Derivative of CARA utility Can someone help explain the passage here? I'm rusty with my linear algebra so the derivate of these transpose matrices isn't making any sense to me. A detailed explanation would be very much appreciated. What happens to the 1/2, and that whole second term in general?

All you need for this particular question is the following. Let $\mathbf{X}$ be a $T \times K$ matrix, $\mathbf{w}$ a K-dimensional vector and $\mathbf{y}$ a T-dimensional vector, then

$$
\begin{eqnarray*}
\frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{y}}{\partial \mathbf{w}}&=&
\mathbf{X}^{\prime}\mathbf{y}\\
\frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}}{\partial\mathbf{w}}&=&2\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}
\end{eqnarray*}
$$
Back to your original post, we will first rewrite it in order to apply the rules provided. The first element in the expression we want to differentiate is the same as $\phi'(\mu-R_f\mathbf{1})$, hence
$$
\frac{\partial \phi'(\mu-R_f\mathbf{1})-1/2\alpha\phi'\Sigma\phi}{\partial \phi}=
(\mu-R_f\mathbf{1})-\alpha\Sigma\phi
$$
and equating to 0 you get the desired result.


How to decide the best bid in an auction? I am going to participate in a decision committee of a governmental auction in energy market. I want to learn more about how to decide which bid is the best in short/mid/long term.

Can you suggest me a solid source to read/learn about it?

Thanks!

Edit: Private sector is going to bid to rent a geothermal field to construct a power plant and produce electricity.

Edit2: Maybe my question was not clear. If one party is offering high amount of cash + low annual payments and the other body is offerıng low amount of cahs + high annual payments, which one I should accept? How can I do the risk analysis? So on ...

To understand the economics of auctions a little better, I'd recommend taking a look at "Auctions: Theory and Practice" by Paul Klemperer. This book has a number of advantages:

with the exceptions of some non-essential appendices, the book is non technical and therefore more accessible to lay-readers. Other texts on auctions tend to be aimed at economics graduate students or above.
the book was motivated by Klemperer's experience designing radio spectrum auctions in the UK and discusses applications for public sale/procurement auctions at some length.
the book emphasises the main considerations and intuition behind the analysis and design of auctions, rather than the details that are more likely to be of interest to academics rather than participants.

The book can be read free of charge on Klemperer's website here. Alternatively, it is available in print; here is the publisher's page.


Constant Returns in a Production Function $\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$ ($R$ = Resource) In his 1977 article (from which has developed a considerable literature on the Hartwick Rule for maintaining long-term constant consumption given depletion of non-renewable natural resources), Hartwick uses (on p 973) this aggregate production function:

$$x = k^{\alpha}y^{\beta}1^{\gamma}$$

Here (see p 972) $x$ is per capita output, $k$ is per capita reproducible capital, and $y$ is per capita use of an exhaustible resource. $1$ is just the number one, labour being assumed constant (so the term $1^{\gamma}$ seems redudant).  So in more familiar notation (output $Y$, capital $K$, use of exhaustible resource $R$, labour $L$), and glossing over the difference between population and labour, this is:

$$\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$$

Hartwick then assumes (p 972) constant returns to scale in the form (explicitly on p 973) $\alpha+\beta=1$.

Question: What reasons might justify the assumption of constant returns in the above form?  Isn't it more plausible to assume constant returns when all factors are increased, that is to assume $\alpha+\beta+\gamma=1$ in a production function of the form:

$$Y=K^{\alpha}R^{\beta}L^{\gamma}$$
This does imply the above function since:

$$\frac{Y}{L} = \frac{K^{\alpha}R^{\beta}L^{\gamma}}{L^{\alpha}L^{\beta}L^{\gamma}}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$$

However, given $\gamma>0$, it is inconsistent with $\alpha+\beta=1$: instead $\alpha+\beta<1$.

The purpose of the paper under consideration is to examine/show the "investment rule" that leads to "intergenerational equity", which with constant population translates into constant consumption.

The investment rule under examination is (last line of p. 973) "invest all net returns from exhaustible resources in reproducible capital" (and consume the rest).

Earlier, eq. $(1)$ of the paper (the law of capital accumulation) tells us that gross returns to exhaustible resources equal $f_yy$: this implies that we assume that the return per unit of exhaustible resource equals its marginal product.

But this in turn implies pricing and the existence of markets. So there must be a market for capital also. If the capital market is also characterized by marginal pricing, then , if we assume that the production function has decreasing returns to scale in capital and exhaustible resource ($\alpha + \beta <1$), then

$$f_kk + f_yy < x$$

and some part of output would have been left unaccounted for.

So the author assumes constant returns to scale in these two so that he can also assume competitive markets and marginal pricing, and per capita output exhausted in the rewards to these inputs.

This of course begs the  question: what happens to the labor market?
Well, we can get away with murder making the following assumption: There is no leisure-labor choice, labor is offered inelastically, and moreover, there is no market for labor, it is subsumed to the other factors of production, i.e. it is offered together with them and it is not paid separately: think capital owners that also work in their business without paying themselves a wage.

This of course means that the formulation with unitary labor and an irrelevant exponent $\gamma$ is sloppy and problematic, it should be absent (it would not affect the paper), and it rightfully led to the OP's question.


All else being equal, What is the value of job creation? Job creation value
Just to clarify, I have no experience, academic background or deep understanding of economics. Apologies for such a basic question but I couldn't find a real answer.
I hear a lot about job creation:

"This will create X jobs"
"Losing jobs overseas"
Etc.

I have trouble understanding the inherent value of the current amount of jobs to society.  If jobs are lost together with the productivity they created then I get why there is a problem.  For example: A factory was destroyed in a fire, the workers lost their jobs, and society lost the productivity of the factory. But if all other things remain equal, what is the value of a job?  I'm going to add my very naïve analysis below to help illustrate my thinking.
Naïve interpretation
Let's say we have a country that has some people working right now. Suddenly 10,000 stopped working without affecting productivity. Maybe there was a temporary need that was filled or perhaps some technology made their jobs obsolete.
If current amount of jobs is an inherently positive thing, I should now be at a worse state than before. As I see it, I just got 10,000 potential workers. Modern humans, even not well trained ones, produce vastly more than they cost to support.
I can certainly understand that it can be traumatic to lose your job but that's an argument for job stability or something, not job creation.
The argument that all those people are not retrainable at all, seems a bit far-fetched. They can't all be 60 year olds who trained their entire life in this field and are uneducated in absolutely anything else. And if that was the case, it would be better treated as a surprising early retirement. Tragic for people who weren't prepared, yes, but a completely different problem.

You are correct that creating jobs in themselves, generally speaking does not have any economic value (save some exceptional situations - see below). In fact this is not novel observation, and you will find this stated in any 101 economic textbook for example see Mankiw Principles of Economics 8ed pp 33. As written in the textbook (in an excerpt from Economics article):

Little in the literature seems more relevant to
contemporary economic debates than what usually is called the broken window fallacy. Whenever a government program is justified not on its merits but by the jobs it will create, remember the broken window: Some teenagers, being the little
beasts that they are, toss a brick through a bakery window. A crowd gathers and laments, “What a shame.” But before you know it, someone suggests a silver lining to the situation: Now the baker will have to spend money to have the window repaired. This will add to the income of the
repairman, who will spend his additional income,
which will add to another seller’s income, and so
on. You know the drill. The chain of spending will
multiply and generate higher income and employment. If the broken window is large enough, it
might produce an economic boom! . . .

Most voters fall for the broken window fallacy,
but not economics majors. They will say, “Hey,
wait a minute!” If the baker hadn’t spent his
money on window repair, he would have spent it
on the new suit he was saving to buy. Then the
tailor would have the new income to spend, and
so on. The broken window didn’t create net new
spending; it just diverted spending from somewhere else. The broken window does not create
new activity, just different activity. People see the
activity that takes place. They don’t see the activity that would have taken place.
The broken window fallacy is perpetuated in
many forms. Whenever job creation or retention
is the primary objective I call it the job-counting fallacy. Economics majors understand the
non-intuitive reality that real progress comes
from job destruction. It once took 90 percent of
our population to grow our food. Now it takes 3
percent. Pardon me, Willie, but are we worse off
because of the job losses in agriculture? The
would-have-been farmers are now college professors and computer gurus. . . .
So instead of counting jobs, we should make
every job count

The above being said there is some value to creating/preserving jobs in some situations. For example, during recessions some public works program might be part of a mix that is intended to stimulate the economy, or during short but severe unpredictable crises such as the current Covid-19 situation.
For example, the Furlough/Kurzarbeit schemes  are generally considered by conventional economists  to be valuable as they can help boost aggregate demand and prevent even more severe recession (Giupponi & Landais; 2020),however such support is exceptional.


Inferior goods, monotonic utility and strict concavity Is it possible to have an inferior good under the assumption of a utility function which is strictly monotonically increasing and strictly quasiconvex?

I do not think so, but I could not find a formal proof.

*My underlying idea is that whit an inferior good, there exist a point where the utility consuming less of one good is better and then its derivative negative at that point, moreove, I think that this behaviour would break the concavity shape.

I could not add images in the comments, therefore I used this method.

I think that the convexity definition does not hold for the function, since there are pairs of points that when they are conected, produce a segment which are out of the set. Because of it, the set is not convex.

Here there is an example of such situation with this function, which does not show a convex set:


Question on significance of different ways of measuring Price Elasticity of Demand I'm a pre college student, self studying economics, I have a question regarding the different forms of measurement of Price Elasticity of Demand.

The basic equation for PED with respect to a point $(x_1,y_1)$ on a demand curve (qty x axis, price y axis), given another point $(x_2,y_2)$ is:
$\frac{(x_1-x_2)/x_1}{(y_1-y_2)/y_1}$
This gives us the responsiveness of qty demanded relative to price at an initial point $(x_1,y_1)$.

However, my text mentions that depending on the choice of initial point, either $(x_1,y_1)$ or $(x_2,y_2)$, the PED is different. Hence, in order to overcome such a problem, we utilize arc elasticity of demand, which uses the midpoint as a initial value.

My question is this:
Isn't PED supposed to vary along a demand curve? If we rearrange it as (Change in qty/Change in price)(Price/Quantity), we can see that it does depend on which point on the curve it is. Hence, why is it regarded as a "problem" that depending on which initial value we use, we get different PED?

And by extension, I read that the definition of arc elasticity is the elasticity of one variable to another between two given points. I don't fully understand what is the significance of using the midpoint as an initial value, other than it netting the same PED for two distinct points. What does it actually mean when we take the difference of two points and divide it by the midpoint?

And on a final note, is point price elasticity of demand recommended when we have the functional relation of price to quantity?

I've been looking on the internet, and I haven't really been able to find/just don't understand anything about this. Sorry if this is elementary, but I haven't got anyone I know to ask.

Ideally the points $(x_1,y_1)$ and $(x_2,y_2)$ are very close. The assumption is that that they are so close that the elasticity of the demand function on the curve between them is nearly constant. This is what elasticity is trying to measure, local changes, because on the far side of the range of the demand function consumer behavior may be unobserved and totally different. E.g. to my knowledge no one has tried to sell a bottle of Coke for \$40,000 so we don't know what value the demand function would take. If you would choose a very distant point $(x_3,y_3)$ instead of the local $(x_2,y_2)$ you would get a very different value of elasticity.

So what does the book mean by initial value? Ideally the elasticity measured in points $(x_1,y_1)$ and $(x_2,y_2)$ would be constant because they are very close. Of course in practice this will not always be the case, meaning
$$
\frac{(x_1-x_2)/x_1}{(y_1-y_2)/y_1} \neq \frac{(x_1-x_2)/x_2}{(y_1-y_2)/y_2}.
$$
Instead of taking one of these values as the elasticy of the curve we can improve our estamite by taking the arc elasticity which usually yields a value between these two.


Extensive form: pareto inefficiency? The question I'm dealing with is:

  Suppose A plays bf, Which of B's strategies would lead to an outcome that is not pareto efficient?

The answer is apparently ei as $(bf,ei)=(0,5)$, but I don't understand why. If this answer is correct, could you please explain why?

My understanding of pareto efficiency is that by moving from $(x,y)$, we cannot make any player better off without making the other worse off. If this is the case, then $(x,y)$ is pareto efficient.

$(x,y)$ would be pareto inefficient if there exists a movement away from $(x,y)$, where a player can be made better off without the other being made worse off.

From $(bf,ei)=(0,5)$, we can move to:

$(4,4), (5,0),(2,2)$

I don't see how we can move to any of these without making B worse off, I think that $(bf,ei)$ represents a pareto efficient outcome.

Any chance you could offer some advice?

Thanks.

Thanks to denesp, I realised that I was only considering outcomes resulting from the following subgame:

If you consider all outcomes within the extensive form, then it is clear that $(2,6)$ pareto dominates $(0,5)$.


Balance of payments and the free floating exchange rate system I am a beginner economics student and recently, I have been learning about how balance of payments surpluses are important for a country to control the value of its currency using its accumulated foreign reserves. In the case of the free floating exchange rate regimes (e.g. Russia), by default, the foreign reserves account is empty. This makes perfect sense since the country does not need any reserve assets to manipulate the value of the currency. Does that mean to say that the balance of payments position is always "neutral" (i.e. neither deficit nor surplus) for a country with a freely-floating exchange rate regime and that the BOP position is irrelevant for such a country since the current account and capital and financial accounts automatically balance each other?

The Balance of Payments are a form of double-entry bookkeeping and so in theory should always balance overall

If official reserves do not change because the currency is floating freely and the Central Bank or Treasury is not intervenening, then a country's current account balance should be offset exactly by the financial/capital account balance.  But measuring this is difficult, which is why you often see a "statistical discrepancy", for example in row 100 of page 10 of the US Balance of Payments

But the individual flows matter, and it might then be wrong to say "the BOP position is irrelevant", at least in the medium to long term. A country with a persistent current account deficit will need to see net inward financial flows, which may be desirable if it leads to increased real investment and later to increased output, but may be less desirable if it merely leads to greater indebtedness and the need to pay for that debt

Meanwhile a country with a persistent current account surplus is effectively continually exporting more than it needs to pay for imports, so reducing its consumption and standard of living; this may make sense if it later intends to increase consumption with the income from its investments abroad, but otherwise is unnecessary and potentially wasteful and can lead to a failure of these unbalanced investments, as happened with German banks' speculative assets in the USA during the financial crisis


What keeps minimum wage rates in balance? I would posit that the minimum wage is determined more on a political than an economic basis.  The fact is that there are more wage earners than business owners and, generally speaking, the wage earners prefer an increase while the business owners prefer a minimum wage decrease (perhaps even to 0).  There are more wage earners to vote than business owners, and this makes it politically expedient, on the balance, to vote for a minimum wage increase.

First, is this true?

Second, if you find it to be true, why doesn't the minimum wage increase substantially more than it has over the years (focusing on the US)?  What is the counter balancing force that causes minimum wages to seek equilibrium at some something between "poverty" and "living wage" level (again, in the US)?  Or is even that assumption false?

How is the minimum wage determined?

Yes and no. It is a political choice, but with economic reasoning. In the end, that's the case for every economic rule, maybe with the exception of monetary rules, as we try to preserve independence for central bankers (but it isn't completely true for the US FEDs).

When setting the new minimum wage, you compare minimum wages with other states, and with your own (earlier) state, to get a feeling what's possible and what is not. In the end, the minimum wage is a trade-off between raising earnings for some parts of the population, and decreasing employment chance for others.

How does the minimum wage affect employment?

The degree of trade-off (if you will, the transition rate between the two) is unclear. In the data, we tend to see no employment effect of minimum wages, but an increase in earnings (Dube et al (2010): (Ungated version). In fact, we see a large number of estimates around 0:

Many economists suspect (based on a strong prior that there should be an employment effect) that this is due to poor data (Matt Rognlie's comment), or long-term effects of minimum-wages (adjustment costs in technology) that we are not capable of measuring/estimating precisely (Sorkin (2015) (IDEAS)).

Why doesn't the minimum wage affect employment?

The counter-effect to some part is the political process. People are not stupid, they will accept raises in the minimum wage to some degree, but not if you go bananas. Also, there are natural counter-forces in the economy:

minimum wages are nominal
minimum wages are lower bounds

To that end, if you increase the minimum wage, eventually inflation will wash off the real impact. Also, raises in the minimum wage tend to become substitutes for (non-forced) increases in the wage level.

Have a look at the following graph, which depicts the relative impact of minimum wages. The author plots the maximum of minimum wages at a time period, deflated by average hourly wage in the private sector among production and non-supervisory employees (no managers). It really looks as if any spike (raise in the minimum wages) gets deflated quite quickly, doesn't it?


Driving down the price of a good I've been thinking about this situation lately:

A normal good $x$'s price has become unreasonably high. Would it be a good idea for consumers to boycott (i.e. stop buying) $x$ in order to drive down its price, then continue purchasing at the lower price? Why or why not?

This is actually an interesting question that I've never really thought of. What you are essentially describing is a monopoly on the side of the consumers to artificially drive down quantity demanded in order to lower prices.

You didn't state many details in your question but I'm assuming that the good in question is inelastic, is subject to competitive pricing, and has it's current price in competitive equilibrium.

The outcome of consumers collectively refusing to buy a product above a certain price would have the same effect of monopolistic pricing of a firm but in reverse, there would be a shift of some producer surplus into consumer surplus as well as the creation of dead weight loss as some producers are priced out of the market creating shortages.

This could also be empirically the same as the creation of a price ceiling for the good. If collectively consumers refused to buy a product above a certain price then that product would have a price ceiling and producers would produce at that price in accordance with their own supply curve.

As for your question about whether it would be a "good idea" that depends on what your definition of a positive outcome is. This type of action would have the effect of transferring some of the producer surplus to the consumers which would benefit the consumers but it would simultaneously lead to less production on the part of the producers due to the lower price creating dead weight loss and shortages. Empirically this would be bad for the aggregate economy as a whole due to the loss of surplus to dead weight loss, however, depending on the producer supply curve this may or may not lead to an aggregate gain in consumer surplus. An inelastic supply curve would lead to a gain in aggregate consumer surplus while a elastic supply curve would lead to a loss.


Why, in such a debt-heavy era, is inflation still such a Bad Thing(TM)? Going through the Wikipedia article on inflation, a number of pros and cons to inflation are covered. Most of the "cons" of inflation (cost-push inflation, hoarding, hyperinflation, diminished allocative efficiency, etc.), don't become a problem until inflation is well above the levels currently targeted by The Fed.

A number of "pros" are also covered, but one I don't see addressed anywhere I've looked is this: the average U.S. household debt is 3x the average annual income. Our government is also in debt to the tune of \$21T, which comes at a cost of \$400B per year in interest paid. I do see acknowledgement that higher inflation eases debt, but nothing on why increased debt hasn't incentivized us to increase inflation.

Wouldn't a higher inflation rate be a good thing? It would devalue those (household and federal) debts, making them easier to pay off, and saving the interest paid for other budget items. If the Government (and most Americans) were running closer to the black, I can see why you'd want to keep interest low, but given the unprecedented situation, why not allow inflation to float higher?

While many points can be made about the pros and cons of inflation, the following is one of the most fundamental.

Suppose A borrows \$100 from B and inflation is 5% per year. To keep things simple, assume the terms of the loan require repayment after one year. When the repayment falls due, A pays B \$100, but because of the inflation it is only worth (at the original price level) $100/1.05 \approx \$95.2$.  So A's debt has been somewhat eased by the inflation (and if the term of the loan were much longer the easing could be much greater).  But what about B? In real terms B has lost the same amount that A has gained.

Other things being equal, therefore, inflation benefits borrowers and costs lenders. And in total, amounts lent must equal amounts borrowed. It may be that the the borrowers (government, struggling households) are more visible than the lenders (better-off households, foreigners).  Nevertheless, the effects on lenders, and on incentives for future lenders, cannot be ignored.  Where inflation is expected, lenders will as suggested in Henry's comment demand a higher interest rate to offset their expected loss on the principal repayment.


Why recent crises all started with a housing bubble? I was writing some notes about the last economic and financial crisis and I noticed that all these recent crisis start with a housing bubble. Beside the 2007-08 US's one, also Japanese crisis in the beginning of the 90s and Spain's crisis, just to cite a couple of them, all have started by housing price boosting.

Why are all recent crisis starting with a housing bubble? Is something related to modern economies or just a coincidence?

In a fractional reserve monetary system loans create money and loan repayments destroy money. Most bank lending in modern times is for the purpose of purchasing real estate. So the amount of money that exists in the economy is closely tied to the enthusiasm for purchasing real estate. You could not really make this kind of statement for any other class of asset - this is why the housing market in particular has such an impact on the economy. During the upswing of a housing bubble the money supply will be growing fast, but when the bubble bursts the money supply will fall (as existing mortgages are repaid) leading to recession. The whole point of QE is to counteract this fall in the money supply.

Confusingly economists use the word "credit" to describe (most of) the money we use in the economy, so many people do not realise that the "credit crunch" was in fact a "money crunch".

Hyman Minsky is the most famous economist that would subscribe to these kind of views. You might also want to look up the term credit cycle.

By the way, none of these views are mainstream. Mind you the mainstream economists did not see the last crash coming, so you should not really expect them to explain what happened.


Mathematics in modelling: Issues? Are there some known examples where the use of mathematics in Economics (as opposed to economic intuition or experience) has led to flawed models or some other kind of issues?

I often hear people say that they don't like the use of mathematics in economics .... but why? So I was wondering maybe it lead to flawed models, perhaps because human behavior isn't as nice as some mathematical equation?

Math is a tool. You can hardly blame a saw when you fail to sharpen your knife with it.

So is math necessarily a poor tool? In my view, it's a wonderful language through which concepts can be expressed concisely and unambiguously. It's quite powerful to express an economic problem as a math problem because oftwn this means the problem is already solved in full generality (e.g. Recognizing an incapacitated lot problem as an integer programming problem which has easy algorithms for solution).

I've heard some complaints that the temptation to pretend a problem is accurately modeled by a solved math problem is quite strong - this is probably one of the issues you're getting at. Again, nothing wrong with using a tool its proper place. The fault is unambiguously with the user.


Examples of Covariance Stationary Time Series I seem to be having trouble in comprehending what it means for a time series to be covariance stationary. Specifically, with the third condition that for any $t,h$ the $cov(x_t,x_{t+h})$ only depends on $h$ and not $t$.

Would anyone have any examples of how a time series might be covariance stationary or any examples of non-covariance stationary time series?

Let $\epsilon_1, \epsilon_2, ... ,\epsilon_t,...$ denote i.i.d. stochastic variables, each having a standard deviation of $\sigma$.

Let
$$
x_t = \sum_{i = 1}^t \epsilon_i.
$$
Then
$$
cov(x_t,x_{t+h}) =
cov\left(
\sum_{i = 1}^t \epsilon_i,
\sum_{i = 1}^{t+h} \epsilon_i
\right).
$$
Because of the i.i.d. property
$$
cov\left(
\sum_{i = 1}^t \epsilon_i,
\sum_{i = 1}^{t+h} \epsilon_i
\right)
=
\sum_{i = 1}^t cov\left(\epsilon_i,\epsilon_i\right) +
\sum_{i = 1}^t\sum_{j \neq i} cov\left(\epsilon_i,\epsilon_j\right)
$$
and furthermore because of independence
$$
\sum_{i = 1}^t cov\left(\epsilon_i,\epsilon_i\right) +
\sum_{i = 1}^t\sum_{j \neq i} cov\left(\epsilon_i,\epsilon_j\right)
=
\sum_{i = 1}^t \sigma +
\sum_{i = 1}^t\sum_{j \neq i} 0
=
t \cdot \sigma.
$$
Now we have shown that for this time series
$$
cov(x_t,x_{t+h}) =
t \cdot \sigma
$$
depends on $t$ hence it is not covariance stationary.

For a covariance stationary time series, let
$$
z_t = \epsilon_{t-1} + \epsilon_{t}.
$$
By performing the same analyis, you can conlude that for $t > 1$,
$$
cov(z_t,z_{t+h}) =
\left\{
\begin{array}{ll}
\sigma & \text{ if } h = 1 \\
0 & \text{ if } h > 1.
\end{array}
\right.
$$
Since this depends on $h$, but not on $t$, the time series $z_t$ will be covariance stationary (after $t = 1$).


Testing for heteroskedasticity in panel data vs time series? I watched this video on how to check for heteroskedasticity using Stata, and it helped me a lot. But the data example in the video was time series data.

He used the Bruesh-Pagan test.

hettest dependntvar1 dependvar2 dependvar3 ... dv6

And the output was like

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
     Ho: Constant variance
     Variables: dependntvar1 dependvar2 dependvar3 ... dv6

     chi2(6)     =  86.56
     Prob > chi2  =   0.0000

The Ho had a p-value of 0.0000 so it had heteroskedasticity.

Does it work the same for panel data?

If not, how do I test for heteroskedasticity on panel data?

You can regress residual squares (from RE or FE depending on your estimation) on $X_{it} \hat\beta$ and its square using the clustered standard errors (the vce(cl id) option), and read the F statistic and the associated p value. This is basically the same as Het test for cross sectional models (White's simplified test).

xtreg y x1 x2, re
predict uhat, ue
predict xb, xb
gen uhatsq = uhat^2
reg uhatsq c.xb##c.xb, vce(cl id)
testparm c.xb##c.xb

Another test is for groupwise heteroskedasticity proposed by Greene (2000). This test assumes that the variance of the error term is $\sigma_i^2$ (no heteroskedasticity over $t$) and then test whether $\sigma_i^2$ is the same for all $i$. You can do it using the xttest3 module. The xttest3 manual says:

  In terms of small sample properties, simulations of the test statistic have shown that its power is very low in the context of fixed effects with "large N, small T" panels. In that circumstance, the test should be used with caution.


Empirical set-up for measuring elasticity with respect to quality in an environment without price I want to answer the broad question: In the absence of prices, how are decisions made?

I have data on the determinants of market share in an environment without prices. So, imagine, person A chooses technology C at time X of a certain size, speed, and resolution (all measurable). And I have data for thousands of people and the technologies they choose over a time period of about 10 years. So again, imagine, changes in quality (a newer high-quality technology is developed) and this shocks the system, therefore demand changes as technology A is replaced by technology B, and then a few years later, technology C is available, etc., in an environment where everything is free and only 3 technologies are available, for example.

How do I model elasticity with respect to quality?

The data is stored in MySQL and I typically use Stata for statistical regression. Can you link me to a good example of an empirical test for this type of problem, or, at least, a step-by-step description of what info I need?

You might have to create your own set-up for a question like this. Elasticity of quantity with respect to some variable is can generally be measured by

$$\frac{\frac{\partial Q}{Q}}{\frac{\partial x}{x}}$$ where x is the variable you are measuring. In this case, you can do three separate elasticities w.r.t size, speed, resolution. Since you have data on thousands of people, just aggregate it all together and run a regression to measure the quantity of people who buy a tech with such and such size, speed, resolution across your whole time period. So you'll have something like

$$y = \beta_1 x_1 + \beta_2 x_2 + \beta_1 x_1 + \epsilon$$

Where $$x_1 = \frac{\frac{\partial Q}{Q}}{\frac{\partial \text{size}}{\text{size}}}, \quad x_2 = \frac{\frac{\partial Q}{Q}}{\frac{\partial \text{speed}}{\text{speed}}}, \quad x_3 = \frac{\frac{\partial Q}{Q}}{\frac{\partial \text{resolution}}{\text{resolution}}}, y = \text{total techs of all sorts sold}$$

Although you probably want to tweak that a bit, as I'm not sure why you'd want to specifically use elasticity.

If I am fundamentally misunderstanding the model you are trying to do, please let me know.


Reverse auction formula I am studing a little bit of auction theory. I found the optimal bid value in the Milgrom paper for the first price auction that is
$$
P=v \frac{n-1}{n}
$$
where $P$ is the optimal bid, $v$ is the true value and $n$ is the number of bidders.
Now I am wondering if there exist an equivalent formula for reverse auctions.
What is the formula for the optimal bidding in reverse auctions?

A first price standard and reverse auction are formally equivalent to each other, and the same method can be used to solve both:

First Price Auction

In a first price auction, $n$ bidders choose their bid, $b_i$, as a function of their value $v_i$ (distributed according to $F$. They seek to maximise their expected payoff:
$$[v_i-b_i(v_i)]\Pr(b_i\geq\max_j b_j).$$
If we look for a symmetric equilibrium in which bidders with higher values bid more then the probability that I have the highest bid is just the probability I have the highest value:
$$\Pr(b_i\geq\max_j b_j)=F(v_i)^{n-1}.$$

A useful trick is to think of a bidder as choosing what kind of value to "pretend to have". If we call their choice $\widetilde{v}$ then a bidder's probablem is to
$$\max_{\widetilde{v}}[v_i-b(\widetilde{v})]F(\widetilde{v})^{n-1}.$$

Differentiation yields the first-order condition:
$$(n-1)F'(\widetilde{v})F(\widetilde{v})^{n-2}[v_i-b(\widetilde{v})]-F(\widetilde{v})^{n-1}b'(\widetilde{v})=0.$$

In equilibrium, we know that a bidder should not want to pretend to be anyone other than his true type, so this first-order condition must hold at $\widetilde{v}=v_i$:
$$(n-1)F'(v_i)F(v_i)^{n-2}[v_i-b(v_i)]-F(v_i)^{n-1}b'(v_i)=0.$$
This is a differential equation that can be solved for the equilibrium bid function, $b(v_i)$. In, particular, if values are uniformly distributed on $[0,1]$, we have $F(v_i)=v_i$ and the equation becomes
$$(n-1)v_i^{n-2}[v_i-b(v_i)]-v_i^{n-1}b'(v_i)=0,$$
which can be simplified to
$$(n-1)\frac{1}{v_i}[v_i-b(v_i)]=b'(v_i).$$

The solution to this differential equation is
$$b(v_i)=v_i\frac{n-1}{n},$$
which is what you have in your question.

Reverse Auction

In a reverse auction, $n$ bidders choose their bid, $b_i$, as a function of their cost $c_i$ (distributed according to $F$. They seek to maximise their expected payoff:
$$[b_i(c_i)-c_i]\Pr(b_i\leq\min_j b_j).$$

If we look for a symmetric equilibrium in which bidders with higher costs bid more then the probability that I have the lowest bid is just the probability I have the lowest cost:
$$\Pr(b_i\leq\min_j b_j)=[1-F(c_i)]^{n-1}.$$

A useful trick is to think of a bidder as choosing what kind of cost to "pretend to have". If we call their choice $\widetilde{c}$ then a bidder's probblem is to
$$\max_{\widetilde{c}}[b(\widetilde{c})-c_i][1-F(\widetilde{c})]^{n-1}.$$

Differentiation yields the first-order condition:
$$-(n-1)F'(\widetilde{c})[1-F(\widetilde{c})]^{n-2}[b(\widetilde{c})-c_i]+[1-F(\widetilde{c})]^{n-1}b'(\widetilde{c})=0.$$

In equilibrium, we know that a bidder should not want to pretend to be anyone other than his true type, so this first-order condition must hold at $\widetilde{c}=c_i$:
$$-(n-1)F'(c_i)[1-F(c_i)]^{n-2}[b(c_i)-c_i]+[1-F(c_i)]^{n-1}b'(c_i)=0.$$
This is a differential equation that can be solved for the equilibrium bid function, $b(c_i)$. In, particular, if costs are uniformly distributed on $[0,1]$, we have $F(c_i)=c_i$ and the differential equation can be simplified to
$$-(n-1)[1-c_i]^{n-2}[b(c_i)-c_i]+[1-c_i]^{n-1}b'(c_i)=0.$$
$$(n-1)\frac{1}{1-c_i}[b(c_i)-c_i]=b'(c_i).$$
The solution to this differential equation is
$$b(c_i)=\frac{1+c_i(n-1)}{n}.$$

Comparing the two

The two solutions,
$$b(v_i)=v_i\frac{n-1}{n},\qquad b(c_i)=\frac{1+c_i(n-1)}{n}$$
are equivalent in the sense that a bidder in the first price auction shades (bids below his true valuation) by exactly the same amount as a bidder in a procurement auction bids above his true cost. You can confirm this by, for example, plotting these two bid functions for some value of $n$. Here's an example with $n=2$:


Variance of a rational forecast In Chapter 24 of Richard Thaler's book Misbehaving, he writes:

  An important property of rational forecasts---as a stock price is supposed to be---is that the predictions cannot vary more than the thing being forecast.

He then goes on to explain this using an example of weather forecast. If the temperature in a region typically varies between 85°F and 95°F, then it would be wrong to predict 50°F one day and 115°F the next day (although this would be approximately right on average).

What exactly is underlying mathematical principle for when he says "the predictions can't vary more than the thing being forecast"? Do forecasts have to get both the mean and the variance right to qualify as rational?

A colleague of mine pointed me to Steven M. Sheffrin's Rational Expectations, where I found the explanation (on p.122):

  [L]et $P_t^*$ be the variable to be forecast. Then the optimal forecast is given by
  \begin{equation}P_t=E[P_t^*|I_{t-1}].\end{equation}
  Let $u_t$ denote the forecast error, which, by the orthogonality principle, must be uncorrelated with the forecast $P_t$. Thus $P_t^*=P_t+u_t$ and, taking the variance of both sides,
  \begin{equation}
\mathrm{var}(P_t^*)=\mathrm{var}(P_t)+\mathrm{var}(u_t)
\end{equation}
  or
  \begin{equation}
\mathrm{var}(P_t^*)\ge \mathrm{var}(P_t),\quad\text{since $\mathrm{var}(u_t)\ge0$}.
\end{equation}


Production Possibility Frontier for 3 goods Is it possible to construct a Production Possibility Frontier (PPF)  for 3 goods? Would this require a 3D graph and a part of a sphere, in one quadrant (as opposed to the usual 2D plot, with a part of a circle in one quadrant).

Is this something used in more complex analysis of production?

It is possible. In fact you can construct PPF for arbitrary number of goods.

When it comes to plotting it for a case of 3 goods it is easy to make 2D projection of the 3D space. You can see the image below for an example.

Also note PPF is not necessary spherical or circular. The shape of PPF depends on the opportunity cost. In textbooks you might sometimes see it represented like a portion of a circle or elipse but it does not necessarily have to have that shape.


How is bartering (goods for goods) conceptually related to trading (goods for money) and monetary (money for money) exchange? My terminology may be off on this one but I can't get a definitive answer on how the above are related. To me, it seems like bartering is a superset of trading which is itself a superset of monetary exchange.

The goal of every individual using money is to acquire things of value. Money is the means of acquiring valuables but of no value in themselves. Money indicates that at some point in time I produced, acquired, discovered etc something of value which became my possession but I later gave up receiving a numeric representation, which I deemed to accurately represent my possessions value at the time. I now have the "value" but not the valuable (effectively storing it for later use or subdivision).

Bartering is the exchange of something of value for something else of value (the ultimate goal).

Trading is the exchange of a valuable for a (perceived) representation of its value.

Monetary exchange is the exchange of two tokens of value accepted in different regions (or by different organizations, countries etc...)

It appears to me that the purpose of any exchange is ultimately between things of value (bartering), on top of which has been added an additional layer of abstraction (money) in order to have a constant (and consequently always desirable "something" for which to make an exchange) and abstracting the trade even further is to make an exchange between mediums of exchange.

Yet, most sources I view, trading is distinguished from bartering as bartering is an exchange between valuables but money is a medium of exchange.
Is my assumption of a direct hierarchical link between the three forms of exchange correct?

Consider an economy where there are $N$ goods and assets. How do you purchase a good? If $n \cdot (n-1)$ forms of exchange are possible because all $n$ goods can be all other $n-1$ goods then this is a barter economy. If there is only a single good that can be exchanged for all other goods then only $n-1$ forms of exchange are possible then this is a monetary economy where that single good is "money". See, for example,  Kiyotaki and Wright (1993). Used this way, Trade Equilibrium $\supset$ Barter Equilibrium $\cup$ Monetary Equilibrium. A no trade equilibrium is sometimes called autarky.


Understanding distortions in investment in human capital I am currently reading Economics of The Colour Bar by WH Hutt, and I'm having trouble understanding this bit he has written: “But when the price of any commodity is raised above the free market level, by strike threat or action or by state edict, the effect is always regressive, harming poorer consumers more than wealthier consumers. Moreover, such a process distorts investment in human capital. It causes, inter alia, relatively more people to be trained for and employed in occupations if low productivity and remuneration, and hence relatively few to be trained for and employed in occupations of high productivity and remuneration.”

I don't get the part where he talks about more low productivity jobs will be created and HENCE relatively few high productivity jobs will be created. How does this work? If I'm selling some commodity above free market price, if I'm making a profit, why would I hire more low productivity folks than high productivity folks? Is it because I'm assumed to be losing money for going above the free market price?

I assume WH Hutt is referring to long-term effects. Here's what I understand. If "the price of any commodity is raised above the free market level by strike threat or action or by state edict," so that the price is above marginal cost. Then firms earn a "free" mark-up (=price - marginal cost) and need not be as productive as in a competitive market. As a result, they may hire less productive workers or offer less productive jobs, distorting investment in human capital in favor of low productivity and low-paying jobs.


Do real life economists and financial analysts actually use calculus in their jobs? From what I understand a lot of calculus is used at university level when studying economics and some finance courses (correct me if I'm wrong) but I was just wondering if economists and financial analysts actually use calculus in their actual jobs on a day to day basis.

Yes, whether you do research and you need to study a new model and characterize optimal behavior, or if you are studying data and you are estimating any model that is not linear you use calculus on a regular basis.

Even though computers are very powerful and can save the time and energy of doing calculations (including finding derivatives), often times having a firm knowledge of the maths behind what the computer is doing can save tons of time and improve the performance of algorithms.

That being said, it is also true that many economists can get by only knowing the minimum of calculus, since a lot of the tools that the market values the most from economists require, instead,  a firm grasp of probability, statistics and good economic intuition.


How does demand curve for labor change when there is a proportional payroll tax on employer In the textbook "Borjas, G.J. Labor economics." (Boston, Mass.; London: Irwin/McGraw-Hill, 2009)
fifth edition [ISBN 9780070172708]. Below is a screenshot of the example of payroll tax of 1 dollar on the employer. And the labor demand curve shifts down by 1. What happens if the tax is not 1 dollar but, say, 10% of wage? I know that the labor demand curve will shift downward. But does it also rotates in a way that it has a steeper slope?

Your guess that somewhat is happening with the slope was a good one. The slope will indeed change. If you want to draw that new Demand line ($D_1$) it has to start at the exact same point of the old demand line (if wages are zero the percantage tax will be zero too). The correct intersection with the Y-Axis would be: $(1-t) W_0$ (this simply means one minus the tax rate ($t$) times the old intersection point ($W_0$). I also made a little graph:


How do efficient credit markets allocate capital properly? I've learned that efficient credit markets should theoretically allocate capital efficiently, which means that all possible investments should have the same marginal product of capital (assuming some type of concave production function).

How exactly does that work? On the surface level, it makes sense that we should allocate more money to projects that have higher returns, but what would be the mechanism by which that happens? It seems like banks would have to charge a higher interest rate in Africa - and does that happen? I think I'm having a lot of trouble developing the correct intuition here.

Under the usual assumptions of decreasing MPK and complete markets in equilibrium $MPK_{Firm_1} = MPK_{Firm_2} = r$. If firm i's MPK is larger than r and firm i does not face constraints with respect to scale, if will simply borrow more capital as it can make a net return of $MPK_{Firm_i} - r$ on every unit for which  $MPK_{Firm_i} > r$. If $MPK_{Firm_i} < r$ firm i will divest until $MPK_{Firm_i} = r$.

There are many reasons for which this nice result can break down in reality (Deviations from the nice assumptions above). For example agency costs (Bernanke, Gertler, and Gilchrist (1996)), where an entrepreneur can only borrow up to the present value his net worth at end of the loan contract. If an entrepreneur does not have enough wealth he cannot borrow enough capital to achieve $MPK_{Firm_i} = r$. Hence when talking about this topic you should make your assumptions known in advance.

With regard to you question about Africa, from the point of view of the setting described above too few capital is flowing to developing countries. This fact has been known for a while and is referred to as the "Lucas paradox" or the "uphill flow of capital" a recent summary on the topic can be found here:

enter link description herehttps://voxeu.org/article/revisiting-paradox-capital


Calculating Real Budget Deficit, given Nominal Deficit and Inflation I am doing this course on Coursera. In the context of explaining what Budget Deficits are, the professor puts up the following formula:

  Real Deficit = Nominal Deficit - (Inflation*Total Debt)

My confusion is, where is the Total Debt coming from? Shouldn't Real Deficit be just a function of Nominal Deficit and Inflation? i.e.,

  Real Deficit = Nominal Deficit - (Inflation*Real Deficit)

My understanding is that Total Debt is just the accumulated debt over the years (including the current year's budget deficit incurred so far). Is that understanding wrong?

The professor goes on to give a numerical example to explain the concept of inflating the debt. That's when the effective burden of debt falls due to inflation.

If Nominal Deficit = \$100 billion, inflation =10% and Total Debt = $5 trillion, then

Real Deficit = 100 - (0.10*5000) = -$800 billion.

I don't get it.

Edit: This is the direct link to the 6 minute lecture where he talks about it.

Edit: based on the video I get the logic that the person was trying to get across.

Basically the argument is that if in year $t$ deficit was 100mil but the same year the total debt 5billion and inflation was 10% that means that in real terms the value of debt is 500mill less and the teacher of that course treats this as 500mill revenue so under that definition even though you had 100mill deficit you would end up with 400mill surplus because thanks to inflation the net real debt decreased.

However, I must say it’s quite unusual to define real deficits in this way (see Barro’s Macroeconomics).


Validity condition for Instrumental Variables My professor states the validity condition in the following way:
$W$ (the matrix of instrumental variables) must be such that
$ \text{plim} \, \frac{W'u}{N}=0$, where $u$ is the error term.
Intuitively, I can see that in some way, this relates to the covariance between $W$ and $u$, but am not entirely convinced.
Moreover, why do we look at this condition in an asymptotic setting? Why not impose mean independence, i.e., $E(u|W)=E(u)$?

Essentially, $Cov(u,W) = 0$ is implied by $E(u|W) = E(u)$ by the Law of iterated expectations


Taylor Rule: Relationship between interest rate and inflation Q. According to the Taylor Rule, what should the central bank do to stabilize the economy after an increase in oil price?

My points:

Increase in oil prices (1970's) raises the price of production for producers and hence shifts the AS curve to the left.
With lower AS and a constant demand the price for goods is raised and inflation increases.
Taylor rule suggests that in order to counter the effect of inflation and lower it back down to its target (usually 2%), for every percentage point that inflation is above its target, the Fed funds rate should be raised by 0.5%. For example if inflation is at 8% and the target is 2%, 8-2=6% above target so Fed funds rate should be raised by 6x0.5 = 3%.

I think I am correct in saying this but if am not please let me know. What I don't understand is why raising interest rates will actually help?

A higher Fed funds rate means banks will have to charge higher interest to its customers which will surely discourage consumption?

I understand the relationship between inflation and Real interest rate:

i = r + pi (Fisher Equation)

I'm just finding it hard to explain why increased interest will lower inflation in practice, i.e. what this means for consumers.

Does anyone have any pointers?

Thanks again in advance

It's true that in response to an oil shock, the Taylor rule could recommend increasing the interest rate to reduce inflation. In practice it would mean that as interest increases, consumption falls. This could be from less credit financed spending, or because the opportunity cost of holding money has increased, therefore people invest more in illiquid assets (bonds in an IS-LM context) and therefore consume less. As consumption falls, aggregate demand falls, therefore putting pressure off prices causing a reduction in inflation.

Note: an oil shock might also increase the output gap, so its not inherently clear what the taylor rule would say in this situation.


Why do people trade derivatives? If I have information that an asset will increase or decrease in value, I can profit on that information by buying or shorting the asset. Why, then, would anyone trade derivatives when they can just profit by directly trading the asset? How does this work from a theoretical perspective (game theoretic, etc)? This is a very simple question, I'm not sure if this is the right place for it, but thought I would try asking.

There are number of reasons to use derivatives. How practical the following are? It, in turn, depends on the costs compared to the importance (utility) of the case. The costs can be substantial.

Hedging purposes, which comes in many forms. E.g. when you buy stocks in foreign currency, you may want to take only the equity risk, not currency risk. Or you buy a fund or etf that comes with country or industry risk that you want to get rid off. Or you believe equity in the long run but don't like the volatility, so you (try to) hedge volatility away.
Limit controlling. E.g., if you are working in asset management industry and take care of funds of customers, the portfolio should have set limits: the derivatives may give you an easy way to comply with limits ("you are happy with the instruments you have chosen to your portfolio and want to keep them but without derivatives, say the minimum amount of equity risk would not be attained").
Leveraging a (sub-)portfolio. "Sometimes you cannot apply loans. But want to have more exposure than is otherwise possible."
Bearing a particular type of risk, e.g., you may believe that volatility is going to increase from the current level. This particular example can work as a hedge to your portfolio at the same time, since usually the increase of volatility means that portfolio performs badly. Moreover, this is an example of risk that is hard to get in to your portfolio without derivatives.
Quick change of allocation for tactical reasons. Say, evolving crisis either forces you to move quickly, e.g., because of limits as in case 2, or because of the capacity to bear risk (economic reason). And this case includes also the panic moves.
Sometimes, there are other motives behind applying derivatives that somehow build upon the first four cases. E.g. you have a complex company structuring case in your hand and you may have to transfer particular types of risk from one party to the other. To the other this is hedging and to the other risk taking: the derivative use could be classified to be "enablers" of transactions.
In the spirit of previous case, derivatives can be used in product design. E.g. some hedge funds promise to apply certain derivatives strategies. These products (hedge funds) may be targeted to professional investors and retail customer. Another example include index loans, typically marketed to retail customers, and may include exotic derivatives like binary options where the underlying is determined e.g. by the month end values of the last 12 months for a basket of securities. The motivation for options reduces to the first one listed above (hedging): the index loan may include a promise ("a guarantee") that, say 95% of nominal will be preserved in every possible scenario to be faced. So hedging appears to be provided to customers.
"Personal characteristics" like willingness to gamble may apply as reason. These may reduce to the third item.


Calculating the growth rate of a stock over a year with imperfect information I am trying to find the growth rate of a stock over a given year.

Let's say I wanted to find the growth rate from today, June 11, 2015 to June 11, 2014. This is easy enough when you have perfect information and both sets of data exist.

However let's say June 11,2014 stock data does not exist, but June 10th and June 12th do. Which date do I compare to? Do I extend the "year" by a day and go to June 10th, or do I shorten the year and go to June 12th?

I don't think that on yearly returns, a day more or less matters too much. In the unlikely case it actually does, you could interpolate the final day.

A first approximation would be the average over June 12th and June 11.


Free source of large amounts of financial data I am looking for a free source of large amounts of long term financial data on stocks, exchange rates, government bond rates, etc.

I know there is Thomson Reuters, but this requires a costly subscription.

The data doesn't have to be super up-to-date.

I assume that you are looking for macro financial data across countries. So, you can check out:

World Bank Open Data with a specific query on Financial Sector
IMF Financial Statistics or IMF Finances. Last time I used is a free registration was required .
Agricultural Exchange Rate Data Set. This data set contains annual and monthly data for exchange rates (important to U.S. agriculture). It includes both nominal and real exchange rates for 79 countries, plus the European Union (EU), as well as real trade-weighted exchange rate indexes for many commodities and aggregations. All series are updated quarterly. Data series start at the beginning of 1970 and run to the last available data point.


Recover cost function from profit function How do I recover the cost function from the profit function?

Suppose I have $$ \pi(p, w_1, w_2) = p^3 \cdot w_1 \cdot w_2 $$.

How do I get the cost function?

Using Hotelling's lemma I get the supply function: $$ y_s(p) = \frac{\partial  \pi(p, w_1, w_2)}{\partial p} = 3 p^2 \cdot w_1 \cdot w_2$$ and the input demands:
$$z_1(p, w)= - \frac{\partial  \pi(p, w_1, w_2)}{\partial w_l} = - p^3 w_2 $$ and $$z_2(p, w)= - \frac{\partial  \pi(p, w_1, w_2)}{\partial w_2} = - p^3 w_1 $$but what is the next step?

Note that the functional forms are hypothetical and highly probable to be invalid.

I suggest that you ask yourself the following questions:

If $\tilde{z}(y;w)$ is the input demand from cost minimization, can there be any difference between $\tilde{z}\big(y(p);w\big) \equiv \hat{z} (p;w)$ and $z(p;w)$ from Hotelling's lemma and profit maximization?
Even if $\tilde{z}(y;w)$ is a correspondance rather than a function, is there any difference between $ p \hat{z} (p;w) = p \tilde{z}\big(y(p);w\big) = c(y;w) $ and $p z(p;w)$ (where the later are dot products).

Hope this helps,


Solow Model with a Capital Stock Externality Suppose we have the following form of the Solow Model: $$ Y_t=K_t^{a+b} L_t^{1-a} $$ where a,b >0, and a+b<1.

Is it possible to determine the steady state growth rate of k (k=K/L), and the steady state growth rates of MPL and MPK?

There is no steady state in the Solow framework. The system is explosive because $a,b >0 \wedge a+b<1 \Rightarrow a+b + (1-a) > 1$. Thus $(zK_t)^{a+b} (L_tz)^{1-a} = z^{1+b}(K_t)^{a+b} (L_t)^{1-a}$. I.e. this production function has increasing returns to scale. Such capital stock externalities are discussed in the AK-Models.


Is a 6% wealth tax a sustainable source of income? Elizabeth Warren is planning to pay for various things with a 6% wealth tax on those with fortunes in excess of $1B USD.  Now, right or wrong, this seems to be an awfully big annual scoop from a bucket that one intends to be scooping from in perpetuity.  If the bucket gets low, then we'll have to find other as-yet-unknown sources of income to fund these various projects. Is the bucket filling up at a rate large enough to sustain such a 6% tax?

In other words, is the rate of wealth increase of Billionaires at least 6% on average?

I think the big unknown here is how much they'll start to dissimulate and/or offshore their (new) wealth.
According to one paper:

A recent study by Brülhart et al. (2017) gives support to the plausible assumption that the
effect of net wealth taxes on reported wealth is the more pronounced the more integrated
the regions involved are. According to the authors’ estimations, the semi-elasticity of
reported wealth with respect to the net wealth tax rate amounts to 35% in aggregate, i.e. a
rise in wealth taxation by one percentage point decreases reported wealth by 35%.
Moreover, Brülhart et al. (2017) find that financial assets seem to be more responsive to
taxation than non-financial assets. They also interpret their results as suggesting that
wealth holders primarily respond by reducing their wealth holdings, not by moving to
jurisdictions with lower tax rates.

So if that is true, then >3% wealth tax will result in "over" 100% dissimulation of new wealth. But caveat emptor, that's based on the fairly secretive Swiss:

It must be noted, however, that these analyses do not uncover the channels via which
wealth holdings are lowered. As indicated above, reported wealth holdings may be reduced
by real responses (i.e. by lowering accumulation of wealth) or by decreasing reported
wealth through hiding it from tax authorities. Which of these mechanisms is working in the
Swiss case cannot be determined without further analysis, e.g. by exploring whether there
is some relationship between the savings rate and the taxation of net wealth.

The biggest issue with extrapolating much from the Swiss case is that

Wealth taxes are cantonal and municipal; there is no federal taxation of wealth.

It's basically unknown how big that semi-elasticity would be if a whole country (like the US) had a wealth tax. France apparently had one, but it looks like they don't have a clear figure of the aforementioned elasticity, despite the fact that the rate changed a few times. It is known however that for about a decade there was a net outflow of around 300-500 people (per year) who would have qualified for the wealth tax, although they could have left for other reasons (including other parts of the tax code). France being in the EU also offers some special circumstances to migrate to neighboring countries but still (partly) invest in France.

If we ignore the potential flight or dissimulation, there's still the issue that billionaire wealth might decrease even in not so bad times, e.g.

Billionaires around the world saw their total wealth decline by 7% from 2017 to 2018, according to Wealth X’s Billionaire Census 2019. At the same time, the number of these ultra-high-net-worth individuals fell from 2,754 to 2,064.
American billionaires had an easier time.
The U.S. was one of the few countries that saw its UHNW population grow from 2017 to 2018, per the report, and America’s richest people lost only around 5% of their total wealth, far less than their peers did in most other regions. [...] Total U.S. billionaire wealth fell from \$3.2 trillion to \$3.013 trillion.

So, before we talk of sustainability, we'd need to decide on some time frame. Clearly their (total) wealth can go down year-to-year even with no wealth tax, although the trade war was apparently blamed for the recent downward trend. From 2015 to 2016 it actually went up 15%.

In 2016, US billionaires saw their wealth increase from USD 2.4 trillion to USD 2.8 trillion

So it looks like US billionaire wealth is more volatile than the US economy as a whole.


Comparative statics of a monopoly Consider a profit maximising monopolist with linear demand Q(P*) and total production cost C(Q(P*)) who faces a per unit tax t. Suppose the slope of marginal cost is decreasing in some parameter, μ. Let P* denote the monopolist’ profit maximising choice of price. Being careful to explain your method and interpret your result, determine the comparative static:
∂²P*/∂μ∂t
I got to P*= price Q = Q(P*) t=t(Q(P*)) C=C(Q(P*))
πmax = P*(Q(P*))-C(Q(P*))-t(Q(P*))
dπ/dP* = Q'(P*)[-C'(Q(P*))-t'(Q(P*))+P*]+Q(P*)=0 dπ/dP* = μ + Q'(P*)[-t'(Q(P*))+P*]+Q(P*)
Equilibrum: PQ'(P)=t'(Q(P*))Q(P))-μ-Q(P*)
P* = t'(Q(P*))- μ/(Q'(P*)) - Q(P*)/Q'(P*)
This is where I got to, when I solved the comparative static partial differentiation I got an answer of 0, I don't believe this is correct, can anyone help me solve this? Thanks!
If 0 is somehow the correct answer, what does it mean?

The profit function is given by:
$$
\pi = PQ(P) - C(Q(P), \mu) - tQ(P)
$$
Assume demand is linear, s
$$
p = \alpha - \beta Q \to Q = b(\alpha - P),
$$
where $b = 1/\beta$. So:
$$
Q_P = -b,
$$
where I use subscripts to denote the partial derivatives.
The first order condition for profit maximisation gives:
$$
\begin{align*}
    &Q + P Q_P - C_Q Q_P - t Q_P = 0,\\
\iff &Q - b P + b C_Q + bt = 0
\end{align*}
$$
then differentiating this with respect to  $t$ gives:
$$
\begin{align*}
&(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial t} + b = 0,\\
\to &(-2b - b^2 C_{QQ})\frac{\partial P}{\partial t} = -b,\\
\to &\frac{\partial P}{\partial t} = \frac{1}{2 + b C_{QQ}}
\end{align*}
$$
Differentiating the first order condition with respect to $\mu$ gives:
$$
(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial \mu} + b C_{Q\mu} = 0,\\
\to (-2b - b^2 C_{QQ})\frac{\partial P}{\partial \mu} = - b C_{Q\mu},\\
\to \frac{\partial P}{\partial \mu} = \frac{C_{Q,\mu}}{2 + b C_{QQ}}
$$
Then differentiating $\frac{\partial P}{\partial t}$ once more with respect to $\mu$ gives:
$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{1}{(2 + bC_{QQ})^2}b\left(C_{QQQ}Q_P \frac{\partial P}{\partial \mu}+ C_{QQ\mu}\right),\\
= -\frac{1}{(2 + b C_{QQ})^2}b \left(-b C_{QQQ} \frac{C_{Q,\mu}}{2 + b C_{QQ}} + C_{QQ\mu} \right)
$$
Where the last line uses the expression for $\frac{\partial P}{\partial \mu}$ from above.
Now assume that costs take the form:
$$
C(Q, \mu) = \delta + \eta Q + \frac{\gamma(\mu)\, Q^2}{2}
$$
So the slope of the marginal costs depends on $\mu$. Then $C_{QQ} = \gamma(\mu)$, $C_{QQQ} = 0$  and $C_{QQ\mu} = \gamma_\mu$ , so:
$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{b \gamma_\mu}{(2 + b \gamma)^2}.
$$
We have that $\gamma_\mu < 0$ by assumption (slope decreases in $\mu$) then we see that the sign of $\frac{\partial^2 P}{\partial t \partial \mu}$ is positive (at least if I didn't make any mistakes).


New Keynesian IS curve: question about time dependence A simple version of the new-Keynesian IS cuve is given as follows:
$$
\ln Y_{t} =\ln Y_{t+1} -\frac{1}{\theta} r_t.
$$
Here $r_t$ is the real interest rate and $1/\theta$ is the cross elasticity of consuming goods at different times. On the book I am reading (Advance Macroeconomics by David Romer, Chapter 6), the equation is derived based on consumption $C$ rather than income $Y$, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so $C=Y.$
I have some doubt about this equation. The equation shows a direct relation between real interest rate $r_t$ and real GDP growth rate $\ln(Y_{t+1}/Y_t),$ which does not look quite right, as we often say that there is an inverse relation between GDP and real interest rate for an IS curve. Of course, it is the growth rate rather than the GDP $Y$ itself, but this direct relationship is still kind of hard to explain. Why is this the case?
If we keep $Y_{t+1}$ fixed, then there is an inverse relation between $r_t$ and $Y_t,$ as expected, but it seems unrealistic to hold the future $t+1$ as fixed and predetermined.
In summary, the additional term $\ln Y_{t+1}$ adds in some time-dependence, which makes this model difficult to understand in terms of an IS-LM diagram, which is a diagram at a fixed time.
How to explain the observations I make above about the interest rates?
Edit: It is easy to understand the above in terms of consumption $C,$ but it appears harder to believe when we replace $C$ by $Y,$ for the reasons above.

As you already mentioned this equation can be derived from the households consumption rule.

the equation is derived based on consumption C rather than income Y, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so C=Y

So lets take a look at the Euler condition for the household (where $\pi$ represents inflation, $\beta$ the discount factor and $R_t$ the noninal interest rate)  and remember the market clearing condition $C=Y$:
$$\begin{align}C_t^\sigma = \beta^{-1}  E \left[\frac{\pi_{t+1}}{R_t} C_{t+1}^\sigma \right]\end{align}$$
You can directely see, that if the household expects future consumption to increase, current consumption will also increase (assuming no changes in inflation and interest rates). This is called consumption smoothing. The households tries to balance consumption over different periods. A rise in nominal interest rate makes saving more attractive and thus households shifts some of the current consumption into future periods.


Writing the core as the intersection of pareto efficient outcomes of all coalitions I have been reviewing general equilibrium models and was trying to find an efficient method for computing the core of a cooperative game. I was taught this topic in a very poor way so I believe I still have some conceptual errors.

Here is a thought I had:

Suppose we are in an economy with three consumers, $A$, $B$, and $C$, with utility $u_{i}(x)$ defined over bundles $x \in \mathbb{R}^{2}$ and endowments $\omega_{i}$ for $i = A, B ,C$. I want to compute the core for this economy.

I know the core must satisfy:
\begin{align}
u_{A}(x_{A}) &\geq u_{A}(\omega_{A})\\
u_{B}(x_{B}) &\geq u_{B}(\omega_{B})\\
u_{C}(x_{C}) &\geq u_{C}(\omega_{C})\\
\end{align}
i.e. the core must be individually rational. So let $$D =\{ x \in \mathbb{R}^{2} : x \text{ is individually rational for $A$, $B$ and $C$} \}$$ I also know that the core is a subset of the pareto efficient outcomes, so let $$E =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient} \}$$ Now here is the part I am not sure about: I know that the core is also not blocked by any two-person coalition. I think this means that any core allocation is pareto efficient for any two-person game. Thus I define:
\begin{align}
F_{1} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $B$ } \}\\
F_{2} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $C$ } \}\\
F_{3} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $B$ and $C$ } \}
\end{align}

Here are my questions:

Is the above analysis correct?
Can I write the set of core allocations $\mathcal{C}$ as $$\mathcal{C} = D \cap E \cap F_{1} \cap F_{2} \cap F_{3}\text{?}$$
Can this method of solving be generalized to a game with $n$ players and $m$ goods?

Let me know if anything is not clear!

Most of what you write is correct, but the definitions of the $F_i$ sets is imprecise. The problem is that in the core $A$ and $B$ may get goods that do not match their initial endowments. In this case it is not true that the core allocation $x$ is Pareto-efficient in the restricted 2-person economy of $A$ and $B$, because $x$ is not even an allocation in that game.

Edit: (An example)

Consider the initial endowments
$$
\omega_{A} = (1,1),  \omega_{B} = (1,1), \omega_{C} = (2,2)
$$
and an allocation $x$
$$
x_A = (2,2), x_B = (2,2), x_C = (0,0).
$$
$A$ and $B$ cannot Pareto-improve on $x$. But $x$ is not a Pareto-efficient allocation in their 2-person economy, because it is not a feasible allocation of their economy:
$$
x_A + x_B \neq \omega_{A} + \omega_{B}
$$

A better definition for the sets $F_i$  would be something like:

Let us denote the set of feasible allocations of the 2-person economy of $A$ and $B$ by $Y_{A,B} \subset \mathbb{R}^{2}$. Then
$$
F_{1} =\{ x \in \mathbb{R}^{2} : \nexists y \in Y_{A,B} \text{ such that } u_{A}(y_{A}) \geq u_{A}(x_{A}), u_{B}(y_{B}) > u_{B}(x_{B}) \}
$$
There are still some issues with cases when $A$ is better off and $B$ is not worse off, but if the utility functions are continuous then this should not cause a problem.

You can define $F_{2}$ and $F_{3}$ in a similar manner.

A remark: $E$ is not 'special', it is the set of allocations that cannot be improved upon by the three player coalition. This is equivalent to Pareto-efficiency.

Yes. Why not? This is exactly what the core is.
I would not call it solving, because usually you do not get a unique solution, and in extreme cases you may get no solution. But yes, every economy ($n$ players, $m$ goods) has a core, and it is defined in this way. (As indicated, unless some conditions are met the core may be empty. A competitive equilibrium is always an element of the core, so if that exists, the core is non-empty.)


Assumption of sufficient wealth in quasi-linear preferences Whenever we talk about quasi-linear preferences, we assume that the consumer is sufficiently wealthy. As far as I understand is that we need that assumption in order to obtain an interior solution.
But what would the economic intuition behind that premise be? Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?

This depends on the exact preferences, but usually the utility function
$$
U(x,y) = v(x) + y
$$
is such that
$$
\lim_{x \to 0} |MRS(x,y)| = \lim_{x \to 0} \frac{\text{d}v(x)}{\text{d} x} = \infty.
$$
In this case it is the consumption of an additional marginal unit of the nonlinear good $x$ that is infinitely useful compared to the consumption of an additional marginal unit of the linear good $y$. Hence the consumer will consume some units of $x$ before buying any of $y$. If the consumer is not 'wealthy enough', the corner solution will be such that $x > 0$, $y = 0$.


Is this a valid derivation of FWL? $Y = X\hat{\beta} + Z\hat{\gamma} + e\\
(X'X)^{-1}X'Y = (X'X)^{-1}X'X\hat{\beta} + (X'X)^{-1}X'Z\hat{\gamma} + (X'X)^{-1}X'e\\
\hat{\beta}_{short} = \hat{\beta} + \hat{\delta}\hat{\gamma}
$

where $\hat{\beta}_{short}$ is the OLS estimator of the coefficient on $X$ from the misspecified model that omits $Z$, and $\hat{\delta}$ is from the regression $Z = X\hat{\delta} + \eta$.

This seems infinitely more simple than FWL proofs I've seen in econometrics textbooks. So what is this missing or getting wrong?

The result is not the Frisch–Waugh–Lovell Theorem. Rather, the result shows the extent of Omitted variable bias.

The Frisch-Waugh-Lovell Theorem states that the residuals of the regression of $\boldsymbol y$ on $\boldsymbol Z$, regressed on the residuals of the regression of $\boldsymbol X$ on $\boldsymbol Z$, gives $\boldsymbol{{\hat\beta}}$. To put it another way, let $\boldsymbol{M}$ be the residual maker. Then the regression of $\boldsymbol{M_Z y}$ on $\boldsymbol{M_Z X}$ gives $\boldsymbol{{\hat\beta}}$. The equation is:
$$\boldsymbol{\hat\beta} = \left(\boldsymbol{X}'\boldsymbol{M_Z}'\boldsymbol{M_Z X}\right)^{-1}\boldsymbol X'\boldsymbol{M_Z}' \boldsymbol{M_Z y}$$
Note that the residual maker matrix is symmetric and idempotent.

An analogous result holds for $\boldsymbol{\hat\gamma}$.


Why is $\Delta p \Delta q$ in this diagram infinitely small? I am curious why the area $\Delta p \Delta q$ is so small that it can safely be neglected.

It says "for small values of delta p and delta q". So something small times something small is something very small. Basically we are taking a limit and as delta p and delta q go towards zero, the product goes to zero faster.


Did real incomes drop significantly since the 1950s? From what I heard, in 1950s America, you didn't have to go to college to support a large family and a stay-at-home wife. People could buy houses easily, drove giant cars and didn't worry about gas prices.

Did real incomes drop significantly since then? What was the cause?
(My college history professor had this pet theory that America gobbled up almost all the world's gold reserves during WW2, and that was what caused its post-war economic boom. Not sure if that is true, but perhaps this effect went away?)

Most of the US real earnings data which  go only as far as mid 60s. According to the statista data presented in this article by world economic forum the evolution of real hourly earnings in the US for production and non-supervisory workers looked like this:

If we extrapolate to 50s then the real earnings are now higher overall. However, this being said the real wages first peaked in the 70s at $\\\$23.24$ then fallen to about $\\\$20$ and recently in 2019 they finally again reached the previous peak from the 70s. Hence, we cannot really say they dropped significantly but rather they tended to stagnate.
There is no single agreed upon explanation for this. Some authors argue this is due to technological change and decline in demand for low skill workers (see Fernandez 2001), other arguments include declining union membership (see David,  Katz, and Kearney. 2006),  or international competition (see here). Others argue that CPI overstates inflation and that this can be also due to increase in non-monetary benefits such as health insurance (see here).  A good explainer is also provided by this brookings article.


MLE estimation with serially correlated errors I want to estimate the parameters of $a_t = a_{t-1}+\theta+\epsilon_t$ using MLE. Assume errors $\epsilon_t$ are serially correlated, then how would I choose the likelihood function?

This is model is known an GARCH(general autoregressive conditional heteroskedasticity) where the AR component of the observable series $\{a_t\}$ is equal to unity.

The likelihood is determined by your model assumptions on the innovation terms $\varepsilon_t$ and $e_t$.

If you are looking for a canned function to estimate an GARCH see https://www.kevinsheppard.com/MFE_Toolbox. If you are looking for some more theory on how to form the likelihood see http://www.math.elte.hu/probability/markus/AlkmatTS1/ARCH_GARCH_Eload.html.


Germany's current account surpluses: Negative effect? How to solve them? The case for a pernicious persistent current account deficit is easy to build: the home country is borrowing abroad for a long period and that debt may reach a point of becoming unsustainable. There are several ways to put pressure of the debtor to ensure repayment. The home private and government agents may self-regulate their spending, by incorporating their future expectations of what happens when their debt is no longer repayable (not likely). Another way is by market forces, if creditors think the home country is no longer able to service its debt, then funds for home will disappear, or there could be political pressure for repayment.

However, for Balance of trade surpluses, there doesn't seem to exist any 'economic' force at work for a downward pressure... or is it?

For example Germany has been amassing enormous surpluses since the 70's, and recently, their surpluses have reach 8.5% of GDP in 2015. How does one explain this 'economically'?

Any help would be appreciated.

As requested in comments:

German banks lost a lot of money in the US sub-prime crisis and would have lost a lot more in Greece if the Eurosystem had not saved them. This is what might reasonably be expected if you run persistent trade surpluses: you gain foreign assets which may never be repaid or may be less valuable than you expected

The European Commission is supposed to investigate under its Macroeconomic Imbalance Procedure when a country's current account surplus exceeds 6% of GDP, and Germany's is now about 8%. In the end the Macroeconomic Imbalance Procedure might in theory become an Excessive Imbalance Procedure and potentially sanctions for euro-area member states of up to 0.1% of GDP. But it will not happen to Germany, for European political reasons. Wikipedia has a description and you can also read the latest Commission report

In effect Germany is making stuff and sending it to foreigners without exchanging it for a similar value of imports. Eventually people in Germany might notice that they could have a higher standard of living which their mercantilist politicians and economists have been denying them

German savers and politicians are already protesting about low interest rates and perhaps one day might think that in a period of negative interest rates it could be sensible to consume (and import) more


Does stock picking work? Does stock picking work? Why/why not? What does the research say about it?

(I was expecting that this should already have been asked and looked quite extensively for such a question but couldn't find anything)

In general, stock picking does not work.

For more information, see this keynote address by Daniel Kahneman, who found that stock-pickers perform worse than chance when they change stocks: https://forum.amundi.com/the-psychology-of-investors-the-cost-of-ideas/

Here is another reference on the subject:

  Abstract

  Is expertise portable across closely related domains? We investigate this question empirically using a unique dataset of market
  experts---institutional investors with portfolios averaging $573
  million. A striking finding emerges: while there is clear evidence of
  skill in buying, the investors' selling decisions underperform
  substantially--even relative to random selling strategies. This holds
  despite the similarity between the two decisions in frequency,
  substance and consequences for portfolio performance. We present
  evidence that an asymmetric allocation of cognitive resources such as
  attention can explain the discrepancy: investors are prone to a
  systematic, costly heuristic process when selling but not when
  buying.

  Akepanidtaworn, Klakow and Di Mascio, Rick and Imas, Alex and Schmidt, Lawrence, Selling Fast and Buying Slow: Heuristics and Trading Performance of Institutional Investors (May 8, 2019). http://dx.doi.org/10.2139/ssrn.3301277


Trade-off between determenistic and random alternatives in a one-shot game I wonder how to analyze the following problem.

Suppose we play a one-shot game. We have two options. We can either get $x$ dollars or win $y$ dollars with probability 50% (or, probably, win nothing with the same chances). What is the minimal value that the game-designer should assign to $y$ to make you to choose the second option?

The game is one-shot. It means there is no way to play it several times. It works for me in the following way:

if $x$ is 50 cents, I'm ready for \$1 at $y$;
if $x$ is \$100 dollars, \$300 will be enough.
if $x$ is \$100M, I won't choose the second option.

I suppose the decision depends on your income, your tendency to risk and so on. So what factors are essential for rational decision? And how to compute $y(x)$?

Let's assume that the player is an expected utility maximizer, and that the utility they get from a payoff of $\$z$ is $u(z)$, for any $z$ in $\mathbb{R}$ such that $u$ is strictly increasing.

You are essentially asking about certainty equivalents. Call $\sigma$ the gamble that pays $y$ with probability $\frac{1}{2}$ and $0$ with probability $\frac{1}{2}$. Then, the utility of the $\sigma$ (under the EU hypothesis) is $U(\sigma) = \frac{1}{2}u(y) + \frac{1}{2}u(0)$. This means that the player is indifferent between this gamble and getting $u^{-1}(\frac{1}{2}u(y) + \frac{1}{2}u(0)) = u^{-1}(U(\sigma))$ for sure. This is called the certainty equivalent of $\sigma$. Notice, $U$ transforms a gamble over monetary payoffs into utilities, and $u^{-1}$ transforms the utilities back into monetary payoffs. If $x$ is a constant gamble then $u^{-1}(U(x)) = u^{-1}(u(x)) = x$.

If $x$ is larger than the certainty equivalent of $\sigma$, the player will choose $x$. You are asking the inverse (if $CE: \sigma \mapsto u^{-1}(U(\sigma))$ you are interested in $CE^{-1}$). Nonetheless the underlying mechanics are the same, once the utility function has been specified. The $y$ you are interested in satisfies

$$ y = u^{-1}(2u(x) - u(0)).$$

The deeper question here, however, is how is the certainty equivalent related to the gamble itself. In general (for a very ugly $u$), not much can be said. However, it is common practice to make some structural assumptions on $u$ to answer the questions you are asking. For example, what if we assume the utility index is differentiable and concave, for example $u(x) = ln(x)$. Then it is relatively straightforward to show that $y(x)$ is increasing and convex (loosely speaking, the relative difference between $x$ and $y$ is increasing, as suggested in your post, although it does not capture the complete reluctance to choose the risky prospect in the third example (although this can be captured by other, more complicated models)).

It is worth noting, all of this assumes EU, which while extensively used in applications, especially game theory, is not without its empirical failures. There are many other "psychological" models which might provide a different resolution to your inquiry.


Specific question about some figures in national accounts I have highlighted four highlighted numbers in the UN System of National Accounts (p. 31, Table 2.13). My question is: What are these numbers?

My guesses:

6 = Compensation paid by foreign firms to resident employees.
2 = Compensation paid by resident firms to foreign employees.
44 = Property income paid by foreigners to residents.
38 = Property income paid by residents to foreigners.

I'm sure the answer is somewhere in the document (700 pages and not exactly user friendly), but I can't find it.

When answering my question, please cite where in the document you found the answer.

Also, how is the number 1864 obtained from the numbers above?

Para 2.43 explains the difference between the ‘uses’ and ‘resources’ sides of the account. The columns then refer to different sectors, where you are highlighting transactions with the ‘rest of the world’ sector.

For the numbers you refer to, the left-hand side is uses:

6 = compensation of employees paid out to the ‘rest of the world’ sector.
44 = property income paid out to the rest of the world sector.

The right-hand side is resources:

2 = compensation of employees paid into this economy by/from the rest of the world sector.
38 = property income paid into this economy, by/from the rest of the world sector.

Edit: you also asked how the 1,864 is derived. The balancing item for this account should be calculated as total resources less total uses (although I just tried it and got 1,854). There are separate gross and net balances, where the difference is consumption of fixed capital (depreciation of assets).


Axiom of Minimal Liberalism & Sen's Theorem of Paretial Liberal Suppose that a person believes that all humans are guaranteed a set of rights that cannot be taken from them in any situation or circumstance (for example, the right to marry a person of your choice, the right to practice any religion...). Is this moral belief compatible with Harsanyi's axioms? (This is part of an essay prompt that I wanted some guidance on. Thanks!)

Your question is a bit confused, because it mixes together several different things.  For example, in the title, you mention Sen's Minimal Liberalism, but in the actual question, you don't mention Sen at all --- instead you talk in more general terms about "rights that cannot be taken ... in any situation".  You mention "Harsanyi's axioms", but it isn't clear what you mean by this ---or what you mean by "compatible".  So I will try to answer a couple of different interpretations of your question.
Presumably, by "Harsanyi's axioms", you mean the axioms in Harsanyi's Social Aggregation Theorem.  (These are: (1) All individuals and the social planner have preferences over lotteries that satisfy the von Neumann-Morgenstern axioms, and hence admit expected utility representations; and (2) The social preferences over lotteries satisfy the Pareto axiom with respect to individual preferences over lotteries.)  Harsanyi's Social Aggregation Theorem is an axiomatic characterization of utilitarianism. (Or at least, that is how Harsanyi interpreted it.)  So I will take your question to really be asking:  "Is utilitarianism compatible with an absolute, non-negotiable respect for individual rights?"
The answer is "No".  And this has nothing to do with Harsanyi, or even modern  axiomatic social welfare theory in general.  The incompatibility of utilitarianism with absolute individual rights has been understood for two hundred years --indeed, this is one of the major philosophical objections to utilitarianism.  Of course, utilitarianism generally promotes human rights, because utilitarianism wants to maximize sum-total utility, and violating people's rights tends to cause a massive loss of utility, so it is something you generally want to avoid, if possible. However, because it aggregates over the preferences of many people, utilitarianism has the feature that the desires of a very large group of people can override the desires ---or even the "rights" ---of a small group of people, in some situations.  The moral philosophy literature is full of thought experiments where utilitarianism leads to some egregious violation of individual rights.  (Of course, the same is true for almost every other social welfare function.)
But it is important to nuance this point.  We want to maximize social welfare over a "feasible set" of possible social outcomes.  Whether or not this will involve violating individual rights depends upon what is contained in this feasible set.  One way to prevent utilitarianism from violating rights is to just exclude rights-violating outcomes from the feasible set.  In the terminology of Robert Nozick, we can impose "side constraints" on the social optimization problem, so that certain policies or social outcomes are just off-limits.
Most moral philosophers who endorse utilitarianism are not fans of using "side constraints" to protect individual rights, because it ruins the beautiful conceptual simplicity and unity of pure utilitarianism.  Most utilitarians don't actually believe that "rights" are part of the fundamental structure of morality ---except in the limited, instrumentalist sense that I described two paragraphs back.  Jeremy Bentham (often seen as the originator of utilitarianism) once described the idea of "natural rights" as "nonsense upon stilts".
However,  I am raising this issue because it has a bearing on whether or not Harsanyi's Theorem is "compatible" with some notion of rights.  The reason is that Harsanyi's Theorem does not make any assumptions about the feasible set.  It simply supposes that there is some set $X$ of "social alternatives", and every agent (i.e. the social planner and every individual) has preferences over the space $\Delta(X)$ of lotteries over $X$.  So the question of whether Harsanyi's theorem is compatible with rights is simply the question: what is contained in the feasible set $X$?  If  $X$ contains outcomes that violate rights, then under some conditions, a maximization of utilitarian social welfare will choose these outcomes, so Harsanyi's Theorem is incompatible with rights.  But if $X$ does not contain outcomes that violate rights, then  Harsanyi's Theorem is incompatible with rights.
Turning to Sen:  Sen's Minimal Liberalism axiom can in fact be seen as an assertion about what is in $X$.  It stipulates that $X$ contains a pair of outcomes $x$ and $y$ such that (on the grounds of "liberty" or "rights") the social preference between $x$ and $y$ should be entirely determined be the preferences of some individual $i$.  For example, it might be that $x$ and $y$ are two outcomes which are identical in every way except for what colour shirt $i$ is wearing.  If we believe (plausibly) that people should have the "right" to chose their own clothing, then the social preference between $x$ and $y$ should be determined by $i$'s preferences between $x$ and $y$.  This is what Sen's Minimal Liberalism Axiom is saying.  Sen's Impossibility Theorem says that even this minimal liberalism condition is incompatible with the Pareto axiom.
However, notice that this is only an issue because somehow the set $X$ of social outcomes included information about what colour of clothing people can wear. So Sen's Minimal Liberalism is really saying, "$X$ contains outcomes which bear upon individual liberties, so that it is possible to violate individual rights by choosing the wrong element of $X$".  In other words, Minimal Liberalism is saying something about what is contained in $X$.  If we simply restrict $X$ to social outcomes where rights violations are not possible (i.e. impose "side constraints"), then the problem goes away.


Does it make sense to talk about Bitcoin 'inflation' when little to no goods and services are listed in Bitcoin currency? I think of inflation as a generalized increase in the price levels of a basket of goods and services.
I should add that my understanding presumes that inflation should be measured in the same currency as that in which the goods and services that make up the basket are listed and sold in the market, e.g. when measuring inflation in France, the basket price is in Euros not in US dollars or rupees.
Measuring inflation for a currency in which few if any goods are listed in, such as Bitcoin, seems erroneous as there is not Bitcoin-denominated basket anywhere in the world.
Am I onto something here?

The answer depends on what inflation we are talking about since the definition you mention in your question is not the only type of inflation in use. There are several different inflations used by economists. When economists talk about inflation without qualification we mean inflation of goods and services, as you mention in your question, but beside that in economics we have concepts such as asset price inflation, or even house price inflation etc.
In principle, you can view bitcoin as a speculative asset not as a currency Baur et al (2018), there is actually not clear consensus on what bitcoin is because it has some characteristics of speculative asset. If you decide to view bitcoin as an asset, and construct some price index for price of an asset (what would be bitcoin exchange rate if you want to view it as a currency) then you can call change in that price index 'bitcoin price inflation', the same way as some call change in the housing price index house price inflation.
If we are talking about inflation more specifically about just goods and services inflation, then you cannot talk about bitcoin inflation, but note you cannot talk about USD or euro or pound inflation either (even if I seen people loosely talking about USD inflation, it is a misnomer).
Inflation is by definition  (see Mankiw Principles of Economics pp 13):

an increase in the overall level of prices in the
economy

In turn economy is by definition (ibid, pp 5):

Whether we are talking about the
economy of Los Angeles, the United States, or the whole world, an economy is just
a group of people dealing with one another as they go about their lives.

Hence inflation is about measuring change in prices in an economy (which could even be a household or just some region) and it does not matter how prices are quoted. As a matter of fact inflation is completely unitless quantity (in fact even CPI change of which gives you inflation is itself unitless, but if you would measure inflation observing growth in actual prices the units would still cancel out).
Consequently, it is misnomer to talk about USD or euro or bitcoin or any other currency inflation.  Euro is good example to illustrate this. The 'euro inflation' in France is different than 'euro inflation' in Euro-area and 'euro inflation' in Portugal. You can see that yourself by looking at Eurostat's inflation statistics for both EU and individual Euro-zone members (see Eurostat here). Heck, as Eurostat shows you can even calculate inflation for EU and not whole EU even uses the same currencies. More properly you should be talking not about USD, euro or bitcoin inflation but about inflation of USA, Euro-zone and so on. Inflation in USA is measured by change in CPI derived from prices quoted in USD because that is the official currency of US economy, but if half of USA are would use different currency you would still be able to calculate inflation for USA. Moreover, if you want to measure inflation of goods and services in some economy (e.g. some community of people) where people use bitcoin you can do that as well, although again it would be misnomer to call it bitcoin inflation, it would be inflation of that economy (and again the economy could be as small as you want).


Why labour, capital, and output levels cannot be pinned down in perfect competition? Consider a firm producing with the following technology:

\begin{equation}
 Y = AL^{\alpha}K^{\beta}
\end{equation}

Assuming that factors are paid their marginal contribution to output, it can be shown that the optimal factor choice for this firm is:

\begin{equation}\label{eq:k-l ratio}
 \frac{L^*}{K^*} = \frac{\alpha}{\beta}\frac{r}{w}
\end{equation}

(this comes from finding each factor's payment, and combining them)

Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, $L^*$, $K^*$, or $Y^*$.

This is not the case if there is just one input. For instance, if $Y=AL^\alpha$, then

$$L^*=\left(\alpha A\frac{p}{w}\right)^{1-\alpha}$$

The reason is not related to the degree of returns to scale, as this is not constrained. It is not related to elasticity of substitution either, as in the case of a CES, it is also the capital-labour ratio which is solved for. Here, the solution is

$$ \frac{L^*}{K^*} = \left(\frac{\alpha}{1-\alpha}\frac{r}{w}\right)^\sigma $$

So, the question is, why can we not pin down the level of optimal inputs and output? Is this because we have treated factor prices as exogenous, so we are missing the supply side of each factor market?

Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, $L^*$, $K^*$, or $Y^*$.

I don't think this is true. You combined two equations into one, and thereby lost information. The optimization problem is

\begin{equation}
 \max_{K,L} p\cdot AL^{\alpha}K^{\beta} - w \cdot L - r \cdot K
\end{equation}
Taking derivatives w.r.t. $L$ and $K$ yields two first order conditions and the only two unknowns are $L$ and $K$.
$$
\alpha \cdot p\cdot AL^{\alpha-1}K^{\beta} = w
$$
$$
\beta \cdot p\cdot AL^{\alpha}K^{\beta-1} = r.
$$
As you noted this implies the cost minimization equation of
$$
\frac{L^*}{K^*} = \frac{\alpha}{\beta}\frac{r}{w},
$$
but why go that way? You could get
$$
L = \left(\frac{r}{\beta \cdot p\cdot A K^{\beta-1}}\right)^{\frac{1}{\alpha}}
$$
and plug this into the first equation, yielding
$$
\alpha \cdot p\cdot A\left(\frac{r}{\beta \cdot p\cdot A K^{\beta-1}}\right)^{\frac{\alpha-1}{\alpha}}K^{\beta} = w.
$$
And then
$$
K^{\frac{\alpha\beta - (\alpha-1)(\beta-1)}{\alpha}} =
K^{\frac{\alpha + \beta - 1}{\alpha}}
= \frac{w}{\alpha \cdot p\cdot A\left(\frac{r}{\beta \cdot p\cdot A}\right)^{\frac{\alpha-1}{\alpha}}}.
$$
Clearly this is not very pretty but it is very solvable.

Note: I did not go into second order conditions but this is only a maximum if $\alpha + \beta < 1$.