Meaning of assumptions regarding continuum of agents in economics models I am reading an economics paper which contains a model with households and banks. In the model, banks are owned by households. The authors make this assumption when they discuss households (section 2.2, page 8):

  There is a continuum of risk-neutral households denoted by $h \in [0,1]$ ...

When they move to discussing banks, they make a similar assumption (section 2.3, page 9):

  There is a continuum of banks denoted by $b \in [0,1]$ ...

I have seen countless papers in economics with such assumptions regarding continuum of agents. Most of the time, I just gloss over this but I don't really understand what these assumptions mean, what they are useful for and why there are there in the first place. It seems like it's something basic and nobody discusses it but I really want to grasp the meaning and purpose of this assumption.

Economic models often have a finite number of agents. In such models, agents are commonly named using the positive integers (i.e. 'natural numbers'): so the set of agents is $\{1, 2, ..., n\}$.

Other economic models have an infinite number of agents. For example, the number of agents might be assumed to be the same as the number of positive integers. In that case, the set of agents is $\mathbb{N}=\{1, 2, ...\}$. It is said to have a 'cardinality' (or size) of aleph-null.

Finally, some economic models have even more agents. In the examples you mention, each agent is named using a real number in $[0, 1]$; so the set of agents is the set of real numbers in $[0, 1]$. This set is said to have cardinality aleph-one and can be shown to be 'larger' than the set of natural numbers (in the sense of 'larger' defined by Cantor).

Of course, this is not very realistic -- the justification, presumably, is to make the mathematics easier. In particular, it allows for integration (e.g. to find total consumption in the economy) as opposed to summation.


Effect on Large Company on Redistributing Profit from Shareholder to Employees Looking over the information in the tour I'm aware this might be too opinion based or require too long answers, so I apologies in advance if this is off topic.

In an attempt to keep the question tightly focused I'll restrict it to one specific example.

Recently Bernie Sanders has been taking aim at large corporations that make significant profits while leaving employees struggling to buy food and healthcare. One of the more prominent examples is the Disney Corporation.

According to online information in 2017 Disney made a net profit of 9.8bn USD and employed approx 200,000 people. If we assumed that 80% of these employees were the low wage earners referred to by Bernie Sanders, taking approximately 10% of this net profit and redistributing it as wages would allow for a pay rise for each individual of over $2 an hour.

A net profit of $8bn would still seem to indicate a very healthy company that would presumably be attractive to shareholders and investors. What other economic impacts would be expected if Disney reduced it's net profit margin by 10% purely by increasing low wage employee remuneration? Or what is wrong with the publicly available numbers/my understanding that means these figures are inaccurate?

What other economic impacts would be expected if Disney reduced it's
  net profit margin by 10% purely by increasing low wage employee
  remuneration?

The reduction of net profit margin might exceed the percentage of wage increase if benefits, insurance premiums, and any [Disney's] other obligations are tied to wages.

Likewise, a reduction of net profit margins might prompt shareholders to dump Disney stocks and invest in other companies they consider more profitable and/or unlikelier to act to the detriment of shareholders.

Companies with significant levels of debt might experience an increase in financial costs (higher interest rates) if --for instance-- the reduction of profit margin prompts a rating agency to lower the company's grade a notch.

This answer is neither an exhaustive list of possible impacts nor an assessment of whether or not wages at large corporations currently are at fair levels.


Why are simple returns additive over securities? I've found a few sources for instance this one, that makes the claim that over a portfolio, simple returns are additive. I can see why it doesn't make a difference if we use net or gross returns as long as the weights add to one, but I can't see why we should get this additive behaviour for a quantity that has a division in it. Perhaps I am using a bad definition of weight?

If someone could share a derivation for a portfolio of two assets that would be sufficient.

We have a portfolio of two assets, with prices $P_1(0), P_2(0)$ at $t=1$. In the next period, their prices are $P_1(1), P_2(1)$.

The portfolio market value at $t=0$ is $V(0) = W_1 P_1(0) + W_2 P_2(0)$.

The portfolio weights $\omega_i$ (for $i =1,2$) are:
$$
\omega_i = \frac{W_i P_i(0)}{W_1 P_1(0) + W_2 P_2(0)}.
$$

(We can easily see that $\omega_1 + \omega_2 = 1$.)

The return for an asset $R_i$ is given by:
$$
R_i = \frac{P_i(1)}{P_i(0)} - 1.
$$

The market value of the portfolio $V(1)$ is equal to:
$$
V(1) = W_1 P_1(0)(1+R_1) + W_2 P_2(0) (1+R_2).
$$
Then,
$$
V(1) = (W_1 P_1(0) + W_2P_2(0))(1 + \frac{W_1P_1(0)R_1}{W_1 P_1(0) + W_2P_2(0)} +
\frac{W_2P_2(0)R_2}{W_1 P_1(0) + W_2P_2(0)}),
$$
$$
V(1) = V(0) (1 + \omega_1 R_1 + \omega_2 R_2).
$$
If we define the portfolio return $R$ as $\omega_1 R_1 + \omega_2 R_2$, then
$$
V(1) = V(0) (1 + R).
$$

This is easier to see if we normalise prices.


Fair and efficient allocation of "family goods" Consider an exchange economy with two goods, e.g. home furniture (x) and electrical equipment (y). The interesting thing about these goods is that, when a family owns a bundle, all members of the family enjoy the same bundle (it is like a "club good" but only for the family).

There are two families. In each family, there are different members with different preferences over bundles. Assume that all preferences are monotonically-increasing and strictly convex.

An allocation is a pair of bundles, $(x_1,y_1)$ for family 1 and $(x_2,y_2)$ for family 2.

An allocation is called envy-free if:

All members of family 1 believe that $(x_1,y_1)$ is at least as good as $(x_2,y_2)$;
All members of family 2 believe that $(x_2,y_2)$ is at least as good as $(x_1,y_1)$.

An allocation is called Pareto-efficient if there is no other allocation of bundles to families such that all members of all families weakly prefer and at least one member of one family strictly prefers.

Under what conditions does a Pareto-efficient envy-free allocation exist?

If each family has a single member, then a Pareto-efficient envy-free allocation exists; this is a famous theorem of Varian. Has this theorem been generalized from individuals to families?

Right now I'm not sure about the equivalence of the relabeling, and therefore the usefulness of this anwer -- see comments below.

This is the beginning of an answer and an attempt to demonstrate how strong the necessary assumptions would have to be to guarantee existence.

Let's transform the problem into one that's equivalent but a bit easier to work with. Instead of indexing over families, let's instead index over the agents (members of families). The key to this relabeling is understanding that families can be written as constraints: If agents $i$ and $j$ belong to the same family, then $x_i=x_j$ and $y_i = y_j$.

Now we're back in the standard environment with individual agents (not families) but with these familial constraints. Recall the proof of Varian's theorem, which you link in the question. It uses the existence of a competitive equilibrium from equal incomes. In this context, we would need the existence of a competitive equilibrium from equal incomes in which the familial constraints were also met. This is going to be very difficult to do. For instance, consider $i$ and $j$ are in a family, and
$$
u_i=x_i + \varepsilon y_i \:\: \text{ and } \:\: u_j = \varepsilon x_j + y_j
$$
where $\varepsilon>0$ is tiny. These preferences are monotonic and convex. Basically, one family member cares about $x$ and the other cares about $y$. If each of the two agents is purchasing $x$ and $y$ to maximize his or her utility, you would not expect $x_i^* = x_j^*$ or $y_i^* = y_j^*$ in the competitive equilibrium (see addendum at end).

This is why you certainly need some assumption on preference similarities within families (at least to use a version of Varian's proof). My sense is that if you give me any arbitrarily small difference in preferences between family members, I can construct an example around it where there exists no CEEI in which they choose the same allocation. And then, at the very least, you can't use Varian's proof.

Two questions:

Do you agree that my reformulation of the problem is formally equivalent to yous?
Can you think of any assumption weaker than assuming preference homogeneity within the family that I can try to invalidate with a counter-example?

Addendum: Remember that in a competitive equilibrium, each agent's marginal rate of substitution (MRS) equals the price ratio. Here, my agents have constant and different MRS's, so there can exist no competitive equilibrium with a price ratio that equals both of their MRS's. If each agent has an MRS that varies, then perhaps they could happen to be equal at the equilibrium price ratio. So maybe you could get away with some notion of local homogeneity of familial preferences. But you need to have them be locally homogenous at the competitive equilibrium, which is exactly what you're trying to prove exists, so it would be a bit circular.

Important note: As mentioned previously, I'm assuming that the only way to prove existence is how Varian did it, via CEEI. There may be other proof techniques that skirt these issues, but I suspect not.

Beyond CEEI: As the OP points out in the comments, proving existence of PEEFs through CEEI as Varian does is somewhat restrictive. I do not have a lot to say about proving existence of PEEFs directly, but the following is readily apparent: For any allocation satisfying your Pareto efficiency condition (ignore envy-freeness for the moment), for any $i,j$ such that $x_i, x_j, y_i, y_j > 0$,
$$MRS_i = MRS_j$$
If this weren't true, there would be a Pareto improvement. Competitive equilibrium essentially equates the MRS's through the price ratio, but you still need to equate these MRS's just to find a Pareto efficient allocation. I think the familial constraints will make this very difficult -- it's not hard to come up with an environment and familial constraints such that there exists no Pareto efficient equilibrium satisfying those constraints. In any case, this could be another partial step towards an answer: Forget about envy-freeness. First try to come up with an assumption on preferences (and maybe on familial constraints) that guarantees the existence of a Pareto efficient allocation that satisfies familial constraints. Then worry about envy.


How do you define the market cap of an ETF? I'm hesitating between these 2 solutions:

The conventional calculation:

MCAP = #shares * SharePrice

Perform the weighted average of the MCAP of each stock in the ETF

MCAP = sum(iStockWeight * iStockMCAP)

Thanks for your help.

An ETF’s assets will fluctuate based on both changes in the value of
  the underlying securities and the creation of new shares or redemption
  of existing shares. It is worth noting that there may be a difference
  between an ETF’s market capitalization and the net asset value (NAV)
  of its underlying securities. The difference results in either a
  discount or a premium in the trading price of the ETF.

ETF DB: Largest ETFs: Top 100 ETFs By Assets

The market capitalization at time $t$ is therefore  the number of shares at $t$ times the price of those shares at $t$. The net assets are the total value of all the assets in the fund at 't':
$$ Net\ Assets_t = U_{t} \cdot NAV_t = U_{t} \cdot \sum_{i} (S_{i,t} \cdot weight_{i,t})$$
where $S_{i,t} $ is the value of a share of stock $i$ at time $t$, $weight_{i,t}$ is the fraction of the index invested in stock $i$ at time $t$, and $U_{t}$ is the number of shares of the index.
Or equivalently:

$$ Net\ Assets_t = \cdot \sum_{i} (S_{i,t} \cdot q_{i,t}) $$
Where $q_{i,t}$ is the number of shares of stock $i$ at time $t$ held by the ETF. This sounds simpler but it is often easier to find index weights than exact share holdings measured in number of shares.

But for open-ended funds that allow creation and redemption or have minimal expenses these NAV and market price based approaches will give similar answers.


Diagram with axes $r$ and $S,I$ In the figure below, I have a coordinate system where $r$ depends on $I$ or $S$, but I do not completely understand why I can have multiple functions inside this coordinate system.

Why can I write $S(r)$ and $I(r)$? Normally, in an x-y-diagram, I interpret it as $y(x)$, since $x$ varies and $y$ depends on $x$.

The diagram below is a quite normal diagram in economics, so I know it is correct, but why isn't the figure 'transposed', i.e. with $r$ on the $x$-axis and $S,I$ on the $y$-axis?

It doesn't make sense that $r = S(r)$ or $r = I(r)$.

It was Alfred Marshall's doing, but based on a premise that still prevails in many parts of Economics: if prices are assumed totally elastic, then, although it may be the case that they are taken as given by the individual buyers or sellers, at market level it is quantity that determines price. So price is indeed the dependent variable, if you care about the market and not about the individual.

And here is page 96 from Marshall's Principles of Economics (8th ed.1920), (the 1st edition was in 1890), the funny thing being that here Marshall talks about... individual demand! (but it would be totally confusing to swap axes between graphing individual demand and graphing market demand). Read the annotated footnote.


Where can I find data related to foreign remittances? I am doing research related to remittances, where can i find a good source of data?

Thanks

In the World Bank website you can find data on remittances such as

Annual and Monthly Remittance Flows to Selected Countries,
Bilateral Remittances Matrices,

and

Household survey data on migration and remittances.


Impact of news coverage on ecomomy I was wondering whether there exists a study which measures the impact of economic news coverage to gdp or something like that

I have not heard of such a study and I have a feeling studying how a daily news item would affect an annual measurement value would be a pretty futile experiment.

However, there are some studies on how announcements from central banks or other financial authorities might affect expectations in the short run. My collaegue actually did her licensiate thesis on how ECB board announcements affected the growth path of euro inflation.

A pretty recent article would be Moessner, R.: "Effects of ECB balance sheet policy announcements on inflation expectations." in Applied Economics Letters and that might be a good place to start looking.


When estimating VAR(1) by OLS, I get transpose of AR matrix. Why? Model: $y_t = \Phi y_{t-1} + (\mathbb{I_n}-\Phi)\mu + \Sigma \varepsilon_t$
s.t. $y_t$ is a $(n \times 1)$ vector, and $\varepsilon_t \sim N(\vec{0},\mathbb{I}_n)$. $\mathbb{I}$ is the identity matrix. $\Phi$ and $\Sigma$ are $(n \times n)$ matrices.

Consider the following MATLAB code.

Sigma = eye(2); %sqrtm(Covariance matrix)
Phi{1} = [0.9,0.12;0,0.9]; %Autoregressive matrix
mu = [0;0]; %Constant vector
nTs = 1000; %Length of simulation

%Specify Model
mdl = varm(2,1);
mdl.AR = Phi;
mdl.Constant = mu;
mdl.Covariance = Sigma^2;

%Simulate Model
simTs = simulate(mdl,nTs); %Simulate VAR(1) (nTs x 2)

%Estimate Model
X = simTs(1:end-1,:);
Y = simTs(2:end,:);
hatPhi = (X'*X)\(X'*Y)

>> hatPhi =
       0.8939  -0.0057
       0.1721   0.8732

Notice the output is the transpose of the AR matrix supplied. Why is this? I feel dumb, lol.

I see that $n=2$ and $\mathbf \mu = \mathbb0$.
At single (2-dimensional) observation level, we usually write (ignoring the disturbance vector)

$$\mathbf y_t = \Phi \mathbf y_{t,-1}$$

where dimensions are for your case $(2 \times 1) = (2 \times 2) \times (2 \times 1)$.

But notice that the standard regression formulation has the coefficient vector to be "to the right" of the regressor matrix... so in order to be able to apply the standard OLS estimation formulas, we transpose and obtain

$$\mathbf y'_t = \mathbf y_{t,-1}'\Phi'$$

and at full sample level we get

$$\mathbf Y' = \mathbf Y_{-1}'\Phi'$$

where with $T+1$ 2-dimensional observations available, the dimensions are ($T \times 2$) = $(T \times 2) \times  (2 \times 2)$

So, when applying the standard OLS formulas as your code does, the estimator estimates
$\Phi'$, not $\Phi$.


Which of Arrow's four desirable properties' is violated in this scenario? So the scenario is as follows: there are 3 agents and 4 alternatives, $a,b,c,$ and $d$. Society's ranking of the 4 alternatives is such that

  the highest-ranked alternative is agent 1's highest-ranked
  alternative,

  The alternative that agent 2 ranks highest among the three remaining alternatives is ranked second highest by society

  The alternative that agent 3 ranks highest among the two remaining alternatives is ranked third highest by society.

  Society ranks the 4th alternative lowest.

So my interpretation of the ranking would look something like this:

Which of Arrow's four desirable properties is violated?

I'm guessing that the "Unanimity" preference would be violated, because agent 2 prefers $b$ over $c$, but this is not reflected in society's ranking?

The four desired properties I am referring to are:

1/ Complete and transitive preferences

2/ Respect unanimity

3/ Non-dictatorial outcomes

4/ Independence of irrelevant alternatives

Note that an alternative being highest-ranked by an agent involves comparisons with all other alternatives, which raises doubts that your method satisfies independence of irrelevant alternatives. Indeed, it does not.  Independence off irrelevant alternatives is actually the only property that is violated.

Here are two profiles, only the preferences of the first two agents matter in them, so I will not specify the preferences of the other agents. I only assume the other agents preferences are the same in both profiles.

Agent 1, Profile $1$: $a\succ b\succ c\succ d$

Agent 2, Profile $1$: $b\succ a\succ c\succ d$

In Profile 1, the social ranking of $a$ and $b$ is $a\succ b$.

Agent 1, Profile $2$: $c\succ a\succ b\succ d$

Agent 2, Profile $2$: $b\succ a\succ c\succ d$

In Profile 2, the social ranking of $a$ and $b$ is $b\succ a$.

Since the relative ranking of $a$ and $b$ is the same in both profiles, independence of irrelevant alternatives is violated.


Consumer surplus with isoelastic demand Let $q(p) = \frac{1}{p}$ denote the demand function and $p^*$ some equilibrium price. Consumer surplus is defined as
\begin{align}
CS = \int_{p^*}^\infty\frac{1}{p} dp= \ln(\infty) - \ln(p^*) = \infty
\end{align}
which is quite unsatisfactory. What's the convention here?

Edit: Since I wanna compare welfare under price and quantity competition I found a workaround. Let $n$ the number of firms in the market, $\bar \pi^j$, equilibrium profits of the firm, $\bar p^j$ equilibrium prices and $j \in \{b,c\}$ is the index for the compeition type.
Now we may calculate the welfare difference as
\begin{align}
\Delta W := W^c - W^b &= n(\bar \pi^c - \bar \pi^b) + \int_{\bar p^c}^\infty\frac{1}{p}dp - \int_{\bar p^b}^\infty\frac{1}{p}dp\\
& = n(\bar \pi^c - \bar \pi^b) + \ln(\bar p^b) - \ln(\bar p^c)
\end{align}

Suppose we get the following relationship
\begin{align}
\Delta W = \frac{(n-1)(\sigma-1)^2}{\sigma n^2(\sigma(n-1)+1)} + \ln\left(\frac{\sigma(n-1)+1}{n \sigma}\right)
\end{align}
with $n \geq 2$ and $\sigma \in [0,\infty)$ is a parameter.
Clearly $\Delta W = 0$ for $\sigma = 1$ and for all $n$.
I wanna further show that $\Delta W > 0$ for $\sigma < 1$ and $\Delta W < 0$ for $\sigma > 1$. Any ideas how to proceed?

This simply tells us that the mathematical object $f(x) = 1/x$ is not friendly to all the economic concepts we want to associate/obtain from its use as a demand function.
So, it is not the right tool to represent a demand function, we should use something else.

Mathematically, this relates to the harmonic series which is divergent (think of the integral as a discrete sum).


Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable? Suppose we have a model that is:

$$
y = X_1\beta_1 + X_2\beta_2 + X_3\beta_3 + \varepsilon
$$

where $X_1$ is independent of $X_2$ and $X_3$ but $X_2$ and $X_3$ are correlated with each other.

Suppose we don't have data on $X_3$ so we omit it and run the regression:

$$
y = X_1\gamma_1 + X_2\gamma_2 + \nu
$$

Econometric theory on OVB states that $\gamma_2$ will be biased.

My questions:

Will $\gamma_1$ be biased in this case as well? I ran a simulation with some data I created and I do get a biased result.
If it is unbiased then how do I prove it? I use FWL theorem to find $\gamma_1$ and show that its expectation equals $\beta_1$ but get stuck here:

$$
\hat{\gamma_1} = \beta_1 + (X_1'M_{X_2}X_1)^{-1}(X_1'M_{X_2}X_3\beta_3) + (X_1'M_{X_2}X_1)^{-1}(X_1'M_{X_2}\varepsilon)
$$

The third term will become 0 when we take the expectation but not sure how to make the second term 0.

I would suspect that if  $X_1$ is independent of $X_3$, then it must imply $E[X_1*X_3]$ is equal to $E[X1] * E[X_3]$, given this you can conclude that the Covariance is 0. $Cov(X_1,X_3) = E[X_1*X_3] - E[X_1]* E[X_3]$. If the Covariance is 0 between these two variables then so is the correlation. Thus I would believe that removing $X_3$ from the regression and having it's effect enter the error term would not result in having $γ_1$ be biased as the $X_3$ in the residual term does not have correlation with $γ_1$.

Are you looking for a rigorous mathematical proof?


A question about the property of quasi-linear preference In case of quasi-linear preference, why would one unit more of the numeraire good (good 1) give the same additional utility as spending an additional amount of wealth equal to the cost of one unit of good 1 on all other goods no matter how much unit of this good 1 have already been present in the consumption bundle? How would one prove it? (Interpretation would also be welcome.)

It seems that it has something to do with the fact that utility maximization implies identical marginal utility per unit currency spent on each good.

This seems like a homework question, so I'll just give hints.

By definition, a quasilinear utility has the form $u(x, y) = x + v(y)$ where $y$ is a vector of all other goods and $v(\cdot)$ is strictly concave. In this case, $x$ is called the numeraire.

From utility maximization, what's the first order condition that relates $MU_x$, $MU_y$, $p_x$, and $p_y$? Write down the equation.
What's the utility gain from consuming one more unit of $x$?

You should be able to work things out from there.

Edit

$MU_x$ will be constant... more precisely, $MU_x$ is always one (by definition).

You can then plug this into FOC and get that $MU_y = 1/z$. where $z = p_x/p_y$ is simply how many $y$'s you could buy with "additional amount of wealth equal to the cost of one unit of good 1 on all other goods".

Since one more unit of good $y$ will get you $1/z$, you need (approximately) $z$ more of good $y$ to get one more unit of utility increase (which is the same as what you get when you increase $x$ by one unit).


Purpose of a monotonic transformations in utility functions Based on my economics book, monotonic transformations for a utility function can look something like this:

$f(u) = u + 17 $ or even like this: $f(u) = u^3$

That being said what it purpose in the case of the utility function, as far as I know, the number of the utility doesn't matter to us (the only thing that matters is if it's bigger or smaller than the utility of another bundle)?

The book explains it in the following way:

  monotonic transformation is a way of transforming one set of numbers into
  another set of numbers in a way that preserves the order of the
  numbers.

Though it the order stays the same why would we do it in the first place?

Certain calculations using utility require utility functions to have some particular properties. The point of bringing up the invariance of preferences represented by different utility functions that are just monotonic transformations of each other is to show that these utility-based calculations are not entirely dependent on the utility functions that you happen to choose. The same results should at least hold up to order-preserving transformations.


Which mathematics are required for fully understanding the theories of the firm? When I say "theories of the firm", I'm referring, in particular, to the theories exposed in the next works: 1) Transaction Cost Economics exposed in "Transaction Cost Economics" by Tadelis and Williamson (2013) (chapter 4 of "The Handbook of Organizational Economics" by Gibbons and Roberts); 2) Property Rights Theories exposed in "The Costs and Benefits of Ownership" by Grossman and Hart (1986), "Property Rights and The Nature of The Firm" by Hart and Moore (1990), and "Firms, Contracts, and Financial Structure" by Hart (1995); and 3) Incentive System Theories exposed in "The Firm as an Incentive System" by Holmstrom and Milgrom (1994), and "Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design" by Holmström and Milgrom (1991).
When I say "Which mathematics", I would appreciate if a list of chapters of a highly recommended math book is given, due to I need to be prepared for analyzing those theories as soon as possible. In this regard, it should be noted that I have studied the first eight chapters of the first volume of Apostol's Calculus. In advance, thanks so much.

I haven't gone through all the papers so I just sampled 'The Firm as an Incentive System'. Browsing through it, it relies on Linear Algebra and Real Analysis. Again, I warn that I have not gone through the entire paper but seeing some of the terminology used there, I could guess that these two a clearly involved (maybe they use some bit of topology as well). They use a concept of sublattice somewhere and that pertains to group theory.
A good starting place for real analysis could be Steven R. Lay's 'Analysis'. There you can learn some of the basic terminologies that they have used like supremum and infimum. For Linear Algebra, it has got to be Gilbert Strang's text 'Introduction to Linear Algebra'. Linear Algebra is used extensively in Econometrics as well. You can find his lectures at: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ .
Hope this helps!


Why hasn't Quantitative Easing surged inflation, or caused hyperinflation? Quantitative Easing (QE) is the buying of bonds and other financial assets by the Central Bank with high-powered money, or money created by the CB for this purpose.

Isn't this ad-hoc creation of money  the main ingredient for hyperinflation?
Why hasn't there been a surge in inflation?

High-powered money is another term for the monetary base or MB.  MB represents all money created by the government.  However, that is not all the only source of money.  Banks are allowed to create money too.  Which is why higher aggregates like M2 and M3 are greater than MB.

Because bank money is greater than government money, it (and not MB) is more of a primary determinant to monetary inflation.  Pretend there is an economy of blue dollars and yellow dollars.  The blue are created by the government and the yellow by banks.  There are many more yellow dollars than blue dollars.  So when the supply of blue dollars goes up it doesn't matter as much because of the number of yellow dollars.

Bank money was being destroyed in the crisis which was actually creating deflationary pressures.  The collapse in higher monetary aggregates was offset somewhat by the surge in the monetary base.

Now typically base money is used to create and leverage higher aggregates, but this was not the case during the crisis.  Because of liquidity concerns, banks curtailed monetary creation so this is why inflation did not happen.


Price Producers Receive as a Result of Indirect Taxes In my economics class, we were introduced to the idea that if the government imposes an indirect tax on a good, the price consumers have to pay increases, but along with this increase in price for consumers, the price producers receive decreases. I understand the former perfectly, however, don't understand why the price producers receive decreases.

$P^*$ is the original price prior to the imposition of taxes, and $Q^*$ is the quantity demanded prior to the imposition of taxes. $P_c$ is the price paid by consumers after the government's decision, and $P_p$ is the price received by firms, once again, after the imposition of taxes.

My question is, why isn't $P^*$ equal to $P_p$?

The idea is that both the consumer and the producer of a good take on some of the burden of a tax, even if you don't intend to levy the tax on one or the other. So if you have a sales tax that consumers have to pay, what will happen is that consumers won't take the full brunt of that tax. They will demand less and producers will have to make the product cheaper in response, so they get less revenue overall anyway.

How much tax burden each side will have to pay depends on their relative elasticities, which I imagine you'll be going over soon if you're going over tax incidence. Elasticity is a measure of sensitivity to a change in price to in this case, quantity supplies or demanded.

So $P* \neq P_p$ because the producer will have to shift from the equilibrium just like the producer will have to. It's the same idea for the consumer as for the producer, since the tax burden falls on both of them.


How would banking in UK change if minimum required capital to open own bank would be £100k instead of €5m? DIRECTIVE 2013/36/EU

  Article 12

  Initial capital

  Without prejudice to other general conditions laid down in national law, the competent authorities shall refuse authorisation to
  commence the activity of a credit institution where a credit
  institution does not hold separate own funds or where its initial
  capital is less than EUR 5 million.

Assumption

Let's assume that at some point people realized how great the banking profession is and a lot of entrepreneurs were willing to take the burden off the bankers by starting their own bank. The only problem they had was lack of required capital, which for the UK, or any bank in an EU member state, was about €5m. In the following parliamentary election people voted for a liberal (not just by name) party that promised to lower the minimum required capital to £100k, so that banking would no longer be a privilege of the super rich. Every middle class entrepreneur can now easily gather the required capital and start competing in the free market of banking services.

I know that there is an exception to the minimum required capital for particular categories of credit institutions which the UK used for building societies but let's focus on fully-fledged banks exclusively.

  Member States may grant authorisation to particular categories of credit institutions the initial capital of which is less than that
  specified in paragraph 1, subject to the following conditions: (a) the
  initial capital is no less than EUR 1 million; (b) the Member States
  concerned notify the Commission and EBA of their reasons for
  exercising that option.

Question

Given that banking, Fractional Reserve Banking especially, heavily relies on support from taxpayers, mostly through Deposit Guarantee Schemes (DGS) without which we can safely assume that FRB wouldn't be possible, banks would have to find other ways to build trust than simply rely on always-solvent taxpayers, and the fact that the minimum capital to open one's own bank in the UK is €5m, what can we reasonably expect to happen if that minimum required capital was lowered to, say, £100k?

In other words, is there any justification, other than we are a cartel go away, that banking, or FRB in particular whose biggest asset is taxpayers guarantees through DGS, be restricted to super rich people that can afford the €5m minimum required capital?
Why can't we have real competition in the banking sector by allowing for creation of micro banks?

My own prediction

Competition in the banking sector would take an unprecedented turn and, as with every liberalisation of market rules, quality of the services would rise.
Regular as sine wave financial crises would then be occurring with higher frequency but a lot smaller amplitude.
No bank will be too big to fail as small banks would take a substantial part of the market.

Community bank

An example of small lending company which is not a bank in regulatory terms but despite its micro capital it successfully competes with big banks on the lending market -
Burnley Savings and Loans. Not directly the kind of changes that I am requesting but it shows that it is possible for a small company to succeed on the lending market.

References

http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32013L0036
http://www.fsa.gov.uk/pages/doing/how/deposit_taking/index.shtml
http://www.bankofengland.co.uk/publications/Documents/joint/barriers.pdf
http://www.fsa.gov.uk/
https://en.wikipedia.org/wiki/Capital_requirement
https://en.wikipedia.org/wiki/Capital_Requirements_Directive

The minimum required capital matters, but in the opposite direction: the more capital you have, the more likely you can challenge big banks. Because the costs of a new bank look like this:

(http://www2.deloitte.com/content/dam/Deloitte/pt/Documents/financial-services/dttl-fsi-uk-Banking-Disrupted-2014-06.pdf)

You need $15 mn just to keep it alive. Recall that, meanwhile, you are supposed to provide credit, so you'll have to borrow more money for this.

With 150K GBP in capital and fractional reserve rules, a new bank will be wiped out with losses. And it has no impact on the competition in banking, which is already quite tough.


Self-selection bias during the course of experiments Suppose you are running a randomized experiment to assess the effect of $X$, say some training program for unemployed people, on $Y$, say the chance of finding a job in the coming year. Suppose also that $X$ takes time : maybe it lasts for several month.

Because you randomize, you do not need to worry about self-selection bias initially. But during the course of $X$, some people will likely realize that $X$ is beneficial to them, and others may realize that they are wasting their time.

As a result, one might expect that among people who drop from the program, there is a higher proportion of agents for which the treatment effect would have been smaller. This might induce an over-estimation of the treatment effect.

My questions are :

Is this kind of bias discussed in the literature on randomized experiments?
Does it have a canonical name ?
Do researcher try to control for this, and if yes, how?

Apparently this is called attrition bias. It's very similar to survivorship bias. This paper suggest correcting for it using Heckman correction. Propensity score matching may also help somewhat. My experience with both has been mixed, but they are commonly used. You should figure out what exact approach is most appropriate for your setting.

One last edit: These two papers, which talk about bounding the average treatment effect, may also be of use to you.


Tax wedge in labour market I'm reading a book ( Macroeconomics - Institutions, Instability and the Financial System )  and on page 62, the authors define tax wedge as the "difference between the real consumption wage and  the real product wage". Should I understand this as a subtraction?

The real consumption wage is measured as $\frac{W}{P_c}$, where $W$ is the after-tax wage that employees receive, and $P_c=P(1+t_v)$ with $t_v$ being the indirect tax rate (VAT, etc).

The real product wage is the wage paid by firms to workers, and is measured by $W^{gross}/P$ where $W^{gross}=W(1+t_d)$ with $t_d$ being the direct tax rate(social securities contributions, etc).

Any help would be appreciated

From OECD website,

  Definition of Tax wedge

  Tax wedge is defined as the ratio between the amount of taxes paid by
  an average single worker (a single person at 100% of average earnings)
  without children and the corresponding total labour cost for the
  employer. The average tax wedge measures the extent to which tax on
  labour income discourages employment. This indicator is measured in
  percentage of labour cost.

So here, it is a ratio, but not between wages, but rather of the direct tax burden (including social security contributions), as a percentage of total labor cost (i.e of total reward of labor). But it does not include indirect taxes.

As you can see, there are different ways to define the concept (extending it to include indirect taxes makes sense -the problem would be to measure it at that level).But it appears that in general we are talking about a ratio rather than a difference.

To reconcile, it is the "difference (in absolute terms maybe) between the real consumption wage and the real product wage" as a percentage of real product wage.


Data Source on US Wage Earnings Does any guys know how to find the data source of US individual wage earnings from 1920 to 2018?

For annual individual-level earnings data:

From 1962: try Current Population Survey (CPS) https://cps.ipums.org/cps/intro.shtml
From 2000, try American Community Survey (ACS) https://usa.ipums.org/usa/acs.shtml

For decadal individual-level census data:

From 1850: IPUMS-USA https://usa.ipums.org/usa-action/faq#ques0


Is MRS supposed to have a minus sign or not? What's there to say about convexity of preferences? If we have a utility function $U(x_1, x_2) = x_1(x_2+1)^2$ of some consumer, then

$$MRS_{x_1, x_2} = \frac{\color{red}{-}(x_2+1)}{2x_1}$$

Some books have a $\color{red}{-}$. Others and Wiki don't have a minus. What exactly does this mean?

Bundle/basket/portfolio(is that only for finance?)/whatever $(3,5)$ and has a utility of $108$.

Bundle/basket/portfolio(is that only for finance?)/whatever $(27,d)$ has the same utility of $108$ for $d=1$

The MRS's are as follows:

$$MRS_{x_1, x_2}(3,5) = -1$$

$$MRS_{x_1, x_2}(27,1) = -1/27$$

  Question: What can you say about the convexity of the preferences of said consumer?

I don't quite know what to say here. The two bundles lie on the same indifference curve ($U = 108$). As we move to the right of the curve, the slope becomes less steep, as expected...I think?

This says MRS supposed to decrease if we move down a standard convex-shaped curve. So this curve is not convex? Or not standard?

The Marginal Rate of Substitution is not just the "ratio of the partial derivatives": it represents the slope of an indifference curve. In order to obtain it, you must guarantee that you remain on the same indifference curve. How do we do that? One way is by taking the total differential of the utility function and requiring this total differential to be equal to zero:

$$dU(x_1,x_2) = 0 \implies \frac {\partial U(x_1, x_2)}{\partial x_1} \cdot dx_1 + \frac {\partial U(x_1, x_2)}{\partial x_2} \cdot dx_2 =0$$

and rearranging (and putting in this case $x_2$ on the vertical axis)

$$\implies \frac {dx_2}{dx_1} = -\frac {\partial U(x_1, x_2)/\partial x_1}{\partial U(x_1, x_2)/\partial x_2} = MRS_{x_1,x_2}$$

If we look at two goods and (not say, one good and one bad), and we have non-satiation, then the partial derivatives are strictly positive. So the minus sign is the one that tells you that the slope of the indifference curve should be algebraically negative, so a "downward slopping" curve. And this is  an argument in favor of keeping it. But indeed sometimes, people omit the minus considering the negativeness of the slope to be "understood". But this can create confusion.

But hey, the symbol $dx_2/dx_1$ can also be read as "the derivative of $x_2$ w.r.t. $x_1$", which is the first derivative of the indifference curve equation. Imagine now a convex to the origin indifference curve. As you increase the value of the variable measured on the horizontal axis ($x_1$ in our case), a convex indifference curve becomes flatter. This means that in absolute terms the slope becomes smaller. But in algebraic terms it becomes larger, because its value moves from a negative number, closer to zero.

I prefer to keep signs visible at all times (many short-lived scientific discoveries have sent their discoverer into depression five minutes later, when (s)he realized that it was just a sign mistake).

So, if I want to determine whether the indifference curve is convex, I want to see the MRS (sign included) increase in value, i.e. become less negative, and so also decrease in absolute value. So we see that "MRS decreases as we move down and to the right in a convex indifference curve" uses "decrease" in the absolute value sense.

And the confusion may get worse.

In any case, one should calculate

$$\frac {d^2x_2}{dx_1^2}$$

But with, or without the minus sign?

Assume that, like me (and unlike many others) you want to keep the minus sign in front of the $MRS$. In that case, you want to see the MRS increase. For this to happen, the derivative of the MRS must be positive. This in my eyes is the 2nd argument in favor of including the minus sign, because we know that positive 2nd derivative means convex function (referring here to the indifference curve equation).

So with the minus sign in front when calculating the derivative, positive $MRS$ derivative $\implies$ $MRS$ increase in algebraic value $\implies$ the slope of the $MRS$ gets larger in algebraic value, $\implies$ it gets smaller in absolute value, $\implies$ the curve gets flatter, and so it is convex indeed.
I leave the chain for the case of ignoring the minus sign as a for anyone interested. You will certainly get "negative 2nd derivative $\implies$ convex indifference curve", which will tend to mess with pre-existing knowledge in your brain.

And check this post.


Does the current situation of excess reserve imply failure of QE? Excess reserve @ Fed

In the past it was assumed that excess reserves should be kept around 0 and banks should be lending to each other as much as they are borrowing. The central banks would correct the negative or positive balance at the end of the day by lending or borrowing at premium rates, but that was only a short term solution.

Now we have 5 years of excess reserves at the Fed and a lot of articles about how banks don't care about cash deposits.

Is this the failure of QE? The money being lent and given frivolously circles right back into the Fed in the form of excess reserve it seems.

Is this making it categorically impossible to raise real interest rates? Maybe interest rates could go up if a devaluation process was launched and inflation started climbing, but I'm more interested in understanding what makes the real interest rate move.

Am I misunderstanding something or everything?

You ask a few questions. I'll try to generally answer them:

  Is this the failure of QE? The money being lent and given frivolously circles right back into the Fed in the form of excess reserve it seems.

No, the existence of excess reserves is not a reflection on bank lending. A significant fraction of excess reserves originate in money market mutual fund holdings. This is very clearly explained in Singh (2015):

  The interest rates in most of these markets are in the range of zero and 25 basis points (bps). The Federal Funds (FF) rate or policy rate has been around 10–15 bps since the crisis, and is largely a negotiated rate stemming from the excess cash balances of nonbanks such as GSEs (i.e., Fannie Mae, Freddie Mac etc.), and banks. Only banks have access to the 25 bps interest on excess reserves (IOER) and thus arbitrage by depositing nonbanks cash at the Fed.

The reason for nonbanks' extra cash showing up as the excess reserves of banks is that it flows through money markets to the Fed because it allows banks and nonbanks to split the 25bps IOER. Singh again:

  The Fed’s balance sheet increased from roughly \$1 trillion (end-2007) to over \$4 trillion by end-2014, owing mainly to some \$3.4 trillion of asset purchases that sit on its asset side. The approximate corresponding entry are excess reserves of \$2.9 trillion on the liabilities side— these are deposits of nonbanks (who sold assets to the Fed) at banks, who then place them as deposits at the Fed. Since October 2008, the Fed offers banks 25 basis points per annum for their deposits (including excess deposits over the required reserves), but pays zero interest on deposits from nonbanks, especially GSEs.

As I note in the discussion in the comments, zero dollars of QE were "lent" or "given"— QE consisted entirely of purchases of government and agency securities on the open market, at market prices. What QE does is change the mix of "safe assets" versus cash. There's nothing "frivolous" about it.

  Is this making it categorically impossible to raise real interest rates?

Due to the payment of interest on excess reserves (IOER) and the creation of the Reverse Repurchase Program (RRP), the existence of excess reserves does not prevent the Fed from raising real interest rates. As noted above, IOER effectively provides a reason for banks to continue to borrow funds from money markets by subsidizing them at a fixed rate, while RRP provides a floor under risk-free rates (by providing a place to park excess collateral at a positive rate, independently of the supply of collateral at dealers banks) that is under the control of the Fed.

Real interest rates are low because the FOMC has decided not to raise them yet for policy reasons.


Understanding minimums in the calculation of Human Development Index In the calculation of HDI, three indices are involved. One of them is the GNI index. In calculating this index, there's some scaling done. Quoting from the UNDP Technical notes

  The low minimum value for gross national income (GNI) per capita, $100, is justified by the considerable amount of unmeasured subsistence and nonmarket production in economies close to the minimum, which is not captured in the official data.

I don't understand what this justification means. Isn't \$100 too low? The index goes on to put an upper cap at $75000 which makes sense - because beyond that income there's not much impact on the standard of living.

The calculation of this index is fairly straightforward:

$Dimension\ index = \frac{actual\ value – minimum\ value}{maximum\ value – minimum\ value}$

This means that if a country would have most of its population living out of subsistence farming, which in practice could be registered as zero income, this would lead to an estimated/assumed GNIpc equivalent to \$100 per year per capita, which no country would ever go below.
Having this value makes sense when reading the starting notes of that section:

  Minimum and maximum values (goalposts) are set in order to
  transform the indicators expressed in different units into indices on a scale of 0 to 1. These goalposts act as the “natural zeros”
  and “aspirational targets” (...)

Said in other words, having this minimum means that \$100 would be the GNI per capita that would provide the lowest or minimum standard of living possible.

Answering to the "isn't \$100 too low":

  For income, it is set at \$100 per capita GNI, which is lower than the lowest value attained by any country in recent history (Zimbabwe in 2008). Should any country’s per capita GNI fall close to or below $100, the minimum will be changed accordingly.

I can recommend to read the source from UNDP, gives interesting inputs on the other dimensions too.

There are mathematical reasons to chose a low minimum

It is also convenient to choose minimums that make the dimensions non-zero, or not too close to zero even for countries which would score close to the minimum. Since the HDI is all about the combination of several aspects, and the formula is:

$HDI = (I_{health} \cdot I_{education} \cdot I_{income})^\frac13$

If any country would score a zero in any dimension this would dominate the whole HDI (make it zero), which defeats the purpose of the index. The index aims at acknowledging average human development balancing the three aspects.


When the Central Bank sells a bond, where does the money go ultimately? As the accepted answer points out, the premise of my question was flawed. Thank you for the dispelling of my wrong assumptions. The original question follows.

There are two related questions (and feel free to assume the Central Bank in the questions is USA's Fed):

When the Central Bank sells bonds, where does the money go ultimately?
How separate is the Central Bank budget from the government's budget?

To clarify: If the Central Bank sells \$1000M worth of bonds in order to repay \$800M old maturing debt and to raise an extra \$200M, does this \$200M simply become available for use to the government? Or is there some extra process in order to transfer money between the government and the central bank? Who decides (central bank, executive branch, legislative branch) how much new debt the central bank will be selling?

Firstly, the central bank doesn't issue bonds. The treasury (in the executive government) does, as a way to finance government expenditure, and make real investments in the economy.

In modern economies, the central bank's primary focus is the interest rate at which they lend money to private banks. If they require more money in order to finance those loans, they will simply create more money, as they have the legitimacy to do that. (Note that most money that is created is not physically printed, but rather it is just the electronic entry of that currency, in the initial transaction. An economy requires a fraction of electronic money to exist as cash, so a fraction of created money is printed.) However, fiat money requires some security to give it legitimacy. To this end, the central bank holds securities, stocks, precious metals, cash and foreign currency in its reserves. You can see here that bonds form part of those reserves, so that is why central banks buy treasury bonds. Plenty of other domestic and foreign financial institutions also buy treasury bonds, as part of their reserves to give legitimacy to their "demand deposits" (which is fiat money), or as part of a strategy to lower the risk of their investment portfolios. Private banks also create money by loaning out their demand deposits.

Analogously, if the central bank raises the interest rate (in order to curb inflation), they will be slowing down the growth of money (which can be thought of as a "leakage"), and that is a situation in which they would buy fewer bonds.

Central banks need to operate autonomously from the executive government, and this autonomy is usually enshrined in the respective country's constitution. When this doesn't happen, the executive government could force the central bank to buy their bonds, which creates a huge injection of money into the economy. This money that is created belongs to the government, so they spend it in order to achieve policy goals, however, after a while the currency becomes worth less and less (as inflation is high). That is the main cause of high inflation. When the central bank is autonomous, they have control over the monetary system, and keep inflation in check with the central bank interest rate.


Is education a part of the U.S. exports component of GDP? I was searching for this but could not find an answer.

My question is whether education of the U.S. universities provided to foreign students counts in exports component on GDP.

And if so, can we call the U.S. "the largest education exporter in the world"?

From Unesco's website:

Top 10 destination countries:

United States (18% of total mobile students)
United Kingdom (11%)
France (7%)
Australia (6%)
Germany (5%)
Russian Federation (4%)
Japan (4%)
Canada (3%)
China (2%)
Italy (2%)

This is about headcount, not value-of-services. For 2nd-in-rank UK to nevertheless generate more income in terms of value, it should be the case that average fees/cost of living in UK is ~$64$% higher than in the USA. Otherwise, USA will be also the biggest exporter of education.

This, without taking into account the size of each country's economy.

The link mentions that Unesco's statistical data base provides further statistics.


Finland plans to give every citizen 800 euros a month and scrap benefits. Will it become successful? Finland plans to give every citizen 800 euros a month and scrap benefits. Will it become successful?

http://www.independent.co.uk/news/world/europe/finland-plans-to-give-every-citizen-800-euros-a-month-and-scrap-benefits-a6762226.html

It will be successful in increasing economic growth. Removing benefits and giving everyone a fixed sum of money per month will disincentivize voluntary unemployment, remove moral hazard from benefit schemes, and encourage poor people to seek ways out of poverty. Also there should be less pressure on public infrastructure, including healthcare.
It will be successful in reducing the number of migrants seeking refuge in Finland.
It will NOT be successful in making life easier for the most poor and unfortunate.

From economic perspective, it should certainly be successful. It is hard to judge it from ethical point of view though.


Continuity Axiom in Expected Utility Theory Take the following definition of continuity.

  The preference relation $\succsim$ over the space of lotteries
  $\mathcal L$ is continuous if for any $L,L',L''\in\mathcal L$, the
  sets $$S_1=\{\alpha\in[0,1]:\alpha L+(1-\alpha)L'\succsim L''\}$$ and
  $$S_2=\{\alpha\in[0,1]:L''\succsim \alpha L+(1-\alpha)L'\}$$ are both
  closed.

Is it necessarily true that $S_1\cup S_2=[0,1]$? If so, why?

It is.
Prior to continuity, which is a property of the preference relation,  the preference relation $\succsim$ itself has been defined to be a binary relation that is characterized by transitivity, and, to begin with,  by completeness.
  Then if $S_1\cup S_2 \neq [0,1]$, it means that there exist some values of $\alpha$ somewhere in $[0,1]$, call them $\tilde \alpha$ for which

neither

$$\{\tilde \alpha L+(1-\tilde \alpha)L'\succsim L''\}$$

nor

$$\{L''\succsim \tilde \alpha L+(1-\tilde \alpha)L'\}$$

In words, for these $\tilde \alpha$'s, the pair cannot be ordered at all. But this contradicts the completeness foundation that is needed to even obtain a preference relation (as of course used in our theory. Psychologists I guess would disagree).

Also, note that completeness is defined over all conceivable pairs, even if, in a specific situation, we chose to restrict the space of lotteries to something smaller. Whether the lotteries under consideration belong to the specified lottery space, is really irrelevant. The person having the preferences has to be able to order them in any case, even as a "hypothetical" scenario (although strictly speaking, for a specific problem we have the "luxury" to impose completeness only as regards the lotteries available, while "remaining agnostic" as regards completeness if we expand the lottery space. Still this "weakening" on the imposition of the completeness axiom, does not really bring any gain).


Expected payment in a second price auction with affiliation The symmetric bidding strategy in a second-price auction with affiliation is given by $\beta(x)=v(x,x)$, where $v(x,y)=E[V_1|X_1=x,Y_1=y]$ (here $Y_1$ is the highest ordered statistic among the remaining $n-1$ bidders, and bidder 1 is assumed to be the winner).

Given this, the expected payment by the bidder is given as $E[v(Y_1,Y_1)|X_1=x,Y_1<x]$. This, I suppose, should be equal to $\int_0^x v(y,y)g_{Y|X}(y|x) dy$. The same expression is given in Introduction To Auction Theory by Menezes and Monteiro. However, Vijay Krishna in his book writes $E[v(Y_1,Y_1)|X_1=x,Y_1<x]=\int_0^x v(y,y)dK(y|x)$, where $K(y|x)=\cfrac{ G_{Y|X}(y|x)}{G_{Y|X}(x|x)}$.

My question is, are the two expressions same?

Note: There is a notational difference in the two books, while Vijay Krishna uses $g(.),G(.)$ for the density and the distribution, Menezes and Monteiro use $f(.),F(.)$, respectively.

I see nothing wrong with Krishna's expression. If I had to take a guess, it would be that either Menezes and Monteiro define their $f_{Y|X}(y|x)$ differently or they simply forgot to adjust the density on $Y_1 < x$. Both expressions are supposed to be the same.

If you have some random variable $Y$ with cdf $F$, density $f$ and support $[a,b]$
, the following two notations say the same thing
$$\int v(y) f(y) dy  = \int v(y) d F(y) \quad  \mbox{for any function } v(y).$$
If you condition $Y$ on being $Y<x$, you have to adjust the cdf because the new support only goes up to $x$,
$$ F_{Y|Y<x} (y) = \begin{cases}
0 \quad &\mbox{if } y <a, \\
\frac{F(y)}{F(x)} \quad &\mbox{if } y \in [a,x],\\
1 \quad &\mbox{if } y >x.
\end{cases}$$
That way you have an expression that is one for $y=x$.

Hence,
$$E[v(Y_1,Y_1)|X_1=x,Y_1<x]= \int_0^x v(y,y)d\cfrac{ G_{Y|X}(y|x)}{G_{Y|X}(x|x)}
= \int_0^x v(y,y)  \cfrac{ g_{Y|X}(y|x)}{G_{Y|X}(x|x)}   d y.  $$


Are prices part of total factor productivity? I am trying to understand how production is related to income/profit and where do prices enter. Suppose there is a single firm with a Cobb-Douglas production technology:
$$Y=AK^{\alpha}L^{\beta}$$
Where do prices come into this? I see this being related to whether $Y$ is expressed in units of output or in monetary value (total revenues?). If the former, I would think this means that only $Y\times p$ gives the firm's profit, if $p$ stands for price of output. But if this is the case, can we "absorb" prices into total factory productivity and rewrite everything as:
$$\pi = Y\times p = pAK^{\alpha}L^{\beta} = \tilde{A}K^{\alpha}L^{\beta} $$ where $\tilde{A} = p\times A$?

No actually by default when you use Cobb-Douglas function  the output is not even measured in monetary units but rather as output per unit of time.  This output per time can be still called income without assigning it any 'monetary' value. For example, in Robinson Crusoe economy if you catch 5 fish then those 5 fish are your income from economic perspective.
Moreover, let us suppose we measure $Y$ by GDP. In that case for calculation of TFP we will always use real GDP not nominal GDP. This is done precisely to get rid of any influences change in price level may have on the measurement.  Even though nominal GDP is calculated involving prices and we can express nominal GDP as $YP$ (where $P$ is the aggregate price level) when we calculate real GDP we deflate the nominal GDP by dividing it with the aggregate price level. Hence real GDP is $\frac{YP}{P}=Y$.
The above being said when it comes to TFP it is incredibly hard to do dimensional analysis (see this working paper) and generally $A$ does not have consistent units. To be more specific, of course in each specific case there will be some units but the units will depend on parameters of the model (unless we impose some restrictive assumptions as mentioned in that working paper). However, note this is not problematic from a statistical perspective as statistical analysis generally does not require dimensions to be consistent (like regression analysis for example). Consequently, in empirical work we usually just avoid assigning any particular units to TFP.


Is there a commitment mechanism which allows players to achieve pareto optimal solutions? For any non-cooperative game, is there some commitment mechanism which will can guarantee that players chose a pareto-optimal solution?

Or, are there known conditions for this to obtain?

EDIT: I elaborate what I mean by "commitment mechanism" below:
Informally, I mean something like a "contract" that players can sign and are guaranteed to follow in the future.
So there is an additional round at the start of the game where each player, independently, decides to sign this contract (or not).

For instance, coordination in the prisoner's dilemma can be achieved by offering each player the option of signing a contract which states:
"If both players sign this contract, then I will cooperate".

Expanding on @VCG's comment, which pretty much answered the question:

You can model this by adding a simultanous first move (to sign or not to sign) for both players. If they both choose to sign the contract, a predesignated outcome $A$ is realized and they get the corresponding payoffs. If they do not both sign, the original game is played.

Whether they will sign the contract or not will depend on their beliefs about what payoff they could expect in equilibrium. If a player's expected payoff from the original game is larger than what they would get in $A$ they will not sign.

(There are similar considerations in Aumann's correlated equilibrium, though there is no enforcable contract there.)

An example for when you would not sign the enforcable contract:
Consider a game of chicken with the payoffs

.
(Source: Wikipedia)

Suppose players could sign a contract forcing them to chose the "Chicken" strategy. A player will not do so if for some reason she expects that the other player will play chicken anyway, allowing her to collect a payoff of 7. In fact, using forward induction not signing may serve as a signal: The only reason he would not sign the contract is because she expects to win a larger than 6 payoff. The only way she can do this is by playing "Dare". Thus if the other player was willing to sign the contract, she has effectively signaled that she intends to play "Dare" by not signing the contract. Unfortunately according to the logic of forward induction the other player will also not sign.


Example of a (not quasi-linear) production function whose inputs are not perfect substitutes but are not asymptotic at the axes I'm looking for an example of a family of production functions indexed by, say, rho, where the inputs become closer and closer to perfect substitutes as rho approaches 1, and yet, the marginal product of, say, L, approaches a finite limit as L approaches 0.   Of course one can construct quasi-linear production functions that exhibit this property for one input, but their behavior is really awful. One would think that CES production functions might exhibit this property, since their limit is an affine function,  but they don't.   Indeed, clearly, any production function in which L is raised to a power < 1 is going to give me  the same problem.    Any suggestion would be very much appreciated.

You could, for example take the function $f: \mathbb{R}^2_+ \times[0,1] \to \mathbb{R}$.
$$
f(L,K, \rho) = L + K + (1-\rho) L K.
$$
For $\rho = 1$, we have $f(L, K, 0) = L + K$ which is a production function with perfect substitutes.
The marginal product (of say $L$) is given by:
$$
\frac{\partial f(L, K, \rho)}{\partial x} = 1 + (1-\rho)K,
$$
which equals $1$ if $\rho = 1$.


What does it mean if a economic model is internally consistent? What does it mean if a economic model is internally consistent? What happens if it is not? Is it not closed anymore? Can it still be solved? Does it still have an equilibrium?

In theoretical modeling the consistency is applied in the same way as in philosophy/logic. Internal consistency simply means that the argument is consistent with itself and has no contradiction within itself (as opposed to external consistency where its not enough for argument to be valid on its own but it should also not contradict other facts).

A simple verbal example of internal inconsistency would be statement: "I always lie". The statement is internally inconsistent because if I always lie then the statement above must be also lie. But if the statement is lie that implies I do not always lie so it is not correct and it contradicts itself.

Math can reveal if a model is internally inconsistent. If you have model relaying on following system:
$$a+b =1$$  $$5a+5b=15$$

You would get that $5 = 0$ which means that the model is clearly  inconsistent with itself. However, sometimes the model might be internally inconsistent because the assumptions made in the model are not consistent with each other but the model might still solve and even have equilibrium (if the logical fallacy is made in a step where you build equations based on the assumptions - or make assumptions that are not consistent with each other but the consistency of assumptions is not checked) and finding the inconsistency is harder.

Moreover, note that internal consistency of model has also some special meaning in statistics, but from your question I assume that was not the meaning you were looking for.


Modifying Hotelling's lemma: Is this valid? Hotelling's lemma is stated as:

$$\frac{\partial \pi}{\partial p}=y$$

knowing however that on the more basic level, output $y$ is determined by the input(s) $x(p,w)$,let the profit function be defined as:

$$\pi=py(x(p,w))-wx(p,w)$$

taking the derivative with respect to $p$

$$\frac{\partial\pi}{\partial p}=y(x(p,w))+p\frac{\partial y(x(p,w))}{\partial x(p,w)}\frac{\partial x(p,w)}{\partial p}-w \frac{\partial x(p,w)}{\partial p}$$

Wouldn't this be a more accurate definition of hotelling's lemma?

I'm speculating considering some of the critsisim I read on hotelling's lemma in applied work. Namely: Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher

You are basically on the right track but might have been carried away by not writing down function arguments. The profit function $\pi$ has arguments $p$ and $w$; don't drop them.

In Hotelling's lemma, $y$ is the output. On the other hand, you write $y$ for the production function. Let's make things clear by writing neutrally $f$ for the production function. The profit-maximizing input choice is $x(p,w)$, so the quantity produced is $f\big(x(p,w)\big)$, the correct meaning of $y$ in Hotelling's lemma.

Hotelling's lemma then becomes
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big).$$
The profit function is given by
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$

Taking the derivative with respect to $p$ gives us
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big)+pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}.$$
This is pretty much what you got there above.

Now $x(p,w)$ is by definition a maximizer of $pf(x)-wx$. We have the first order condition
$pf'(x^*)=w$. Therefore,
$$pf'\big(x(p,w)\big)-w=0,$$
which implies $$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}=0$$
This reduces our expression for ${\partial \pi(p,w)}/{\partial p}$ to Hotelling's lemma.

This tell's us also what the important message of Hotelling's lemma is: Indirect effects don't matter. A marginal change in prices has a direct effect on profits since the output gets multiplied by a different number. There is also an indirect effect that comes from firms adjusting how much to produce. Hotelling's lemma just tells us that the latter effect is zero for a profit-maximizing firm, which need not do much adjusting. At a profit maximum, the marginal effect of adjusting production on profits is zero.


Aggregation theorems hold if endowments are endogenous? I have a specific question about considering a representative agent. According to Rubinstein 1974, in an inter-temporal economy with $N$ individuals each one with initial endowment $e_i$ and everything else homogeneous. The intertemporal constraints of the investor are:
$c_{0i}=e_{0i}-\eta_i p$
$c_{1i}=e_{1i}+\eta_i X$
Where $e_{0i},e_{1i}$ are present and future endownments, $c_{0i}, c_{1i}$ represents present and future consumption, $p$ is the price of the investment technology, and $\eta_i$ is the number of units bought. Finally $X$ is the payoff of the investment technology.

Assuming investors differ only on their endowments, we can solve for the equilibrium using a representative agent with the average endowment $\bar{e}=\frac{\sum e_i}{N}$.

Suppose now that the present endowment of each agent depends on how many units he/she decides to buy.
$c_{0i}=e^i(\eta_i)-\eta_i p$

The function $e^i(\eta_i)$ is not necessarily the same for all investors.

My question is: Can we solve for the equilibrium using a representative agent with endowment
$\bar{e(\eta)}=\frac{\sum e^i (\eta)}{N}$ ?

Thank you

Reference: Rubinstein, M., 1974. An aggregation theorem for securities markets. Journal of Financial
Economics 1, 225-244.

The answer is it depends on more assumptions on preferences. The short answer is no. To provide as much intuition we will look at the optimal conditions in each of these problems.

First, for the descentralized problem you have that individuals choose $(c_{0i}, c_{1i}, \eta_i)$ to maximize:

$\max \qquad \sum_t \beta^t u(c_{ti}) $

$s.t \qquad c_{0i} = e_{0i}(\eta_i) - \eta_i p$

$ \quad \qquad c_{1i} = e_{1i}(\eta_i) + \eta_i X$

The solution, for each individual $i$ will be optimal if they follow the Euler equation derived from the first order conditions and that markets clear:

$\frac{1}{\beta} \frac{u'(c_{0i}^*)}{u'({c_{i}^*)}} = \frac{p - e_{i}'(\eta_i^*)}{e_{1i}'(\eta_i^*) + X}$

Now look at the representative agent problem for whom you say the endowment is given by $\bar e(\eta) = \frac{1}{N}\sum_{i} e^i(\eta)$. I will drop the $N$ because scaling does not matter (you are assuming the mean endowment, I am using the total endowment so all of my solutions are $N$ times what you want). This agent's problem is to choose $(c_0, c_1, \eta)$ to maximize:

$\max \qquad \sum_t \beta^t u(c_{ti}) $

$s.t \qquad c_{0} = \sum_{i} e_{i}(\eta) - \eta p$

$ \quad \qquad c_{1} = \sum_{i} e_{i}(\eta) + \eta X$

Now the first order condition says:

$\frac{1}{\beta} \frac{u'(c_{0}^*)}{u'({c_{1}^*)}} = \frac{p - \sum_{i}e_{i}'(\eta^*)}{\sum_i e_{i}'(\eta^*) + X}$

These two need not be equal. Do you have more assumption on the preferences/endowment functions?


Profit maximizing, Quantity and Price with Dominant Firm and Competitive Fringe A dominant firm operates in various markets. In one of these markets, it is a monopolist and produces with the following cost function: C(q1) = 5q1. The market demand is P = 1005 - Q.

(a) Find the profit maximizing P and q1 and profit.

(b) A new fringe firm with C = 28,900 + (qf^2)/4 enters the market. The fringe firm acts as a price taker and maximizes its profits. The dominant firm, after the entry , subtracts the fringe firm's supply and behaves as a monopolist for the residual demand. Find the new profit maximizing P, q1, qf and profits as well.

My attempt for (a):

P = 1005 - Q.

MR = 1005 - 2Q.

MR = MC.

1005 - 2Q = 5

Q = 500

P = 505

Profit = 505 x 500 - 5 x 505 = 249975

Hopefully I'm right.

My real problem is I'm stuck on (b). I'm not familiar with price taker. I did read my course book but it wasn't that helpful. I'd appreciate if anyone can help me, thank you. Please let me know if (a) was right as well

Your answer for part (a) is correct.

For part (b) you need to first determine what's called 'residual demand', which can intuitively be thought of as 'leftover' demand given how much the competitive fringe will produce. Denote residual quantity demanded as $Q_r$.

Marginal cost for the competitive fringe is:
$$MC_f=\frac{d}{dQ_f} C_f = \frac{Q_f}{2}.$$
To find the supply function for the competitive fringe set $MC_f=P$
$$Q_f=2P.$$

To find residual demand, subtract fringe supply from direct market demand (not inverse market demand). Direct market demand is given by
$$Q=1005-P$$
so residual demand is
$$Q_r=Q-Q_f=1005-P-2P$$
$$Q_r=1005-3P.$$

Rearranging residual demand gives inverse residual demand of
$$P=335-\frac{Q_r}{3},$$
which gives residual marginal revenue of
$$MR_r=335-\frac{2Q_r}{3}.$$
First find the profit-maximizing output for the leader firm by setting $MR_r=MC$:
$$335-\frac{2Q_r}{3}=5$$
$$Q_r=495.$$
To find the leader's profit-maximizing price, plug $Q_r$ into inverse residual demand to get
$$P=335-\frac{Q_r}{3}=335-165=170.$$

Finally, we can find the quantity supplied by the fringe by plugging the leader's price into the fringe supply function
$$Q_f=2P=2*170=340.$$


Why isn't high inflation along with robust welfare a 'good' system? High inflation, by definition, means that the demand for goods and services is higher compared to the demand for currency. In other words, people are more willing to give up currency for goods and services.
Therefore, high inflation brings opportunities for able people to provide goods and services and earn their livelihoods.
The downside of course is that, those who aren't able to produce goods and services may suffer, if their savings are insufficient. However, they can be covered by a robust and guaranteed welfare program.
Therefore, it seems to me that issuing large amounts of currency and assuring welfare with it makes for a fairer distribution of economic resources and opportunities.
Why, then, is high inflation considered undesirable?
[I have learnt from this forum and elsewhere that a small positive inflation is considered desirable. My question is why doesn't the same apply to high inflation.]

There are several reason for that.
First, issue here is that you are talking about high inflation. Most economists are in favor of small inflation (around 2% per year for reasons you can read about in this and this old Economics.SE answers).
Second, inflation does not generally have the beneficial properties you ascribe to it. Inflation can stimulate output in the short-run but in the long-run the same level of inflation does not provide stimulus to the economy anymore (see discussion of this in any standard macro textbook such as Blanchard et. al. Macroeconomics: A European Perspective or Mankiw Macroeconomics). Consequently, in the long-run inflation does not increase opportunities for employment. Inflation, by itself, is not able to permanently raise real output.
Third, inflation has various costs. A non-exhautive list includes:

increase in uncertainity and volatility. This holds exclusively for high levels of inflation since modest levels of inflation are not associated with increase in uncertainty and volatility.  Higher uncertainity and volatility is generally bad for the economy as it leads to less investment and lower economic growth (see discussion in Mankiw Macroeconomics or Wilson (2006)).

next there are shoeleather costs of expected inflation. As explained in Mankiw Macroeconomic:

One cost is the distorting effect of the inflation tax on the amount of money
people hold. As we have already discussed, a higher inflation rate leads to a higher
nominal interest rate, which in turn leads to lower real money balances. If people
hold lower money balances on average, they must make more frequent trips to
the bank to withdraw money—for example, they might withdraw \$50 twice a week rather than \$100 once a week. The inconvenience of reducing money
holding is metaphorically called the shoeleather cost of inflation, because
walking to the bank more often causes one’s shoes to wear out more quickly

Of course, the above is written for undergraduates, it is not really the cost of new shoes that is biggest cost here, but opportunity cost of wasted time and general costs of adjusting to the inflation.

High inflation causes menu costs. These are costs firms incur to change prices (see Mankiw Macroeconomics).

High inflation tends to be more volatile and this complicates financial planning and distorts peoples intertemporal choices (again have look at Mankiw Macroeconomics).

The list above is not exclusive, the point is that high level of inflation has many costs associated with it. These costs increase disproportionally as inflation becomes higher, while at some point very high inflation won't even necessary have any short run benefits because the increase in uncertainty will depress economic activity more than the output get stimulated by the effect of short run effect of increase in prices.
Consequently, maintaining high level of inflation is considered very inefficient by most economists.  This, however, does not hold for low inflation (1, 2 or even 4%) as explained in the beginning.


Kuhn Tucker Maximization I have to maximize following expected utility function using Kuhn tucker conditions -

Since expected utility function are increasing $C_{1,t}$ and $C_{2,t}$ so constraints (i) and (ii) will hold with equality. Thus, I substituted these constraints into the objective function.

Note: Here, only $q_t^e$ is the only variable and expected utility function is maximized subject to $q_t^e$

For simplification, I denoted M= (1-$q_t^e$)$w_t$ + $q_t^e$$w_t$$z_t$

This made $C_{1,t}$ = nM and $C_{2,t}$= (ARθ/$z_t$)M

I also broke down constraint iii into two constraints $q_t^e$≤1 and -$q_t^e$≤0

I set up the legrange:

L = (1-π)[${(nM)}^{-γ}$]/(-γ) + π[${((ARθ/z_t)M)}^{-γ}$]/(-γ) + $λ_1$(1-$q_t^e$) + $λ_2$(0+$q_t^e$)

Differentiating with respect to $q_t^e$

(1-π)[${(nM)}^{-γ-1}$][n($z_t$-1)] + π[${((ARθ/z_t)M)}^{-γ-1}$][(ARθ/$z_t$)($z_t$-1)] = $λ_1$ - $λ_2$

$λ_1$(1-$q_t^e$) =0;   $λ_1$≥0

$λ_2$(0+$q_t^e$) = 0;  $λ_2$≥0

Case 1: $λ_1$=0 and $λ_2$

I got (1-π)[${(nM)}^{-γ-1}$][n($z_t$-1)] + π[${((ARθ/z_t)M)}^{-γ-1}$][(ARθ/$z_t$)($z_t$-1)] =0

Which boils down to M=0

which gave me - - > $q_t^e$ = 1/(1-$z_t$)

But solution provided is of the following form -

My answer doesn't match the solution provided. Can someone please look at my solution and tell me what did I do wrong?

After substituting for $c_{1t}$ and $c_{2t}$, the problem can be rewritten as :
\begin{eqnarray*} \max_{q_t^e} \ & \frac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} + \frac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}

which is same as
\begin{eqnarray*} \max_{q_t^e} \ & \alpha \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma}  \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}
where $\alpha  = \dfrac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma}  + \dfrac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} $

Observe that this is equivalent to solving :
\begin{eqnarray*} \max_{q_t^e} \ &  (1-q_t^e)  + z_tq_t^e  \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}

This is an objective that is linear in choice variable $q_t^e$, which is increasing in $q_t^e$ when $z_t > 1$, decreasing in $q_t^e$ when $z_t < 1$, and is a constant for $z_t = 1$. Consequently, the solution is
\begin{eqnarray*} q_t^e \in \begin{cases} \{1\} & \text{if } z_t > 1 \\ \{0\} & \text{if } z_t < 1 \\ [0, 1] & \text{if } z_t = 1 \end{cases} \end{eqnarray*}


Intuition behind risk premium In Lecture 20 of MIT's Microeconomics course, a situation is proposed where a 50/50 bet will either result in losing \$100 or gaining \$125 with a starting wealth of \$100. It is stated that a person would be willing to insure themselves for \$43.75 (the difference between \$100 and \$56.25). What is the intuition behind this?

Thanks in advance!

The name for the amount $56.25 is certainty equivalent.

The expected utility for the individual from taking the bet is calculated as follows:
$$E[U]=\frac12U(100+125)+\frac12U(100-100)=75$$
Suppose the individual can pay an amount of money $x$ so that she can avoid taking the bet (which leads to expected utility $75$). What's the maximum amount of money $x$ she's willing to pay? Well, she would pay up to a point where she's indifferent between taking and not taking the bet.

If she takes the bet, expected utility is $75$. If she pays, her utility is $U(100-x)$. We want her to be indifferent, so that $U(100-x)=75$. Reading off from the blue curve in your graph (the curve describing $U$), we see that
$$U(56.25)=75$$
which means $100-x=56.25$, or $x=43.75$.

So we can interpret 43.75 as the maximum amount of money that an individual is willing to pay in order to avoid the (risky) bet.


Do all mixed strategy Nash Equilibria satisfy Trembling Hand perfection? Moreover, does it fulfill other criteria automatically, like Myerson's "Proper equilibrium" refinement?

No. Trembling hand perfection would be an additional consideration.

Consider this case:

Two agents engage in a two stage game. In stage one, player one selects Rock, Paper or Scissors. In stage two, player two selects Rock, Paper or Scissors. They receive payouts depending on whether or not they match. They both have identical payout matrices.

Payoffs (Match, Unmatched)

Rock (1,0)
Paper (1,0)
Scissors (1,-1)

In a mixed strategy nash equillibrium consider player one. He can choose any mixed strategy such that

p(Rock)+p(Paper)+p(Scissors)=1

and they would all be equally rational. Since even if he chooses scissors, a rational player two would match him. However, taking into account trembling hand perfection, the subset of strategies where

p(Scissors)=0

dominate the previous strategy set. This is the case because we are concerned that player two has a chance of making a mistake if we choose scissors and not matching us. Therefore, mixed strategy and trembling hand mixed strategy give us different sets of strategies which fulfill our Nash equilibrium criteria.


Conventional Monetary Policy and Income Inequality In almost every textbook, it's stated that monetary policy is preferred to fiscal policy, since the latter is more politically difficult to implement due to smaller consensus on its redistributive effects. Monetary policy would not suffer, since it has no redistribution effect.

Has any empirical study ever tried to verify/assess this claim that Monetary Policy has not redistribution effect?

Well, certainly you'll see literature saying it is possible to make monetary redistributive. This paper by Brunnermeier and Sannikov (2012), Redistributive Monetary Policy, says that deflation can cause redistributive effects in an economy, and that monetary policy can be used to correct that, but "shouldn't" be used as a redistributive tool beyond that correction.

  In general, conventional monetary policy focuses primarily on the short end of the yield curve. Expectations about future policy indirectly affect the long end of the yield curve. Unconventional monetary policy directly targets the long end of the yield curve and prices of specific assets. All these measures can redistribute wealth across and within sectors.

Notice how they emphasize the unusual nature of monetary policy meant to create (anti-)redistribution.

This paper by Faust (1996), Whom can we trust to run the Fed? Theoretical support for the founders' views, shows that while generally any policy that affects inflation will create redistributive effects, it's better that the Federal Reserve control monetary policy to mitigate those effects rather than let policy be determined by voting, because of conflicts of interest that can arise from voting.

The final paper I'd like to present is by Romer and Romer (1998). Their main findings in their paper, Monetary Policy and the Well-Being of the Poor, is that expansionary monetary policy is better for the poor in the short run, but less so in the long run. Low inflation and stable aggregate demand growth is better for the long run, and that usually means tighter monetary policy.

So it seems that monetary policy can be redistributive very easily. Also consider that optimal monetary policy usually wishes for the nominal interest rate to be close to zero. What does that do for capital owners who want to borrow for investments? How does that change long run growth and affect income inequality? Characterizing that can be a bit nebulous.


What's the point of adding security features to new bill designs? What's the point of creating new bills with more security measures to prevent counterfeiting? (For example in the USA the new $100 bills have more security features).

If the old bills are still acceptable as currency, why would anyone even try to counterfeit new banknotes with all these security features?

Why don't people counterfeit bills from the 1800s (which are still legal tender - USA) and have NO protection from counterfeiting at all?

Bills don't last that long. The Federal Reserve estimates that $100 bills last about 15 years. Therefore, old bills are fairly rapidly rendered quite rare. Bills that look really odd (like the rare ones) are likely to warrant a second glance from the acceptor and later, if the counterfeiting is uncovered, much more likely to be traceable back to the forger. In The Great Paper Caper, Wells Tower tells the story of Frank Bourassa a counterfeiter who focused exclusively on modern \$20 bills because the enormous scrutiny on \$100 bills in general.


Criticism of Math in Economics I've been reading and speaking to a number of educated economists and economics PhDs who are against the use of intense mathematics and mathematical proof in economic theory. Specifically I've been speaking to those of Marxist and heterodox persuasion and reading their work in an attempt to become more open minded.

They emphasize that the study of work by classical economists (like Adam Smith, Karl Marx, and David Ricardo) is still relevant and that the practice of how mainstream economics uses mathematics is abusive and is an attempt to fool the masses regarding the "science" economists practice.

I have difficulty understanding this argument. What is a reason to be against mathematics in economics?

Note: I'm pretty mainstream and like how economics is taught and structured. I'm not anti math-in-economics, I just want to know why this is an argument.

I find that the essay "The New Astrology" by Alan Jay Levinovitz (an assistant professor of philosophy and religion, not an economist) makes some good points.

  ...the ubiquity of mathematical theory in economics also has serious downsides: it creates a high barrier to entry for those who want to participate in the professional dialogue, and makes checking someone’s work excessively laborious. Worst of all, it imbues economic theory with unearned empirical authority.

  ‘I’ve come to the position that there should be a stronger bias
  against the use of math,’ Romer explained to me. ‘If somebody came and
  said: “Look, I have this Earth-changing insight about economics, but
  the only way I can express it is by making use of the quirks of the
  Latin language”, we’d say go to hell, unless they could convince us it
  was really essential. The burden of proof is on them.’

The essay also makes a (more or less adequate—which, I leave up to you) comparison with astrology in ancient China to show that excellent math can be used to prop up ridiculous science and grant status for its practitioners.


Does the Federal Reserve buy and sell stocks? Can the Fed buy and sell stock in publicly traded companies? Is there evidence of this and, wouldn't this behavior drive the price as opposed to actual market forces?

No, the Fed is not allowed to buy stocks, they are allowed to buy government securities in open market operations in order to achieve the target rate for the federal funds rate. The guidelines for this are explained in the Section 14 of the Federal Reserve Act. You can find the Fed holdings in the Federal Reserve Statistics.

However other central banks, like the Bank of Japan, started buying stocks as a measure to support their financial institutions (their banks were subject to too much market risk because of their stock holdings). The have detailed their stock purchasing plan in their website.


Is the concept of limited liability in corporate law beneficial to economy The concept of limited liability in corporate law is widespread all over the world. Limited liability means, that owners of a company can only loose their capital invested into the company - the shares - as opposed to being liable with their whole private equity. My question is simple:

Does this benefit the economy as a whole or is this just a legal trick by corporate crooks?

It is beneficial (at least, mostly beneficial). Investments are risky, and business projects fail all the time at no fault of the entrepreneur. By bounding the risk profile of corporations, you give incentives for more people to start a business and look out for funding. Let me explain myself with a super simple example.

Suppose there was a project that has a 60% chance of being successful and create 10k jobs, and economic welfare to the community with a value estimated to be 500K Dollars. However, with  40% probability, it will fail and the initial investment of 150K Dollars will be lost. The entrepreneur lacks sufficient funds for the initial investment (let's assume it has to be fully funded by investors). If there was no limited liability, it is clear that most entrepreneurs will not do the project. Only entrepreneurs with sufficiently low risk aversion or sufficiently large pockets will be willing to raise the money (or use their own) and do the risky project with those odds.

You see that a project that is arguably welfare enhancer is not made, because it is too risky. Limited liability smooths things out by distributing the risk of a project among all the investors.

Clearly, this example is super simple and I have abstracted from principal-agent problems, entrepreneur fraud, situations where the success of the project depends on the entrepreneur's risk aversion, etc. There are many papers dealing with these issues (and others), but the main argument for limited liability is captured in this toy model. So yes, a very strong case can be made to conclude that it is beneficial.


Restricting "money creation" to the central bank Iceland is considering removing "money creation" from commercial banks.

[Despite the date of the article, I don't think this is a joke, as it has been picked up in other places since without anyone claiming it was a joke.]

I guess that what they are proposing is that borrowing short and lending long based on fractional reserves wouldn't be allowed, and only the central bank would be allowed to issue unbacked long-term loans in exchange for short-term deposits. Commercial banks could fund their own lending with these loans and would make short-term deposits at the central bank to reflect deposits made with them.

What are the advantages and drawbacks of this approach? If it makes sense, why isn't everyone else already doing so?

The proposal is to prohibit fractional reserve banking. So if enacted, Icelandic banks would have to have at least as much in reserves as they have in checking deposits. To do any lending, the bank would need to raise funds either from it's equity holders or by borrowing from investors. A number of US economists, such as Chicago economist John Cochrane, have endorsed this idea.

The main reason for abolishing fractional reserve banking is to eliminate the potential for bank runs. Standard banks don't keep enough money in reserve to pay out everyone's deposits, but they are obligated to let anyone withdraw their deposits at any time. Thus, if too many people withdraw at once the bank can go bankrupt, and that possibility of loosing their deposit in turn causes everyone to try to withdraw at once. It's a self-fulfilling panic that can happen at anytime, and doesn't necessarily even need to be caused by anything in particular. In economics terms, fractional reserve banking has two equilibria at all times: the normal one with deposits and lending, and the bank run equilibrium.

The US and most countries have tried to elimintate the bank-run equilibrium by insuring deposits--this is what the FDIC does. The problem, as countries like Cyprus found, is that if depositors don't believe the government really has the resources to pay out the full insured deposit amount for everyone, then you can still get a bankrun, this time not merely wiping out the banks and everyone's savings, but also the government's finances as well. Small countries that can't borrow and lend in their own currency are especially vulnerable here. Iceland does have its own currency, but often does business in US dollars or Euros anyway.

Additionally, there is concern that fractional reserve banking could lead to increased risk-taking by banks, since part of the risk is born by the depositors who (it is assumed) aren't as sophisticated as equity investors at measuring the banks' risks.

The drawback, of course, is that this likely makes checking more expensive to depositors since the bank can no longer lend the money out to make interest. Presumably it won't lead to higher borrowing costs because the central bank would offset the inability to lend out deposits with a larger base money stock, but it could exacerbate inequality as depositors would no longer share the returns to banks' investments (think Mr. Potter vs Bailey Savings and Loan from the movie "It's a Wonderful Life").


Refinements in signaling games with three sender types What are the pros and cons of the Intuitive Criterion and the Criterion of Divinity in signaling games with three types?
I find that separating equilibria often survive the intuitive criterion, whereas only the "Riley"/Pareto-efficient outcome survives divinity. But what about potential pooling equilibria?

For example, I have difficulties investigating whether the asset-pooling equilibria (for positive correlation and no synergies, i.e., section 2.1) in Edmans and Mann's "Financing Through Asset Sales" satisfies the criteria, if I solve for three types.

I hope someone can help me.

Both of these criteria refine equilibria where there is an unused message or action. Notice that if there is one action that is not used by any of the types, it can be either because they are all pooling on some other action, or because they are all separating, but there are more actions than types, or anything in between (for example some types are pooling and some other are separating, etc.).

Now, unless the unused action is dominated by all types, for most games you will see that no type is using that action because if they do, they anticipate that the other player(s) will react and choose an outcome that no type would rather have.

The idea behind these two criteria is to realize that perhaps the only reason why the other player(s) react in such a bad way is that they have an (off-equilibrium path) belief about the type of the first player that is un-reasonable.

Because no type is actually choosing that action, the second player's beliefs are not restricted in a perfect Bayesian equilibrium. The intuitive criterion, however, imposes the restriction that the second player should not base her strategy after that unused action based on a belief that she is facing a type that would not possibly benefit from deviating to said action.

In the typical signaling game, this means that an employer should correctly believe that if a worker has sufficiently more years of education than what expected in a pooling equilibrium, it is not the low type worker since he has no incentives to signal their type by acquiring more education.

Similarly, the criterion of Divinity imposes stricter restrictions on off-path beliefs. Not only the second player should not believe she is playing against a type that cannot benefit from deviating to this action, but also it should assign a higher probability in her beliefs to be playing against a type who has a higher incentive to deviate.

Due to these extra restrictions, it should not be surprising that while many separating equilibria survive the first criterion (included the pareto-efficient outcome), with stricter restrictions, only the Riley outcome survives. Of course, the result that the efficient outcome is the one that survives is not obvious at all.

I hope this helps you form an intuition about whether the pooling equilibria you are considering survives these criteria or not. I am not familiar with the model you reference, so I can't help much with that. But I'm happy to give further thoughts if you introduce the model here, or expand on what you have been able to figure out.