be used? next quarter (q+1) or the same quarter of next year (q+4)? and why?

Your formula isn't quite right. If $r$ is the nominal interest rate, and $\pi$ is the inflation rate, the real interest rate is $\frac {1+r}{1+\pi}-1$. The formula $r-\pi$ is approximation for small rates. (Note: for all of this, I'm using "rate" to mean the multiplicative factor minus one. So if the balance is multiplied by 1.2, the rate is 0.2.)
The inflation should be over the period over which the interest rate is being calculated. It's like to like: the real interest rate at a particular time is given by the nominal interest rate at that time compared to the inflation at that time.
The fact that you think you are wondering whether you should use the future inflation suggests that you may be confusing the spot inflation rate with the Consumer Price Index. You can get the real interest rate by taking the total nominal interest times the ratio between the CPI at the end of the period to the CPI at the beginning. That is, if $r$ is the total nominal interest, $CPI_i$ is the starting CPI, and $CPI_f$ is the final CPI, then the total real interest is $(1+r) \frac {CPI_f}{CPI_i}-1$.


Elucidation needed on textbook's description of the effect of increased Interest rates on the money supply curve? A textbook, when explaining the effects of higher interest rates, claims that:

  The initial equilibrium is with real output at $Y_0$,the price level
  at $P_0$ and the rate of interest $r_0$. An increase in the rate of
  interest to $r_1$ will need to be balanced by a decrease in money
  supply to maintain money-market equilibrium.

The illustration below accompanies the claim.

How/why does the money supply shift leftwards to accommodate a return to "money-market equilibrium", and what is such an equilibrium to begin with?

Additionally, is this shift automatic or is it one where the government must intervene?

The graph is a little misleading.

Imagine the interest rates increase without $MS$ shifting to $MS1$. Here the demand for money is less than the supply, so there is a surplus. If the MS does not shift, eventually the interest rate will fall again until the shortage is no longer (we are back in equilibrium). If the Fed wants to maintain the higher interest rate, $r_1$, it also must shift $MS$ to $MS1$.


What is the public spending / government expenditure share of GDP in Europe? I have found data that seems to be contradicting:
statista gives public spending at country level between 40% and 60%, but
eurostat gives an aggregate figure of about 20%, as shown in the following pie chart:

Which is it? Is there a difference in the definition of public spending and general government expenditure?
Edit: I am comparing different years, 2017 and 2019, but the percentage change of each category is relatively stable over the period, giving the same order of magnitude.

The data provided by Statista are estimates of public expenditure to GDP ratio for year 2017 while the Eurostat data are for year 2019. Consequently I do not see any a priori reason why the two series would be expected to match. They are not necessarily contradictory as they are for different years.
Furthermore, I can't see the information on Statista on the methodology/sources as it is hidden behind a paywall. It is quite possible that Statista's definition of public spending does not match the Eurostat definition of government expenditure. The definition provided by Eurostat is:

Final consumption expenditure by general government includes the value of goods and services purchased or produced by general government and directly supplied to private households for consumption purposes.

Where general government is defined thusly:

of institutional units which are non-market producers whose output is intended for individual and collective consumption, and are financed by compulsory payments made by units belonging to other sectors, and institutional units principally engaged in the redistribution of national income and wealth.

If you happen to know the Statista definition you can compare them.


Choice correspondence notation My question is about the following notation:

I have noticed in several places (for example here (page 15)  and here (page 1)) that different authors use different notations for choice correspondences. Given a set $X$ of objects that might be chosen and a collection of non-empty subsets $\mathcal{A}$ of the power set of $X$, some use

$c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset$ (as in the second link)
$c:\mathcal{A} \rightarrow 2^{X}\backslash \{ \emptyset\}$ (as in the first link)

They both want to express that the co-domain of the choice correspondence must include all the non-empty subsets of the power set of X, i.e from any menu of options, you must choose something.

My question is, which one is the correct way to define it? Clearly, since $\emptyset \neq \{\emptyset\}$, both definitions cannot be equivalent.

My intuition would tell me that from the list above, (1) is the correct one. For example, if $X=\{\text{pasta (p)},  \text{chicken (c)}, \text{sushi (s)}\}$, then the power set of $X$ is:

$$
2^{X}=\Bigg\{\emptyset, \{p \},  \ \{c \}, \{s \},     \{p,c \}, \{p,s \} , \{c,s \}, \{ p,c, s\} \Bigg\}
$$

Since I want to “take out” the empty set from $2^{X}$ so that the choice from any non-empty subset of $X$ is itself non-empty, and since $\{\emptyset\} \notin 2^{X}$, it should be

$$
c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset
$$

Thank you for any answers!

"1." is wrong and "2." is correct, because (see e.g. Wolfram):

$$A \setminus B = \{x: x \in A \text{ and } x \notin B\}$$

Say for example we have $S = \{a, b, c\}$ and $T = \{b,c\}$.

Then we'd write $S \setminus \{a\} = T$. We do NOT write $S \setminus a = T$.

And so in general, for any set $S$, $S\setminus \emptyset =S$.

We write $2^{X}\backslash \{\emptyset\}$ because $\emptyset \in 2^X$. In contrast, we have $2^{X}\backslash \emptyset = 2^{X}$.

Example. The set $S$ contains four elements: the empty set, an element called $a$, the set containing the empty set, and the set containing $a$.

$$
S=\Bigg\{\emptyset,a,\{\emptyset \},\{a \}\Bigg\}.
$$

Then we have:

$$ \begin{align} S \setminus \emptyset & = S
\\ S\setminus\{\emptyset\}& = \Bigg\{a,\{\emptyset \},\{a \}\Bigg\},
\\ S\setminus\{\{\emptyset\}\}& = \Bigg\{\emptyset,a,\{a \}\Bigg\},
\\ S\setminus\{a\}& = \Bigg\{\emptyset,\{\emptyset \},\{a \}\Bigg\},
\\ S\setminus\{a,\{\emptyset\}\}& = \Bigg\{\emptyset,\{a \}\Bigg\},
\\ S\setminus\{\{a,\emptyset\}\}& = S.
\end{align} $$

The last equation is true because $S$ does not contain the set $\{a,\emptyset\}$.

See Enderton, in particular p.3, top paragraph -- it's well worth spending a little time if only on the first chapter.


Why is market power presumably high when goods are fungible and demand relatively inelastic? What does the fact that the goods in an industry are fungible add to a market power analysis if you already know demand is relatively inelastic?

"Fungibility", when referring to goods, essentially means homogeneity -all units of the good are exactly the same from all aspects that matter to consumers (including "exterior" aspects, like location etc), and so one can substitute for another.

If one knows that in a market essentially only one homogeneous good is traded, then one knows that he must look for market-power sources elsewhere and not in product differentiation (since the latter is non-existent), like for example in barriers to entry, cost structure etc.

If "fungibility" is narrowed down to the inherent characteristics of the good, then geographical location or other such aspects, may become possible sources of product-differentiation, and so of market power.

Fungibility is a property coming from the supply side, so by construction is unrelated to the elasticity of demand (which depends on preferences). Therefore it is a useful piece of information on its own.


Interest rate rule in monetary DSGE model I am studying DSGE models and try to solve exercise 2.2 from Gali's (2008) book. In short, consider the simple classical economy where the following approximate equilibrium conditions must be satisfied:
$$y_t = E_t\{y_{t+1}\} - \frac{1}{\sigma}(i_t - E_t\{\pi_{t+1}\} - \rho)$$
and
$$r_t = i_t - E_t\{\pi_{t+1}\}$$
and $y_t$ and $r_t$ determined independently of monetary policy.

Derive an interest rate rule that guarantees full stabilization of inflation (i.e. $\pi_t = \pi^* \;\; \forall t$).

I thought of the simple rule like $i_t = \phi\pi_t$. It is possible to show (using forward iteration) that in this case
$$\pi_t = \sum_{i=0}^{\infty}\frac{1}{\phi^{i+1}}E_t\{r_{t+i}\}$$
and letting $\phi \to \infty$ inflation goes to zero. However, I think that this not what was meant by the author. Any help appreciated.

I've just solved this problem. First of all, your solution does not make too much sense, as in a simple interest rate rule it must hold that the sum of all coefficients must be greater than one. In your case this means that $\phi>1$. Therefore, the series would converge not to zero. Second, an interest rate rule should try to offset fluctuations. This means that the central bank should try to offset gaps from its target (like in the task). Try instead the following interest rate rule.
\begin{equation}
i_t=\rho+\phi_{\pi}(\pi_t-\pi)+\sigma(\Delta y_{t+1})+\pi
\end{equation}
Now what is the idea? Assmung that the output does not change, this term is zero. Using the Fisherian $i_t=r_t+\mathbb{E}_t\{\pi_{t+1}\}$ we get
\begin{equation}
\rho+\phi_{\pi}(\pi_t-\pi)+\sigma(\Delta y_{t+1})+\pi \overset{!}{=}r_t+\mathbb{E}_t\{\pi_{t+1}\}
\end{equation}
which can be summarized as
\begin{equation}
\phi_{\pi}^{-1}\left(r_t-\rho+\mathbb{E}_t{\bar{\pi}_{t+1}}-\sigma(\Delta y_{t+1})\right)=\bar{\pi}_t,
\end{equation}
where $\bar{\pi}_t=\pi_t-\pi$.
Solving forward yields
\begin{equation}
\bar{\pi}_t=\sum_{k=0}^{\infty}\phi_{\pi}^{-(k+1)}\mathbb{E}_t\{r_{t+k}-\rho-\sigma\Delta y_{t+1+k}\}
\end{equation}
Notice that for $|\phi_{\pi}|>1$, the series converges. However, your log-linearized Euler equation can be rearranged such that it holds that $r_t=\rho+\sigma\mathbb{E}_t{\Delta y_{t+1}}$. Finally, this can be plugged into $\bar{\pi}_t$ such that $\bar{\pi}_t=0 \quad \forall t$. Notice that this expression finally is your answer, as $\bar{\pi}_t=\pi_t-\pi=0$ means that the central bank always achieves its target.


Is the Trans Pacific Partnership good or bad for the U.S? I've explored a number of different articles and some say it's good because american consumers/business aren't impacted by the tarrifs, however some other sources say we barely export anything to the counteries part of the TPP and hence we're giving them too much benefit for nothing.

I have found plenty of sources saying it's 'good' or 'bad' and they cite numbers such as '50000' added jobs, 5% increase in gdp, etc but there is no actual discussion of how they came up with these estimates.

These estimates are almost certainly bs.
The TPP would benefit consumers because easier imports may make consumer goods cheaper directly and through more competition on product markets.
The TPP would hurt some workers, because increased import
competition would force some American companies out of business.
The TPP would benefit other workers, because their companies would
export to Asia and grow.

I think economists consensus is that on average and in the medium run, free trade will benefit people in both countries. However, it has also become pretty clear that free trade hurts some people quite badly in the short run because they'll lose their jobs and possibly won't find new ones. It has also become pretty clear, that even though in theory you could put policies in place that help people and places that are hurt by trade at the expense of those who benefit from it, this is almost never done in practice. If you are for more or less free trade depends largely on what weights you apply to this trade-off.