Why does state-owned company so aggressive in making profit? Take China National Petroleum Corporation(CNPC) as an example, almost 86% of its stock is stated-owned. Its senior managers' promotion are related to the company's profit. And since there are investors who own 14% percent of its stock, maybe the company will face revenue pressure from them. But I think this is not sufficient to explain why they are so aggressive, I must overlook  something more important. Any ideas are going to be  appreciated.

You have to distinguish between state owned public service companies and state owned for-profit companies, the for-profit companies are really commercial companies with governments being major shareholders. I would not use the term "state owned" in case of the latter form. Governments can form joint-ventures with private and multinational companies. Governments in this instance would have a special investment management agency or just a fund that are formed to attract investments from abroad. And, in every new joint venture, the government is the main shareholder. So, in CNPC's case, government is a proper shareholder just like institutional shareholders. Since Chinese government is the main shareholder it has the majority of voting rights and would certainly want a good return on investment and they are "aggressive in making profit" as you suggest.

In the former case where the companies, if it can be called a company, are formed to provide public services (think of nationalised train companies), they are not there to make profit, but provide low cost high quality services.


Cournot equilibrium question There are two firs in the market. They produce perfect substitutes at cost $c(y_i)=y_i/3$ for i=1,2. The demand function is $p=1-(y_1+y_2)$

Consider the Cournot competition where firms simultaneously produce their respective outputs. However firm 1 has the opportunity to announce the output it will produce to firm 2 before the firms have produced any output. How can I find the equilibrium quantities?

————————

What I have done...

For basic case, Cournot equilibrium

For firm 1,

$$max[(1-y_1-y_2)y_1-(y_1/3)]$$

FOCs

$$1-2y_1-y_2-(1/3)=0$$

$$y_1={2-3y_2\over 6}$$

For firm 2,

$$max[(1-y_1-y_2)y_2-(y_2/3)]$$

FOCs
$$1-2y_2-y_1-(1/3)=0$$

$$y_2={2-3y_1\over 6}$$

So,

$$y_1=(1/3)-(1/2)(2-3y_1/6)$$
$$y^*_1=2/9$$
$$y_2^*=2/9$$

For Stackelberg eqn in basic cases

First firm is mover first
$$max[(1-y_1-y_2)y_2-(y_2/3)]$$

FOCs
$$1-2y_2-y_1-(1/3)=0$$

$$y_2={2-3y_1\over 6}$$

For firm 1,

$$max[(1-y_1-y_2)y_1-(y_1/3)]$$

$$max[(1-y_1-({2-3y_1\over 6}))y_1-(y_1/3)]$$

FOCs

$$1-2y_1-(1/3)y_1-y_1-(1/3)=0$$

$$y_1^*=1/5$$

$$y_2^*=7/30$$

I just find only Stackelberg equilibrium and Cournot equilibrium in basic cases.

But I cannot find the part that I write above. How can I solve this part?

Thank you.

Edit: (I just post the original version of my question)

To answer your question I will assume that

$\bf{A1}$ firm 2 always "trust" in the output reported by firm 1, and

$\bf{A2}$ firm 1 always "keep its word" if announces an output.

Without $\bf{A1}$ and $\bf{A2}$ I think you need more information to solve the game.

Based on how the question is written, at the first period firm 1 has the opportunity to whether announce or not its output. So, lets say that firm 1 must to chose between $\{A, NA\}$ ($A$ for announce and $NA$ for not announce) at $t=1$.

Then, if firm 1 chose $A$ the resulting game will be the same as point i) of your question. In the same way, if firm 1 chose $NA$ the resulting game is the point ii) of your question.

Finally, you only need to obtain the utility of firm 1 for both cases to know what is the best action for firm 1 at $t=1$. The subgame perfect equilibrium must to include the action of firm 1 at $t=1$ and the quantities of both firms at $t=2$.

PS: In point ii) you have a mistake, the Cournot game is symmetric so there is impossible to get different quantities in a Nash equilibrium. (sorry for my bad english)


Theories about how lack of supply can feed into lack of demand Are there any economic theories supporting the possibility that lack of economic supply can result in a reduced economic demand?
As an example, let's say a product exists, an electronic product let's say.  Due to supply chain shortages, suppose the end  product cannot be effectively created in large supply.  An example is how coronavirus has impacted the supplychain and thus product manufacturing.
I'm searching for theories that may lead demanders to decide to back away as a result of the lack of supply.  Perhaps the unit cost may be the same on the product, but demanders may have more difficultly in finding the product, and realize that there is an additional hassle to obtain said product.  This effectively amounts to an additional marginal cost on top of the quoted price on the product.

Lack of supply of a network good - a good associated with a network effect - may result in lack of demand for that good.  The greater the number of people who have or use such a good, the greater its value to any one person.  Hence a lack of supply of such a good to some people will probably reduce demand from those to whom it is available.  For example, if there were a temporary lack of supply of phone use in city A (eg due to technical connectivity problems) making it impossible for people to make or receive calls, then demand for phone use in other cities might be expected to reduce by the amount of calls that would normally have been made to or from city A.  More fundamentally, some goods with the potential to create network effects may never be supplied and used in sufficient quantity to establish such effects, so that demand for them remains minimal (even though it might, if supply had been large enough and other circumstances such as first-mover advantage had been favourable, have become very large).
Possibly not what you had in mind, but if X and Y are complementary goods, then lack of supply of X can result in lack of demand for Y.  For example, a shortage of pasta might result in reduced demand for pasta sauces.  Or closure of sports centres (eg due to Covid-19) might reduce demand for sports clothing.


What happened to the money multiplier after QE? QE meant that the monetary base increased dramatically, yet we don't see such a large reaction in money supply (I am thinking of M4 for the UK) nor in inflation. Why is this money not being lent out? Are banks keeping large excess reserves for fear of a crash?

I know this is a quite common question but I still don't understand what the accepted answer is here.

You may want to look at the following paper by Cukierman (2017). The author compares inflation for a period of identical base expansion in post Lehman US and hyper-inflationary Germany in the 1920s. They mention exactly what you said, that banks have increased their reserves at the Fed. In the US this has not translated into a heightened demand for goods and services as was the case in Germany. Furthermore the author cites fundamental differences in the economical and institutional circumstances as a reason for the difference in transmission. I suggest reading the paper for a more in depth explanation.


Examples of how economists come up with a production function for a firm? I am a young man studying economics on his own. In every microeconomics text I find, they teach you what a production function is, but they never show examples (or practice problems) on how one comes up with a production function of a firm on his/her own. Even in practice problems, a production function is already given. I can never find practice problems that help you practice finding an actual production function (ex: A firm has 3 ovens, 2 computers, 10 special extractors, and 15 workers in an olive oil plant. Find the production function for this firm).

I personally want to see how an economist makes up his/her own production function of a firm he/she is observing. I know this takes some sort of mathematical ability. I personally am not that good at modeling real life situations into functions yet (I am practicing that as well). This is why I want to practice this particular skill.

Can anyone please show me examples of this? Does anyone have any good websites or books that show great examples of this, along with practice problems? I have found nothing in almost every book/website I've found. Please help. Thank you.

The "production function" in economics and econometrics is not an engineering concept. It is a highly abstract construction where the myriads of inputs used are lumped together in very few categories: "capital" and "labor" being the most usual ones, while sometimes inputs critical to an industry like say "Energy" may be singled out and appear in the production-function formulation. At such an abstract level, it is easily implementable without the need to go into particulars of each firm.

Moreover, in many if not most of the cases, what is modeled is the production of value added, not the total output (either in quantities or in values).

Applied work commissioned by specific (usually very large) firms, and also, applied research into "adjustment costs" or what is called "insider econometrics", may go some steps towards creating much more complex production functions, that essentially try to take into account the actual physical relations and interactions between the various inputs. But these are very specialized areas -remember that the Economics discipline ultimately is about markets and economic systems -the "individual" (firm or person) level is researched mainly in order to have sound foundations for the aggregation that follows.


In log-linearized New Keynesian model, what do $Y$, $Y_t$, $y_t$ actually mean? This may be a weird question, but I am unfortunately confused by terms. Let us suppose log-linearized New Keynesian model as suggested by Gali here: http://crei.cat/people/gali/pdf_files/monograph/slides-ch3.pdf

My first question is, apparently steady value $Y$ is assumed to do log-linearization of $Y_t$, output, but is this $Y$ a constant value, or the whole steady output path? Equivalently, is $Y$ about how output will evolve if $Y_t$ evolves without stochastic factors and errors according to long-run natural rate?

My second question, related to the first question, is whether $Y_t$ refers to total output or normalized output. That is, if economy has positive output growth rate, will $Y_t$ grow? Or is it normalized output that does not change without stochastic elements?

My third question is, what $y_t$ actually mean. As I understand it, it is just $\log Y_t$. Is this correct?

The fact that consumption euler equation exists seems to support the intuition that $Y$ is steady output path, not constant steady value, as real interest rate is often positive for economy. My confusion all rises from here, and I am not sure if this is correct understanding.

The following post explains in a somewhat easier way what exactly is happening when we log linearize a model.

http://economictheoryblog.com/2012/06/22/latexgx_t/

Going through the provided example should make it clear what the single steps are.


How do I choose the correct model for a regression? So the central question of my project is to what extent does a country's level of export contibution towards GDP (i.e. exports as a % of total GDP) affect its GDP growth.

I'm comparing this hypothesis for two different countries, China and India.

I have annual time-series data for exports as a % of total GDP for both countries and their GDP growth for the last 30 years, but I have no idea how to choose a correct model to begin to anaylse it with.

I was thinking of pehaps using Autoregressive model one, any thoughts? If this is correct then I how would I begin to model it using my data?

Plot things. Do the raw data look like they have cycles? (I assume so, given that you are talking about economic data and want to use an autoregressive model.) Next, try to get the order of the autoregressive model by taking differences until the cycles disappear. This can also be tested formally using (a version of) the Durbin-Watson test. There will likely be further patterns. What do you see? Does the variability stay constant over time (likely when you have percentages) or is there a tendency for the variability to grow? What smooth (and simple) function follows the general distribution of the data? Fit it, assess the fit, repeat with different functions, until you have something reasonable.

Model selection is far more of an art than a science, and there often are several reasonable models that you can use.


Can the stock market grow faster than GDP indefinitely? My understanding of conventional wisdom is that average long-term growth of advanced economies is expected to be significantly lower than the average long-term growth of the value of the largest firms in the stock market (e.g. the total value of the FTSE 100).  I also accept Kenny LJ's argument that GDP (and the stock market) can grow exponentially, indefinitely.

But if the largest firms in the stock market, taken as a group, always grow faster, doesn't that suggest either

an unsustainable trend, or
a fundamental restructuring of the economy is in progress?

Is there any limit to how many times larger FTSE 100 firms could be than the output of the entire economy, for instance?

EDIT: To be completely clear, I am not using the FTSE as a proxy for the wider stock market or private sector more generally, so I'm pretty sure survivor (or selection) bias is nothing to do with the answer.  I am literally asking about the size of the group of the largest $x$ companies listed today relative to the total economy.

There are trends that has allowed stock markets in advanced economies to grow faster than GDP for a long time:

Branching out abroad. This gives access to faster growing markets in developing countries. This trend will end when all countries are advanced economies.
Fewer private companies. This trend will end when most of GDP is generated by companies listed on the stock market.
Multiple Expansion. Reduced interest rates and increased money supply can reduce returns expectations and increases PE multiples. This trend has worked both ways through time but for the last 10-20 years it has generally acted in an expansive capacity.

In addition, there is a way for the stock market to generate a higher return  without an increase in stock market value:

Dividends. Dividends increase stock market return without increasing its value. Though not really considered growth, dividends are often included when comparing stock market growth with GDP growth.


Is inequality good for growth? I was reading about the debate about income inequality and I would like to know your opinion on the matter.

Basically, the positions are two: the classical one, which states that inequality is good for growth because it fosters innovation and investments and the most recent one, the one of IMF and OECD, which states that inequality causes a decrease in human capital investment among lower-income households, which prevent them to become earners or consumers in the future.

I read that both are supported by empirical evidence but there is skepticism about them (e.g. Chris Giles on FT).

What's your idea on the topic? Do you have any suggestion about literature/articles I can read about it?

It depends upon how and why the inequality exists.  Equality can also be bad for growth.

In the old Soviet system, most people were approximately equal and this was done by the use of police power.  It did permit rapid industrialization but then stagnated.  Not all innovation was good either.  Look at the Trabant.  An amazing vehicle.  Watch https://www.youtube.com/watch?v=No1-4GsQa-g

The US used to be very flexible.  That was good for growth.  If you were born poor, then you could become rich.  If you were born rich, then you could become poor.

Some income inequality is due to changes in technology and required education.  Some are due to legal changes that transfer resources from ordinary people to the rich.  It is now very difficult to become poor if you are rich and to become rich if you are poor.  That damages the incentives to build a business to grow the system.

Inequality is neither good nor bad, by itself.  It is a consequence of something else.  It is smoke; it is not fire.

There are several papers by Peter Temin that are very good on the topic, but he has quite a few and I don't know exactly what you are searching for or have already read.


Understanding the linear expenditure system I've been reading up on the uses of the linear expenditure system via a the papers which apply them.

One paper which gives a good introduction to the topic in a paper called:
Modeling household behavior in a CGE model:
linear expenditure system or indirect addilog? by Paul de Boer. (pages 5-6) It goes through the basic theory behind the linear expenditure system fairly well.

However in terms of working out the mathematics myself i'm having some difficulty, below is an abridged version of what is found in the paper.

  To derive the linear expenditure system we have:
  $$\max \ \ \ \ \ \ U(x_1,...,x_n)=\sum_{i=1}^n\gamma_i ln(x_i-a_i)$$
  $$s.t.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m=\sum_{i=1}^n p_ix_i$$

  After computing this we should get our linear expenditure system which is defined as:

  $$p_ix_i=p_ia_i+ \gamma_i(m-\sum_{j}p_{j}a_j)$$

  where
  $(m-\sum_{j}p_{j}a_j)$ is  supernumerary or discretionary expenditure.

This is not what I got.

when solving this I end up with this (like how it was previously derived on Linear Expenditure System of Demands, Derivation Help):

$$\sum_{i=1}^nx_i=(a_1,...,a_n)+\left(\frac{\gamma_1}{\lambda p_1},...,\frac{\gamma_1}{\lambda p_1}\right)$$

How do i get the result like that in this paper?

For every $x_i$ separately you get

$$\frac {\gamma_i}{x_i-a_i} = \lambda p_i \tag{1}$$

Re-arranging,

$$\frac {\gamma_i}{\lambda} = p_ix_i - p_ia_i \tag{2}$$

Sum over $i$

$$\frac 1{\lambda}\sum\gamma_i = \sum p_ix_i - \sum p_ia_i \tag{3}$$

Re-arrange to solve for the optimal $\lambda$ taking into account the restrictions

$$\sum p_ix_i = m ,\;\;\; \sum\gamma_i=1$$

$$\lambda^* = \frac {1}{m-\sum p_ia_i}\tag {4}$$

Insert into $(2)$, re-arrange and you 're done.


Compound interest ($r$ in terms of $\frac{dA}{dt}$) Suppose the amount of money in bank account that is compounded annually is given by $A(t)$. The annual rate of interest is $r$. Find a relation between $\displaystyle\frac{dA}{dt}$ and $r$.

My attempt:

$A(t)=A(0)(1+r)^t$

$\displaystyle\frac{dA}{dt}=A(0)(1+r)^t\ln (1+r)$

Dividing, we get

$\displaystyle\frac{1}{A}\frac{dA}{dt}=\ln(1+r)$

However, I think the answer should be $\displaystyle r=\frac{1}{A}\frac{dA}{dt}$.

What am I doing wrong?

You are not doing anything wrong. Since
$$
\lim_{r \to 0} \frac{\ln(1+r)}{r} = 1
$$
(to see why: use l'Hôpital's rule or Taylor approximation) holds, for small numbers $r$
$$
\ln(1+r) \approx r.
$$


Would this hypothetical firm exit in the long run? I'm trying to work out the following problem for my microeconomics course, using the standard method we have been taught.

Problem: Output is a function of inputs $x_1$ and $x_2$, $y=x_1^{1/2}x_2$. Total cost is a function of inputs $x_1$ and $x_2$ and their corresponding prices, $w_1=3$ and $w_2=2$, $c=3x_1+2x_2$. The price at which output can be sold is $p=6$. What is the profit maximizing level of $x_2$ for the firm to use in the long run?

Attempt: In the long run, the profit maximizing level of $x_2$ is found by solving $p\times MP_2=w_2$. That is, $6x_1^{1/2}=2$. But, $x_2$ is no longer a variable in this equation! This leads me to believe the firm will exit, meaning $x_2=0$, but I cannot think of how to relate the economic theory to the mathematics.

The firm will not exit: for example inputs $x_1=1,x_2=1$ lead to $y=1, c=5$ and profit $py-c=1$

In general profit is $6x_1^{1/2}x_2-3x_1-2x_2$

As you found with $x_1 =\frac19$, profit is $-\frac13$ which is independent of $x_2$
But with $x_1 \gt \frac19$, profit is an increasing function of $x_2$ and is positive when $x_2 \gt \frac{3x_1}{6\sqrt{x_1}-2}$
For example with $x_1 =1$, output is $x_2$, profit is $4x_2 -3$ and is positive for $x_2 \gt \frac34$
For example with $x_1 =4$, output is $2x_2$, profit is $10x_2 -12$ and is positive for $x_2 \gt \frac{6}{5}$

So there appears to be no  profit maximizing level of $x_2$ because (providing $x_1 \gt \frac19$) profit is an increasing  function of $x_2$. So a profit maximiser will want to increase $x_2$ without limit.  Your partial derivative method did not find this infinite solution

For what it is worth, given a particular $x_2$, the optimal value of $x_1$ seems to be $x_2^2$ making the profit $x_2(3x_2-2)$ which is clearly positive when $x_2>\frac23$ and then again an increasing function of $x_2$


MLE using multivariate normal distribution I am reading econometrics lecture notes of a Japanese professor. To explain MLE estimation he uses multivariate normal distribution.

As you can see, the first line is the loglikelihood function and the second line is proportional to the first line equation. I cannot get my head around this. How and why the first equation can be written as the second one using trace of a matrix. I heard about "trace trick" and this must be the application of this trick, but I need help to understand it and possibly use it.
Thanks

The "trick" you are referring to is a property of the trace of a product of matrices, namely

$${\rm tr}(ABC) = {\rm tr}(BCA)$$

assuming the dimensions are conformable of course.

Now, note that the dimension of $(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)$, for each observation, is $1 \times 1$. So trivially,

$${\rm tr}\Big[(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)\Big] = (x_n-\mu)^T\Sigma^{-1} (x_n-\mu) $$

But since also, from the mentioned property of the trace,

$${\rm tr}\Big[(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)\Big] = {\rm tr}\Big[\Sigma^{-1} (x_n-\mu)(x_n-\mu)^T\Big]$$

combining one gets the alternative expression for the mutlivariate normal log-likelihood.


How do debt-free governments control the money supply? From what I read in my introductory macroeconomics textbook, central banks can control the money supply by selling government bonds in the market (decreases money supply), or by buying government bonds from the market (increases money supply). Countries that have no debt have no need to issue bonds. In the case of these debt-free countries, how do their central banks control the money supply?

There are several ways how central banks can still control money supply even if we assume the government does not issue any bonds.

First, there is the discount rate at which other banks can borrow money form the central bank. The nominal interest rate throughout the whole economy depend on the discount rate as the rate at which banks are willing to borrow to households and businesses also depend on the rate which they can borrow from the central bank.

This is what the Norway (a country with negative net debt) is doing. According to the Norges Bank

  Instruments and implementation
  The main monetary policy instrument is the policy rate, which is the interest rate on banks' deposits in Norges Bank up to a specified quota.

  The policy rate is set by Norges Bank's Executive Board. Normally, the Executive Board makes monetary policy decisions eight times a year. The Executive Board meetings where monetary policy decisions are made are called monetary policy meetings.

Second, there is the reserve requirement. In principle central banks could increase money supply by decreasing the reserve requirement which would increase the size of money multiplier  $1/RR$ where $RR$ is the reserve ratio. This is quite crude way ho to conduct monetary policy though and its quite uncommon.

Furthermore, as was correctly pointed out in the comments even government with no net debt might still issue bonds, so the assumption that they do not is not necessarily correct. Hence in practice that option is still there at least in principle - although as correctly pointed by @BrianRomanchuck your question excludes it, its worth while to keep this in mind.

Hypothetically, there could also be other ways how this could be done but they are not used too much in practice. For example. there are also precedents for central banks buying up private bonds but its not common. Central banks could in principle also directly change the monetary base with something like helicopter money (depositing money directly to private accounts and hence increasing the monetary base - or in cases like ECB where central bank also has power to directly coordinate printing of the money by literary printing them and throwing them out of helicopter), but since your question is more about how its actually done not how hypothetically it could be done I wont go into them.


Contribution to change of a ratio with 3 terms I have a micro dataset of firms' financial liabilities for 15 years. I would like to plot the contributions to change in leverage, decomposing the single contribution of bank loans, bonds and equity variations.
Leverage is built as:

Leverage=(Loans+Bonds)/(Loans+Bonds+Equity)

Therefore contributions to growth are positive if financial debt (Bank Loans or Bonds) increases or if its sum with capital decreases and vice versa.

I would be able to decompose with logarithm a simple ratio a/b but I have no clue on how to do it with 3 components. Any idea? Thanks

Suppose we want to find the contributions of the variables to the given function:

$$f(x,y)= {x\over{y}}$$

To find the growth rates of the given variables first take the logarithm of the function $f(x,y)$:
$$ln(f(x,y))=ln\left({x\over{y}}\right)=ln(x)-ln(y)$$

And then take the partial derivative of the variable you would like to analyze:

$${\partial ln\left({x\over{y}}\right)\over{\partial x}}={\dot{x}\over{x}}$$

$${\partial ln \left({x \over{y}} \right)\over{\partial y}}=-{\dot{y}\over{y}}$$

Where $\dot{x}$ and $\dot{y}$ are the derivatives of $x$ and $y$ respectively

Thus in your case if you would like to find the contributions to growth to leverage you can simply do the same thing as above.

i.e.

$$\mathfrak{L}={{l+b}\over{l+b+e}}$$

Where:

$\mathfrak{L}=$ leverage

$l=$ loans

$b=$ bonds

$e=$ equity

Thus

$$ln \left( \mathfrak{L} \right)= ln(l+b) -ln(l+b+e)$$

you can thus find the contributions to growth by the variables involved by taking the partial derivative of the variable you wish to examine.

Hope this is helpful!!


Why do the gross return of wealth and wealth move in opposite direction? I have a short question:

Why  does the gross return of wealth decrease as wealth increases?

Why do the gross return of wealth and wealth move in opposite direction?

$$R=1+(a/w^2)$$ where a is non-negative parameter.

Per our comments above, I believe the expression you have is incorrect.  Your confusion is worsened by some poor terminology use.

Gross return can be written as $R = 1 + \frac{a}{w}$ where $w$ is initial wealth (or more accurately, the initial investment) and $a$ is new cash flow generated by the investment.  So for example, with $w = 100$ and $a = 5$, the gross return is $R = 1 + \frac{5}{100}$ = 1.05, representing a 5% return on $100.

What you call the "gross return of wealth" is, I think, just the derivative of $R$ with respect to $w$: $\frac{dR}{dw} = -\frac{a}{w^2}$.  The question is, why is this negative?  Because what this quantity tells you is how your gross return changes when your initial investment grows, holding the new cash flow constant.

It really just follows from the arithmetic of proportions. Return is a proportion while wealth is a number.  As wealth $w$ increases, the return from a new cash flow $a$ is reduced, because it is a smaller proportion of a larger whole.  So in the example above, if we double our investment to $w = 200$ but we keep $a = 5$, then the return is $R = 1 + \frac{5}{200} = 1.025$.  Because we had to invest more to get the same amount of new dollars, our return is smaller.


Does the contract curve always have to connect the initial points on an edgeworth box? Why or why not? What if the contract curve goes out of bounds? In that case, do I assume it superimposes itself on the axes it is closest to?

I hope this was clear. Not sure how to explain this other than visually...

The contract curve is the locus of Pareto optimal points in an Edgeworth box. What we get from that:

To be P.O., an allocation must be feasible. So, the contract curve does not extend beyond the edges of the box (opposed to indifference curves, which we can draw as extending beyond the edges of the box) because points outside the Edgeworth box are not feasible.
Because the Edgeworth box does not allow for destruction/disposal, we know that the initial allocation point will account for all resources in the economy. This may make you think that the initial endowment should be part of the contract curve. However, this is not true. That is, the initial endowment point is not necessarily part of the contract curve because there are easily imagined cases where trading happens without making either agent worse off and makes at least one agent strictly better off.
Yes, there are certain circumstances that cause the edges of the box to form part or all of the contract curve. If you want to explore the idea more I suggest you spend some time reading about what happens when one agent has (or both agents have) linear indifference curves.


What happens to a country with lot of debt? As far as I know, debt is a way to get hands on extra money which will burden the future governments.

What if every successive government keeps adding to the national debt?
Isn't it like free money? You can win by developing the country and keep adding to the debt as well... maybe forever?
The US has a debt of $18 trillion. How has it affected United States? Why should the government even bother to repay?

If a government owes money to domestic or foreign entities and does not repay that debt then the government's credibility — just like any other borrowing entity's credibility — as a reliable economic entity takes a hit. As a consequence when it wants to borrow more money in the future, the lending entities either charge a higher interest rate to cover the risk or refuse to lend to avoid the risk altogether.

Here's the only difference — governments can print their own currency unlike other entities. But too much currency in circulation will cause inflation, which is bad for the economy. So it is in the government's own interests to repay its debt, as it is to borrow money in the first place.


What is the difference between comma and plus in the bellman equation? Bellman equation:

$V(x) = max \{F(x,y)+ \beta V(y)\}$

$V(x) = max \{F(x,y), \beta V(y)\}$

When to use the plus and when to use the comma?

Would you mind give me an example to explain such difference?

Thank you very much !

You should be careful about the way you wrote your two equations because the $\max$ operators are different. In the first equation, you maximize an objective function on a set, meaning you look for the best value of $y$ given that $y$ belongs to a known set. You did not define this set. In the second case, you choose the maximum among two values. Contrary to the first equation, $y$ is predefined and is not a parameter to find given your notations.

If I abstract from the notation mistakes, I think you want an example to understand the difference between a classical cake-eating problem (your first equation) and a stopping-rule problem (probably your second equation).

Consider the following framework. Time is discrete. There is one good in the economy.
We are interested in the problem of a consumer.
At each period $t$, the consumer has access to a stock of good $x_t$. She faces a standard inter-temporal saving problem; she has to decide how much to consume, $c_t$, and how much to save for the next periods, $x_{t+1}$.
The consumer enjoys a flow utility $u(c_t)$ and discounts future consumption with a factor $\beta$.
To close the model, a saving technology must be defined. We can consider two different cases:

(i) There is an interest rate $r$. The consumer can freely withdraw any amount from the saving account.
(ii) There is an interest rate $r$. The consumer, however, cannot withdraw a quantity of good without closing the saving account.

Let's write the optimization problem in the two cases:

(i) $$V(x_t)=\underset{0\leq x_{t+1}\leq (1+r)x_t}{\max} \{u((1+r)x_t-x_{t+1})+\beta V(x_{t+1})\}$$

(ii) $$V(x_t)=\underset{x_{t+1}= (1+r)x_t \text{ or } 0}{\max} \{u((1+r)x_t-x_{t+1})+\beta V(x_{t+1})\}$$
Consumption $c_t$ is equal to $(1+r)x_t-x_{t+1}$. We can make an analogy with what you wrote. My $x_t$ and $x_{t+1}$ are your $x$ and $y$, and $F(x,y)=u((1+r)x-y)$.
In the first case, the consumer has to choose the amount to save $x_{t+1}$ in the interval $(0,(1+r)x_t)$. In the second case, she can only choose whether to keep on saving, $x_{t+1}=(1+r)x_t$, or to close the account, $x_{t+1}=0$. By normalizing $u(0)=0$, the second case can write:

(ii) $$V(x_t)=\max \{u((1+r)x_t),\beta V((1+r)x_t)\}$$

In the first problem, the consumer eats a share of the cake (which is expanding at a rate $r$) and keeps the remaining for the next periods to maximize her inter-temporal utility. In the second problem, the consumer has to decide when to eat entirely the (growing) cake, which defines the stopping rule.


Hypothetical price of a city (e.g. Munich) At lunch we pondered, how much a hypothetical investor would have to pay for a whole city.

What would the combined value of all real estate in a large city (like Munich) be?
Is there a way to estimate this even halfway reliably?

Of course there are buildings, that won't ever be for sale, like public buildings or cultural heritage. These should probably be excluded from the computation.

A rough estimation technic consists in (i) getting the half of the Gross Urban Product (GUP), thinking of it as a perpetual yearly net income flow (i.e. a profit flow) attributable to the entire city as such, and (ii) dividing it by the average cost of capital that is immobilized locally, say, $2\%$, minus the GUP's growth rate, say, $1\%$ (both real and assumed constant over an infinite horizon)

In the case of Munich it would be

$\frac{1}{2}\frac{GUP_{2018}}{.02 - .01} = 105\text{E}^{11}€$

$10.5$ trillions euros, actually calculated using the DD Model.

Note that public infrastructures are not excluded from the above valuation since the city is considered as an indivisible whole.


Is it more costly to raise cattle for beef or raise fish in aquacultures? I can't find any sources comparing the cost of the two methods.
Specifically, I want to know if its more economically efficient to raise fish as a food source rather than beef - not counting government subsidies.
My intuition says raising fish is cheaper since you can raise thousands of them at a time, but I don't know what costs are associated with maintaining an aquaculture environment (water temperature, chemical/pH balance, etc). I know it costs a lot of money to put a pound of beef on the table (without subsidies helping lower that cost), but i can't find a similar cost for aquacultured fish.
Also i'm not sure if this is the best stackexchange community for this question but i couldn't find a better one that makes sense for this question; if there is a better one please let me know!

Cost of production is somewhat difficult to pin down, since much of that information is private.  However, consumer costs are widely available: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2954450/
I have managed to find a comparison of beef, pork, dairy, poultry, and eggs to plants in terms of protein, and broadly beef is the most inefficient of those (See Figure 3): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5899434/
This suggests (but does not prove) beef is probably the more inefficient of the two.


What is a digital platform? In context of European research projects, they invest into development of digital platforms.

Here is a source to give a definition  but not really, and every time people refer directly or indirectly to Apple, Google and Facebook.

Somehow, they try to replicate the business model, but not culture.

But, the question is: is there a formal definition of what digital platform is?

I am not aware of any definition, and believe me, lots of people would like to have one - mostly for regulatory purposes. There are attempts in many countries to regulate digital platforms, but without good definition, you can't create a good legal regulation.

Any definition I have seen so far is a list of features, that is, however, not exhaustive - it would be a good definition if we could say, "it is a platform if it has those 8 features," but then you will find one that doesn't have all of them, and you find something that has all of them, but you won't call it a platform...


Are there any examples where fractal geometry is applied in economics or finance? Can fractal geometry give better insights into the working of the financial markets or give rise to better financial models? If so, are there any existing models?

Fractals applied to finance was a research area of Benoît Mandelbrot. He wrote books and research articles in the matter. Looking at his work could be a good way to start with.


Term for when using a single number is used instead of a function? A cost per unit could be 20, but it could also be 20 + 2q (in the cast of the latter, the cost per unit is a function whose result increases with each unit produced).
What is the term to describe when a single number is used where we could also expect a function?
I have in mind 'scalar', but I'm not sure if that's the best term.

A cost function can be constant (in $q$), but this term is not particular to economics.


AR 1 Process simulation I wanted to use STATA for a kalman filter simulation and wanted to go about it the following way

1. Generate iid error terms

$$
{e_t}, {n_t}, {u_t}
$$

2. Generate three AR1 process like so

$$
             \beta_t  = a*b_{t-1} + u_t
$$
$$
             x_t = c_0 + c_1*x_{t-1} + n_t
$$
$$
             y_t = b_t*x_t + e_t
$$

 3. use Kalman filter to predict b_hat

I have been able to use rnorm() to generate the iid error terms, but I am stuck at the second step. I know how to run a state space model to get the kalman filter estimates but AR1 has really tripped me up.

Since you are stuck at just estimating AR(1) models I can show you some example code for estimating the equations you gave. Keep in mind that I did not set slopes: $a$ and $c_1$ in the random walk and  random walk with drift. Furthermore, the example code is in Python, however it should be self-explanatory and reproducible in STATA.

import numpy as np
import matplotlib.pyplot as plt

# Setting sample size
sample_size = 100

# Vectors with variables of interest
b = np.zeros(sample_size)
x = np.zeros(sample_size)
y = np.zeros(sample_size)

# IID random errors (normally distributed)
u = np.random.normal(size = sample_size)
n = np.random.normal(size = sample_size)
e = np.random.normal(size = sample_size)

# Drift
c_0 = 2

# Simulating each series
for t in range(sample_size):
    b[t] = b[t-1] + u[t]
    x[t] = c_0 + x[t-1] + n[t]
    y[t] = b[t] * x[t] + e[t]

# Plotting the results
plt.rcParams["figure.figsize"] = [10,10]

plt.subplot(321)
plt.plot(b)
plt.title('Random walk: $b_t = b_{t-1} + u_t$')

plt.subplot(322)
plt.plot(x)
plt.title('Random walk with drifft: $x_t = c_0 + x_{t-1} + n_t$')

plt.subplot(323)
plt.plot(y)
plt.title('$y_t = b_t * x_t + e_t$')

plt.show()


Unrestricted domain vs complete Arrow's impossibility theorem states that no social choice rule satisfies a certain list of desiderata. Amongst these are completeness and unrestricted domain. Could someone please explain the difference?

You could have completeness and give up unrestricted domain. A non social choice example:

The relation "larger than" or $\leq$ is not complete over the complex plane $\mathbb{Z}$, but it is complete if it is restricted to a subset of it, real numbers.

Similarly you could have unrestricted domain and give up completeness.

E.g. majority voting is not complete if you allow unrestricted domain.

Therefore these are separate properties. You can argue you need both, but perhaps you are willing to part with one or the other. Personally I agree that giving up completeness rather than unrestricted domain seems strange.


Why, theoretically, is economic growth exponential? Despite the experimentally well-observed exponential nature of GDP growth, I've never actually seen a theoretical reason or "prediction" why it should be so.
(Even if the answer is "actually, it isn't exponential, growth rates increase/decrease ever so slightly", the question "why is it almost exponential?" still remains.)
Similarly, what is the standard economic explanations for the big "breaks" in economic growth rates ~10,000 BC and ~1700 AD? Historians explain it to be the result of specific technologies, but this seems to be a post-hoc explanation to me (i.e. you couldn't actually predict the effect on GDP growth from just information about technological changes -- is there any theoretical reason why "agriculture" and "the steam engine" would create a break in economic growth rates but "the assembly line", "3D printing", "computers" or "women entering the workplace" wouldn't?). What is the economists' explanation? Changes in political/economic systems?

There are two main theories of economic growth, the Solow-Swan growth model and Romer endogenous growth model. Both of these models allow for exponential growth.
The economic output (GDP) can be modeled through standard Cobb-Douglas production function. For example, a classical production function is given by:
$$ F(K,L) =  AK^{\alpha}L ^{(1-\alpha)},$$
where, $F(K,L)$ is the output (which we can measure by GDP), $A$ is a stock of technology, $K$ stock of capital and $L$ is a stock of labor.
In the standard Solow-Swan model we can show that in the steady state the GDP will grow at the combined rate of growth of technology and population $(g_A + g_L)$, and GDP per capita at the growth rate of technology $g_A$ (I omitted all derivations of this result for brevity but you can see it in Romer's Advanced Macroeconomics chapter 1). Hence Solow-Swan model offers solid theoretical reasons why growth should be exponential in approximately last 200 years.
According to Gordon's The Rise and Fall of American Growth the technology in last few centuries grew extremely rapidly and had more profound effect until the recent decade which he argues can explain both the unprecedented U.S. growth since industrialization and recent decline of U.S. growth. Gordon's work focuses on US but cited therein you will find sources that discuss other countries. Furthermore, population also grown rapidly in last few hundred years so if you care only about GDP not GDP per capita the growth of world's population is also part of the story.
Next we can also look at growth through the lenses of the endogenous growth theory. The main distinction between the above Solow-Swan exogenous growth model and endogenous growth model is that in the latter the rate at which technology itself grows is not just exogenously determined by outside forces (i.e. the inevitable march of human knowledge) but also endogenously determined within the model depending on how many resources are devoted to the discovery of new knowledge.
As with the Solow-Swan model I will skip any formal derivation, which you can review at your leisure in Romer's Advanced Macroeconomics chapter 3. In this model the relationship between technology and economic growth is more complex and depending on what parameters you assume for the model (especially how new technology is produced) the theoretical predictions can be virtually identical with Solow-Swan model or they can also suggest various positive feedback loops that can accelerate growth. For example, in one version of endogenous growth theory if world's population increases the number of people devoted to producing new knowledge (i.e. scientists) also increases further boosting growth. This offers some additional explanation in sense that it also tries to explain where the technologies come from instead of treating them as just exogenous shocks to the system.
The Solow-Swan model and its predictions have been rigorously tested in the past and especially the newer augmented versions of the theory (e.g. Mankiw, Romer and Weil (1992) augmented Solow model) are empirically quite successful (see the discussion of this in Romers's Advanced Macroeconomics). The endogenous growth theory is not well empirically tested, as it is difficult to do so, but it is becoming more popular. Paul Romer even recently earned Nobel prize for his contributions to endogenous growth theory.
Furthermore, aside from the above growth theory when we talk about economic development from historical perspective it is also important to acknowledge the role of institutions. One reason why prior to industrial revolution so little progress was done was that most countries had practiced extractive institutions. For example, countries without secure property rights where warlords can just take any possession of their subjects at will create environment in which people have no incentives to spend time creating new technologies, as doing so takes a lot of time and effort and without having secure property rights why would anyone bother if their inventions can easily be expropriated. Hence as argued by Acemoglu and Robinson (see their book Why Nations Fail), technological and economic growth could really start to grow rapidly only once proper inclusive institutions were adopted (i.e. secure property right, ban on slavery/serfdom/unfree labor, free entry into occupations, lift of arbitrary restrictions on commerce etc.).
Hence those historians are more or less correct that the growth can be attributed to technology, with a necessary caveat that having proper institutions is a pre-requisite for sustainable growth. The historians might explain the growth in post-hoc fashion because they might have no economic training but the above mentioned economic models offer both clear mechanisms of how growth depends on technology and offer testable predictions that can be applied equally to past, present and future. Also it is possible that historian's are actually aware of this (at least economic historians definitely are) and you just misread them - since you never provide any citation for your assertion it is hard to judge.
In addition the assertion in your title that growth is exponential is not generally valid. It has been so in recent history, but it wasn’t always exponential and it might not be exponential in the future if technological progress slows or countries start adopting worse institutions or if you buy into endogenous growth theory if people start investing less in new technologies and as population growth halts or even if it declines in the future.


Why does supply (and demand) shift left and right as opposed to up and down? I've wondered about this for a long time. In the past I rationalized it by saying that up and left were the same (or at least have the same overall effect), and that thinking in terms of left and right works for increase/decrease in both supply and demand curves. But things like taxes raise prices which is most intuitively viewed as an upward movement (although saying left and right are useful regardless). Then when I encountered Keynesian aggregate supply (which makes it obvious that left and up are not interchangable), I was astounded when I learned that decrease in supply, including supposedly taxes, actually moved the curve left, not up. Shouldn't taxes exist which increase price levels of all firms in order to pay for them, moving (Keynesian) aggregate supply up because whatever quantity of goods firms produce they have to set prices higher to pay for the cost of taxes?

Both ways of thinking are correct. They are two equivalent and complementary ways of thinking about the same thing:

Shift left (right): At each price, the quantity that producers are willing and able to supply has gone down (up).
Shift up (down): To get producers to supply each quantity, the required price has gone up (down).

Example where the supply curve shifts left or (equivalently) up:


Labour Productivity Growth and Reallocation of Labour Using a shift-share methodology, labour productivity growth can be arise from productvity growth within a sector (within effect) or reallocation of labour to sectors with higher productivity (reallocation effect).

1)

However we know that Productivity = Real Value-added / Employment.

So won't an increase in employment in a sector result in a fall in the sector's productivity and hence overall productivity?

2) Based on
http://www.ilo.org/wcmsp5/groups/public/---ed_emp/---emp_elm/documents/publication/wcms_143163.pdf

  employment intensity is the direct inverse of labour productivity

So does this mean the government should not employ policies to reallocate labour to higher productivity sectors?

It will be good to refer me to some paper that address these questions

Ad 1)

You make a hand waiving claim. In these scenarios making concise mathematic statements helps improving your arguments.

Denote employment as $L$, define real value added as $F(L) - wL$, and productivity as real value / employment:

$$ A = \frac{F(L) - w(L)}{L}$$

The labor share should be some constant $\beta$ of total value added. Without going deep into mathematics, straight away assume $w(L) = \beta\alpha F(L)$. Hence, we have that

$$ A = \frac{F(L) (1 - \beta\alpha)}{L}$$

Now, what happens if $L$ increases? Let $L_2 > L_1$

$$ A_2 - A_1 = (1-\alpha\beta)(\frac{F(L_2)}{L_2} - \frac{F(L_1)}{L_1})$$

Assume $F$ to be concave. Then we have that the change of productivity, $A_2 - A_1$ is negative - as you say.

However, $A$ here is average productivity. Total productivity is $AL$.

$$ A_2L_2 - A_1L_1 = (1-\alpha\beta)(F(L_2) - F(L_1))$$.

Independent of the curvature of $F$, for any strictly increasing production function, this will be positive.

tl;dr

Don't confuse average productivity with total productivity.
Whenever you find something weird, use algebra to make your statements precise

Ad 2)

Again, you are not being precise enough. What is the government's objective? If government is only about maximizing average productivity, the solution is to let $L\to 0$, as then $F'(L) \to \infty$ (and for Cobb-Douglas, $F'(L)$ is proportional to average productivity).

Maximizing total output implies allocating workers to the sectors with highest productivity. I do not understand how the functional relationship between employment intensity and labor productivity affects that premise. Furthermore, unless there is a market distortion, the role of any government intervention is not clear at all.


Heating oil more expensive than gasoline? I see that "heating oil", which I presume to mean No. 2 fuel oil, is more expensive than gasoline:

My understanding was that fuel oil is much easier to refine than gasoline and more of it is produced per barrel of crude than gasoline is produced. Why then is it more expensive?

I would expect the gasoline market to be more competitive than heating. Transporting heating oil into homes requires large infrastructural investment and high fixed costs, so you'd expect there to be more market power concentrated in a fewer number of firms, driving the price of it up. Creating gasoline for sure requires a lot of investment too, but not for nearly a large area I would imagine.

So basically, consider heating oil an input for a "utility" sort of company, which acts as a natural oligopolist/monopolist. I'm not exactly sure what the market looks like here in the United States, but this is my best guess. Someone else can probably elaborate on this answer or provide an alternative suggestion.


Monotone transformation of utility We have learned that any "strictly positive monotonous transformation" of utility functions is okay, as long as they preserve the ranking of choices implied by the underlying preferences.

Consider $U(x, y) = (x/y)^\alpha$

The cross derivative is $\frac{\partial^2}{\partial y \partial x} U(x,y) =
-\alpha^2 (\frac{x}{y})^{\alpha - 1}$

Now, consider the monotonous transformation $V(x,y) = \log U(x,y) = \alpha \log x - \alpha \log y$. The cross-derivative of $V$ is $0$.

The cross-derivative incorporates important information on how the ranking of choices of $x$ changes, as we change $y$. Clearly, these two different cross-derivatives cannot be generated from the same underlying preference ranking - or am I mistaken?

The preservation of ranking is over consumption bundles, not individual goods. So the one and only question is whether the ranking of bundles $(x_i,y_i),\; (x_k,y_k); \forall i,k$ is preserved or not.

If preferences are rational (complete and transitive), and continuous, then they can be represented by a continuous utility function. So if these hold and

$$(x_i,y_i)>_{pr}\; (x_k,y_k) \implies \exists \;U, U_i > U_k$$

The logarithm is a strictly monotonic transformation of any amenable to it function as a mathematical fact, irrespective of what we use the function for. So it follows that
$$U_i > U_k \implies \ln U_i > \ln U_k ,\; \forall i,k$$

and the ranking of bundles is preserved. Whether due to the transformation we lose other information or not (that we may have wished we could have kept) does not concern the representation/transformation theorem.


Formalizing unorganized Sector This is a fairly basic question. What are the real benefits of formalizing the unorganized sector? What does integrating these into the organized sector achieve other than maybe increased tax revenue for the government?
It just seems like a hassle for the people who are used to working in the unorganized sector. They have to learn new skills and spent time and energy to get used to new ways of working. I can think of labor safety to be another good reason as government's administration can be streamlined. But for some reason none of these feel compelling to change the entire system when millions of people are employed in unorganized sector. I am surely overlooking a lot of things here. Please enlighten me. Thank you for your time.

One important reason why it is better to have activity be done formally rather than informally (grey or black markets) is access to proper institutions.
Property rights, ability to enforce contractual obligations, above mentioned safety laws etc, are vital for economic activity (see Stiglitz Economics of Public Sector 3rd ed).
Some economic activity does not require institutions. You might hire someone for mowing your lawn, where you can easily observe if the job is done properly. At the same time the worker has little in stake since even if they can't force you to pay without contract they will not do business with you in future. But when it comes to more complex economic interactions which are often necessary as economic activity expands, such as establishing supply chains, having several workers at different locations etc it becomes more and more limiting if people do not have access to proper instiutions.
What even worse, in absence of government imposing institutions, gangs, mafia or local paramilitary might. For example, one reason why gangs exist in drug trade is that drugs are expensive and easily portable, so as all expensive things they attract thieves. If selling drugs was legal firms would be just able to either rely on police or in worst case scenario private security forces to defend their labs, storages or shipments. However, without access to this institutions they have rely on hiring thugs and gangsters. Moreover, in absence of access to court & legal system any disputes between drug producers about quality of drugs or payments between producers and dealers will be settled by gang conflict, intimidation and so on.
Second, you mention that formalizing sector brings in more government revenue, but then you just brush this point off as if it would be nothing even though this is crucial argument.
If millions of people work in informal economy (as you argue in your question), this seriously retards government ability to extract revenue. Even if you would happen to libertarian and reject any redistribution concerns government is still necessary to provide wide range of public goods (or in some cases quasi-public) goods such as national defense, infrastructure, law and order enforcement. If government can't efficiently raise enough revenue to properly fund these functions then whole economy will suffer. As a matter of fact provision of  public goods and access to well designed institutions,  economic development is thought to be virtually impossible  (see Acemoglu & Robinson. Why Nations Fail).
Consequently, it is vital for the country to have most of its economic activity be done in a formal way. That does not mean all grey economic activity should be completely eliminated, it would probably not be worth while to require even occasional babysitting should be only done by formally organized business, and have some space for economic activity between relatives. However, grey economic activity should not be encouraged and if as you put it millions of people are employed in informal economy then that would be a problem.
Lastly, you seem to assume that formalizing economic activity has to be costly but that is not true.  Formalizing economic activity does not mean that people there have to follow some draconian regulations. Maybe in your place government plans that, but many advanced countries will have lower regulation/administration/tax burden imposed on self-employed people or small businesses. Consequently, there is also no reason to think that formalizing informal sector should involve large costs a priori.


Non-Bayesian Mechanism Design Suppose we have a mechanism where a finite number of agents possess private information that is not drawn from a probability distribution.

The agents' types are given and fixed but agents only know their own type. As there is no probability distribution, players do not form any beliefs.

The agents' behavior is modeled as utility-maximizers and the utility functions are quasilinear in nature.

Does it make sense, from the above, to use as a solution concept the non-Bayesian Nash equilibrium (Nash, 1950)? If yes, could you please point me out to the right literature (some paper references)?

Thank you in advance.

Most prominently, it was Bob Wilson who criticized the role of common priors in game theory. Starting from the "Wilson doctrine", some work in mechanism was done in that direction. Note that this research endeavor is not hopeless: in a popular auction format, the second-price auction, it is a (weakly) dominant strategy to report your true value. That is, your belief about others' types or strategies does not matter.
Chung and Ely (ReStud 2007) have a paper on this topic:

Robert Wilson criticizes applied game theory's reliance on
common-knowledge assumptions. In reaction to Wilson's critique, the
recent literature of mechanism design has adopted the goal of finding
detail-free mechanisms in order to eliminate this reliance. In
practice this has meant restricting attention to simple mechanisms
such as dominant-strategy mechanisms. However, there has been little
theoretical foundation for this approach. In particular it is not
clear the search for an optimal mechanism that does not rely on
common-knowledge assumption would lead to simpler mechanisms rather
than more complicated ones. This paper tries to fill the void. In the
context of an expected revenue maximizing auctioneer, we investigate
some foundations for using simple, dominant-strategy auctions.

Another popular approach is by Bergemann and Morris (2005) (several papers).
However, I guess this is not so much what you have in mind, but they have to be mentioned here.

The mechanism design literature assumes too much common knowledge of the environment among the players and planner. We relax this assumption by studying mechanism design on richer type spaces.
We ask when ex post implementation is equivalent to interim (or Bayesian) implementation for all possible type spaces. The equivalence holds in the case of separable environments; examples of separable environments arise (1) when the planner is implementing a social choice function (not correspondence) and (2) in a quasilinear environment with no restrictions on transfers. The equivalence fails in general, including in some quasilinear environments with budget balance.
In private value environments, ex post implementation is equivalent to dominant strategies implementation. The private value versions of our results offer new insights into the relationship between dominant strategy implementation and Bayesian implementation.

Usually in mechanism design, there is private information to be extracted and you need to take a stand on how to model it. That could be Bayesian or not, but the classical Nash equilibrium concept is suited for full information settings. The "prior-free" mechanism design literature pushed by computer scientists is concerned with worst-case analysis.


Log log regression with fixed effects and cross elasticity of demand I have a time series of units sold, and price. I'd like to calculate elasticity of demand wrt to price and a few other variables, some of them are fixed effects.
Qp = b0 + b1 * log(Price) + b2 * Location + b3 * log(Income) + b4 * log(HouseHolds)

I am using log-log regression to calculate the elasticities, and the independent variable of interest is Price. Qp is # of units sold.
My questions are

Can I use Median Income, and Households are fixed effects for the time period in log-log regression model.

Interpretation of Log - Log regression.
1% change in price changes the units sold by b1%(coefficient of price), Ceteris paribus, all else being equal.

Cross Elasticity.
What I add a term for # of available substitutable units? How would I interpret the term?
Qp = b0 + b1 * Price + b2 * Location + b3 * Income + b4 * HouseHolds + b5 * substitutable units

I also have additional categorical variable, i.e type of unit sold. type 1 and type 2. Does it make sense to include that variable in the equation or fit a regression separately for each type? How would you interpret different coefficients of categorical variable?

I'd like to find the point where increasing the Price negatively affects units sold.

Note that you have an identification problem here because price is an endogenous variable. A way to deal with this problem would be to find an instrument (e.g. prices  of the same good in other markets as in Hausman (1996)).
Regarding your questions.

Yes (although fixed effects are probably not the right wording here).
Yes.
If you want to capture a cross elasticity effect you need to introduce the own price as well as price of substitutes into the equation. You may face a dimensionality problem due to the large number of parameters to be estimated. The effect substitutable units would be interpreted as the effect of the availability of substitutes on demand.
I would think about the status of these types. Is the same product with different attributes? Are these types substitutes? Is it a kind of aggregation? Regarding interpretation: you interpret one category with respect to another (or to a benchmark category in case of more than one category).


What is the economic meaning of a differentiated variable cost function Given the total cost function $C(Q)=Q^3-5Q^2+12Q+75$, write out a variable cost function. Find the derivative of the variable cost function and interpret the economic meaning of that derivative.
Note: the derived variable cost function is 3Q^2-10Q+12

HINT

What is the difference between variable and fixed cost? Try separating your total cost into fixed and variable cost. To do this, think about what costs the firm faces even when they have no output (let quantity produced be zero.) For a bit of intuition, think of these costs perhaps as rents paid by a firm during the initial setup stage (time spent fitting a factory with production equipment). They have a certain amount of bills to pay even though they produce nothing.

After doing this it should be a straight forward thing to differentiate and interpret.

MEANING

Let's take your example. We can see that we have a cost of 75 (let's assume American dollars) regardless of how much output we produce. So Q=0 still incurs a cost of $75. Then by definition, this is the fixed cost portion of your total cost function.

Then since any cost function is the sum of fixed and variable costs, we have that your variable cost function is $$C(Q)=Q^3-5Q^2+12Q$$

differentiating with respect to $Q$ yields: $$3Q^2-10Q+12$$

This is your marginal variable cost. Notice this is the same function achieved by simply differentiating the total cost function. This is because fixed costs are constants (and not functions of Q) and therefore disappear when C is differentiated with respect to Q.

Then we can just refer to the equation above as marginal cost

Marginal cost equations tell you how much your total cost rises when you decide to produce an additional unit of output. Notice here that making the production decision incurs a new sort of "fixed" cost.

So let's examine what happens when Q increases:

$$Q=1 \implies MC=5$$
$$Q=2 \implies MC=2$$
$$Q=3 \implies MC=9$$

and this is enough for us to tease out another important part of this function! Notice that marginal costs decrease as production increases over our initial range and then, at a certain point, marginal costs begin to increase as production continues to increase. This is a example of increasing marginal returns and decreasing marginal returns (respectively)

In general, you can picture a marginal cost curve as a differentiable nike symbol or a U where the right side of the U is drawn much higher than the left side(and where it is a bit slanted to the right This would tend to infinity as Q tends to infinity).

Alright - hope that cleared some things up. And to top it off, you can do a bit of reading about marginal cost.


Are prepaid cards, and gift cards considered a form of money? Are the followings considered a form of money

prepaid (debit) cards
gift cards (both bank-issued such as American Express gift cards, and store-issued such as Walmart gift cards) ?

Are their balances considered demand deposits, and therefore in M1?

Taking gift cards for example. They seem a form of money, but not exactly. Bank-issued gift cards are more likely to be a form of money than store-issued ones.

They can be used to buy products.
But gift cards are usually not allowed to resell, but I don't understand why they are resold on ebay and many websites if reselling them is illegal.
Cashing the balance of new or used gift cards are usually difficult. Cashing bank-issued gift cards seem more difficult than store-issued ones.
bank-issued gift cards are usually sold above their face values, while store-issued ones are usually not.
bank-issued gift cards can be used to purchase anywhere the bank is accepted (e.g. Amex), while store-issued gift cards can be only in specific stores.

Prepaid cards are definitely money by any definition— they are essentially demand deposits. An article from the SF Fed states this clearly:

  In contrast to purchases with credit cards, debit card purchases transfer money electronically from individual and business bank accounts to the sellers’ accounts. Spending with a debit card would affect demand deposits and the money supply in the same way that purchases with a check or cash does. Because a debit card transfers your existing financial assets—the financial assets that you may access with a debit card are included in the money supply.

Gift cards, in contrast, aren't generally considered to be part of the measured money supply, but they are in some sense also a form of money. They can be used as a medium of exchange; they are tied to a broader unit of account, and they're (in a manner that depends significantly on the terms of the gift card) a store of value. Rather than being liabilities of banks, as deposits are, they're liabilities of non-financial firms. As a result, gift cards are not quite as fungible as currency is; as you note, they typically trade at a discount to face value. This isn't terribly unusual for money— money during the free banking period often traded at discounts to its face value, particularly as the distance from the issuing bank increased. One of many discussions of this phenomenon can be found in Gorton (1999).


Apply Ito's Lemma to exponential martingale $\newcommand{\dd}{\, \mathrm{d}}$
Consider the exponential martingale,
$$
\xi_t^\lambda = \exp \left\{ - \int_0^t \lambda_s \dd z_s - \frac 12 \int_0^T \lambda_s^2 \dd s \right\},
$$
that is used in the statement of Girsanov's theorem (this martingale represents the Radon-Nykodym derivative $\frac{\dd \mathbb Q^\lambda}{\dd \mathbb P}$.).

Exercise 2.4 in Munk's book Financial Asset Pricing Theory deals with applying Ito's lemma to this process.

Suppose
$$
X_t = \frac 12 \int_0^t \lambda _s^2 \dd s + \int_0^t \lambda_s \dd z_s.
$$
Part (a) asks us to argue that $\dd X_t = \frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t.$ Part (b) asks, "Suppose that the continuous-time stochastic process
$\xi = (\xi_t)$ is defined as $\xi_t = \exp\{-X_t\}$. Show that $\dd \xi_t = -\lambda_t \xi_t \dd z_t$."

Informally, we can argue part (a) by applying Ito's lemma,
\begin{align*}
\dd X_t &= \left( \frac 12 \lambda_t^2 + \lambda_t \dd z_t\right) \dd t + \lambda_t \dd z_t \\
  &= \frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t,
\end{align*}
and arguing that $\dd z_t \cdot \dd t = 0$.

How do we solve part (b)?

$\newcommand{\dd}{\, \mathrm{d}}$
If we apply Ito's lemma,
then
\begin{align*}
\dd \xi_t &= -\xi_t \dd X_t + \frac 12 \xi_t (\dd X_t)^2\\
  &= -\xi_t \left(\frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t\right) + \frac 12 \xi_t \lambda_t^2 \dd t \\
  &= -\xi_t \lambda \dd z_t.
\end{align*}


What is the reason behind the demand function of a perfect complement good? so I know that usually the income curve is equal to:
$$x_1p_1 + x_2p_2 = m$$
if we rearrange this equation we get that the demand for good one ($x_1$) is equal to:
$$x_1 = \frac{m-x_2p_2}{p_1}$$

None the less in my book it is said that:

  The demand for good 1 is $x_1 = \frac {m}{(p_1 + p_2)}$

if anyone could explain me why $x_1 = \frac {m}{(p_1 + p_2)}$ instead of $x_1 = \frac{m-x_2p_2}{p_1}$ that would be greatly appreciated. many thanks in advance.

Suppose you have $M=20$ to spend on shoes. Left shoes cost $p_L=5$ and right shoes cost $p_R=5$. A bundle, $(x_L,x_R)$ consists of $x_L$ pairs of left shoes and $x_R$ pairs of right shoes.

How would you figure out how many of each type of shoe to buy? Probably the reasoning would go like this:

You only ever want to buy shoes in pairs (consisting of a left and a right shoe). So instead of asking how many left shoes and right shoes to buy, you can define a new good called "pairs of shoes" and ask how many units of this new good to buy.
The price of a pair is $p_P=p_L+p_R=10$. Thus, you can afford to buy up to $\frac{M}{p_P}=\frac{M}{p_L+p_R}$ pairs (two pairs in our example).
Since you need one left shoe to make each pair, the number of left shoes you buy is simply the same as the number of pairs calculated in step 2: $x_L=\frac{M}{p_L+p_R}$, or two left shoes. Likewise, you need to buy $x_R=\frac{M}{p_L+p_R}$ right shoes.


How does buying/selling houses affect the GDP? Suppose I buy a house and sell it off in the same year for the same price, would the GDP for that year increase by twice the value of the house, or only once, or not at all?

There is only a change in GDP to the extent there are market goods and services used in the sale and only those goods and services are counted. The actual sales revenue are irrelevant. For example, the home inspection, appraisal, brokerage fees, and, I believe mortgage closing costs, would be in GDP. If you and your sister swapped houses in as-is condition without getting the market, bank, or tax authority involved, there would be no change in GDP.


The notions of Bayes Correlated Equilibrium and Bayesian Nash Equilibrium in a game with one player I would like your help to use the correct terminology to define a Bayes Correlated Equilibrium (BCE) and a Bayesian Nash Equilibrium (BNE) in a "game" with one player. The notion of BCE in an $N$-player game has been introduced in this paper.

Let me describe the game and give the notion of BCE and BNE in an $N$-player game.

There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.

There is a finite set of states $\Theta$, with $\theta$ denoting a generic state.

A "basic game" $G$ consists of

for each player $i$, a finite set of actions $A_i$, where we write $A\equiv A_1\times A_2\times ... \times A_N$, and a utility function $u_i: A\times \Theta \rightarrow \mathbb{R}$.
a full support prior $\psi\in \Delta(\Theta)$.

An "information structure" $S$ consists of

for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.
a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.

A decision rule of the incomplete information game $(G,S)$ is a mapping
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

A mixed strategy of player $i$ of the incomplete information game $(G,S)$ is a mapping
$$
\beta_i: T_i\rightarrow \Delta(A_i)
$$

Definition of BCE: The decision rule $\sigma$ is a BCE for the game $(G,S)$ if, for each $i=1,...,N$, $t_i\in T_i$, and $a_i\in A_i$, we have
$$
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \sigma(a_{-i}, a_i|t_i, t_{-i}, \theta) u_i(a_i, a_{-i},\theta)
$$
$$
\geq
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \sigma(a_{-i}, a_i|t_i, t_{-i}, \theta) u_i(\tilde{a}_i, a_{-i},\theta)
$$
$\forall \tilde{a}_i\in A_i$.

Definition of BNE: The strategy profile $(\beta_1,...,\beta_N)$ is a BNE for the game $(G,S)$ if, for each $i=1,...,N$, $t_i\in T_i$, and $a_i\in A_i$ such that $\beta(a_i|t_i)>0$, we have
$$
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \Pi_{j\neq i} \beta_j(a_j|t_j) u_i(a_i, a_{-i},\theta)
$$
$$
\geq
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \Pi_{j\neq i} \beta_j(a_j|t_j)  u_i(\tilde{a}_i, a_{-i},\theta)
$$
$\forall \tilde{a}_i\in A_i$.

Question: when $N=1$, I've understood from the various versions of the paper I linked that the following terminology may be appropriate. Could you check and provide better suggestions, if any?

$S$ is called "experiment" in the sense studied by Blackwell (1951; 1953), rather than "information structure".
$G$ is called "decision problem", rather than "basic game".
The player is called "decision maker".
The BCE concept is referred to as "one-player BCE".
Also, how would you refer to the BNE concept with one player?
Lastly, are there other crucial papers in the literature introducing the concept of BCE when $N=1$? Or the paper linked is still a valid reference?

The terms experiment and information structure are used interchangeably by the literature, so both terms are valid regardless of $N$.
I agree if $N=1$ there is really no game, and it is rather a single-agent decision problem (or decision problem with uncertainty), and yes, you don't have a player, rather a decision maker.
For the same token, you don't want to say the solution concept if BNE, instead you want to say that the agent maximizes his expected utility given the information she received.
I was not aware of the term "one-player BCE" but Bergemann and Morris coined the term BCE, so if that is how they call it, it is correct.
In terms of other references, this is an earlier version of that same paper. However, I think that the paper you linked is the right paper for citing the concept of BCE.

In this later paper, the same two authors made the remarkable connection that the problem of Bayesian Persuasion popularized by Kamenica and Gentzkow could be interpreted in light of the concept of BCE when $N=1$. So perhaps, this later reference of BM can be useful if you want to explore more the case of $N=1$.


conditional mean and conditional median In Wooldridge's book (Page 452), it says

  When linear absolute deviation (LAD) methods are applied alongside OLS, thre are often reasons to think a priori that OLS and LAD will not produce similar slope estimates. (in fact, it is unlikely that the conditional mean and conditional median are both linear in $x_i$).

I understand that LAD estimates and OLS estimates are generally different, but the words in parentheses confuse me. Why they cannot be both linear? Could anyone give me an example?

In Alecos Papadopoulos's answer, both the conditional mean and the conditional median are linear in $X$. In the following example, the conditional mean is linear in $X$ while the conditional median is quadratic in $X$. This example is a constructed toy but it touches the heart of the issue.

Example: Suppose that $y = \beta_0 + \beta_1 x + u$ and $u = x^2 (w - 1)$, where $w \sim \chi_1^2$ and $x$ and $w$ are mutually independent. Then, as the R command qchisq(0.5,1) gives, $med(w) \doteq 0.455$ so that $med(u|x) = x^2 (-0.545)$, while $E(u|x) = 0$.  We thus have
\begin{equation}
  E(y|x) = \beta_0 + \beta_1 x, ~~\text{ while }~~ med(y|x) = \beta_0 + \beta_1 x - 0.545 x^2.
\end{equation}

Note: Asymmetry of the error distribution and the dependence of $u$ on $x$ are both important. If the error distribution is symmetric, then the conditional mean function and the conditional median function coincide. Also, if the distribution of $u$ is independent of $x$, then $med(u|x)$ is nonzero but constant and thus $med(y|x)$ is again linear in $x$.


Lending and interest rates It states in my textbook that as interest rates are lowered it says lenders are the ones who gain from the decrease in interest rates. But isnt this not true? As lower interest rates imply the amount you net from lending decreases as interest decreases?

Yes. That's tricky for newcomers to figure out. The concepts to keep in mind are:

a) to buy a bond is essentially to lend to the bond issuer

and

b) bonds are tradable.

So let's say that, in 2010, you bought a 30-year bond at \$10,000 face value that was yielding a 5% rate (known as the coupon rate) and today, in 2020, a 20-year bond is being issued also at \$10,000 par value but only offers a 4% return ($400 annual coupon).

Because bonds are tradable, that means your 30-year bond with 20 years remaining and an annual coupon of \$500 should now have buyers willing to pay a bit more than the $10,000 you paid for it.

Specifically, they can pay up to around $1.135 and still get the equivalent yield to maturity as the newly issued 20-year bond, a 13.5% profit for you.

The reverse also applies. If your 2010 30-year bond had a 4% coupon and the 2020 20-year bond is yielding 5%, you would only be able to sell the 2010 today at a price lower than $10,000.


Change vs Difference I have data on "share in labor employed in manual occupations" for two years (i.e. 1980 and 2005) and looking to create the following dependent variable:
change in the share of labor employed in manual occupation over the period 1980-2005
Which is the appropriate method based on the definition in bold above?

Share in 2005 - Share in 1980

(Share in 2005 - Share in 1980)/(Share in 1980)

My best guess: given that I am dealing with "shares", method 1 would be appropriate.

1 gives you the percentage point (p.p.) difference,
2 gives you the percentage difference.
There is no rule that says that if you're dealing with shares, you have to use 1. The reason 1 is preferred is that, when the measurement unit is in percentage, things get confusing quickly (e.g. going from a 2% to a 1% interest rate is a 50 percent reduction in the interest rate, but of course only a 1 p.p. reduction).
In your case, 2 could be meaningful, though. For example, "the share in labor employed in manufacturing decreased by half". A compromise could be to express the change in percentage points to avoid confusion, and then add the percentage change if it is meaningful (e.g. the labor share decreased by 20 p.p., a reduction of 50%).


Why labor-augmenting technological progress increases employment and wages? My textbook says that advances in technology typically raise the marginal product of labor and shift the labor demand curve to the right and calls it labor-augmenting technological advances. The book then claims that this labor-augmenting advancement is responsible for the simultaneous growth in wages and employment.

I am struggling to make sense of this as the way I understand it is that if fewer workers are needed to produce a certain output quantity, then why would someone hire more workers and pay them higher wages when they can hire fewer workers and pay them lower wages (since more applicants will be per one spot)

Thank you for your response.

The book is Principles of economics by Gregory Mankiw, 8th edition pg 367 btw.

I found this excellent Economic Letter from the Federal Reserve Bank of San Francisco that speaks to this observation. It is from Carl R. Walsh (july 16,2004) (https://www.frbsf.org/economic-research/publications/economic-letter/2004/july/the-productivity-and-jobs-connection-the-long-and-the-short-run-of-it/  )

In this letter he notes that, newspaper articles blame strong productivity growth for a “jobless recovery,” as economic output grows yet employment does not.

While it appears that faster productivity growth allows businesses to increase production without increasing employment.

His answer lays in the difference between macroeconoimcs and microeconomics perspective on productivity and between the short-run and long-run effects of changes in productivity.

From a micro perspective, productivity growth and new technological innovations are constantly leading to structural changes in the economy, causing one industry to expand in terms of both production and employment, while other industries contract.   However, jobs in other industries also expand.

For example, the manufacturing sector productivity increase but there share of total employment declined while employment and output in areas such as the computer industry have grown rapidly.

These shifts in the economy cause jobs to disappear in some sectors while jobs are created in others.

In the short-run, if demand of products rise this will lead to expanding product. And if labor productivity is unchanged, then typically they need to hire more workers to do this.
But, if overall demand in the economy has not expanded, then an increase in labor productivity could lead to a fall in employment in the short run.

In the long run, there is an emphasis that an increase in labor productivity increases potential GDP. It does so directly by allowing more output to be produced with the same level of employment, but it also increases employment because it decreases the cost of labor to firms and promotes the creation of new industries.

The cost of labor is noted as not just wages and benefits but also cost relative to the output the workers are able to produce.

A rise in labor productivity lowers the cost of labor at a given level of wages and benefits. If higher productivity makes labor less costly, firms will find it profitable to expand employment.


China GDP Data - Base year I had been searching for the GDP data of China and finally found it here.

The dataset gives the log real GDP data from 1992 (annual, quarterly and monthly). It doesn't specify the base year of the data, all it says is
"Implicit price deflator for GDP by value added, 1992Q1-", for the GDP deflator. Is there a way to find the base year for the real GDP data?

In this paper, page 8, it says

  Quarterly seasonally adjusted nominal GDP and its deflator
  are constructed by Higgins and Zha (2015).

That paper can be found here (google did not help to find this paper. I had to look for the personal website of one of the authors, here).

This document states in section II.1, Step 1:

  Annual
  real GDP-va –
  RGDP_va_A
  – is set to
  NGDP_va_A
  in 2008 ...

This is confirmed later in the next page, when they say:

  Except for the base year (chosen to be 2008) ...

So it seems to be 2008.


How to calculate PPP$ from LCUs? Is there one single formula which allows to calculate the income of a person in PPP\$ from her income in LCUs (local currency units), given for example the  PPP conversion factor, GDP (LCU per international $) as published by the World Bank.

Assume for example an Indian farmer with a monthly income of ₹50,000, given India's PPP conversion factor of 18 and exchange rate ₹70 = $1. What is his monthly income in PPP\$ and by which formula? Is other or further information needed?

Vice versa: What is the farmer's monthly income in rupees when his income is known to be PPP\$30?

References:

GapMinder's Dollar Street
Detailed income calculations for Dollar Street
The Economic Lives of the Poor by Abhijit V. Banerjee and Esther Duflo

It works like any other exchange rate:

Divide ₹50,000 by 70 to get about US$714 / month at market exchange rates
Divide ₹50,000 by 18 to get about PPP$2778 / month at PPP rates (so not a poor farmer)
Multiply USD$30 by 70 to get ₹2100 at market exchange rates
Multiply PPP$30 by 18 to get ₹540 at PPP rates


If a call may expire OTM, why won't a \$1 increase in the underlying's price necessarily increase the call's price by \$1? Zvi Bodie, Alex Kane, Alan J. Marcus. Investments (2018 11 edn). p 723 scanned.

         Figure 21.9 verifies that the slope of the call option valuation function is less than 1.0,
  approaching 1.0 only as the stock price becomes much greater than the exercise price. This
  tells us that option values change less than one-for-one with changes in stock prices. Why
  should this be? Suppose an option is so far in the money that you are absolutely certain it
  will be exercised. In that case, every dollar increase in the stock price would increase the
  option value by \$1. But if there is a reasonable chance the call option will expire out of the
  money, even after a moderate stock price gain, a \$1 increase in the stock price will not necessarily
  increase the ultimate payoff to the call; therefore, the call price will not respond
  by a full dollar.

Pls elaborate the embolding? The authors didn't spell out all the reasons. If a call has a chance of expiring OTM

why won't a \$1 increase in the underlying asset's price "necessarily increase the ultimate payoff to the call"?
why won't the call price increase by a full $1?

The reason that a call option price does not increase one-to-one with spot price of the underlying stock is simple: A 1 dollar increase in stock price only increases the likelihood that the option will be ITM. It is not a guarantee of 1 dollar increase in payoff at maturity. Naturally the price of call does not increase by 1 dollar.

More precisely, the price of the call should increase by the probability that the option will be ITM, which is less than 1. This fact is model-independent. (Try buying calls at one dollar plus current price whenever the underlying increases by one dollar, and see what happens.)

Accordingly, any model of option prices will have this property. Consider, for example, two basic models---the binomial tree and the Black-Scholes model.

Binomial tree

Suppose the stock price today is $S$. Tomorrow the stock price can be $S+\Delta S$ with probability $q$ and $S-\Delta S$ with probability $1-q$.

The price $C$ of a European call option with strike $K > S - \Delta S$ is therefore equal to the expected payoff $(S + \Delta S - K)q$.

(To be correct, assuming risk-free rate $r$, $q$ should be the risk-neutral probability given by
$$
q e^{-r} (S + \Delta S) + (1- q) e^{-r} (S - \Delta S) = S.
$$
But this is not central to the discussion.
)

If $S$ increases to $S + 1$, the price of the call becomes
$C = (S + 1 + \Delta S - K)q$.
The increase in the price of the call is exactly the probability $q$ of the call being ITM.

Black-Scholes

In the context of the Black-Scholes,
the sensitivity of option price with respect to the underlying is called the delta of the option. It is a standard quantity that arises in delta hedging.

Let $C(S_0, K)$ denote the Black-Scholes formula for price of European call with strike $K$ maturing at time $T$.
The delta of the call is the partial derivative $\frac{\partial C}{\partial S_0}$. The delta is the increase in $C(S_0, K)$ when $S_0$ increases by 1 dollar.

As you would expect, it's immediate from the Black-Scholes formula that the delta $\frac{\partial C}{\partial S_0}$ is equal to the (risk-neutral) probability that the option will be ITM at maturity.

(In Black-Scholes, under risk-neutral dynamics, the price of the stock $S_T$ at maturity $T$ is given by
$$
S_0 e^{( r + \frac12 \sigma^2 )T + \sigma \sqrt{T} N}
$$
where $N$ is a standard normal random variable.
So $\frac{\partial C}{\partial S_0}$ is equal to
$$
P( S_0 e^{( r + \frac12 \sigma^2 )T + \sigma \sqrt{T} N} > K)
$$
where $N$ is a standard normal random variable.
)


What would be the effect on the economy if the United States legalized counterfeiting? If today the United States legalized counterfeiting, what would be the effect on the economy? The counterfeit must look authentic to be legal, thus requiring some skill.

The initial effect would be great inflation, likely hyperinflation, devaluing the US dollar.

Some counterfeiters have had extensive operations, running in the tens or hundreds of millions of dollars. Legalizing these operations would likely lead to a great increase in the number and extent of these operations.

Having more and larger "counterfeiting" operations, or under the proposed law, simply additional printing presses, would cause the US to lose all control over its monetary policy.


How to increase price gradually? I sell so called "tokens" for so called Ether.

I receive an arbitrary amount of Ether in a transaction. I sent back a calculated amount of tokens.

I want the price of a token increase as more tokens are sold.

At first I introduced the formula $t=\frac{e}{T+e}$ where $e$ is the amount of Ether received in this transaction, $T$ is the total amount of Ether received before this transaction, and $t$ is the amount of tokens sent back in this transaction.

This formula has the deficiency that the very first payment (that is when $T=0$) always sends back $1$ token, no matter how much Ether we receive in this transaction. This is silly.

Please propose any alternative formulas with less silly results.

Note that it is impossible to make the total amount of tokens received to be independent of whether the purchase is done in one big transaction or several smaller transactions, as I proved in my answer to https://math.stackexchange.com/q/2709460/4876.

Despite the project is nonprofit (noncommercial), the main purpose of this is to receive as much Ether as we can.

The formula $t = \frac{e^2}{T^2 + e}$ does not scale linearly:

$\frac{(ke)^2}{(kT)^2 + ke} \ne c \frac{e^2}{T^2 + e}$ for any constants $k$ and $c$. (Linear scaling is OK as this is just a denomination of money. Nonlinear scale is a kind of nonsense as it introduces the need to add arbitrary coefficients, without a hint how to choose particular numerical values.)

So I choose the formula $t = \frac{e^2}{T + e}$.


what is the most comprehensive History of Economic Thought book for an undergrad? Which Book or books would you recommend for an undergrad to learn about the history and evolution of economic thought and theory, and why?
There Just so many of them out there and I just don't know which one will be best.
the one I was thinking of getting was Ekelund & Hebert's A History of Economic Theory and Method but am not sure about it.

My suggestion is not necessarily the most comprehensive, but "The Worldly Philosophers" by Robert Heilbroner is very enjoyable.


How does real gdp affect the domestic currency? When the domestic real gdp per capita increases then it has a negative effect on domestic currency (depreciation)? Why?

Source: The real exchange rate of euro and Greek economic growth, Gregory T. Papanikos, The Journal of Economic AsymmetriesVolume 12, Issue 2, November 2015, Pages 100-109.

file:///C:/Users/ACER/Downloads/1__Papper_exchange%20rate-gdp%20growth%20(15).pdf

The real exchange rate of euro and Greek economic growth
Gregory T. Papanikos - So i got the information from this source


Goldfeld - Quandt test statistic equal to 1 I wonder what it means if the statistic (ratio) of this test is one?
The statistic is built from sample splitting. Then you have to calculate the ratio RSS2 / RSS1, which are the Residual Sum Squares of the new subsamples.
I think it means that both subsamples have the same variance but I am not sure at all.
Thank you in advance.

The situation implicitly described in the question assumes that the two sub-samples have equal size (which is not necessary for the test to go through).

In such a situation, if the ratio $RSS2/RSS1$ equals unity, then the null-hypothesis of homoskedasticity won't indeed be rejected for any significance level up to $0.5$, (to be compared to the conventional levels of $0.1,\; 0.05,\; 0.01$)  because the statistic used, under the null hypothesis follows an F-distribution (exactly or asymptotically), and more over, one that has equal degrees of freedom for the numerator and the denominator (due to the equal sub-sample sizes).

This also makes intuitive sense: if the obtained estimates for the sub-sample variance are equal, then, in order to reject the null, we should accept that it would be more probable than not, that the rejection will be mistaken.


How to find the corresponding APR to a given APY? Suppose that $6000 is invested in a 3-month CD with an APY of 1.5%
I want to find the corresponding APR to a given APY. If 6000$ is invested in a 3 month CD with an APY of 1.5% then according to my calculations, the corresponding APR is 1.491%
I used this relation between APY and APR to get this result: $$APY=(1+ \frac{APR}{n})^{n} -1$$
Where $n$ denotes the $n$ compounding periods per year, I'm not sure what value $n$ should take, but in my case, I guessed that $n=4$ (based on the 3 months CD). So according to this formula by writing the APY in decimal point I got the result that the APR is 1.491% but I'm not really sure if that's tru.

The length of the CD is not relevant. The point of APR and APY numbers is that they annualize the rate you pay over a time frame that may be smaller or larger than a year. What matters is the compounding frequency. CDs generally compound on either a daily or monthly basis, so n will be 12 or 365, depending on the CD.
Is this a real-life problem or an homework/practice question? If it's the latter, you should probably assume n=12 because teachers seldom want you to mess with daily compounding without also telling you how many days you should assume there are in a year for compounding purposes.


Is free trade transitive? If country A has a free trade agreement with country B, and country B has a free trade agreement with country C, does that mean country A effectively has a free trade agreement with country C?

If the answer is “it depends what you mean by free trade”, what characteristics of the A-to-B agreement would be be conducive to it being “transitive” with the B-to-C agreement?

If A-to-B and B-to-C were perfectly “compatible” in this sense, could A ship goods directly to C or would they still have to go via B?

Free trade agreements are not transitive. Each country in a free trade agreement needs to ratify it with a legislative or regulatory body. Otherwise there is no legal basis for it to function and customs officials won't be able to observe it.

In practice if A and B have a free trade agreements and B and C do too, businesses can transship in the best case scenario. Or, most likely, ship from A to B, unload-unpack-repack-load using an intermediary company in B, and ship from B to C. That might cost less or more than paying import/export tariffs from A to C.


Financial investment in the composition of GDP In the production function Y = C + I + G + NX

Does foreign investment in domestic assets (i.e. foreign buying of domestic bonds) - and vice versa - come under the Net Exports variable? Which denotation does domestic investment in domestic assets come under? The Investment variable (I), one would think, however I seem to recall being told that the Investment variable does not include financial investment, so does it come under saving?

What you're describing is a change in the capital account, not in GDP. They're related through the balance of payments, in that if a country is running a current account deficit (usually arising through being a net importer), they'll have a capital account surplus (i.e., foreigners will on net buy more domestic assets or domestic owners will be net sellers of foreign assets, or both), and vice-versa. So if you take foreign investment in domestic assets for example, it'll be associated with being a net importer, but it does not appear directly in the national accounting identity $Y=C+I+G+NX$.


How can this be proved? (Convex optimization) Consider the following maximization problems:

$\max_{x} x -\gamma p(x)$
subject to $x \in \Omega_1$

$\max_{x} x-\gamma (p(x) + q(x) )+K$
subject to $x \in \Omega_2$

where $\Omega_1 $ and $ \Omega_2$ are convex sets, $p(x) \geq 0 $ and $q(x) \geq 0$ for all $x\in \Omega_2$. Also, $p''(x)>0$ and $q(x)$ is linear in $x$ and $K>0$ is a constant.
If for given $\gamma = \bar{\gamma}$, the optimized objective value for problem 1 was greater than the optimized objective value of problem 2, does the optimized objective value for the problem 1 always greater than that of problem 2 for all $\gamma > \bar{\gamma}$?
Prove or provide counter example for (1) $\Omega_1= \Omega_2$ and (2) $\Omega_1 \subset \Omega_2$.
Since the higher penalty proportional to $\gamma$ is imposed to the objective function of problem 2, this claim seems right. I tried using contradiction, in which assuming there exists $\gamma'>\bar{\gamma}$ such that optimized value for problem 2 is greater than that of problem 1, but struggling. How can this be proved?

Take $\Omega_1=\Omega_2=[0,0.5]$.
Let $p(x)=x^4$, so that $p''(x)=12x^2>0$ on $(0,0.5)$.
Let $q(x)=0.5x$, which is linear in $x$, and $K=0.2>0$.
For $\bar\gamma=1$, both objective functions attain their respective maximum at $x=0.5$. As the following figure shows, objective function $(1)$ (blue curve) has a higher maximum than that of objective function $(2)$ (red curve).

But for $\gamma=5>\bar\gamma$, objective function $(1)$ attains a maximum value of approximately $0.276$, while objective function $(2)$ attains a higher maximum value of approximately $0.309$, contradicting the claim that $(1)$'s maximum would always be greater than $(2)$'s maximum for $\gamma>\bar\gamma$.