Transition Matrix: Discrete -> Continuous Time I have the code corresponding to Tauchen (1986) (Python equivalent of this), which generates a discrete approximation of a discrete time AR(1) process.

For example, if you set up grid size as 3, it gives you a vector of productivities

[A_1, A_2, A_3,]

and a matrix of transition probabilities

A_11, A_12, A_13
A_21, A_22, A_23
A_31, A_32, A_33

Where row i, column j gives you the probability of transitioning from i to j, and it satisfies that the sum of each row is approximately one.

I am wondering how I can transform this to a continuous time equivalent of the transition matrix; a set of poisson probabilities controlling the flow rates between the states.

All I remember in this regard is that we can get the linear approximation to the poisson probabilities using

$$Prob(i \to j) = \lim_{\Delta\to0} \exp(-\lambda_{ij}\Delta) \approx 1-\lambda_{ij}\Delta  $$

But I can't see how that helps me transforming that former matrix to the $\lambda$s... I'm looking forward to any suggestion.

Suppose $B$ is an $n\times n$ matrix of Poisson transition rates, where $B_{ij}\geq 0$ for $i\neq j$ denotes the rate at which state $i$ transitions to state $j$, and $B_{ii}\leq 0$ gives the rate at which state $i$ transitions to all other states. Each row of $B$ sums to 0.

Then if $p(t)$ denotes the probability distribution at time $t$, by definition of $B$ we have the ODE
$$\dot{p}(t) = Bp(t)$$
We know what the solution to this kind of ODE looks like: $p(t)=e^{Bt}p(0)$, where $e^{Bt}$ is the matrix exponential of $Bt$. So, if we want $B$ to generate the Markov transition matrix $A$ after $t=1$, we need to have $e^B=A$.

In principle, to get $B$, we need to invert the matrix exponential, taking the matrix logarithm of $A$. The problem is that each matrix has many matrix logarithms - the logarithm in one-dimensional complex space has infinitely many branches, and this is compounded when we're talking about matrices in $n$-dimensional space. Most of these logarithms will not be satisfactory Poisson transition matrices: maybe they won't be real, or the entries won't have the right signs. Yet it is possible that more than one of them will be: in some cases there is more than one Poisson $B$ corresponding to a Markov $A$, just as in some cases there is no Poisson $B$ corresponding to $A$. It's messy.

Fortunately, there is a situation where life is relatively simple, and it almost certainly includes your own case: when all the eigenvalues of $A$ are positive, distinct reals. In this case, there is only one logarithm of $A$ that will be real, and it's easy to compute: you just diagonalize the matrix as $A=V\Sigma V^{-1}$ and take the real logarithm of the eigenvalues, getting $B=V\Omega V^{-1}$, where $\omega_{ii} = \log(\sigma_{ii})$. Indeed, you don't need to do this yourself: if you use the command $\text{logm}(A)$ in Matlab (presumably Python too), it will give you precisely this $B$.

Given this $B$, all you have to do is to verify that it's actually a Poisson matrix. The first requirement, that the rows all sum to zero, is satisfied automatically due to the construction of $B$.** The second requirement, that the diagonal elements are negative and the off-diagonal elements are positive, does not always hold (I think), but it's easy for you to check.

To see this in action, I'll consider an $A$ for a 3-state Markov process that resembles a discretized AR(1).
$$A = \begin{pmatrix}0.5 & 0.4 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.4 & 0.5\end{pmatrix}$$
Now, if I type $B=\text{logm}(A)$ into Matlab, I get
$$B = \begin{pmatrix}-0.86 & 0.80 & 0.06 \\ 0.40 & -0.80 & 0.40 \\ 0.06 & 0.80 & -0.86\end{pmatrix}$$
This is indeed a valid Poisson transition matrix, as we can easily check that the rows sum to zero and have the right signs - so this is our answer.

The case with positive eigenvalues is pretty important, since it spans all cases where there is not some kind of oscillatory behavior in the Markov chain (which would require negative or complex eigenvalues), presumably including your discretized AR(1).

More generally, the $\text{logm}$ command on Matlab will give us the principal matrix logarithm, an analogue of the principal scalar logarithm that takes all eigenvalues to have imaginary part between $-\pi$ and $\pi$. The problem is that this is not necessarily the logarithm we want, and by looking at it we might miss a Poisson $B$ that does generate $A$. (That's why the positive eigenvalue case, where we didn't have to worry about this, was so nice.) Still, even in these other cases it can't hurt to try and see if it works.

By the way, this problem of seeing whether there is a $B$ that generates some Markov matrix $A$ has been studied extensively. It is called the embeddability problem: see some overview and references in this excellent survey article by Davies. I'm not an expert on technical aspects of the problem, though; this answer is based more on my own hackish experience and intuition.

I feel obligated to close by seconding ecksc's comment and saying that there might be better, more direct ways to convert a discretely fitted AR(1) into a finite-state continuous time process - rather than just taking the matrix obtained via the Tauchen method and making it continuous. But I don't personally know what that better way is!

**Explanation (though I'm rusty): $A$ has a unique Perron-Frobenius eigenvalue of 1, and since $A$ is stochastic the right eigenvector of this eigenvalue is the unit vector $e$. This is still the right eigenvector, now with an eigenvalue of 0, when we take the matrix logarithm.


Solve for the bundle component that makes one as 'well off' as earlier Suppose we have a (presumably time independent?) utility function $U(x_1,x_2)$ for consumer Rita.

1.

  What is Rita's MRS of $x_2$ for $x_1$?

$$MRS_{x_1, x_2} = \frac{MU_{x_1}}{MU_{x_2}} = \frac{\frac{\partial U}{\partial x_1}}{\frac{\partial U}{\partial x_2}}$$

(or in some strange books $-\frac{MU_{x_1}}{MU_{x_2}}$)

Is that right?

2.

  If Rita consumed $a$ of $x_1$ and $b$ of $x_2$, what was her MRS?

$$MRS_{x_1, x_2}(a,b) = \frac{MU_{x_1}(a,b)}{MU_{x_2}(a,b)}$$

Is that right?

3.

  3.1 If Rita is currently consuming $c$ units of $x_1$, how many units of $x_2$ must she consume in order to leave her just as well off as she was in #2?

  3.2 What is the MRS at $(c,x_2)$?

What's the equation to solve here?

$$U(a,b) = U(c,x_2')$$

?

Or

$$\frac{MU_{x_1}(a,b)}{MU_{x_2}(a,b)} = \frac{MU_{x_1}(c,x_2')}{MU_{x_2}(c,x_2')}$$

?

I'm thinking the former because 'well off' seems to call for 'utility' and not 'MRS', and the latter answers 3.2 easily...

Anyway, if the former then for 3.2 I just compute

$$\frac{MU_{x_1}(c,x_2')}{MU_{x_2}(c,x_2')}$$

?

Converting my comments to an answer:

I think your answers to 1 and 2 are correct. For 3.1, your intuition that "as well off as" should be interpreted "indifferent" is correct, and indifferent is about comparing utility levels, not MRS. You answer to 3.2 looks correct.


Application of Laffer Curve on the price of Goods and Services The Laffer Curve shows the relationship between Tax-Rates and the Amount of Tax Revenue Collected. Can't a similar principle be used on the price of goods?
To elaborate: The total revenue of a firm will only go up to a certain extent when they increase the price before it starts falling, just like the Laffer curve shows for tax revenue.
I couldn't find much on this on the internet; So wanted to see if there is anything wrong with this assertion.

Trivially this holds, in many cases, but it is not considered to be Laffer curve or application of the Laffer curve. This is just standard profit/revenue maximization. Let us focus on the revenue part. The total revenue is given by price times quantity $TR=pq$, now quantity will depend on some demand, for example the demand could be $Q = 100 - p$, so the total revenue will in terms of price be:
$$TR=100p -p^2$$
Which in this case will also give you something that looks like Laffer curve, as you can see on the plot from simulation I made in R:

However, note two things:

Demand curves come in various shapes and they critically influence what the shape of total revenue is - consequently the above might not hold for any demand curve you can imagine.
Both this result for revenue and the result that you get zero tax revenue at both $0%$ tax rate and $100\%$ tax rate was known to economists long before Laffer. Laffer curve is called Laffer curve just because Laffer popularized already existing idea and he did so in the realm of taxation. Hence we talk about Laffer curves mostly just when we talk about taxes not other quantities where there will be some 'optimum peak'.


"For a market to exist there must be scarcity" (T/F) The answer provided was 'true', that scarcity was necessary in the existence of a market. But isn't that not always true? For example, there's a market for canned air, and air is most definitely not scarce.

Is the statement true or false? And if true, what about in the case of canned air?

I agree that air is not, in any meaningful sense, scarce. But there is no market in air! There is a market in canned air, but that's different because canned air is most definitely scarce.

The can is made of metal, which is not in infinite supply. If you use metal to produce a can of air then you must take that metal away from production of, say, airplanes or paperclips. Alternatively, you could just mine more metal. But then you would need to hire more miners, so you are taking workers away from some other industry.

Moreover, the can is manufactured by scarce labour. To manufacture the can, you also need machines and tools that are in limited supply abd could be used to make soup cans instead. The machines stand in a building on land that is in scarce supply and could be used for housing or farming. To get the metal to that building, you need scarce trucks, drivers, and fuel.


Theories that complement/contradict prospect theory? Kahneman and Tversky's prospect theory, which they developed to contradict expected utility theory, is obviously an interesting result.

But, after their experiments has anyone tried to completement the theory? Or to contradict it?

A good place to start would be the book Prospect Theory: For Risk and Ambiguity, by Peter Wakker (2010, Cambridge University Press).  There is an entire area of economic theory called decision theory which studies models of decision making under uncertainty --typically, models which deviate from the classical expected utility maximization model in some way.  Decision theory involves both theoretical results and experimental tests.  Prospect theory is only one among many models that decision theorists consider.


Are overlapping generation (OLG) models extensions of a DSGE model? Are overlapping generation models (OLG) extensions of a dynamic stochastic general equilibrium (DSGE) model? Or aren't these DSGE per se?

You can find OLG models that do not classify as DSGE (in particular, the model might not be stochastic) as well as DSGE with overlapping generations (contrary to those with infinitely lived agents).

You can find more detail on this on this working paper by Assous and Duarte (2017), as they note

  In the early 1980s, when the real business cycle macroeconomists
  brought one single model (a perfectly competitive growth model with
  infinite-lived agents, flexible prices, and perfect information) to
  bear on any macroeconomic issue, several macroeconomists were working
  with OLG models and addressing business fluctuations matters. Besides
  the efficiency issue, the model seemed to have much more to offer. New
  classical economists such as Wallace and Lucas saw in the OLG model
  the possibility to develop new microfounded models of fiat money
  without postulating that money balances enter the utility function of
  agents. At about the same time, other macroeconomists discovered that
  OLG models give room for either deterministic or stochastic
  oscillatory trajectories. Endogenous cycles and chaos as well as
  sunspot equilibria were then shown to occur in the presence of
  perfectly competitive product market devoid of any nominal price
  rigidities. Gradually over time, important contributors to that
  literature — most notably, for our interests here, is Woodford —
  strove to transfer OLG conclusions to infinite-lived agents models. In
  this context, new dynamic models with market imperfections were
  developed, initially with flexible prices, that eventually became the
  hallmark of the sticky price, DSGE macroeconomics (earlier referred to
  as the new neoclassical synthesis).


Where did Robert Lucas say "market is always in equilibrium"? Does have anyone link to the reference where Robert Lucas said "market is always in equilibrium"?

I think this is a poor oversimplication of Robert Lucas's maxim. It's one thing that Robert Lucas models markets as if "always in equilibrium", it's quite another Robert Lucas claims it's an empirical truth that  "market is always in equilibrium".

Lucas(1978) explained briefly why he chose such methodology which doesn't faithfully reflects reality:

  As always, there are innumerable ways for the economy to be out of equilibrium, so we must expect any treatment of out-of-equilibrium behavior to have considerable arbitrariness, not resolvable by economic reasoning.


Market structure and price setting: Imperfect competition in the final good sector I'm looking at an economy with two sectors. The intermediate good is produced under perfect competition and Cobb-Douglas production.

There is a fixed-cost for producers to enter the final market. Hence, there is a measure of firms that produce here. There is a search friction between consumers and producers, such that ultimately, each consumer will by from one producer only. Final-goods producers produce "on demand" using the intermediate good only, hence, some of them will need more of the intermediate good and some will need less.

This is the first time I'm looking at a market like this. There is a measure of firms in both sectors. This means, there are countably infinitely many firms in both sectors, right? How does this make sense, given that there is perfect competition in one and imperfect competition in the other?

I would like to final sector firms to all pay the same price, despite having different ex-post demands. I guess some sort of insurance prior to period start would take care of that. However, how would then prices be set here?

One solution to this problem is to use monopolistic competition (sometimes called the differentiated goods monopoly) in the final good sector.  For example, households might consume a continuum of final goods, each different, and aggregated with a constant elasticity of substitution utility function. This is the setting in the hugely influential Dixit and Stiglitz  (1977) paper. The approach of Dixit and Stiglitz is heavily used directly or as motivation in new Keynesian modeling because monopoly pricing power allows for sticky prices.


What's the term for elastic pricing? What I've called "elastic pricing", has a true economics name that I believe I once heard.

Here's the notion:
I walked into a gas station to buy a fountain drink. I only brought a dollar with me, but when I went to purchase the drink, I found out it was actually \$1.89. The cashier accepted that as payment - making a real-time decision that he was able to capture my demand at only $1. I ended up trying this at a few other gas stations, sometimes bringing 90 cents, and it always works.

An alternative to this is that my University just started pricing football tickets at a variable rate based on the location of the seats. Invariably, they were trying to capture the additional value that the second market was instead capturing.

A second alternative is that a former company of mine negotiated every price. So we would usually price the product at $500/user/month. But, we would always assess the situation, and with negotiation offer discounts, or increase the pricing. Or, provide value added services for free or at an increased rate.

So how do I talk about this flexible pricing concept – where a business sets it's pricing based on some kind of average demand, but considers an additional spread based on an individual's demand?

Edit:
In my first example, the bargaining was only noteworthy because it informed a new mental model of mine – that the pricing was flexible. I don't believe it's an inherent component in the term I'm seeking.

As yet another example, consumer package good brands are known to alter their products slightly for online marketplaces – so that value savvy customers can't compare products directly online with those in the store. E.g. Duracell might offer a 6pk of batteries on Amazon vs. a 4pk at a retail outline. Or, they might package the item differently so it has a different SKU.

Perhaps you may want to look into Price Discrimination? This however is the concept of companies varying their prices on different consumer markets.

1st degree price discrimination, at it's most optimal, is when companies charges the price of each individual consumer's demand. Those willing to pay the most will pay the most. Your description resembles this procedure, but it is not quite exact. Yours implied bargaining, which may or may not lead to 1st degree price discrimination.

2nd degree price discrimination is setting variable prices depending on the product or the quantity sold, playing off of the consumer's preferences.

3rd degree price discrimination is the segmenting of a market into smaller markets (based on some characteristic), and charging a uniform price in that market. The price between these segmented markets vary.


How is India able to increase its foreign exchange reserves despite historically running a trade deficit? Balance of Trade: https://tradingeconomics.com/india/balance-of-trade

Foreign Exchange Reserves: https://tradingeconomics.com/india/foreign-exchange-reserves

Remittances do help (avg. \$4B/mo) but are not enough to offset the trade deficit (avg. \$15B/mo).

India Remittances: https://tradingeconomics.com/india/remittances

Please note that the time scales in the above charts are not the same.

The central bank's accounts are not the same as entire country's accounts.

The foreign currency reserves statistic comes from the Reserve Bank of India, the central bank, when it discloses its account. The trade deficit is not a property of the central bank's accounts, although the central bank might have economists producing that statistic. The trade deficit is a property of the entire country in aggregate, which includes household, business, and government trade activity.


Demand Curve Price not perfectly elastic I'm trying to make the argument that our Sales go up when prices for all our Products are 30% off. Which I believe should be our new Regular Price. but our Demand Curve doesn't support this. Should I be setting up the demand curve differently? on a per product basis? or package size?

I sell our food product in different package sizes
2oz, 4oz, 9oz , 16oz 3.5lb
Normalized the time our products were at that price.

Counting your quantity in ounces makes sense, I wouldn't change that (I assume each point is a total sales/price combination at a given time?).

If you want to look at the impact of a percentage change in your price, you should take the logarithm of price. A .3 change in log(price) is roughly equivalent to a 30% change in price.

That said, you do in fact see a downward slope on your demand curve, which indicates that a drop in price is correlated with an increase in sales, although it's a general negative relationship and there's nothing special about 30% in particular (I wouldn't expect there to be!). So you're pretty much showing what you want (discounting issues about statistical significance, or causal identification - if these prices aren't set randomly this isn't technically showing you a demand curve).


Net Neutrality and Recent Telecom Projects It has come to my attention that the big telecom providers are launching some big infrastructural projects recently (at the time of post, Jan. 2018). I presume by that they mean talking laying cables and related activities. Supposedly this is being at least partially attributed to the repeal of net neutrality.

Question: What is it about net neutrality that would impact the profitability of these projects? Would it be fair to compare this to the price of gold and mining? If the costs to mine gold (per unit) outweigh the trading price of gold (per unit), the miners close up shop; and likewise if net-neutrality hurts the big telecom companies' margins enough for them to forego big projects?

Or what is really behind the incentives here?

My theory:
With the ever-increasing demand for bandwidth, there is a need for capacity.

With the repeal of net neutrality, providers can charge for a premium level of service, as well as refusing traffic that does not bring in revenue.

I think a better analogy would be an apartment building.
With net-neutrality: everyone would expect the pay the same rate and get the same level of amenities. Since no one can be turned away the front door of the building would also be unlocked 24/7.

Without net-neutrality: There will be multiple levels of service and price. The people that want a really fast service without having to share resources will have access to that level of service (if they want to pay for it), Those willing to have a little latency will pay less.


How central banks "create" negative interest rates? How can the Fed push Fed Funds rate to negative? I think I understand how it works with >= 0 -- they just lend reserves for bonds until they hit their targets, but what about <0? Has this something to do with purchasing bonds until their yield becomes negative, making banks which hold bonds as cash to have negative interest on them?

I am not sure about the institutional limitations on policies, but this is easily done in principle. Banks hold deposits at the Federal Reserve banks (reserves/excess reserves). If the Fed sets the interest rate on those deposits to a negative level, that pricing would ripple out everywhere.

For example, assume the rate was -1%. Banks would buy Treasury bills until their yields were comparable to -1%, as they could replace excess reserves with bills to reduce the cost associated with the negative rates. These lower rates then affect the traded yields of other instruments traded in the bond/money markets.


Standard Deviations and Elasticities I am interested in two variables $x, y$. Their (observed empirical) standard deviations are $\sigma_x$, $\sigma_y$. I know the elasticity of $x$ w.r.t. $y$ is $\eta_{x,y}$.

Let $x$ be fully determined through $y$, which is the only source of randomness. Is it true that

$$ \frac{\sigma_x}{mean(X)} = \eta_{x,y} \cdot \frac{\sigma_y}{E[y]}$$

?

I can't find anything on this relationship, but it appears that appropriately scaled  standard deviations of two variables should be connected through their elasticity.

Let $X = h(Y)$. A first-order Taylor expansion around $E(Y) = \mu_y$ gives

$$X \approx h[\mu_y] + h'[\mu_y]\cdot [Y-\mu_y]$$

This easily leads to

$$\sigma^2_x \approx \big(h'[\mu_y]\big)^2\cdot \sigma^2_y$$

Since

$$\eta_{x,y} \equiv h'\cdot \frac Yh \implies \eta_{x,y}(\mu_y) = h'(\mu_y)\frac {\mu_y}{h(\mu_y)} \implies h'(\mu_y) = \eta_{x,y}(\mu_y) \cdot \frac {h(\mu_y)}{\mu_y}$$

Substituting,

$$\sigma^2_x \approx \left(\eta_{x,y}(\mu_y) \cdot \frac {h(\mu_y)}{\mu_y}\right)^2\cdot \sigma^2_y$$

Taking the square root we are led to

$$\frac {\sigma_x}{h(\mu_y)}  \approx \eta_{x,y}(\mu_y) \cdot \frac {\sigma_y}{\mu_y} $$

The difference of the above from the expression in the question is that

a) The elasticity must be evaluated at the center of the Taylor expansion used
and
b) $h(\mu_y)$ is usually equal to $E(X)=E[h(Y)]=\mu_x$ only to a first approximation, due to Jensen's inequality.

Of course if the relation between $X$ and $Y$ is linear, and/or the elasticity is constant, things specialize.

In any case, it is valid as an approximation to write

$${\rm cv}_x \approx \eta_{x,y}(\mu_y) \cdot {\rm cv}_y $$

where "cv" stands for "coefficient of variation".


why do we analysis the housing market from 'short run' and 'long run'? As we can see the factors impacting the housing market, particularly house price, are always analysed from the long run and short run, even some researchers analysis those factors from cyclical and structural approach. How can we understand the long run and short run? And what are cyclical and structural influences on house price? Thank you.

Housing markets are complex and the following are just some basic points.

Key features of houses as economic goods are: 1) their very long life; 2) their differentiation by type and location. In the short run housing supply (the total existing stock, plus any new houses completed) is price-inelastic because the number of new houses can be only a tiny proportion of the existing stock.  Hence a short-run analysis of house price changes will focus primarily on short-term changes affecting demand (eg changes in interest rates or in housing-related taxes).  In a long run analysis, however, consideration also needs to be given to changes in supply, and to long-term changes affecting demand (eg population growth, rising incomes).

Cyclical factors are those that tend to reverse over a period of a few years. The cause of a cycle in a housing market could be internal to that market, eg a self-fulfilling expectation at one time that prices will rise, leading to a price bubble and eventual collapse, as described here.  The cause could also lie in the wider economy, eg house prices tending to rise when the economy is booming and fall in a recession.  Structural factors are permanent or at least long-lived, eg land availability and planning constraints, housing-related policies that are considered unlikely to change, and the distribution of the existing stock between types and locations.


How could Greece-related worries drive US stocks down about 2% in a day? Major US stock indices are down ~2% on Jun 29, and it's mostly associated with investor's worries related to the Greece crisis.

I understand about possible "chain reaction" and that a lot of companies may be exposed indirectly to Greece debt.

But given that, it seems silly that investors think that because of this average US companies lose 2% in values. I mean, Greece is relatively a small country and it doesn't have a very close link to the US economy (unlike, say, Canada). Even if all Greek government and banking bonds evaporate, the impact of all those would still be way less than 2% of an average US company, right?

Because Greece sets a precedent for Italy, Spain and Portugal (and others).
And a default of Italy (and Greece) might will cause financial difficulties in France.

If the interest rates for Italy and Portugal rise (they do now), then those countries are even more unlikely to repay their debt.
And meanwhile, somebody  has to write them off.

Greece in itselfs really is kind of unimportant (except for France).
That the debt really cannot be repaid is at this point kinda obvious to everyone. And while a write-off of 80*109 € (or part thereof) is painful for Germany (and others), it's simply not the size that can seriously jeopardize an economy of 8 times the polulation and 15 times the GDP of Greece. The risks of political fallout in Germany are another story (and the problem of Merkel).

Greece mainly is kind of an experimental laboratory of things to come (Syriza in Greece = Podemos in Spain, Italy => ? ...)

And the experiment seems to spell disaster.
Meanwhile, nobody likes to be the labrat.

As always, the stock-exchange is the place where the expectations of the future generate the prices of today (together with the panic of the moment).


Who really pays transaction taxes like VAT or sales tax? In Europe, shop prices are typically displayed including VAT. This gives the impression that the seller is paying the VAT. They have to offer a competitive price, then a percentage of that is taken out as tax. But, if VAT didn't exist, sellers could offer a lower price. So in another sense, buyers pay the tax.

In the US, sales tax is typically added at the till. This makes it feel like the buyer is paying the tax. But ultimately, VAT and sales tax are the same (at the point of final sale) - does it really make a difference how the prices are displayed?

The same issue exists with income tax, stamp duty, and presumably many other taxes. It's quite integral to Western life that income tax is paid by the employee - you hear people say "I pay my taxes, I'm part of this society". But the need to pay income tax means that employers need to pay higher wages, so the employer is bearing the cost as well.

Does it make a difference what happens when the rate changes? For example, if income tax rises, typically salaries do not rise to match. But if sales tax rises, the the amount added at the till immediately rises too.

The issue you are asking about is tax incidence. This Wikipedia entry provides a good introduction. In short: it depends on how easily supply and demand respond to price changes relatively to each other, the so called price elasticity of demand and supply.

According to standard theory, it doesn't matter who pays (in the legal/fiscal sense) the taxes, as the burden of the taxes depend on economic circumstances. If demand falls rapidly with higher prices while supply does not, sellers/producers will lower their price (before tax), and therefore carry a relatively large part of a tax.

Of course, there may be all kinds of administrative costs and other 'frictions', that make it more or less efficient to have one group or another pay taxes. Also, there is evidence that people do change their behavior according to how the taxes are paid; see for example this paper, or this document.


Linear Regressions: When to expect the residuals to NOT have conditional mean 0? Suppose that $Y, X,$ and $U$ are random variables such that the regression $$Y=\beta_0+\beta_1X+U$$ is the best linear predictor of $Y$ given $X$.

My question is, when can we expect $E(U|X)=0$? I know that this conditional expectation is usually assumed to have mean $0$ for OLS and under other specific cases this is also true (such as when $X$ is binary). But why is $E(U|X)=0$ be generally not true (for example, when $X$ takes on other values besides $\{0, 1\}$) and are there any ways that we can modify it to make $E(U|X)=0$?

The assumption could be violated due to counfouding variables or endogeneity for example. In fact the assumption fails any time there is correlation between $X$ and $U$ (hence any time $COV(X,U) \neq 0$. This actually happens all the time - most of the whole field of econometrics exist to help us deal with this problem.

An concrete example when the assumption would fail would be omitted variable bias. For example if the population model is:

$$y = \beta_0 + \beta_1 x + \beta_2 z + u$$

but you try to fit just simple regression $\hat{y} = \hat{\beta_0} + \hat{\beta_1}x$ and there is correlation between $x$ and $z$ then the zero conditional mean assumption will be violated because the regression model will attribute any changes in $y$ even those caused by $z$ to $x$. For example, suppose you want to examine the effect of education $(x)$ on wages ($y$) and also suppose education correlates positively with education of parents ($z$) and you exclude the education of parents from your model. In that case the regression model applied to the sample would systematically overestimate the effect of education as it would attribute portion of the effect of parents education to the persons education.

Also note the zero conditional assumption is assumption about true model - it cannot be verified with data, you need knowledge about the subject matter and make some logical argument to argue its valid. In sample the  expected mean of error will be always zero just because OLS is calculated by minimizing the sum of squared residuals.


Robust predictions in single-agent decision problem with uncertainty I would like your help to better understand the possibility of using the notion of Bayes Correlated Equilibrium (BCE) in a single-agent decision problem with uncertainty to make predictions on optimal strategies that are robust to the belief environment. The notion of BCE is provided in this paper for a generic $N$-player game.

Consider the following single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{v})$. First, let nature draw a realisation $v$ of $V$ from $P_V$.
Then, let the decision maker choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the decision maker gets a payoff $u(y,v)$.

For example, suppose that $\mathcal{Y}\equiv \{1,2,3\}$. $V$ is a $3\times 1$ random vector, $V\equiv (V_1,V_2,V_3)$. $P_V$ is the 3-variate standard normal distribution. $u(y,v)\equiv v_y$.

Before choosing an action, the decision maker can receive at least one  signal to refine her prior (minimal amount of information). I assume that this minimal signal is completely uninformative (degenerate information structure).

Now, I want to use Theorem 1 in Bergemann and Morris (2016) to characterise the set of optimal strategies under minimal assumptions on the amount of information that is processed by the decision maker (degenerate in this case). To do that, I introduce the notion of one-player Bayesian Correlated Equilibrium (BCE).

A one-player BCE of the game described is a probability distribution $P_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ such that:

1) $\forall v \in \mathcal{V}$
$$\sum_{y\in \mathcal{Y}}P_{Y,V}(y,v)=P_V(v)$$

2) $\forall y\in \mathcal{Y}$ and $\forall \tilde{y}\neq y$
$$
\sum_{v\in \mathcal{V}} (v_y-v_{\tilde{y}}) \times P_{Y,V}(y,v)\geq 0
$$

Question:

1) Is it obvious that a BCE exists and is unique in my example?

2) Suppose now that I know for sure that the decision maker does not have any information in addition to the degenerate information structure. How would I characterise an optimal strategy in such a case (without the necessity of being robust to possibly richer belief environments)? How would that definition differ from the BCE definition given above?

1) Given that $\mathcal{Y}$ is finite, it is obvious that the BCE is not an empty set (or it exists). Remember that the set of BCE's is the set of decisions that are optimal for some belief about $\mathcal{V}$ that could be formed (as a posterior belief) after observing some signal and having a prior $P_V$.

To show there is at least one distribution satisfying the definition if BCE, you can consider an uninformative signal. Given that there is only a finite set of actions, one of them must maximize the expected utility given the prior. If there is more than one, randomize among them as you please. Let the optimal action be $P_Y^*$ (in this notation I capture both situations where the maximizer is unique and when there is more than one maximizer) and define $P_{Y,V}=P_Y^*P_V$, then such a distribution constitutes a BCE.

Now, you should not expect uniqueness because if there was only one distribution that constitutes a, or if BCE was a singleton (since we usually think of BCE as a set), then information is essentially irrelevant. It would mean that there is an action in $\mathcal{Y}$ that is the best action regardless of any extra information you might receive.

2) To characterize this, simply find the optimal strategy when the belief is equal to the prior, $P_V$. If you want to  put it in terms of the two expressions you have, then $P_{Y,V}$ must  be a product distribution, i.e $P_{Y,V}(y,v)=P_Y(y)\cdot P_V(v)$ for some $P_Y(y)\in\Delta(\mathcal{Y})$ (This will capture the fact that the player has no further information beyond the prior and so it cannot correlate his actions with the state of the world). The second inequality will pin down $P_Y$. In fact, using the notation from the previous bullet point, it should be equal to $P_Y^*$.


When can the fiscal multiplier be negative? I am looking for a case study or data which shows a negative (fiscal/spending) multiplier effect.

Does anyone know a study where this is analyzed?

The paper How Big (Small?) are Fiscal Multipliers? by Ethan Ilzetzki, Enrique G. Mendoza, Carlos A. Végh (2010)should give you useful information.

  We contribute to the debate on the macroeconomic effects of fiscal
  stimuli by showing that the impact of government expenditure shocks
  depends crucially on key country characteristics, such as the level of
  development, exchange rate regime, openness to trade, and public
  indebtedness. Based on a novel quarterly dataset of government
  expenditure in 44 countries, we find that (i) the output effect of an
  increase in government consumption is larger in industrial than in
  developing countries, (ii) the fiscal multiplier is relatively large
  in economies operating under predetermined exchange rates but is zero
  in economies operating under flexible exchange rates; (iii) fiscal
  multipliers in open economies are smaller than in closed economies;
  (iv) fiscal multipliers in high-debt countries are negative.


Which data on GDP to use? Wiki's article provides with 3 sources for GDP data: IMF, World Bank and CIA.

I wonder which of them is better (i.e. more accurate)? Also, what about reliability of so called Penn tables?

If anyone is aware of the source which is better than the above mentioned, please, share it.

The answer depends on whether you want accuracy with respect to original (national) sources, more comparability, and how much time you want to spend compiling data.

If you want to focus on a very few countries, or even just one country, I suggest to always go to the primary source, which is the national statistical office, or central bank (depending on the country). There you will have ample documentation about what is being measured. Albeit countries are urged to use the latest UN System of National Accounts, there might be particular issues that are nation-specific, which you will find in the national documentation. Additionally, you will find references to GDP measured in different ways (output, income, or expenditure approaches), which do not always give the same result.

To compare a few countries, you can use primary sources and then decide yourself how to compare them. For example, you can compare growth rates, transform GDP into an index (e.g. GDP=100 in 1990), or into a common currency, where you can decide which exchange rate to use.

This approach on primary sources gives you more accuracy with respect to national data, and also greater transparency in what you are actually measuring or comparing.

Now, if you want to compare many countries, or entire regions (e.g. OECD, EU, NAFTA, etc), using a consistent definition, or perhaps have little time for this comparative exercise, it might be better to use some international database like World Bank or Penn Tables. Here, you depend on the particular arrangements of each institutions (in terms of assumptions regarding exchange rates, price indexes, etc).

For example, the World Bank data states its source as World Bank national accounts data, and OECD National Accounts data files.  And yet, it indicates some limitations of the calculations:

  Limitations and Exceptions: Gross domestic product (GDP), though widely tracked, may not always be the most relevant summary of aggregated economic performance for all economies, especially when production occurs at the expense of consuming capital stock. While GDP estimates based on the production approach are generally more reliable than estimates compiled from the income or expenditure side, different countries use different definitions, methods, and reporting standards. World Bank staff review the quality of national accounts data and sometimes make adjustments to improve consistency with international guidelines. Nevertheless, significant discrepancies remain between international standards and actual practice. Many statistical offices, especially those in developing countries, face severe limitations in the resources, time, training, and budgets required to produce reliable and comprehensive series of national accounts statistics. Among the difficulties faced by compilers of national accounts is the extent of unreported economic activity in the informal or secondary economy. In developing countries a large share of agricultural output is either not exchanged (because it is consumed within the household) or not exchanged for money.

The Penn Tables are also based on national accounts. As the Documentation states:

  One of the major inputs into PWT is National Accounts (NA) data on gross domestic product (GDP)  at  current  and  constant  prices, in  local  currency  units.  [...] We  primarily  source  these  data  from  the  UN  National  Accounts  Main  Aggregates Database,
  see http://unstats.un.org/unsd/snaama/Introduction.asp

There are difference between these two sources. On the one hand, the WB database offer data in local currency units and US dollars (either in current or constant prices), and also in PPP. Additionally, the WB dataset includes all the exchange rates used to calculate its series. Penn Tables only include data in constant USD and PPP.

On the other hand, Penn Tables include data on PPP GDP measured in terms of output and expenditures. With WB, it is not clear which they are using for each country. Additionally, Penn Tables data is usually longer (some series start in 1950).

To check consistency across these databases, I made a few comparisons between the WB and Penn Tables data, for constant USD dollars, and for all countries I tried (France, India, Brazil, Canada), the correlation between the two series was 0.999). So clearly, there might not be much of an issue which one you use (as long as the one you want is available).

Finally, regarding IMF, I would not use these. Their comparative advantage is not to be a database of GDP statistics but of financial and government variables. Its database is not intuitive to use, and GDP documentation is in my opinion of low quality. If you want other alternatives to the above two, you can check UN stats, OECD, Eurostat, or other regional bodies. Finally, you can also use FRED.

(there are other benefits of Penn Tables, not related to GDP itself, which is that you can compare components of GDP (e.g. investment, government spending, etc) or TFP in a consistent fashion across countries)

TL;DR

For single or few country analysis, use primary source (national statistic office or central bank offices). For comparative analysis across many countries or macro-regions, the World Bank is the most comprehensive database. Penn Tables include just a few definitions of GDP. In any case, for common definitions, the two datasets are practically identical.


Does this price elasticity depend on p? I'm studying for a test I have this week and a practice question asks whether or not the price elasticity of demand for the demand function $\ q = Ap^{-x} $, where A and x are positive constants depends on the price p.

I've attempted to use the equation, $ \frac{dQ}{dP}*\frac{P}{Q} $ in order to solve the problem and I am getting the result,

$ \epsilon = \frac{-xA}{p^{x+1}}*\frac{p}{q} = \frac{-xA}{p^{x}q}$

Which would mean that the elasticity does  depend on p, however the answer in my practice exam says that it does not depend on p. Why is that the case/where did I go wrong?

That type of demand is called an Isoelastic demand function, because the elasticity of demand is constant, so it does not depend on prices.

You have not done any mistake. But you just need to move forward one more step. Notice that at the moment your derivative has both p and q in it. Try to get rid of q. This is, try to have $\epsilon=f(p)$, and not $\epsilon=f(p,q)$. This hint should be enough for you to solve the question.


Taking observed preferences seriously You often hear economists (proudly) say that they don't take stated preferences seriously, but instead rely on observed preferences. Correct?

To me both are problematic if you want to understand preferences, since people often do things they don't want to do. Example: I prefer to go to bed at 22:00 but I observe that I very often get caught up and don't do this.

Decision making of individuals is complicated and I think everyone should be careful interpreting observerd preferences?

My question is: Can anyone provide further points thinking about this issue and/or articles about the interpretability of observed preferences?

I am aware there exists micro-models with "Hyperbolic discounting" making preferences time-inconsistent. I am looking for some more general discussion and frameworks.

In my view, it is useful to distinguish two phenomena:

individuals have time-inconsistent preferences (this fits your example).  There is a huge literature on that. The common practice is to evaluate welfare from the ex ante perspective, thereby assuming that people have well-defined preferences over the long-run, but that their choices might deviate from it because of immediate impulses. In that case, the true preference can be elicited by asking the individual to choose in advance. Some authors question the ex ante perspective, see for instance "The Welfare Economics of Default Options in 401(k) Plans" by Bernheim, Fradkin & Popov.
individuals make mistakes because of cognitive limitations. In this situation, it is very hard (impossible?) to recover the true preferences from the choices. At best, we can have a measure of the extent to which the individual makes consistent decisions. You can have a look for instance at "The Money Pump as a Measure of Revealed Preference Violations" by Echenique, Lee & Shum.

Overall, it is true that economists usually reject stated preferences, and there are many good reasons for that. It does not necessarily imply that we assume that observed choices are perfect; just that we do not have a better alternative to elicit people's preferences. However, some researchers are now studying how we could take advantage of non-choice data: (i) from a neuroscience perspective (many recent papers by Antonio Rangel and Colin Camerer for instance); (ii) using stated preferences, like "Non-Choice Evaluations Predict Behavioral Responses to Changes in Economic Conditions" by Bernheim, Bjorkegren, Naecker & Rangel.


Money supply and Capital Flows I have a super basic question on international macro.
I read that an increase in the capital inflows (say because of favorable macro enviroment) can increase the domestic money supply. However, it is only the central bank that can engineer an increase in the money supply. How would this look like in practice?

Thanks!

I do not believe that foreign capital investment (at least in the short-run) would directly increase the domestic money supply.

When foreign investors purchase domestic currency, the investors merely transfer that currency from domestic accounts to their own accounts. Then, when the investors purchase capital in the domestic market, the purchased currency is transferred back to the domestic sellers. Thus I do not see how net circulation of domestic currency rises in the event of a capital inflow directly.

There are, of course, more complicated indirect effects such as domestic banks issuing an increased number of loans based off new demand deposits that may increase the money supply, but the direct effect that I believe you're alluding to isn't there. As you say, the domestic money supply cannot be increased without banks playing a role in fueling domestic credit.


Consumer optimum in an economy with a continuum of commodities Consider an economy with a continuum of commodities, with one commodity for each point in $[0,1]$.

Suppose a consumer wants to maximise
$$U = \int_0^1 c_i^\theta\,di\qquad 0<\theta<1$$
subject to
$$\int_0^1 p_i c_i\,di = M$$
where $c_i$ is the amount of the $i$-th commodity consumed, $p_i$ its price and $M$ the consumer's money income.

This kind of problem arises for example in applying the Dixit-Stiglitz model to macroeconomics or international trade.

The solution to this problem is supposedly
$$c_i = Ap_i^{1 \over {\theta-1}}$$
where $A$ is a constant chosen to ensure that the budget constraint is satisfied.

I am not very satisfied with derivations of this result which use Lagrange multipliers in analogy with the case of a finite number of commodities. What would be a completely mathematically rigorous method of deriving the above result?

It seems clear that there isn't a unique solution since arbitrarily changing the values of $c_i$ for a finite number of values of $i$ will leave the integrals in the utility function and budget constraint unchanged. I am expecting that a completely rigorous derivation would also correctly pinpoint this degree of nonuniqueness.

EDIT: In response to the comments by @BKay, @Ubiquitous. My problem with starting out with economies with $n$ commodities and taking the limit as $n \to \infty$ is that this needs to be accompanied by an argument which shows that the limit of optima is an optimum of the limit problem. I would appreciate a reference to a result which shows this either for this particular problem or a general result which is applicable to this problem.

In response to @AlecosPapadopoulos. The proofs of the Langrange multiplier method that is taught in math for economics courses is usually for a finite number of choice variables. I would appreciate a reference to where the method is justified for a continuum of choice variables. Also, the nonuniqueness I mention above shows that the method cannot be exactly right. Then what exactly are the qualifications required for its validity?

As the OP noted in a comment, Theorem 1 in section 12 of Kolomogorov and Fomin's Calculus of Variations seems to provide some comfort that we can indeed use the Langrange Multiplier method when the number of our variables is infinite. Still, the authors do that in a footnote, writing "the reader will easily recognize the analogy with Langrange multipliers". So no, this does not rigorously show what we want.

I think that what we need is a paper like Craven, B. D. (1970). A generalization of Lagrange multipliers. Bulletin of the Australian Mathematical Society, 3(03), 353-362. which in its summary writes:

  The method of Lagrange multipliers for solving a constrained
  stationary-value problem is generalized to allow the functions to take
  values in arbitrary Banach spaces (over the real field). The set of
  Lagrange multipliers in a finite-dimensional problem is shown to be
  replaced by a continuous linear mapping between the relevant Banach
  spaces.

This is math-speak but it says what we wanted to hear (one can also find a short exposition in wikipedia to the degree that it trusts the content).

Then, we can form the Lagrangean of the problem

$$\Lambda = \int_0^1 c_i^\theta\,\text{d}i +\lambda\left(M - \int_0^1 p_i c_i\,\text{d}i\right)$$

and calculate the first-order condition(s) by , informally speaking, "looking at the integral and seeing a sum",

$$\frac{\partial \Lambda}{\partial c_i} = 0 \Rightarrow  \theta c_i^{\theta-1} = \lambda p_i, \;\; i \in [0,1] \tag{1}$$

...a continuum of conditions. For later use we  define

$$\sigma \equiv 1/(1-\theta), \Rightarrow 1-\theta = 1/\sigma, \;\; \theta = \frac {\sigma-1}{\sigma}$$

The constant $\sigma$ can be shown to be the elasticity of substitution between any two goods.

Writing $(1)$ for commodity $j$ and equating through the common lagrange multiplier we arrive at

$$c_i = \left(\frac {p_i}{p_j}\right)^{-\sigma}c_j
 \tag{2}$$

Multiply both sides by $p_i$ and take the integral over the commmodity space with respect to $i$:

$$\int_0^1 p_i c_i\,\text{d}i = \int_0^1 p_i^{1-\sigma}p_j^{-\sigma}c_j\,\text{d}i$$

$$\Rightarrow M = p_j^{\sigma}c_j\int_0^1 p_i^{1-\sigma}\,\text{d}i$$

$$\Rightarrow c_j = p_j^{-\sigma} \cdot M \cdot \left(\int_0^1 p_i^{1-\sigma}\,\text{d}i\right)^{-1} \tag{3}$$

which is Marshallian demand for commodity $j$.


What are the advantage of chain-linked prices over constant prices? Some variables in economics like GDP are computed in chain-linked prices or constant prices.

What are advantages of chain-linked prices vs constant prices?

Thank you

The advantage of chain linking is that component weightings drift away from their weight in nominal GDP as the reference period recedes.

This is a St. Louis Fed document on the methodology: link to website.

I am not familiar enough with the details to summarise it better than that document.


If assets = liabilities + equity, what do we mean by overcapitalization? Overcapitalization doesn't seem to have a clear-cut definition.

For some it seems to deal with the ratio of current output (Y) over optimal output (Y*). A company would then be overcapitalized if Y < Y*, meaning it could raise its output without having to recapitalize.

Others define it as Equity + Debt > "True Value of assets"

I like the second definition but it is subjective.

I am trying to understand the concept of overcapitalization in the context of a firm's valuation.

A firm's Output $Y$ (in value-added terms, i.e. over and above the value of purchases of third-party services and materials) is distributed as reward to the factors of production. Denote $K$ the book value of the company's assets.  Use the standard notation $L$ for payments to the labor/human capital input to production. Denote $r$  the capital rate of return as measured with respect to the book value of firm's capital and denote $w$ the average remuneration of human capital services. Then we have

$$Y = rK + wL$$

Depending on the market, the industry, and a few other things, we can estimate  $r^*$, the rate of return that the firm "should" have on average, if it should be judged as an efficient user of capital allocated to it, for given output and human capital costs. $r^*$ is usually estimated by what are the returns on the market and the industry (we do not go outside the industry since we recognize the existence of "asset specificity"). Associated with $r^*$ goes the efficient level of capital, $K^*$. So we also have

$$Y = r^*K^* + wL$$

leading to

$$rK = r^*K^*  \implies r = r^*\frac {K^*}{K}$$

If we have that

$$r < r^* \implies r^*\frac {K^*}{K} < r^* \implies K^* < K$$

and we can say with some certainty that the inequality does not show a tendency to change in the near future, then we can argue that the company does not make efficient use of the current amount of capital it appears to have, either due to "functional inefficiency" (how things are organized and work inside the company and in the delivery of of its business, a qualitative aspect), or due to plain under-utilization of the existing capital (a quantitative aspect).

The "over-capitalization" story essentially assumes "functional efficiency" but it sees under-utilization of capital, and so it advocates the reduction of the capital/assets currently under the control of the company.

To measure it we have the relation ($K_E=$ Excess Capital)

$$ K_E = K - K^* = K- \frac {r}{r^*}K = \left (1-\frac {r}{r^*}\right)K$$

$r$ and $K$ is measured from the firms financial statement, while $r^*$ from Industry weighted averages.


Why is price plotted as a dependent variable? Watch the video from 5:35:

law of demand

In the description, the narrator says that "demand depends on price".
However, in the plot, he is keeping price "P" as the dependent variable.

How does that make sense?

In supply and demand model both price depends on quantity and quantity depends on price. These two variables are determined together. They are both endogenous variables. Consequently, here the rule that independent variable goes on $x$-axis and dependent variable goes on $y$-axis does not apply because neither of the two variables is exogenous.
By convention in economics price is always plotted on $y$-axis. This was not historically always the case. As you can see in this answer in past there were several authors that decided to plot price on $x$-axis as well. However, the convention of price going on $y$-axis was eventually established by Marshall who considered price to be the depend variable (although really they are both endogenous), and popularized this convention through his textbook (Principles of Economics) that was widely used.


Tax system is progressive or regressive Let´s discuss a little!

When the direct effect of a tax or transfer policy (compared to what would happen in the absence of the policy) is a reduction in inequality, it is called progressive policy. If a policy’s direct effect is a rise in inequality it is called regressive. Policies that are neither progressive nor regressive are called distributionally neutral.

In the book on the website https://core-econ.org/the-economy/book/text/19.html#1910-redistribution-taxes-and-transfers there is an exercise 19.10.

There are question such as

1) Suppose you learn that the richest 10% of people pay 30% of income tax. Does it mean that the tax system is progressive?
2) Some governments of developing countries give scholarships for some of their best students to go to graduate school abroad. If there are no eligibility restrictions, is this policy likely to be progressive or regressive? What might justify this policy?

My answers
1) From the information given we can not say whether tax system is regressive or progressive. First of all we do not know what percentage do the lower-income earners get. Secondly, if poorer people are paying less percentage we could say that the tax system is progressive. If lower-income earners pay the less or equal percentage as the richest 10% of the society then we could say that the tax is regressive.

2) I do not understand exactly the question even though i translated it to my mother language. I do not understang preciecly this part ¨ If there are no eligibility restrictions, is this policy likely to be progressive or regressive? What might justify this policy?¨. However answering this questiong i would like to say that if students who got scholarship get a good education abroad and then return bqck with good knowledge and will transfer this knowledge to other citizens (for example being teacher or professor researchers) or even politicians or economists then in a long-run situation in a country could prosper.

My question
What is your opinion? Do you agree with my opinion? Or you maybe can say something in contrary?

Suppose you learn that the richest 10% of people pay 30% of income
  tax. Does it mean that the tax system is progressive?

You are right. It cannot be determined without knowing the income tax of lower-income earners.

  If there are no eligibility restrictions, is this policy likely to be
  progressive or regressive? What might justify this policy?

That is even more uncertain than the previous question because it requires assumptions that are not inherent to --or implied by-- the policy. In fact, giving scholarships to some of the best students contradicts that there are no eligibility restrictions, but that is irrelevant.

First, not all degrees (whether earned at home or abroad) lead to a profitable career. For instance, graduates in graphical design usually earn significantly less than physicians.

Second, the effect of scholarships also depends on whether those students become high-income earners and decide sever ties with their home country. This could lead to a "progressive" scenario of lower inequality notwithstanding that the population that remains in the home country is the poorer sector.

It is just impossible to conjecture about the effects of the policy based on the information given.


What happens to the billions in Euros that Greece owes if it leaves the Euro Greeces owes billions of Euros to the IMF and European Union. What happens to those debts if it leaves the Euro? Do the debts get translated into Drachma or can the government just print Drachma to convert to Euros freely?

If you owe euros you have to pay in euros. If for example the British government owes euros, then they get some funds (perhaps take another loan) and in case this is in a currency different from euros they convert it to euros on the market. If the sum is large enough it is wise not to convert it all at once but in smaller installments. This way the exchange rate stays stable and you can also get a better deal.

So nothing will happen to Greece's debt because they leave the Eurozone. But as the whole issue is political, there could be debt relief/partial bankruptcy because of the same reasons that might cause them to leave the Eurozone.

About printing drachmas:
There is a technical issue to be addressed: It is not the government that prints money but the central bank and their goal is maintaining price stability, not providing the government with revenue. (In time of crisis this may change.)

Printing money causes inflation which in turn will weaken the drachma relative to the euro. Usually when the state prints drachmas all drachmas would be worth less, but states wealth may increase because their share of drachmas increases.
However, this is a special case because there are no drachmas at all. So initially all drachmas would be owned by the state. Hence I think if they print more the worsening exchange rate would balance it out.
What they could do would be to print some drachmas, exchange it with whomever was willing for euros, and then print more. This would only work if printing more is not expected.


Productive Allocative Efficiency (Competitive Equilibrium) I am doing an Intermediate Microeconomics class... in a 2*2*2*2 economy, I know that MRS (marginal rate of substitution) is supposed to be equal to MRTS (marginal rate of technical substitution) in order for it to be a competitive equilibrium. Why is this? I understand that supply is supposed to be equal to demand...
Thank you!

Equilibrium conditions will require, among other things,:

1.) $MRS_{a,b}^i=MRS_{a,b}^j \forall [i,j] \in N$ , $N$ the set of agents

2.) $MRTS_{x,y}^i=MRTS_{x,y}^j \forall [i,j] \in J$ , $J$ the set of firms

3.) $MRT=MRS$ (Assuming that 1,2 hold)

And I note that 1,2 are efficiency conditions. 3 is efficiency in the output market.

Why? Suppose 1,2 hold at:

$MRS^i_{y,x}=3$
$MRT_{y,x}=2$

Obviously, condition 3 does not hold.

Can you see why this cannot be an equilibrium outcome? Think about what should happen in the economy and the forces that will drive this economy toward a state s.t. condition 3 holds.

So:

$MRT_{x,y}=\frac{MC_x}{MC_y} \equiv \frac{P_x}{P_y} = MRS$

where I assume 1 holds so MRS is same for all agents.


What does it mean if country X has a higher GDP at Purchasing Power Parity (PPP) than country Y? I understand the implications of having a higher/lower GDP at market exchange rates. I also understand what PPP measures.

But, what does a difference in GDP, at PPP, indicate in terms of differences in levels?

I like this explanation from Callen (2007), published in the IMF's F&D, their quarterly magazine.  For measuring consumption in a country, PPP is probably the way to go, but international economic and financial power is probably better measured with market rates.

PPP versus market rates
So which method is better? The appropriate way to aggregate economic
data across countries depends on the issue being considered. Market
exchange rates are the logical choice when financial flows are
involved. For example, the current account balance represents a flow
of financial resources across countries. It is appropriate to use the
market exchange rate to convert these flows into dollars when
aggregating across regions or calculating the global current account
discrepancy. But for other variables, the decision is less clear cut.
Take real GDP growth. International organizations use different
approaches. The World Bank uses market-based rates to determine the
weights in its regional and global aggregations of real GDP, whereas
the IMF and the Organization for Economic Cooperation and Development
use weights based on PPP rates (although the IMF also publishes a
global growth aggregate based on market rates in the WEO). Each
methodology has its advantages and disadvantages.
Advantages of PPP
A main one is that PPP exchange rates are relatively stable over time. By contrast, market rates are more
volatile, and using them could produce quite large swings in aggregate
measures of growth even when growth rates in individual countries are
stable. Another drawback of market-based rates is that they are
relevant only for internationally traded goods. Nontraded goods and
services tend to be cheaper in low-income than in high-income
countries. A haircut in New York is more expensive than in Lima; the
price of a taxi ride of the same distance is higher in Paris than in
Tunis; and a ticket to a cricket game costs more in London than in
Lahore. Indeed, because wages tend to be lower in poorer countries,
and services are often relatively labor intensive, the price of a
haircut in Lima is likely to be cheaper than in New York even when the
cost of making tradable goods, such as machinery, is the same in both
countries. Any analysis that fails to take into account these
differences in the prices of nontraded goods across countries will
underestimate the purchasing power of consumers in emerging market and
developing countries and, consequently, their overall welfare. For
this reason, PPP is generally regarded as a better measure of overall
well-being.
Drawbacks of PPP
The biggest one is that PPP is harder to measure than market-based rates. The ICP is a huge statistical undertaking,
and new price comparisons are available only at infrequent intervals.
Methodological questions have also been raised about earlier surveys.
In between survey dates, the PPP rates have to be estimated, which can
introduce inaccuracies into the measurement. Also, the ICP does not
cover all countries, which means that data for missing countries have
to be estimated.


How is the GDP growing when nothing else seems to be? We've had strong GDP growth all three quarters this year. To me, that would indicate that, in at least some areas, we should be seeing growth. However, from my extremely ignorant armchair, it doesn't look like that's happening.

Wage growth has held steady, the DOW, S&P 500, and NASDAQ are all roughly even year-to-date, and the deficit has grown 17%. Even the trade deficit is growing. This confuses me. Though not perfectly correlated, I thought the GDP would at least be a strong indicator of these other financial components (or the other way around).

To grossly (pun unintended) oversimplify my point: it sounds to me like the value of America has gone up, but there's somehow less money in America.

Am I missing something? How can the GDP be consistently growing at a rapid rate when nothing else seems to be?

There's a bit to unpack here.

Consistent wage growth seems in line with rising GDP. So does a steadily falling unemployment rate.

For other economic measures, the stock market is a whole other animal. It often responds to outside global events that aren't solely economic in nature. It may also be affected by the Federal Reserve's desire to steadily increase the interest rate, though we would expect that to also affect unemployment somewhat perhaps.

The most interesting point of contention here is the trade deficit. It's a common misconception that high trade deficits should imply slower GDP growth. During the Great Depression we ran a trade surplus. During 2001 and 2009, with some of the weakest GDP growth, we also had small trade deficits. One possibility as to why we are running a trade deficit is that with a booming US economy, domestic production demands more inputs, which we have to get from abroad for at least a little bit.

It's not really a matter of "less money in the economy" at the end of the day. You can also consider the typical textbook expression for calculating GDP:

$Y = C + I + G + (X - M)$

Where $Y$ is in GDP and $(X - M)$ is exports minus imports. From this equation, it looks like the more we import, the lower GDP gets, so if we end up running a trade deficit because imports keep rising or "money leaves the economy", then we should have smaller GDP.

What's easy to forget is that we have to do something with these imports. We either consume them, use them for investments, or the government buys them ($C, I, G$), so subtracting imports from the measure of GDP is just a way of preventing double counting.


Game with an equilibrium in pure, but none in mixed strategies? I came across the following game:

The question is to find potential equilibria in mixed and pure strategies. The solution says that there is an equilibrium in pure strategies (B,N), but none in mixed strategies. Mathematically this makes sense to me, since if you solve for the mixed strategy equilibrium, you get the solution that player 2 would have to play strategy L "-100%" of the time, in order to make player 1 indifferent between strategy A and strategy B.
What I don't comprehend however, is how this result is to be reconciled with Nash's theorem, which states that every game with a finite number of players and a finite number of pure strategies has at least one equilibrium in mixed strategies. In the case at hand, we do in fact have a finite number of players and pure strategies. So how is it possible that there is no mixed strategy equilibrium?

As I said in the comment, Nash's theorem shows the existence of a Nash equilibrium (possibly but not necessarily in mixed strategies). If you are interested in Nash equilibria is proper mixed strategies, i.e., NE in which all players play at least two actions with positive probability, you can easily show the following impossibility result: No such NE can exist if one player $i$ has a stricly dominant strategy, i.e., a strategy that gives a higher payoff regardless of the other players' strategy. Since the strictly dominant strategy is $i$'s unique best response against all strategies of the others, this pure strategy must be a part of each equilibrium. Each other strategy by $i$ can be improved upon by deviating to the dominant strategy.
In your case 1 has a strictly dominant strategy, $B$ - so even if 2 always plays $L$, 1 would not be indifferent between $A$ and $B$, but always choose $B, 9>8$. Without indifference, mixing is never optimal and therefore never part of a NE.
Even if $B$ was only weakly dominant, it would always be the unique best response to every strategy in which player 2 plays $L$ and $N$ with positive probability.


How do low interest rates negatively impact bank earnings? A lot of Banks are focused on bread and butter lending to small businesses and consumers. Many of those are putting aside provisions for loan losses in the wake of the Covid-19 pandemic. Headlines point to the drastic impact of lower rates on banks.
Assuming banks lend at certain rate + spread/margin,  are they not supposed to have hedged the interest rate move (E.g. with interest rate swaps or derivatives) ?
Are bank earnings tied to the interest rate level? How exactly ?

Unless someone can get a bank analyst to weigh in, it will be hard to get a definitive answer on this one.
The first thing to keep in mind is that bank regulations and hedging practices are radically different now than in the pre-1994 era. The Savings and Loan industry was compromised by the Volcker shock (early 1980s), and many people base their views on that episode. However, regulatory practices were overhauled to prevent a repeat.
In any developed country, bank regulators monitor the interest rate risks of banks, as do the banks themselves. Expecting interest rates to rise has been a consensus view for decades, and bankers were not outside that consensus. If the interest rate risk is (largely) hedged, then yes, bank earnings are largely insulated from interest rate movements.
Some people look at yield curve slopes, and claim that they are important for bank earnings. However, they are often working with pre-1994 models. If interest rate risks are roughly hedged, then the effect of the slope is minor.
The general absence of banks facing difficulties during the few interest rate rises are evidence that the regulators are doing what they say they are doing.
One can argue that deposits paying 0% are no longer cheap sources of funding, but this would likely only be material if interest rates were very negative. Banks generate fees off those deposit accounts, which offsets this drag.


Simple Derivation of Maximum Principle Consider the simplest problem of optimal control
\begin{align}
      &\max_u\int^T_0{F(y,u)dt}\\
\text{s.t.} \quad&\dot y = f(y,u)\\
& y(0) = y_0\\
& y(T)~~\text{free}
\end{align}
where $y$ is the state and $u$ the control. I'd like to use a static-like approach to derive the necessary conditions of the maximum principle. Build Lagrangian:
\begin{align}
L^1(y,u,\lambda) = \int^T_0{[F(y,u) + \lambda(f(y,u)-\dot y)]dt}
\end{align}

Necessary conditionds for optimum are given by
\begin{align}
L^1_u &= \int^T_0{[F_u + \lambda f_u] dt} = 0\\
L^1_\lambda &= \int^T_0{[f(y,u)-\dot y ]dt} = 0
\end{align}

Define Hamiltonian
\begin{align}
H(y,u,\lambda) := F(y,u) + \lambda f(y,u)
\end{align}

such that the FOCs can be written as
\begin{align}
H_u &= 0~\forall t\\
H_\lambda &= \dot y~\forall t
\end{align}

For last FOC we write Lagrangian as (integrating by parts)
\begin{align}
L^2(y,u,\lambda) = \int^T_0{[H(y,u,\lambda) + \dot\lambda y]dt} + y_0\lambda(0) - y(t)|_{t=T}\lambda(T)
\end{align}

Note that $L^1=L^2$. Last FOC is given by differentiating over $y$ ($y_0$ drops, cause it's fixed)
\begin{align}
L^2_y &= \int^T_0{[H_y + \dot \lambda] dt} - \lambda(T) = 0\\
\end{align}

and finally
\begin{align}
H_y &= -\dot \lambda ~\forall t\\
\lambda(T) &= 0
\end{align}

So we are done here.

How can I get those conditions without varying the Lagrangian, i.e. stick to either $L^1$ or $L^2$?

After a comment exchange, let's provide the answer under disrete-time formulation. The problem is now written

\begin{align}
      &\max_{\{u\}_1^T, \{y\}_1^T}\sum^T_{t=1}{F(y_t,u_t)}\\
\text{s.t.} \quad & y_{t+1}-y_t = f(y_t,u_t)\\
& y_1 = \text{given}\\
& y_{T+1}~~\text{free}
\end{align}

where $y_{t+1}$ is the value of the stock variable at the begining of period $t+1$/end of period $t$.
The Lagrangean is
\begin{align}
\Lambda = \sum^T_{t=1}{\big(F(y_t,u_t) + \lambda_t[f(y_t,u_t)-y_{t+1}+y_t]\big)}
\end{align}

Note that $\lambda_t$ is associated with $y_{t+1}$.

The FOCS are (to hold $\forall t$ in the horizon)

$$\frac {\partial \Lambda}{\partial u_t} = F_u(y_t,u_t) + \lambda_tf_u(y_t,u_t) = 0 \tag{1}$$

$$\frac {\partial \Lambda}{\partial y_{t+1}} = -\lambda_{t} + F_y(y_{t+1},u_{t+1}) + \lambda_{t+1}[f_u(y_{t+1},u_{t+1}) +1]    = 0 $$

$$\implies F_y(y_{t+1},u_{t+1}) + \lambda_{t+1}f_y(y_{t+1},u_{t+1}) = -(\lambda_{t+1}-\lambda_{t}) \tag{2}$$

which are the conditions we would have obtain if we have used the Hamiltonian formulation. To obtain the terminal value for $\lambda_t$, we set $t=T$ in $(2)$, (since the last variable in the $y$-sequence is $y_{T+1}$), and we obtain:

$$F_y(y_{T+1},0) + \lambda_{T+1}f_y(y_{T+1},0) = -(\lambda_{T+1}-\lambda_{T})$$

$u_{T+1}=0$ because there is no decision to be made in period $T+1$. Moreover $\lambda_{T+1}$ should also be set to zero, since it is associated with $y_{T+2}$ (end of period $T+1$) and we are out of the horizon. So we are left with

$$F_y(y_{T+1},0) = \lambda_{T}$$

Whether now we will get $\lambda_T=0$ or not depends on the formulation of the objective function  $F(y_{t},u_t)$: does it indeed include the state variable? If yes what is the value of its derivative with respect to the state variable when the decision variable is set to zero? Etc


Nash equilibrium of a Bertrand game with different marginal costs Consider the following game of Bertrand (price competition):

There are two players, $1$ and $2$. Each has a publicly known marginal cost, $c_i$.
A strategy is a price, $p_i\in\mathbb{R}$.
Player $i$'s payoff (profit) is $\pi(p_i,p_j)=p_i-c_i$ if $p_i<p_j$, $\pi(p_i,p_j)=\frac{p_i-c}{2}$ if $p_i=p_j$ and $\pi(p_i,p_j)=0$ if $p_i>p_j$.

If $c_i=c_j=c$ the result is the straightforward Bertrand Nash equilibrium: both firms set $p_i=c$ and make zero profit. A higher price results in zero demand/profit; a lower price results in negative profits. There is therefore no profitable deviation.

  Now suppose $c_1<c_2$. What is the Nash equilibrium?

I have previously contented myself with the intuition that the equilibrium is for firm 1 to take the whole market at a price 'slightly below' $c_2$? But looking at the technical details raises doubts in my mind:

We can't have an equilibrium with $p_1=c_2$. The best response for $2$ would be $p_2=c_2$, but then $1$'s profit is $(p_1-c_2)/2$ and $1 $ can profitably deviate to $p_1=c_2-\epsilon$ for some small $\epsilon$.
It seems like we can't have an equilibrium with $p_1=c_2-\epsilon< c_2\leq p_2$ because firm 1 could do better with $p_1=c_2-(\epsilon/2)$.

Do we conclude that the only equilibrium of this game is in mixed strategies?

Yes, there is no equilibrium in pure strategies. For any price charged by firm 2 above $c_1$, firm one could only best respond by charging the largest price that is strictly smaller. which is impossible. If both firms charge at most $c_1$, one of these firms must make a loss, which cannot be a best response. So there is no Nash equilibrium in pure strategies.

There are, however, equilibria in mixed strategies. Consider an equilibrium in which firm 1 chooses a price of $c_2$, while firm two randomizes uniformly over the interval $[c_2,c_2+\epsilon]$ for some $\epsilon>0$. For $\epsilon<c_2-c_1$, this is a Nash equilibrium and, moreover, it uses no weakly dominated strategies. Firm 1 is making a profit of $c_2-c_1$ and firm 2 a profit of $0$. Clearly, firm 2 has no profitable deviation. To see that firm $1$ has no profitable deviation, recall that there can be no profitable deviation in mixed strategies if there is no profitable deviation in pure strategies (a very general game-theoretic fact), and observe that firm $1$ cannot profit from deviating to a price strictly below $c_2$ or weakly above $c_2+\epsilon$. So take any $\delta$ satisfying $0<\delta<\epsilon$ and assume that firm 1 charges $c_2+\delta$ (note that $\delta=0$ would be no deviation.) The probability that firm 2 charges a lower price is $\delta/\epsilon$, so the expected profit of firm 1 is
$$(1-\delta/\epsilon)(c_2-c_1+\delta)+\delta/\epsilon~ 0.$$
Write $K$ for $c_2-c_1$. We have a profitable deviation if
$$(1-\delta/\epsilon)(K+\delta)>K,$$
which we can rearrange to get
$$\delta(1-K/\epsilon-\delta/\epsilon)>0,$$
which is equivalent to
$$K/\epsilon+\delta/\epsilon<1.$$ If $\epsilon< c_2-c_1=K$, this can never be the case, so we obtain a Nash equilibrium if $\epsilon>c_2-c_1$.

One can actually use the same construction to construct equilibria in which firm 1 charges less than $c_2$, but in the resulting equilibrium, firm 2 must play a weakly dominated strategy.

The construction in this answer is taken from the following short paper:

Blume, Andreas. "Bertrand without fudge." Economics Letters 78.2
  (2003): 167-168.


Convexity of the Market Demand Function The market demand function can be either concave or convex. I am looking for conditions under which a general market demand function can be considered convex.
For example would convex preferences or a concave utility function imply a convex market demand curve?

I am not asking about the demand set, but rather the function.

I have a hard time imagining a concave market demand function, but the literature often takes this possibility into account, leading me to believe that it cannot be ruled out without imposing further conditions.

The reason I'm having a hard time imagining concave functions is that when we maximize with a standard utility function, what will come out with typical functions is somthing like:

Cobb-Douglas: $X=(\alpha/(α+β)) M/2p$
Perfect compliments (min): $X=αm/(βp1 + αp2)$
Perfect Substitutes: $X=M/p$; where X is demand for some good, M is the budget, p the price (of good 1 or 2) and α and β are just utility parameters.

which are all convex.

Further, since market demand is most often defined as the sum of individual demands (which above appear to be convex mostly) and the sum of convex functions is itself a convex function, then market demand should most often be a convex function it seems.

Can a more general statement be made here about when the market demand curve is convex?

From a mathematical point of view, a function of the form

$$Q_d = \left( \frac {A-p}{B}\right)^{1/\gamma},\;\; \gamma > 1$$

will have a negative second derivative and so it will be strictly concave (for $\gamma =1$ we get a linear demand function). The function hits the (vertical) $p$-axis for $(Q_d=0,p=A)$, and the horizontal $Q$-axis for $(Q_d = (A/B)^{1/\gamma}, p=0$). Namely this is "concave all the way", not just concave in the middle which eventually becomes convex. Typically, we expect  $A>B$.

Inverting, we obtain

$$p=A-BQ_d^{\gamma}$$

Let's now go down to the individual level, and assume quasi-linear preferences in income, an appropriate formulation when we look at one good "against all others":

$$v(q,y) = u(q) + y,\;\;\; s.t. \;\;pq+y = M$$

The first-order conditions will give

$$u'(q) = \lambda p,\;\;\;\ \lambda =1$$

Combining with the inverse demand function we get

$$u'(q) = A-BQ_d^{\gamma}$$

where $Q_d = \sum q$.

Integrating we have

$$\int u'(q) dq = A\int dq - B\int \left (\sum q\right)^{\gamma}dq$$

from which we arrive at (setting the arbitrary constants of integration to zero)

$$u(q) = Aq-\frac{B}{1+\gamma}Q_d^{1+\gamma}$$

We obtained this result by assuming that the  individual believes that its own demand does not affect any other individual demand. So we have a negative externality: the more "massive" is a good (the more massively demanded), the less utility we obtain from it (so any individual demand does not affect any other individual demand, but their sum, does affect the individual elements). So such a specification would be appropriate for goods that are not characterized by the "safety of the flock" aspect, but appeal to our quest for individualism and personal uniqueness.

Assume now that we do not want to have this negative externality in the individual utility. Then, we assume that all consumers are identical, and we start with the utility specification,

$$u(q) =  aq-\frac{b}{1+\gamma}q^{1+\gamma}$$

This utility specification does imply that there exists a maximum utility level from the good and then utility declines, so it belongs to the general family of "quadratic" preferences.

Indicatively, for $a=100, b=10, \gamma=1.3$ we have

The indifference map is constructed by calculating

$$y = \bar v - \left[aq-\frac{b}{1+\gamma}q^{1+\gamma}\right]$$

for arbitrary values for $\bar v$ (which, in quasi-linear preferences, correspond to the income constraint).For $\bar v = M=400,450,500,550,600$ the map looks like

Such an individual utility specification will give the optimal relation

$$a - bq^{\gamma} = p$$

and the individual demand

$$q = \left( \frac {a-p}{b}\right)^{1/\gamma}$$

Summing over all $N$ consumers we get

$$Q_d = N\left( \frac {a-p}{b}\right)^{1/\gamma}$$

and we want to arrive at the market demand function

$$Q_d = \left( \frac {A-p}{B}\right)^{1/\gamma}$$

This can happen if we identify $A=a$ and $B = b/N^{\gamma}$.


Indirect Impacts Of The Netherlands Cutting Greenhouse Emissions? On 24 June 2015, the Dutch District Court in The Hague ruled that the government must put policies in place to deliver, by 2020, cuts in the country's greenhouse gas emissions of at least 25%, relative to 1990 levels.

The government, for months leading up to the tribunal appearances, had been saying that it was hard for them to cut it down by any more than 17%.

What are the likely economic consequences of this ruling?

global
The Dutch energy system is small enough that the ripples to the global supply chain will be very small. However, it is looking possible that there might be some temporary tightness in the global PV module supply chain coming up, after years of capacity expanding faster than demand, so there might be a price-spike up, there, if the Dutch go for rapid PV deployment 2015-6. Conversely, there is a bit of a dip coming in the global offshore wind pipeline, so the Dutch might help there; they do have a deployment pipeline set out to 2023, so one approach would be to accelerate that to deliver all of those windfarms by 2020, smoothing the work of the global supply chain, and reducing wastage there. But the Dutch energy system is small, and so all of these consequences would be small at the global scale.
The much larger potential impact is that of the ruling itself, on other countries. This successful application of tort in one country may well prompt similar cases in other countries, as tort is a principle found in legal codes around the world. Further successful tort cases might help push the world along the decarbonisation trajectory much faster than it is at present. Now, as there is a non-negligible chance that business-as-usual could result in the destruction of much of the global economy, the option value of reducing the probability and scale of that wipe-out is very high; so the ruling may yet have very high, very positive economic consequences beyond the Netherlands.
overview
In many ways, the Dutch are about to go through, at small scale, what the rest of the world is about to go through at much larger scale. We've known for about 30 years that almost-complete global decarbonisation was necessary, and that the later we left it, the more expensive it would be. And then there was very little action in most countries. So now, not only do we have to decarbonise in about half the time we originally had (which will thus be more expensive), but we'll also have to spend a huge amount on adaptation too, as a lot of global warming is already in the pipeline and almost completely unavoidable over the next 40 years.
costs, benefits, prices, quantities
The negative externalities of the current system outweigh the benefits; so once we've transitioned away from fossil fuels, we'll be more economically efficient that we would otherwise have been; but there are short-term costs attached to that transition, and so resources will be diverted from other activities and into an accelerated transition.
The accelerated action required in the Netherlands 2015-2020 will be to temporarily increase prices for the mitigation measures deployed; and those prices will then, post-2020, be lower than they would otherwise have been, as the supply chains will have expanded.
Regarding demand for oil, coal and gas: they are all internationally traded commodities, so a decrease in Dutch demand will add a little downward pressure on their prices across Europe; however, Dutch demand is a small proportion of total European demand , so this price effect will be very small.
The Dutch have been investigating various forms of road pricing for around 30 years; and planned out to some detail. So we may see some version of that finally implemented. Now, that would be more economically efficient in the long run: fossil-fuelled car travel has a lot of externalities in addition to greenhouse gas emissions. But in the short run, there are additional costs as land-use patterns change, and people change where they shop, where they live, where they work.


Definition of 'Price of anarchy' The classical definition of Price of anarchy is the ratio of decentralised optimal utility to the centralised optimal utility. This concept is used to quantify how inefficient is it when agents behave selfishly, in comparison with a situation where a central planner takes optimal actions.

My question: why cannot Price of anarchy be defined as the difference between the two solutions? In my setting, I have a number of log based functions in the utilities. For example, in a network setting defined on a graph $G$, the central planner's utility function is $U_s=f_s(G)+\sum_i \log(l_i)$ and the individual firm's utility is $U_i=f_i(G)+\log(l_i)$, where $l_i$ are loss terms for node $i$. Having a difference as PoA helps me eliminate idiosyncratic losses and I am left with $f_s(G)-\sum_i f_i(G)$, and I have a means to compare dependence of PoA on network structure.

I am greatly benefited by having a difference in place of ratio. Is it fine to define that way? Is there some other term for the difference?

As hinted in a comment, the problem with using the difference instead of the ratio, is that the difference is not free of the units of measurement while the ratio is. And in general we want to not depend on units of measurement (whatever it is that we measure).

Although I am not familiar with the literature, I understand that here utility is a fully quantitative, "cardinal" concept. Moreover, there exists the network $G$ whose characteristics generate utility, and then there are the "idiosyncratic" losses, which is the component we want to get out of our sight.  Then why not define a $\text {PoA}$ net of idiosyncratic losses, especially since they appear not to depend on the decision making framework, and so they are the same both in the decentralized set up and in the centralized one?
I mean,

$$\text {nPoA} \equiv \frac {\sum_iU_i -\sum_i\log(l_i) }{U_s - \sum_i\log(l_i)}=\frac {\sum_if_i(G)}{f_s(G)}$$

This seems to me as a meaningful concept to reflect the network contribution to utility and how it changes under the two decision-making cases.


How much impact does memory have on risk pricing? Have studies been done that demonstrate the limits of human memory on reputation in the Iterated Prisoner's Dilemma?

i.e. at what point (on average) does a player start to "forget" the actions of a previous opponent?

I'm not sure what you mean by risk pricing, but because you mention a repeated game, you might find the following papers useful. They present some theory about repeated games in which players have limited memory.

One - Memory in Repeated Games
Repeated games with one-memory
Reputation and Bounded Memory in Repeated Games with Incomplete
Information


Why does bond price tend to 100? For example: if a bond got a 1% coupon, and tomorrow is the maturity date, so tomorrow I will have 101.

So, in my opinion a bond price should tend to notional+coupon, because today I want this amount of money to be at par. Where I am wrong?

There’s two prices for a bond:

Invoice or dirty price: what you actually pay; and
the clean price, which is the dirty price less accrued interest.

In market convention, the clean price is the quoted price.

Accrued interest starts off at \$0 at a coupon date, and then rises (roughly) linearly each day until it matches the value of a coupon at the coupon date. The exact formula is a fixed schedule that depends upon market convention.

If you buy a 1% annual coupon bond one day before maturity, you will probably have a clean price very close to \$100. The accrued interest will be very close to \$1, and so the dirty price would be close to \$101.

The deviation of the clean price from \$100 would depend on the deviation from 1% for the yield you bought it at. The exact accrued interest will depend on the daycount convention in the market.

In any event, people use the clean price precisely because there is a “pull to par,” we do not have to worry about the coupon when thinking about the bond price.


Euler's theorem and Cobb-Douglas When I apply Euler's Theorem onto the Cobb-Douglas equation I receive the Cobb-Douglas equation back. That is, there is no change after applying the theorem. Can someone please explain what the intent and point of this is, seeing no difference?

You are probably talking about Euler's homogeneous function theorem http://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html , which shows that you can decompose a function into its partial derivatives based on its homogeneity.

So a typical Cobb Douglass $U(x,y)=x^ay^{1-a}$

This function is homogeneous of degree 1 because when we multply each argument by a scalar:

$U(cx,cy)=(cx)^a(cy)^{1-a}=cx^ay^{1-a}=cU(x,y)$

So if we apply the theorem to this function we get:

$U(x,y)=xU_x+yU_y=xax^{a-1}y^{1-a}+x^ay(1-a)y^{-a}=x^ay^{1-a}$

So the result that we get back the original utility is expected.


Comparing baskets of goods in this exercise Max buys two goods, apple and ketchup. We do not observe his preference but we know it satisfies the three assumptions, completeness, transitivity, and more is better. And as usual, we know his preference does not change with prices or income. Furthermore, we know Max seeks to maximize his utility. Initially, Max’s income is \$40, the price of apple is \$4 and the price of ketchup is \$4, and Max chooses 3 units of apples and 7 units of ketchup to maximize utility. Let A denote this basket. Consider basket B that contains 5 units of apples and 3 units of ketchup. Suppose the price of ketchup becomes \$1.5 and Max’s income becomes \$24.5, while the price of apple remains at \$4. Given the new price and income, is it possible for basket B to be the optimal choice of Max? Briefly explain.

For this question, I drew out the initial and new budget lines. I introduce a new point C (5 units of apples and 5 units of ketchup). Can I say that B cannot be the optimal choice because if we assume by contradiction that B is indeed the optimal choice, then B>C although C is cheaper. However, since the assumption more is better is satisfied, C>B which is a contradiction. Thus, B cannot be an optimal choice. Is my reasoning incorrect?

So we are told that bundle A (3,7) is optimal at prices p=(4,4) with wealth w=40

Consider bundle B (5,3) at prices (4,1.5) with wealth 24.5

By Weak Axiom of Revealed Preference (WARP), we know that if both bundles are affordable under both sets of prices and wealth, and we chose A when B was available, then A is revealed preferred to B.

Both A and B are affordable under those price wealth pairs, so B cannot be optimal.

What you did graphically is equivalent to the math.


What is the difference between shareholder's equity, equity, and book value? I have been doing some reading and I have found that shareholder's equity is equal to the company's total assets minus its total liabilities. Equity is apparently defined in much the same way. Book value is equal to the total assets minus intangible assets minus liabilities. So what is the actual difference between all of them?

Intangible assets seem rather hard to quantify, if I bought a house in a good neighborhood, its location could be called an intangible asset, right? But how do I measure this asset with a monetary value so that I can subtract it from its actual value?

What is the difference between all three of these concepts?

Equity is defined as

  The value of an asset less the value of all liabilities on that asset. [source]

This is a rather broad definition and equity can take on different forms. E.g., for a house, it is the difference between the market value of the house and the mortgage still owned by the owner.

In the context of a company balance sheet, we usually talk about Shareholder's equity, which, as Wikipedia puts it

  represents the equity of a company as divided among individual shareholders of common or preferred stock

Contrary to the house example, the market value of a company, is the sum of all shares. And the shareholder's equity is that value (asset) subtracted from liabilities (creditors, etc.). See also this page.

The book value is the value of an asset. But the difference with the Shareholder's equity is illustrated as

  To find a company's book value, you need to take the shareholders' equity and exclude all intangible items. This leaves you with the theoretical value of all of the company's tangible assets which are those assets that can be touched, seen, and felt as opposed to things such as patents, trademarks, copyrights, and customer relationships. [source]


Dynamic Market Equilibrium: motivation for differential equations I came across this discussion, claiming supply and demand are expressible in terms of its price as $$Q_s=c_1+w_1P+u_1P'+v_1P'',\,Q_d=c_2+w_2P+u_2P'+v_2P''.$$Equating supply to demand for an equilibrium gives an ODE one can solve, but where do these formulae for $Q_s,\,Q_d$ come from?

I apologise if, contrary to the impression the page gives, these equations are not a mainstream model.

As the link you provided mentions:

  "Differential equations can be used to include the dynamic aspects to
  economics into a mathematical framework which takes into account the
  volatility present in economics."

So the author does not derive the differential equations as the result of a theoretical model -the author just wants to obtain the movement through time of equilibrium price (the author assumes market-clearing), as a second order differential equation.

So it is a purely descriptive/forecasting exercise, devoid of any economic theory background. The author obtains the solution, which includes a number of coefficients to be determined, for which the author mentions

  "These values can be estimated using statistics and econometric
  methods"

Some constraints on the parameters may come from observing the past price process (is it oscillating? around a trend or around an horizontal line? etc), and in the end, the hope is that we may obtain a good predictor for the price movements -again, without basing it on economic theory.