A country borrowing from the rest of the world Well I'm having a some difficulty in understanding completely what the following sentence means and I would like to be sure.

«(...) a country running a trade deficit is buying more from the rest of the world than it is selling to the rest of the world. In order to pay for the difference between what it buys and what it sells, the country must borrow from the rest of the world. It borrows by making it attractive for foreign financial investors to increase their holdings of domestic assets- in effect, to lend to the country.»

How does this borrowing(in italic) work? I'm not understanding the italic part. Does the country A sells bonds in domestic currency, receives foreign currency, when the foreign investor buys the domestic currency of A to buy the domestic bond of A?

Also why should the current account plus the capital account be equal to zero (up to a statistical discrepancy)?

The idea is that the world economy is closed and value only comes from production. It is maybe best to forget about money for a moment here. Since money derives its value directly from production, anything we say about production applies to money as well.
Since the world economy is closed, a country that is producing less than it consumes must "borrow goods" from other countries to consume (otherwise it cannot consume more than it produces by definition). We can't have the whole world consuming more than it produces, so someone else must be consuming less, i.e. "lends". This is the case of a trade deficit, i.e. positive net imports. If a country "borrow goods" it effectively also borrows money.
A currency's value (purchasing power) is proportionate to the number of goods in that economy, since all money must be able to buy all goods by definition and legally. (If only 1 loaf of bread and only 1 dollar exist, the dollar has a purchasing power of 1. If (only) 2 dollars exist then each dollar has a purchasing power of 0.5).
How this may work is that the exporting country lends money to the importing country in order to be able to buy more than it sells. In a barter economy this happens directly without money, the extra goods given are loans (as goods in equal value have not been receieved). Since money is not "real" in an economic sense, the same must happen with money. This happens, since each currency's value is related to production and the importing country is producing less (as it doesn't have enough to spare), while the exporting country is producing more (it has more than enough), therefore the respective currency values reflect this.
Very often an importing company will take out a credit from the exporting company and this reflects itself at the country level. Even if money isn't actually lent, it happens through adjustment in exchange rates and interest rates.
Since money's value depends on production value, this is also the  reason the current acount + the capital account must be equal to 0. For any goods bought or sold, money must be paid. Its like saying the value of goods you receive at the store must equal the value of money you give to the cashier. If you get more goods than you have money for you'll simply owe the cashier. Internationally (but alson domestically) you only have money in value of what you produce. Therefore if you produce less than you consume, you don't have enough money (production=money) to pay the cashier for the goods you consume.
Further, the current account is constructed similarly as an account from accounting. It sums up to 0 by (that) construction. Goods leave, money enters, they must have the same value. For more on this you can also consult any introductory accounting book.


Is a lack of "rules" dangerous for cryptocurrency An issue/disadvantage that the gold standard had was a lack of mechanisms that forced economies to abide by the rules of the game. As a result, they could at any time de-monetize gold and therefore hurt the whole purpose behind it.

Can't this same idea be applied to crypto-currency? If one day individuals decide to not put any value behind XYZ-Therum Crypto (using a generic name), won't that cause a collapse?

Also, isn't there a danger in the lack of regulations behind it?

Why can't I invent MikeIsCool-Coins. I make a website, with a wallet and say that the direct quote for MikeIsCool-Coins is 1BTC. I then give myself how ever much I want MICC, trade it to bozos for BTC, then sell the btc for USD, renderring myself a millionaire?

And this scheme, as a result of a lack of regulation puts no consequences on me?

Thank you, I'm just trying to challenge crypto. I think a currency backed by prime numbers (something that itself holds no value, where FIAT money is based by a promise) is complete nonsense.

ELI5: What actually makes makes currency worth anything? : explainlikeimfive

One reason why fiat money has value, is that the government insists that people pay taxes, and the only payment of taxes they will accept is in their own currency. So, everyone has to earn or acquire enough in that currency to pay their taxes.

You asked

If one day individuals decide to not put any value behind XYZ-Therum Crypto (using a generic name), won't that cause a collapse?

Yes is your answer. As at July 29 2020, no governments allow citizens to pay taxes with cryptocurrency, unlike fiat currency. Thus "If nobody is actually using them, that means the platform has no adoption, which means the tokens aren't worth anything and it's all just a dangerous speculative pricing game..."

For more detail, read ELI5:Crypto currencies how are they actually worth anything? : explainlikeimfive.

TL;DR: BTC's intrinsic value is represented by fractional participation in the 'illegal economy.'
You're getting a ton of 'BTC is fiat!', which feels true, but this isn't really true. Its the same situation for the USD, as much as people like to scream "THE DOLLAR IS JUST FIAT!!" over at places like zerohedge (don't go there - that website is trash).
Let's start with the USD (but the same is true about the yen, euro, ruble, etc). The thing about the US dollar is that there is an unequivocal end demand for US dollars, imbuing it with an intrinsic worth: if you want any level of participation in the US economy, you need dollars to buy it (to get access). Think of it this way - the US dollar is the vehicle by which you gain access to the US economy.
"But but but you can just convert....." No you can't 'just convert'. What would you say if your employer told you tomorrow that they were going to pay you in rubles, euros, yen or even gold? You wouldn't say anything - you'd quit. Why? Because you have to convert to USD to do anything within the bounds of the US economy. Paying your rent, going to Kroger, buying a car, getting a computer, hiring a maid, paying your taxes all require USD. In the case of your employer paying you in an alternative currency, instead of quitting, you might demand a raise to cover transaction costs, taxes and hassle as you convert back to USD.....so you can participate in the US economy. As such, there is a floating but measurable demand for USD. It is not fiat in the way that that it is typically described.
In slightly more technical terms: the 'intrinsic value' of the USD is a 'right of participation' within the US economy and all ROPs have an intrinsic value. We can debate the value of the ROP, but we are both gonna agree its there. I come on here all the time and talk about how the size of your money supply needs to be linked to the size of the economy - this is the manifestation of that link and an explanation of precisely why it has to be linked. SIDENOTE: John Nash (Nash Equilibrium & A Beautiful Mind) was working on 'solving' the precise math on this when he died, but he admittedly did not think it was a solvable problem.
Back to BitCoin: BTC's intrinsic value is tied to the size of the 'BTC economy' in that the value of a single bitcoin represents a 'right to participate' in the BTC economy. So what do people use BTC for? Well, let's be honest: mostly illegal shit. Here's a really interesting thought piece by a guy who tried to value BTC by measuring against an estimate of the illegal economies that BTC helps to facilitate. We can argue the particulars in this piece all day long, but I really like the economic/financial construction behind the theory - the guy is definitely approaching it the right way.
Now there are some obvious problems here, with the most obvious two being: 1) you are 'long' illegal activity which is very risky from an investment standpoint (I don't mean BTC holders 'go to jail', I mean you have no reasonable recourse if shit goes against you, such as criminal prosecution of the criminals that comprise the BTC economy); and 2) governments want to maintain their currency monopolies and will be fighting you forever (and again you have no recourse).
So how does BTC become really valuable? If BTC is allowed to edge in on the 'legit' economy, then BTC becomes very valuable b/c it becomes a viable alternative to traditional currencies (eg, you now get global access to local economies.)
However, let's be super fucking honest: the last thing the US gov't is going to allow is for a competitor to the USD, so when you buy BTC keep in mind that the government will be fighting you. And yes, they can stop the growth of BTC by restricting access by preventing any company that does business in the US from accepting BTC in any, way shape or form. Additionally they could prevent any transaction that might have come from BTC via existing money laundering laws. That alone could (close to) zero BTC overnight. If you think that could never happen, you're living in fantasy land.
In summary: BTC is granting you fractional participation in an illegal economy - that's its intrinsic value. However, I would not call it a currency (or an asset or a value store) because you do not have any of the legal protections that are inherent with currencies, assets or value stores. This is why its speculative as fuck.
Source: current quant hedge fund manager with a career on wall st. People like me get drunk together and talk about this shit all the time.


Why should I get a bond with negative interest instead of having a bank deposit account either zero interest or positive interest I don't get why central banks apply negative interest rates. They say that buy our bond and at the maturity, it will worth less than today. What is the policy outcome of such decision?
Why an investor still accepts this or is it just a mechanism to indirectly discourage bond sellings?

It makes little sense to me either, but here are some possible reasons for buying a bond with negative interest rates rather than depositing the same amount in a bank:

The deposit-taking bank may go bankrupt during the lifetime of the bond and your deposit would be too large to be guaranteed, with the cost of default insurance being higher than the the negative interest
The bank may start charging you for holding your deposit at some time during the lifetime of the bond, at a higher cost to you than the negative interest
You may be legally obliged to hold bonds, for example as a pension or insurance company
You run an Exchange Traded Fund or similar investment tracking a bond index and some of the bonds in the index have negative interest rates
You are a foreign investor who only cares about the combination of the current exchange rate, future exchange rate and interest rate, not any individual component of this triangle, or you believe currency appreciation will more than compensate for negative interest
You are a derivatives trader, and the market prices of derivatives lead to an arbitrage opportunity where you can only guarantee a profit by buying the bond
You believe that at some time in the future you may be able to sell the bond for more than you paid for it (i.e. an even more negative yield), perhaps to a central bank's quantitative easing programme


How does the ban on surge pricing affect the companies, customers and the economy? Recently, there has been a ban on surge pricing in two of India's busiest cities, namely Delhi and Bangalore.

I understand that surge pricing is a method to maintain the balance between the supply and demand. So, it's ban would affect the taxi aggregator companies.

However, is it really doing benefit to the consumer, as the Govt. claims?

Overall, how does the ban on surge pricing affects the taxi aggregator companies, the consumers and the economy as a whole?

I have read through articles this this, however, they give a very naive explanation. So, I'm looking for something clearer from a microeconomics point of view.

PS: Delhi has an odd-even rule too (for combating excessive traffic and pollution), which means that cars with an even (registration)number [the one on the number plate] are only allowed in the even dates, and the ones with an odd number o the odd dates. For example, if I am DL 23 BK 1900, I am only allowed on the road on even days, and not in odd days.

It depends.

If customers are currently making informed decisions when they book a surge-priced car, then banning surge pricing punishes customers, drivers, intermediaries, and the wider economy. It deliberately introduces an economic inefficiency.

If, however, a lot of customers are making uninformed decisions and are effectively being conned by surge pricing that they didn't expect or didn't understand, then banning surge pricing stops economically inefficient activity, protects consumers, and can be a net contributor to the economy.

Or if surge pricing increases negative externalities, then banning it could give a net benefit: for example, if the worst-polluting cars only come onto the road when surge pricing is available.


Real and nominal value interpretation When price is deflated according to some base year, what does the real value mean? E.g. if we are deflating \$5.000 from year 2000 (base year, deflator=1) to 1980, using the CPI 1.6 from 1980, we get \$3.125.

Is this real value equivalent to $5.000 in 1980?

The formula for calculating the real value of an amount $x$ in a given year using dollars from a particular reference year is

$$
\frac{x_{baseyear}}{x_{current year}}=\frac{CPI_{base year}}{CPI_{current year}}
$$

So in your hypothetical example (where the deflator is higher in 1980 than in 2000, which indicates deflation over the period), the value of \$5 in 2000 is equal in 1980 to:

$$
x_{current year}=x_{base year}*\frac{CPI_{current year}}{CPI_{base year}}
$$

or

$$
$5*\frac{1.6}{1}=$8
$$

So in your example, a basket of goods costing \$5 in 2000 would cost \$8 in 1980.


How does one derive relative demand? Suppose I have two markets, Home and Foreign. Suppose that

$$\frac{p_1}{p_2} = \frac{c_2^F}{c_1^F}$$

$$\frac{p_1}{p_2} = \frac{c_2^H}{c_1^H}$$

Supposedly I am supposed to be able to show

$$\frac{p_1}{p_2} = \frac{c_2^F + c_2^H}{c_1^F+ c_1^H}$$

But by rules of simple algebra, it would seem

$$\frac{p_1}{p_2} = \frac{c_2^Fc_1^H+c_1^Fc_2^H}{2c_1^Fc_1^H}$$

MY QUESTION:
What step am I missing to get
$$\frac{p_1}{p_2} = \frac{c_2^F + c_2^H}{c_1^F+ c_1^H}$$ ?

Take your two beginning expressions and cross multiply them:

$$p_1 c_1^F = p_2 c_2^F$$

$$p_1 c_1^H = p_2 c_2^H$$

Then subtract each right hand side for both equations. Multiply one of the expressions by $-1$ on each side:

$$p_1 c_1^F - p_2 c_2^F = 0$$

$$-p_1 c_1^H + p_2 c_2^H = 0$$

And set each expression equal to each other:

$$p_1 c_1^F - p_2 c_2^F = -p_1 c_1^H + p_2 c_2^H$$

Put the $p_1$ and $p_2$ terms together:

$$p_1 c_1^F + p_1 c_1^H = p_2 c_2^F + p_2 c_2^H$$

Factor the $p_1$ and $p_2$ terms and rearrange:

$$p_1 (c_1^F + c_1^H) = p_2 (c_2^F + c_2^H) \implies \frac{p_1}{p_2} = \frac{c_2^F + c_2^H}{c_1^F+ c_1^H}$$


$1.25 PPP per day poverty threshold - What precisely does it mean? As Wikipedia states:

  In 2008, the World Bank came out with a revised figure of $1.25 at 2005 purchasing-power parity (PPP).

(Update: In 2015, this was updated to $\$1.90$ at 2015 PPP.)

Wikipedia cites Ravallion, Chen, & Sangraula. "Dollar a Day Revisited", 2009.

Of course, there will always be difficulties with measurement.

But conceptually, what precisely is this $\$1.25$ figure meant to capture?

Does it mean the equivalent of someone having to pay for absolutely everything (food, water, shelter, utilities, transport, education, etc.) with just $\$1.25$ a day, in a place where prices are similar to those of the US average in 2005? Or something else?

A partial answer : I m not sure what measurement the World Bank would recommend, but I suspect that in practice, the expenses that are taken into account to measure whether one is below or above the 1.25$ threshold vary from one study to another.

For instance in their famous The Economic Lives of the Poor, Banerjee and Duflo write

  "From each of these surveys we identified the extremely poor as those living in households where the consumption per capita is less than \$1.08 per person per day, as well as the merely “poor” defined as those who live under \$2.16 a day using the PPP in year 1993 as benchmark.3 The use of consumption, rather than income, is motivated by the better quality of the consumption data in these surveys (Deaton, 2004)."

The use of "consumption per capita" suggest that capital income, such as the benefits one get from owning her own house, would not be taken into account.
Because data quality considerations seem to enter the picture, it opens the door to much variation from one study to the other.

The above quote however suggests that at equal data quality, income data would be preferred.


Paper where an integral in the constraint of an optimization problem is treated as infinite sum I am looking for a paper (or textbook, or even lecture notes example) where there is a problem such as
$$
\max f(x) \\
\text{ s.t. } \int g(x) \leq \int c
$$
and there is at least some exposition/explanation about solving it. I believe that these are sometimes (often? always?) treated as infinite sum, but I may be wrong.

In words, I am looking for an example and solution (or at least a discussion of solution) of a problem where the constraint contains an integral. It is okay (and probably will be the case) that $f(x)$ contains an integral too.

I believe such optimization problems occur in macroeconomics. I'm specifically thinking of integrals over possible states, but the integral doesn't necessarily have to be over states.

Or, if someone could explain solving such problems to me, that would be okay as well.

Thanks.

Some papers that come to my mind are ;

-Calvo and Obstfeld (1988, Econometrica)

-Endres et al (2014, Resource and Energy Economics)

-Traeger et al. (2012, European Economic Review)

In these papers, the objective function has two integrals but as these models are treating an overlapping generations model, there is a such aggregation that one of the integrals disappears because it is not possible to have an integral on an Hamiltonian. Otherwise, it is not possible to solve it by standard methods.

I am not sure about but this so take it with precaution, this kind of problem with two integrals on the objective function could not be solved by optimal control techniques but by calculus of variation.

There is a very good section about constraints with integrals on textbooks of

-Kamien and Schwarz (Dynamic Optimization : The Calculus of Variations and Optimal Control in Economics and Management)

-A. Seierstad and  K. Sydsæter (Optimal Control Theory with Economic Applications)


Beginner wanting to learn about economics/finance to explore career options? I want to explore the banking/financial industries more for potential career options, can you recommend any books/ free online courses and sources in general as a starting point to build on my knowledge and understanding of these areas (which is basically starting from zero), for now I want to read more broadly and have a better understanding before narrowing it down. So would I learn about markets, trading, and also economics concepts, and would you be able to recommend any topics that I should focus on for practicality rather than having to read topics which won't be as relevant?

A great starting point is "Fundamentals of Corporate Finance", Ross, Westerfield & Jordan.

This book will give you a good overview of key concepts in finance. This book is widely used in introductory courses for non-finance majors, to both teach some fundamental ideas, and give a panoramic view of the field.

"Trading" will probably be the topic that is less developed in the book. This is partly because there is no bullet-proof method for trading, to some degree it is an art, and also becuase some of the useful techniques for trading are too advanced for an introductory book. However, after reading this book you should have a good sense of the key idea behind the decisions to buy or sell a financial asset.


What is one dimensional, ordered type? I am reading papers about moral hazard. What is one dimensional, ordered type $\theta\in\Theta$?

What is one dimensional, not ordered type $\theta\in\Theta$?

Could you please give an example?

References:

Riley (1979)

Azevedo and Gottlieb (2016)

Judging from the reference you provide, this refers to whether the set $\Theta$ is ordered or not. For example, natural numbers or the alphabet are ordered sets. In the context of moral hazard problems, examples could be effort, or ability. Since this is a numerical variable, it is an ordered set.
In the paper you mention, the phrase "one dimensional, ordered type..." is not found. However, the paper states that:

... differences among sellers are assumed to be parameterized by a single unobservable characteristic $\theta \in \Theta$. Then the family of utility functions can be written in the alternative form,
$U_i=U(\theta_i;y,p) \text{, } i \in I$
If seller $i$ has $\theta_i$ units of the characteristic we may therefore refer to him as being of "type $\theta_i$.

The first bold statement refers to the one dimensionality of the unobserved type, whereas the second refers to it being a numerical variable, which means $\Theta$ is an ordered set.
Conversely, a non-ordered set contains elements which have no intrinsic order. For instance, you can think of a case where agents do not know the city where they are to be located. The set "cities" is unordered because cities do not necessarily hold a particular order in a one-dimension set. You can add a second dimension (e.g. population), where you could order them based on city size.
I would say that having ordered sets facilitates algebraic analysis, since it enables the introduction of monotonic strategies, whereby an important outcome variable is a monotonic function of the agent's type. In effect, this seems to be important in the paper, when the author states that:

A seller of type $\theta_i$ brings to market a product that is valued equally by each of the set of potential buyers, $J$. This valuation, $V_i$, measured in dollars, is an increasing function of $\theta$ and (possibly) also of the intensity of sales-related activity, $y$:
$V_i = V(\theta_i;y)$

The bold emphasis highlights precisely the monotonicity enabled by the order property of $\Theta$. Another example is given in Athey (2001):
$\hskip1in$

In response to the second reference you give (here), in the phrase

We now show that Riley equilibria may not exist when types are not ordered.

the "ordered" adjective refers to the second interpretation given above (i.e. of monotonicity). First, they assume that

Consumers are uniformly distributed in the unit interval, $\theta in [0,1]$

which is an ordered set. However, because both the utility function and the cost function are quadratic, it results that

... the types in the extreme points have the strongest preference for one of the contracts and are the most costly to serve.

which means that the utility and cost functions are not monotonic over $\theta$.


Import/export by sector/country data source I am looking for import/exports data by sector and country, with partner countries. The goal is to get a picture of overall imports/exports to/from a specific country with (possibly) all of its partners, for any particular sector/year.

So for example: For 2017, oil imports and exports for China with Russia, Angola, Saudi Arabia, etc.

I believe that the UN Comtrade database is the main data source for this kind of query with substantial country coverage.  It's broken down by HS code.

It has its issues.  Reporting is based on national authorities, so you should usually expect data on imports to be more accurate than exports because of the tax implications (so Uganda's imports from Kenya are likely to be more accurate than Kenya's exports to Uganda).


Why does a tariff have no effect to an exporter? I'm reading Principles of Economics, the paragraph about tariffs in a imaginary country Isoland wrote: "The Isolandian economists next consider the effects of a tariff—a tax on imported goods. The economists quickly realize that a tariff on textiles will have no effect if Isoland becomes a textile exporter. If no one in Isoland is interested in importing textiles, a tax on textile imports is irrelevant. The tariff matters only if Isoland becomes a textile importer. Concentrating their attention on this case, the econo- mists compare welfare with and without the tariff."
I don't understand why a tariff have no effect to an exporter.

A distinction should be made between two different types of tariffs.

Let's define good imports as goods that enter into a country, across its border, for commercial purposes. Those imports could be subject to a tax on trade, called an import tariff. Thus, the Isoland country only taxes goods that enter. If no one imports the goods, this tax is irrelevant.

Let's define good exports as goods that move outward across a country's border for commercial purposes. Export tariffs would be a tax that a country imposes on its exports, which makes them more expensive. Isoland does not want to tax these goods because it would be a burden on its own businesses.

Note, however, that these export tariffs can be put in place to encourage domestic consumption of domestically produced goods instead of exports. This could be the case in the context of a food or health crisis if a country wants to discourage exports. This is not the cas in your example.


economics paper with most coauthors A high number of co-authors on scientific papers is not uncommon, but in economics I rarely see one with more than 5 coauthors.
Does anyone know the paper in economics that has the highest number of co-authors (whether wp or journal article)?

Economists also publish in Science or Nature sometimes, and if the study is interdisciplinary, there could involve many co-authors just like other scientific papers.
In the prestigious economic journals, the economics paper with most coauthors is Benjamin, Daniel J., et al. "The Promises and Pitfalls of Genoeconomics." Annu. Rev. Econ. 4.1 (2012): 627-662.  or Coenen, Günter, et al. "Effects of fiscal stimulus in structural models." American Economic Journal: Macroeconomics 4.1 (2012): 22-68.


modelling disutility from over consumption In introductory economics courses the concept of marginal utility is illustrated through simple examples like how much benefit one gets from eating another slice of pizza (i.e first slice provides 100 utils, second slice 50 etc).

I was wondering if there exists a utility function that can allow for the possibility of over consumption (i.e. a function that would produce -50 utils from the 10th slice of pizza ).

I know there would be issues with this since utility functions are only defined up to monotonic transformations, and concavity in our utility functions would violate the "averages preferred to extremes" assumption.

Does such a function exist?

Varian's Intermediate Microeconomics covers a concept called the bliss point.
If the consumed amount of a certain good is under the quantity specified by the bliss point, then the consumer would prefer to consume more of the good, all other things being equal. If the consumed amount is over the quantity specified by the bliss point, then consumer would prefer to consume less of the good, all other things being equal (i.e. it becomes a 'bad').

In case of one good, a utility function that represents this kind of preference is
$$
U(x) = -(x - x_b)^2,
$$
where $x_b$ is the bliss point quantity. For two goods, similar utility functions exist, e.g.
$$
U(x,y) = -(x - x_b)^2 - (y - y_b)^2
$$
or
$$
\hat{U}(x,y) = |x - x_b| + |y - y_b|.
$$
Both $U$ and $\hat{U}$ violates monotonicity, $U$ represents strictly convex preferences, so averages are still preferred to extremes. $\hat{U}$ represents weakly convex preferences.


Why does a homothetic function have constant ratio of marginal products along rays? A homothetic ordering is defined as

$x \succeq y \Rightarrow \lambda x \succeq \lambda y \qquad \forall \lambda >0$

where $x,y \in \mathbb{R}^n$

Then, any differentiable function representing the ordering has the property

$\frac{\partial f}{\partial x_i}(\lambda x)= k \frac{\partial f}{\partial x_i }(x)$

with $k,\lambda >0$

How is this results derived?

I can see how we can derive properties of the function values from homothety, but have no idea how we can say anything about its derivatives.

A homothetic function can be characterized as follows:
Let $f(\mathbf x)$, $\mathbf x \in \mathbb R^n$ be a function homogeneous of degree $r$. Let $g()$ be a function with $g'\neq 0$. Then

$$G(\mathbf x) = g[f(\mathbf x)]$$
is homothetic. Since $f(\mathbf x)$ is homogeneous of degree $r$ we have that

$$f(\lambda \mathbf x) = \lambda ^ rf(\mathbf x) $$

Then

$$G(\lambda \mathbf x) = g[\lambda ^ r f(\mathbf x)]$$ and so

$$\frac{\partial G(\lambda x)}{\partial x_i}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_i}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_i}$$

Evidently, we will also have

$$\frac{\partial G(\lambda x)}{\partial x_j}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_j}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_j}$$

which leads to the "constant MRS along rays" characterization of homothetic functions,

$$\frac{\partial G(\lambda x) / \partial x_i}{\partial G(\lambda x) / \partial x_j} = \frac{\partial G (x) / \partial x_i}{\partial G( x) / \partial x_j}$$

(see Simon and Blume 1994, p. 503).


Solving problem for optimal price (maximize profit) *attempt inside* Let a demandfunction be defined as
$D(p)=B-bp$, where $b,B>0$. A firm has some production cost, $c$, and can set the price $p$ under the constrain given by the Demand.

What is the optimal price?
Is the priceelasticity higher or lower than $1$ (absolute value)
How does price depend on $b$?

How would one do such a maximization problem?
My attempt
The elasticity would be given by
$\frac{p}{q}\cdot(-b)$ since $D'(p)=-b$
Then the problem is
\begin{equation*}
    max_p(p-c)D'(p)
\end{equation*}
which then follows that
\begin{equation*}
    D(p)+(p-c)D'(p)=0 \Leftrightarrow
\end{equation*}
\begin{equation*}
    D(p)\left ( 1+(p-c)\frac{\epsilon}{p} \right )=0\Leftrightarrow
\end{equation*}
and then the optimal price $p^*$
\begin{equation*}
    p^*=\frac{\epsilon}{1+\epsilon}c=\frac{\left | \epsilon \right |}{\left | \epsilon \right |-1}c=\frac{\left | \frac{-bp}{q} \right |}{\left | \frac{-bp}{q} \right |-1}c=\frac{c\left | b \right |\left | p \right |}{\left |  b\right |\left |p  \right |-\left | q \right |}
\end{equation*}
and then we notice that
\begin{equation*}
    \left | \epsilon  \right |>1
\end{equation*}
...............................................................................................
I feel like this is not correct nor the right approach. Help appreciated.

Start with setting up profit equation:
$$\max_p \pi = (p-c)D(q)$$
Then substitute demand in:
$$ \pi = (p-c)(B-bp)$$
Take derivative of profit and equate it to 0:
$$\frac{d \pi}{dp}=B-2pb + cb=0$$
Now solve for optimum price:
$$p^*= \frac{B+cb}{2b}$$
There is also alternative way how to get to the same solution that uses elasticity of demand.
There is well known monopoly optimal pricing rule (see Peitz and Belleflamme Industrial Organization: Markets and Strategies) says that optimal monopoly prices are given by:
$$\frac{p-mc}{p}=-\frac{1}{El}$$
In your case marginal costs (mc) are $c$, and your elasticity of demand is actually:
$$EL= -\frac{bp}{q}= -\frac{bp}{B-bp}$$
In your attempt you forgot to substitute for Q in your elasticity.
Plugging these into optimal pricing rule gives us:
$$\frac{p-c}{p}=-\frac{1}{\left( -\frac{bp}{B-bp} \right)    }$$
Which when solved for $p$ gives us again:
$$p^*= \frac{B+cb}{2b}$$
Both approaches give the same solution - the first one is quicker in this case in my opinion.
Now regarding what we say about price and parameter $b$, we can just rearrange the optimal price formula as:
$$p^*= \frac{B}{2b}+\frac{c}{2}$$
So clearly we can see that when $b$ parameter increases optimum price decreases.
We can also prove it more formally by inspecting derivative of $p^*$ with respect to $b$ which gives us:
$$ \frac{d p^*}{d b} = -\frac{B}{2b^2}$$
Since the derivative is negative we confirmed that the price varies negatively with $b$.
Next, per request in comment on what is elasticity at equilibrium price we can calculate that just by substituting the equilibrium price into elasticity which we already calculated above:
$$EL(p^*)= -\frac{b p^* }{B-bp^*}=   -\frac{b \left( \frac{B+cb}{2b} \right)}{B-b \left( \frac{B+cb}{2b} \right)} \\= - \frac{B+bc}{B-bc}$$
The last expression will be bigger than 1 in absolute value because the numerator will be larger than the denominator ($B+cb>B-bc$) and absolute value will make the value positive.


What are the correct utility functions? It is common to talk about utility functions. For example in a universe with only two goods, we might assume each person (or group of people) carries a function $u(x,y)$ in their heads. When offered some baskets $B_j$ containing $x_j$ amount of good 1 and $y_j$ amount of good 2 the person will select the basket $B_k$ that maximises $u(x_k,y_k)$.
I wonder has anyone ever tried to find what there utility functions look like experimentally? Of course they might vary from person to person, and depend on the type of goods. But if the functions are a realistic model it should be possible in principle to see what they look like. Has there been any attempts to do so?
The closest I can find are regression attempts, where data is gathered and then fit to some class of functions. For example linear regression tries to draw the straight line (hyperplane) that best fits the data. This is not what I am looking for however, since regression assumes the type of function in advance. I'm looking for something as simple as offering a bunch of baskets, plotting the data, and comparing to a bunch of different types of functions.

It would be really tough to find papers (other than structural models) that empirically determine a particular utility function. Because of their subjective nature, it is incredibly tough to obtain observable data to estimate them. So researchers study the primitive preference relations that utility functions represent. Arguably, the preference relation is what you want to study when talking about rationality and choice. Utility functions are useful insofar as it opens up a tool-box of real analysis tools which a binary relation doesnt typically allow.
So you should rather look for papers that test rationality from observable choice data. I have elaborated a bit on this in an answer to a related question: Click Here


How will the value of the pound affect the British economy? Given the recent massive decrease in the value of the pound, does this not equally devalue the British economy by a similar amount? Why does that not lead to an immediate large crash in Britain as spending power is reduced?

You think spending power has been reduced, but in reality inflation has only been 1%. Meaning products inside the UK measured in pounds have only become 1% more expensive.
The devaluation of the pound only exists compared to other currencies.
This means two things:
1. Products from the UK have become cheaper for foreigners. So companies from the UK will sell more.
2. Consumers from the UK pay higher prices outside the country. This will also lead to more sales inside the UK.
UK companies profit from a weaker pound even when measured in other currencies.
Consumers on the other hand are far worse off, and if you were planning on spending your retirement abroad you can end up paying a big pricetag for it.
A lower pound is good for business, not consumers, in the UK, leaving the EU is generally seen as a negative influence.


Software to simulate the effects of interest rates on an economy? Is there open source software that simulates the effects of interest rates on an economy?

Unfortunately, the answer is so broad I'm not sure how useful it may be.

First things first, you'd need a model for the economy. There are various around, in the last few years DSGE models like RBCM and NKM are popular. Pick the one that you think is the most appropriate.

After that, procure data from the past for calibrating the parameters of your model. The interest rate, of course, will be set by you.

Finally, you can use R, Matlab or Octave to simulate, given your model and parameters, a set of outcomes and provide a result for each interest rate.


Why is capital often not included in New Keynesian models? Is there a reason other than modeling difficulty? In New Keynesian models, like the ones in Gali's simple New Keynesian model or even Mankiw-Reis NK model on sticky information, capital is often not included.

Now people do say that there are modeling difficulties and that's why capital ($K$) is not included, but is there another justifiable reason other than modeling difficulties?

Capital is included in all the big estimated New Keynesian models (Smets-Wouters, Christiano-Eichenbaum-Evans), etc. But you're absolutely right that the stylized core NK model does not have capital - which is hard to defend on empirical grounds, since capital investment is a very important part of business cycle fluctuations and the response to monetary policy. Ultimately, the reason does basically boil down to the "modeling difficulties" that you mention.

First, there is an obvious way in which capital makes the NK model more complicated: at an absolute minimum, it introduces at least one additional backward-looking state variable $K$. In contrast, the two core equations (the intertemporal Euler equation and New Keynesian Phillips curve) of the ordinary log-linearized NK model are completely forward-looking. Adding $K$ to the mix eliminates this nice analytical feature. Still, on its own, this is not such a compelling reason to leave $K$ out of the standard presentation of the model, since the increase in complexity would still be tolerable and possibly justified by the added realism.

The additional complications that make capital much more difficult to include are the following.

Capital adjustment costs are needed to avoid absurd results. Suppose that there are no capital adjustment costs, and that firms rent capital each period on competitive markets. Suppose also that there are no shocks today. The real rental cost of capital today will be approximately $r+\delta$, where $r$ is the real interest rate that was expected yesterday and $\delta$ is depreciation. For each firm $i$, we have $MC(i)=(r+\delta)/MPK(i)$: real marginal cost equals the real cost of producing another unit of output by increasing capital.

Now, suppose for simplicity that firms are identical and there is no price dispersion. (Moving away from this, everything I say will still be approximately true.) Then we can eliminate the $i$ and just write $MC=(r+\delta)/MPK$. Furthermore, real marginal cost is just the inverse markup, so we can rewrite this in terms of markup $\mathcal{M}$ as $MPK=\mathcal{M}(r+\delta)$. Finally, if we assume that production is Cobb-Douglas with capital share $\alpha$ this becomes
$$\frac{K}{Y}=\frac{\alpha}{\mathcal{M}}\frac{1}{r+\delta}$$
Pick some reasonable parameters: say, $\alpha=0.3$, $\mathcal{M}=1.33$, $r=0.04$, and $\delta=0.06$. Given these we have $K/Y\approx 2.26$. Suppose that we exist in this steady-state world for a while, and then the Fed pushes down the projected path of interest rates such that the expected real rate declines to $r=0.02$. Then leaving $\mathcal{M}$ constant, the value of $K/Y$ given by the expression above increases to $K/Y\approx 2.82$. The NK model implies that this value should hold in the period after the shock.

This is an immense increase in $K/Y$, and the model tells us that it should happen in one period. If our period is a quarter, and $Y$ was not expected to take a sudden dive, then we'd have to invest well above the entire usual level of GDP to accomplish this. (In the continuous time limit, it becomes simply infeasible.) And although the assumption than $\mathcal{M}$ is constant is not right ($\mathcal{M}$ is determined endogenously in the NK model from the relationship of sticky prices to costs), relaxing this could easily make the puzzle more extreme: if $r=0.02$ is expected to prevail for a while, then it will imply higher-than-usual output $Y$ and lower-than-usual markups $\mathcal{M}$, which raise the implied $K$ even further.

The model simply doesn't work in this form: you need some form of capital adjustment costs. And these, of course, make the model still more complicated. (By the way, the problem here isn't so much the NK model as the fact that an assumption of no adjustment costs is generally absurd: seemingly small changes in the real interest rate must be accompanied by massive swings in the capital-output ratio, which we never see in practice. The NK model simply brings this absurdity, which is found in the basic RBC model as well, into sharper relief because exogenous interest rate shocks are such an important feature of the NK environment.)

Firm-specific capital is needed for strategic complementarity. Even if we fix the problem above by including capital adjustment costs of some form, we run into another awkward feature of NK models: taken alone, the Calvo price rigidity is not plausibly large enough to make the NKPC as flat as we think it is.

The most popular fix is some form of strategic complementarity, where firms try not to set prices too far from the aggregate price level. And the most popular way to get strategic complementarity is to assume that firms face both a high elasticity of demand and a steeply upward-sloping marginal cost curve. That way, for instance, any firm that sets its price too far below the average price will receive a flood of demand that causes its marginal cost to spike - and this discourages the firm from setting such a low price in the first place. (Yes, this sounds a little ridiculous, but it's how the models work.)

When the model excludes capital altogether, it's easy to just write a declining-returns-to-scale production function for each firm with labor as the only input. This makes each firm's marginal cost curve slope upward. But when we include both capital and labor in the model, the firm's production function probably should be much closer to constant-returns-to-scale. And this means, if the firm can rent any amount of capital from a competitive market on demand, that the firm's marginal cost curve is much closer to flat. This limits strategic complementarity.

To get around this, you need to dispense with the assumption of a common rental market for capital, and start talking about firm-specific capital accumulation. But then the model becomes much more complicated, and you're reduced to opaque quantitative exercises like ACEL.

Given all this, you can imagine how economists in insight-building mode often just dispense with capital altogether.


Edgeworth box and identical tastes If tastes are homothetic and identical, does this imply that the contract curve is the line connecting the lower left and upper right corners of the box?

My understanding: no, it does not. Consumers' indifference curves, though identical, could have the same slope along some line to the side of the line connecting the lower left and upper right corners of the box, e.g. in the case of steep indifference curves.

Is this correct?

Take the situation where $u_1(x,y)=x+y$, $u_2(x,y)=x+y$. Then the contract curve would be the entire box.


Proof that for increasing AC MC(Q)>AC(Q) at any Q>0? I know the "common sense" proof, but how can we prove it algebraically?

Let $C(y)$ denote the cost function and $y$ the quantity.

The average cost function is $AC=C/y$. This is increasing if its derivative is positive. Let $C'$ denote the derivative of $C$. The marginal cost function is $MC=C'$. We have:

$\frac{\partial AC}{\partial y}= \frac{C'y - C}{y^2}>0$ by using the quotient rule. As $y^2>0$, the condition becomes:

$C'y - C>0$.

Rearranging yields:

$C' > C/y$

Hence: $MC>AC$.


Find probability that payoff function is in $[10,20]$ In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by:

$$f=(\max_{t\in[0,T]} S_t -110)^{+}$$

where $S_t$ satisfies

$$dS_t=15dW_t$$
$$S_0=95$$

Find probability $P(f\in[10,20])$. Use cumulative density function of standard normal distribution to express it.

Obviously

$$S_t=95+15W_t$$

But i don't have any idea how can I move further. Any help please?

Generally:

$P(f\in[10,20]) = P(120 \leq S_t \leq 130) = P(S_t \leq 130) - P(S_t \leq 120)$
That is, the probability that the option is between 10 and twenty is the same that the stock is between 120 and 130. The probability that the stock is between 120 and 130 is the probability the stock is less than 130 minus the probability that it is less than 120. If you are a more visual person consider this picture:

Source MathWave

If we know that $W_t$ follows a driftless-Wiener process that:

$W_t - W_0 \sim N(0, 15^2t) \rightarrow S_t \sim N(95, 15^2t)$

If $S_t \sim N(95, 15^2t)$ then $P(S_t \leq 130) = $ prob. that a normal random variable with mean 95 and std. deviation 15t which is just the cumulative distribution function (CDF) of that normal random variable.

Here, that CDF is $\Phi( (x-95) / (15 \sqrt(t)))$, where $\Phi$ is the CDF of the standard normal. So the answer is $$\Phi( (130-95) / (15 \sqrt t)) - \Phi( (120-95) / (15 \sqrt t))$$

Update:
I thought that this was a simple option payoff but this is actually a trickier problem because it is a look back option taking the maximum stock value from 0 to 2. The general idea with this more complex problem is we are studying the maximum of a Wiener process motion rather than the Wiener process itself. Stochastic Processes and Related Distributions by Daniel Herlemont, section 2 should be of some help. But I don't have a solution to this problem at this time.


Price discriminating monopolist question A price discriminating monopolist sells in two markets. Inverse demand in market 1 is given by: $$P_1(Q_1) = 80 - (1/2)Q_1$$ and inverse demand in market 2 is given by: $$P_2(Q_2) = 100 - Q_2$$ The monopolist's cost function is $C(Q) = (Q_1 + Q_2)^2$
a. Formulate the monopoly's function as a function of $Q_1$ and $Q_2$.
b. Calculate the monopoly's profit maximizing quantity sold in markets 1 and 2 and the corresponding prices.
Now suppose the government forbids price discrimination, so the monopolist can only set a single price for the two markets.
c. Compute the monopoly price and quantity.
d. How much is sold in market 1 and market 2, respectively?
e. Was the government intervention beneficial for social welfare or not?

a. This one is straightforward. $$= Q_1(80 - Q_1/2) + Q_2(100 - Q_2) - (Q_1 + Q_2)^2$$
b. This one I believed to be straightforward - set MR = MC. Thus, $$MR_1 = 80 - Q_1$$ $$MR_2 = 100 - 2Q_2$$ $$MC_1 = 2Q_1$$ $$MC_2 = 2Q_2$$ which gives $$(Q_1, Q_2, P_1, P_2) = (\frac{80}{3}, 25, \frac{200}{3}, 75)$$
c. This is where I start to get a bit messed up. Setting $P_1 = P_2$ gives a total demand function $$Q_1 + Q_2 = Q = 260 - 3P$$ I then solve for price, and get $$MR = \frac{260}{3} - \frac{2Q}{3}$$ and $$MC = 2Q$$ Thus, I'm left with $$(P,Q) = (75.8333, 32.5)$$

However, I feel as though I must have made a mistake. The profit when the monopolist is able to price discriminate actually ends up being lower than the profit when limited to one price. This surely can't be the case - where is my thinking going wrong here? Is my methodology off?

MC = 2Q1 + 2Q2 for both demand functions. Set MR1 = 2Q1 + 2Q2, and MR2 = 2Q1 + 2Q1 and then you have a system of 2 unknowns and 2 equations. The solution ends up being Q1 = 15, P1 = 72.5, Q2 = 17.5, P2 = 82.5. In this case Profits under PD > Profits Under single price. See you in 468 tomorrow.


Optimal control theory: How to maximize Hamiltonian in this case? The problem is to maximize $\int_0^1 y(t) + u(t)^2 dt$ where $y$ is state and $u$ is control.

Further we have $y' = u, y(0) = 5$.

I set up the Maximum Principle equations, but, in particular, I need to maximize the Hamiltonian in the $u$-variable.

My nstructor's solutions does this by differentiating and letting it equal zero, i.e. he gets $$2u(t) + \lambda = 0.$$

Then he goes on with the other equations, and sovles for $\lambda$, and then $u$.

However, if we differentiate the Hamiltonian again with respect to $u$, then we get $$2 > 0$$
which is not negative, hence surely the 1st equation is not warranted, as we have found a minimum, not maximum?

Hamiltonian
\begin{align}
H(y(t),u(t),\lambda(t)) = y(t)+u(t)^2 + \lambda(t) u(t)
\end{align}

First order conditions read
\begin{align}
&H_u = 0 \quad \Longleftrightarrow \quad u(t) = -\frac{\lambda(t)}{2}\\[2mm]
&\frac{d\lambda(t)}{dt} = -H_y = -1\\[2mm]
&\lambda(1) = 0
\end{align}
Integrate costate
\begin{align}
&\int \frac{d\lambda(t)}{dt} = \int -1\\[2mm]
&\int d\lambda(t) = \int -dt\\[2mm]
&\lambda(t) = -t + C
\end{align}
with
\begin{align}
&\lambda(1) = -1 + C = 0 \quad \Longleftrightarrow \quad C = 1\\[2mm]
\Longrightarrow \quad & \lambda(t) = 1-t\\[2mm]
\Longrightarrow \quad & u(t) = \frac{t-1}{2}
\end{align}
Now integrate state equation
\begin{align}
&\int \frac{dy(t)}{dt} = \int \frac{t-1}{2}\\[2mm]
&\int dy(t) = \int \frac{t-1}{2}dt\\[2mm]
&y(t) = \frac{t}{2}\left(\frac{t}{2}-1\right) + C
\end{align}
with
\begin{align}
&y(0) =  C = 5\\
\Longrightarrow \quad & y(t) = \frac{t}{2}\left(\frac{t}{2}-1\right) + 5
\end{align}

If I'm not mistaken necessary conditions are sufficient if maximized Hamiltonian is concave in state (Arrow sufficiency if I recall correctly (see e.g. Caputo, 2005, Ch. 3))
\begin{align}
H^*(y,\lambda) = \max_u H(y,u,\lambda) = y - \frac{\lambda^2}{2}.
\end{align}
Now we have
\begin{align}
H_y > 0 = H_{yy}
\end{align}
such that concavity is given.


Computing pure strategy Nash equilibria in finite games I am trying to compute the (pure strategy) Nash equilibria of some discrete auctions.

More precisely, let us define the strategy of each player as a function mapping from every valuation that they might have to their bid (i.e. the 'bidding function'). Let us suppose that each player's valuation is drawn from some finite set and that their bid must belong to this same (finite) set. I am interested in finding the set of bidding functions, one for each player, such that each player's bidding function is optimal given the bidding functions of all the other players.

If it makes things easier, we can assume that valuations are symmetric (i.e. each player's valuation is generated by the same probability distribution) and that there are only two players. Ideally, however, we would proceed without these simplifications. I am interested in computing the equilibrium for the first price sealed bid and all pay auctions (in the latter, you pay your bid even if you lose; in the former, you don't.)

I have considered writing down the discrete auction game in normal form and finding the equilibria using software like Gambit. However, this would seem tricky since the strategy space is so large.  For example, if a player chooses bids from $ \{1,...,10\} $ and draws values from $ \{1,...,10\} $, then already they have $10^{10}$ pure strategies.

Does anyone have any ideas about how to proceed here?

If $10^{10}$ pure strategies is too large for Gambit, it'll likely be too large for any other software as well. In the comments, I suggested that you could first compute the expected payoffs for each player in a separate program such as Matlab/R/Excel, and then export the resulting matrix to Gambit where the BNEs can be calculated. This way, you may be able to save some resources for Gambit on converting the game into its Bayesian normal form.

If you're okay with significantly reducing the dimension of your problem, then you can rely solely on Gambit (or, Game Theory Explorer (GTE), which is now a part of the Gambit project). In the following example done in GTE, I assumed that bidder $i$'s valuation and strategy spaces are binary: $V_i=S_i=\{0,1\}$ for $i=1,2$. The values are assumed to be uniformly distributed.

First, you enter the game in its extensive form. In the figure below, the four branches at the top correspond to Nature's choices that determine the valuations. From left to right, the branches represent the valuation profiles $(v_1,v_2)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$ respectively, each occurring with probability $1/4$.

After you finish entering the extensive form game, press the "Matrix Layout" tab to see the (automatically calculated) Bayesian normal form. There's a gear button at the top that will compute the BNE of the game.

The result looks like the following. In this case, the result was returned in seconds (in part thanks to the relatively small dimension of the problem). The unique BNE, in which each player bids zero regardless of their value, was found.

You can of course try to increase the size of the value/strategy space. This method would also allow you to specify asymmetric distributions of bidder values. However, GTE only solves two-player games. But Gambit seems to be able to solve games with more than two players.


Convert USD2005 PPP per-capita GDP to USD2011 PPP per-capita GDP for a number of countries Which conversion factor / formula should be used to convert USD2005 PPP per-capita GDP to USD2011 PPP per-capita GDP for different countries?

I have found an identical question, but without a clear, satisfactory answer here:
Conversion of GDP PPP International \$ 2005 into GDP PPP International \$ 2011

Any real figure is based on a basket of goods, obviously, and here, we have a representative basket that is common to all countries across the world. When purchasing power parity prices are recalculated (as with any price index that is recalculated), the weighting of goods in the basket changes. Goods that have improved in technology will be more valuable than those same goods from a decade ago—this would push up the price index unless the weights on those goods are reduced, to reflect that the nature of those goods have changed, and that people buy them more parsimoniously.

So, PPP figures from 2011 will be based on a basket that contain multiple items inside it, which have different weights to the 2005 figures. Different countries produce goods at different levels of technology—some countries power ahead in technology, whilst others lag behind—so this causes the representation of the PPP basket to diverge from what is actually consumed in those countries (I imagine the PPP basket attempts to find the average representation). Thus, in the recalculated figures (from 2011) you will find different growth rates for the countries compared to the 2005 figures.

In short, you will need to know the weightings for all the goods in the baskets from 2005 and 2011 (and the levels of output in those categories for each country) in order to process the recalculation! It is not simply a matter of changing the base year, because the basket has changed.


regression method for data that contain a number of observation for several years I have a data set about 125 companies. For each company I have the salary and some other variables about the top 5 managers in each company.
One observation contains the top 5 managers in each one of the 125 firms  from 2010 to 2018, so every company has the data for its managers for each year.
For some companies some years are missed or it is for less than top 5 managers.
I want to check the effect of a regulation that occurred in 2016, and my question is :

Is it OK to do linear regression for the observations before 2016 at once ?,
I am not sure about that because every year has the same companies with the data of that year.

Can anyone tell me if it is Ok to do that or help by advising another regression method

Hang on - so, what is the unit of analysis in your data set?  Is it companies, or is it managers?  Think carefully about this.  How are the "top 5" ranked?  Does the cohort change frequently?  What variables are you interested in with respect to policy impact?

It's true that from a company view, you've got a panel, but if there's enough churn in the management roster you could justify a pooled approach.  In addition, if the missing data is non-random (e.g., cyclic, confined to certain firms/sectors/etc), then it would bias a panel approach, and a pooled regression may be the best you can do.

It's down to the particulars of your data, which only you have access to.


Help understanding definition of expected utility function This is the definition of an expected utility function that I'm working with:

The utility function $U:\mathcal{L}\to\mathbb{R}$ has an expected utility form if there is an assignment of numbers $(u_{1},...,u_{N})$ to the $N$ outcomes $\{\omega_{1},...,\omega_{N}\}$, such that the following is satisfied for any simple lottery $L=(\omega_{1},...,\omega_{N};p_{1},...,p_{N})\in\mathcal{L}$:
\begin{equation}\label{eq:vnm}
    U(L)=\sum_{i=1}^{N}p_{i}u_{i}.
\end{equation}
My question: Is it true that the assignment of numbers $(u_{1},...,u_{N})$ are given by the Bernoulli function $u:\Omega\to\mathbb{R}$, where $\Omega$ denotes the sample space. And is it true that $U(L):\mathcal{L}\to\mathbb{R}$ is called the Von Neumann-Morgenstern expected utility function, where $\mathcal{L}$ denotes the set of simple lotteries?

That's true, but it should be constructed the other way round: The assignment of numbers $u_i$ to outcomes is not given by the Bernoulli utility function but defines it. The terminology of Bernoulli and Von Neumann-Morgenstern utility functions is common, but not universally used in the context of EUT.


Where can I find the data for income, equity, assets and sales of a company that is presented in terms of U.S. and outside of US.? I need to find the income, sales, equity and assets of Microsoft, Google and Apple for U.S. and outside of U.S. I see these information in the income stamens as a whole. But I need to know how much it is for the U.S. and how much it is for outside of U.S. I've been able to see in the annual report of Microsoft such a division for income and sales. But the other two isn't there. None of those are also available in Google's annual report. Can somebody please let me know where should I look?

Have you tried the Mergent database?  If you're part of a University system, Mergent is a typical offering through University libraries.  Your University librarian should be able to help you find and navigate it if necessary.


Does returns to scale always lead to economies of scale? I can tell the difference between returns to scale and economies of scale, but then I still don't know if returns to scale ALWAYS leads to economies of scale?
Please could you help me?

Excellent question (I am assuming your intended question is "does increasing returns to scale always leads to economies of scale"):
The two concepts are related but Returns to scale (RS) is much restrictive than the Economies of scale (ES).
The concept of RS is embedded in production function. If $Q=F(K,L)$ then increasing returns to scale simple means:
$$F(\alpha K, \alpha L) > \alpha F(K,L)$$
For example, in Cobb-Douglas production function: $Q=AK^aL^b$, we have that
$a+b>1 \implies \text{increasing }RS$
The concept of ES is much, much wider and goes beyond production function. All it says is that average cost (AC) decreases with $Q$:
$$\frac{dAC}{dQ}<0$$
Notice the use of $d/dQ$ instead of $\partial/\partial Q$. This is where everything changes. In partial derivative we are interested in the mathematical relationship that: ceteris paribus whether changes in quantity produced changes cost.
To illustrate the relationship and the difference consider derivation of $C=f(Q)$ using the Cobb-Douglas production function:
Given, wage rates $w$ and cost of capital $r$:
\begin{align}
C=wL+rK
\end{align}
To first solve for output maximization constraint by cost function:
\begin{align}
\max_{L,K} \;{AK^aL^b} \;\; s.t \;\; wL+rK=\bar{C}
\end{align}
Solving the langrangian would give us:
\begin{align}
K=\frac{bw}{ar}L \tag{2}
\end{align}
Substituting $(2)$ in production fuction gives us:
\begin{align}
Q=A\bigg(\frac{bw}{ar} \bigg)^bL^{a+b} \tag{3}
\end{align}
Re-arranging $(3)$ and getting $L$ in terms of $Q$ and then substituting it back to $(2)$ gives us:
\begin{align}
L=\bigg(\frac{ar}{bw}\bigg)^{b/(a+b)} \bigg(\frac{Q}{A}\bigg)^{1/(a+b)} \tag{4}
\end{align}
\begin{align}
K=\bigg(\frac{bw}{ar}\bigg)^{a/(a+b)} \bigg(\frac{Q}{A}\bigg)^{1/(a+b)} \tag{5}
\end{align}
Substituting $(4), (5)$ in $(1)$:
\begin{align}
C=\eta \cdot w^{\frac{a}{a+b}} \cdot r^{\frac{b}{a+b}} \cdot Q^{\frac{1}{a+b}}
\end{align}
where $\eta$ is a constant in terms of $a$ and $b$.
For average cost:
\begin{align}
\frac{C}{Q}=\eta \cdot w^{\frac{a}{a+b}} \cdot r^{\frac{b}{a+b}} \cdot Q^{\frac{1-(a+b)}{a+b}} \tag{6}
\end{align}
Now you see if there are increasing returns to scale, i.e., $a+b>1$,
$$\frac{\partial AC}{\partial Q}<0$$
So you see, if $w,r$ can be taken as constant then of course
$$\frac{\partial AC}{\partial Q} = \frac{dAC}{d Q}  <0$$
On the other hand, this is rarely true. In a general equilibrium model, $w$ and $r$ are also variables. For instance, say the Labor and Capital market is perfectly competitive:
$w=MP_L\equiv \frac{\partial Q}{\partial L} = a\frac{Q}{L}$ and similarly, $r=MP_K\equiv \frac{\partial Q}{\partial K} = b\frac{Q}{K}$
Substituting these in $(1)$ (or equivalently in $(6)$), we get:
$$\frac{C}{Q}=(a+b)$$
So interestingly, the average cost is constant despite having increasing returns to scale.
So, increasing RS can at best ensure:
$$\frac{\partial AC}{\partial Q}<0$$
But what ES requires is:
$$\frac{dAC}{dQ} = \frac{\partial AC}{\partial w}\frac{\partial w}{\partial Q}+\frac{\partial AC}{\partial r}\frac{\partial r}{\partial Q}+\frac{\partial AC}{\partial Q}<0$$
So it is perfectly possible that $\frac{\partial AC}{\partial Q}<0$ but $\frac{d AC}{d Q}>0$


Pareto efficiency (optimality conditions) in simple New Keynesian model I am looking for the pareto-optimal equilibrium for a central planner's problem in a simple New Keynesian model. The planner's problem is to choose $\{ C_{t}, H_{t}, Y_{t}, \pi_{t}, \{h_{t}(j)_{j=0}^{\infty} \}, \{y_{t}(j)_{j=0}^{\infty} \} \}_{t=0}^{\infty}$ to maximize $$ U_{0} = E_{0} \sum_{t=0}^{\infty} \beta^{t}u(C_{t}, 1-H_{t}) $$

subject to: $$ C_{t} = (1-\frac{\emptyset}{2}\pi_{t}^{2})Y_{t} $$ $$ Y_{t} = \ [\int_{0}^{1}y_{t}^{\frac{\epsilon -1}{\epsilon}}(j)\;dj ] ^{\frac{\epsilon}{\epsilon-1}} $$ $$ y_{t}(j) = A_{t}h_{t}(j) $$ $$ H_{t} = \int_{0}^{1} h_{t}(j) \; dj $$

The first-order conditions for this problem imply that for all $j$ and $t$: $$ \frac{u_{l}(C_{t}, 1-H_{t})}{u_{c}(C_{t}, 1-H_{t})} = A_{t} $$ $$ y_{t}(j) = Y_{t}$$ $$h_{t}(j) = H_{t} $$ $$ Y_{t} = A_{t}H_{t} $$ $$ \pi_{t} = 0 $$ $$ C_{t} = Y_{t} $$

To arrive to the above implications I tried setting up the following Lagrangian:
 $$ L_{0}(j) = E_{0} \sum_{t=0}^{\infty} \beta^{t} u(C_{t}, 1-H_{t}) + \lambda_{t}[(1-\frac{\emptyset}{2}\pi_{t}^{2})Y_{t} -C_{t}] + \mu_{t}[\ [\int_{0}^{1}y_{t}^{\frac{\epsilon -1}{\epsilon}}(j)\;dj ] ^{\frac{\epsilon}{\epsilon-1}} - Y_{t}] + \omega_{t}[A_{t}h_{t}(j)- y_{t}(j)] + \alpha_{t}[H_{t} - \int_{0}^{1} h_{t}(j) \; dj ] $$

to obtain the FOCs:

$$ \frac{\partial L_{0}(j)}{\partial C_{t}}: u_{c}(C_{t}, 1 - H_{t}) - \lambda_{t} = 0    $$
$$ \frac{\partial L_{0}(j)}{\partial H_{t}}: -u_{H}(C_{t}, 1 - H_{t}) + \alpha_{t} = 0 $$
$$ \frac{\partial L_{0}(j)}{\partial Y_{t}}: \lambda_{t}(1 - \frac{\emptyset}{2}\pi_{t}^{2}) - \mu_{t} = 0$$
$$ \frac{\partial L_{0}(j)}{\partial \pi_{t}}: -\emptyset \pi_{t} Y_{t} \lambda_{t} = 0 $$
$$ \frac{\partial L_{0}(j)}{\partial h_{t}(j)}: \omega_{t}A_{t} - \alpha_{t} = 0 $$
$$ \frac{\partial L_{0}(j)}{\partial y_{t}(j)}: \mu_{t}[\int_{0}^{1} y_{t}^{\frac{\epsilon -1}{\epsilon}}(j)]^{\frac{1}{\epsilon -1}} y_{t}^{\frac{-1}{\epsilon}}(j) - \omega_{t} = 0 $$

Now, I have no idea what I am doing wrong or how to get to the implications above. Any help would be very much appreciated!

The Lagrangian is also a function of the multipliers $(\lambda_t, \mu_t, \omega_t, \alpha_t)$ so you are missing their FOC's (these will give you the constraints as FOC's so if you were already taking them into account, omit my comment, please). Also, $-u_{H}=u_l$ since $l=1-H$ (I would re-write the second FOC, in terms of $u_l$ in order to construct the first "target" equation that has the MRS=MRT. Finally, there are a couple of useful facts that should help you. First, note that in your 4th FOC, you have a product equal to zero, so at least one of the elements is equal to zero. Try to convince yourself, that the only possibility (so that you don't run into a contradiction with the other FOC's) is for $\pi_t=0$ as required by your second-to-last target equation, and in turn implies $C_t=Y_t$ (from your first constraint). Second, using the second constraint you can replace the integral in your last FOC with $Y_t$.

This should help for further simplifying the 10 FOC's you should have, into the 6 target equations you want.


Uses of loans during recession I've noticed that certificates of deposit have dropped their interest rates in my country. This seems to be due to lack of trust in economic opportunities and has caused an increase of liquidity in banks that has not been able to be released via loans.

This leads me to think that loans will be favored, although interests rates for them are actually high due (I suppose) to an inherent risk in the local economy and therefore the typical loan is small ($\sim 3000 $ $). So, what are the typical purposes --other than immediate consumption-- that these type of loans have for households (i.e. not for starting a new business) as a whole?

Let me rephrase my question: what incentive would there be for families to borrow during a recession?

Because of the recession some members of the household may have lost their jobs. So maybe they don't have enough money to pay a utility bill? To pay for urgent repairs? To buy a Christmas present? There are a variety of possible reasons.


What is the opportunity cost of doing something in your spare time? My university is using this textbook. The text clearly states, “Economists use the term opportunity cost to indicate what must be given up to obtain something that is desired.”

This leads me to believe that if you are a salaried worker who makes 50 dollars per hour and works a standard five-day workweek, the opportunity cost of you mowing your lawn during the weekend is 0 dollars. You are giving up your unpaid free time to mow the lawn. When I brought this up in the online discussion I was told by the instructor, "the cost of your free time is estimated using your wage rate."

Is free time = wage rate just the standard convention? This hardly seems appropriate since you aren't giving anything up other than your free time. The situation would be different if you were giving up weekend work to mow your lawn.

There is a crucial detail missing in the scenario you describe.  Is the worker working a five day week (and no more) because that is what their terms of employment require?  Or do they have the option of doing additional paid work (at the same rate, and for as many hours as they wish) at weekends, but on this occasion have opted to work only five days in the week?  If the latter, then it is clear that the opportunity cost of mowing the lawn is the value of the time taken at the wage rate.

If however, the worker's paid work is limited to five days in the week, then it is not paid work that is being given up in order to mow the lawn at the weekend. Although the concept of opportunity cost seems clear enough, its application in this situation can be quite difficult, because it may be hard to identify what exactly has been given up. Suppose (as suggested in the comment by 1muflon1) that an alternative use of the time was to watch a movie. Fleshing out this scenario, suppose that late on Sunday afternoon, the worker was just about to watch the movie, but suddenly noticed that the grass was getting long, and decided that it couldn't be left for another week, so instead went to mow it. So far, it seems that what has been given up is watching the movie. But suppose that, having mowed the grass, they decide to watch the movie in the evening, displacing some other activity.  Rather than a clearly identifiable alternative that has been given up to mow the lawn, there may be a whole series of activities, each of which has merely been slightly postponed to give priority to another.

A context in which economists make use of the opportunity cost of leisure time is the travel cost method for estimating the value of non-market environmental goods such as parks. In this context the value of leisure time has often been taken to be a fraction of the wage, an influential source explaining the basis for such an assumption being Cesario (1976).  Use of a fraction of the (average) wage can be a pragmatic way to avoid the complexity of identifying what an individual has given up, and for aggregation over individuals in different circumstances.

Reference

Cesario, F J (1976) Value of Time in Recreation Benefit Studies  Land Economics 52(1) Feb 1976 pp 32-41


Capital mobility across industries and equalization of profit rates What are some restrictions to the competitive restoration of proportionality across industries in regards to capital mobility? What is generally impeding the free flow of capital from less profitable sectors into those with higher profitable potentiality and from eroding larger firms market share? Also does the integration of finance capital and industrial capital play any role in this? Lastly, why isn't new capital entering these markets and averaging out profit rates?

This is a massive area of research, revived recently under the title of "capital misallocation".

Some reason why capital do not fully adjust:

grow in intangible capital-intensive sectors (which require more internal financing than other sectors) + low interest rates.
quadratic adjustment costs on investment.
financial frictions lowers access to credit to small yet highly productive firms.
industry-specific physical and human capital.

The are many more papers studying financial frictions and related aspects of the financial market, showing that the role of finance is very important.


Best response correspondences game theory I'm studying game theory and there is a slide where I don't quite understand how exactly each value was deduced particularly:

For $BR_L(q)$ and $BR_c(q)$. I understand why they are functions of $q$ and $p$, since the best response of one player is dependent on what strategy the other player will choose, but i'm not sure exactly why they take values of $1$ and $[0,1]$ and $0$ for the specific values of $q$ and $p$. If someone could explain the derivation of these, that would help a lot!

Let $\pi_L(p,q) = 3pq + p(1-q) + 2(1-p)(1-q)$ denote the expected payoff of the lecturer. The best response is defined as $BR_L(q) = \arg\max_{p \in [0,1]}\pi_L(p,q)$.
Now, note that
\begin{align}
\frac{\partial \pi_L(p,q)}{\partial p}
\begin{cases}
> 0 \quad& \text{for } q > 1/4,\\
= 0 & \text{for } q = 1/4,\\
< 0 & \text{for } q < 1/4.
\end{cases}
\end{align}
For $q>1/4$ the payoffs are strictly increasing in $p$ such that $p = 1$ is optimal.
For $q=1/4$ the payoffs are constant in $p$ such that any $p \in [0,1]$ is optimal.
For $q<1/4$ the payoffs are strictly decreasing in $p$ such that $p = 0$ is optimal.
Same logic applies to The Catalan.


how to measure the funding liquidity, i.e., equivalent U.S. TED spread, in the Australian market? I'm stuck on how to estimate the credit (counterparty) risk in interbank lending in the Australian market. TED spread is usually measured as the difference between the LIBOR rate and the T-bill rate. However, when I use the same framework to calculate the Australian TED spread, it is often negative. I use two methods:
1. 3-month BBSW (bank bill swap rate) -  3-month bank bill rate (or cash rate)
2. 3-month LIBOR in Australian Dollar - 3-month bank bill rate (or cash rate)

The negative value definitely makes no sense as the former should always be larger than the risk-free rate.

Can anyone illuminate how this works? or did I use the wrong proxy?

it is measured by the swap spread, the difference between the 3-month bank bill swap rate (BBSW3M) or 3-month bank bill rate and the 3-month overnight indexed swap rates (OIS3M) captures the inherent bank credit risk and therefore funding liquidity in the inter-bank lending market.


What is the game theory of retaliatory trade tariffs? Suppose no tariffs, or negligible tariffs, exist between nations A and B, and suddenly A introduces substantive tariffs on certain products A imports from B. In theory, this could lead to B and A taking turns imposing tariffs on specific products they import from each other. Given the economic benefits of tariff-free trade, this scenario requires

(i) the optimum tariffs in one direction to depend on those in the other and

(ii) the relationship between these to bear in mind expectations of how (i) would lead to a feedback.

I'm sure economists have tried to quantify this (possibly with an eye to each nation's alternative sources for goods from third parties), but I know nothing about the relevant literature. So what do models predict?

Indeed it's difficult to account for all the real-life complexities, but the basic game-theory model of trade wars is prisoner's dilemma, e.g.

https://leadersatwork.northeastern.edu/management/trump-tariffs-and-the-prisoners-dilemma/

  The concept of the Prisoner’s Dilemma is important to a number of business school subjects, including economics and negotiations. The current Trump administration provides a rich opportunity to explore this concept in the context of important policy deliberations. [...]

  This also happens between countries and is playing itself out in many of President Trump’s recent actions involving tariffs. Bilateral agreements between the U.S. and its trading partners confront the “Prisoner’s Dilemma” when it comes to the issue of tariffs. Consider when the President said on March 8, 2018, “he” will impose tariffs on steel and aluminum imports within 15 days. Tariffs represent a situation where the “rational” decision would appear to be to raise tariffs on imports and hope the other side either doesn’t or cannot. The “rational” decision typically leads to retaliation as it does in many prisoner dilemma situations.

  Of course, President Trump doesn’t see tariffs as posing a prisoner’s dilemma type problem, at least in his public pronouncements: “When a country (USA) is losing many billions of dollars on trade with virtually every country it does business with, trade wars are good, and easy to win. Example, when we are down $100 billion with a certain country and they get cute, don’t trade anymore-we win big. It’s easy!” (March 2, 2018 Twitter)

  Not surprisingly, other countries threaten to retaliate. China, Canada, and the European Union have already said they would respond by enacting tariffs that could result in extensive American export losses affecting industries and farms that the President had claimed to protect. Most economists feel that a trade war is disruptive and will make the world as a whole poorer.

  Perhaps the President sees international trade as similar to the construction industry; i.e., as a forum where he can exert power and his will on the other party. What works in the construction industry where Trump gained his negotiation experience may not work in the very different world of international diplomacy. The President might have had leverage in dealing with contractors. But in international trade, the U.S. accounts for 9 percent of world exports and 14 percent of imports. This would not appear to be a position of domination.

  Of course, another characteristic of President Trump’s negotiation style is to stake out an extreme position (anchor) in order to both intimidate an opponent and to give room to make concessions or compromise, and this may be the case with his tariff pronouncements.

http://blogs.cornell.edu/info2040/2018/09/20/the-trade-war-between-the-u-s-and-china-a-prisoners-dilemma/

  In his [FT] article “Trade wars and the prisoners’ dilemma,” Gavyn Davies draws connections between this tit-for-tat tariff game and the classic example of the prisoners’ dilemma.

  Davies highlights the fact that, if the U.S. and China are truly engaged in a non-cooperative, prisoner’s-dilemma style game, they will likely end up in a “bad Nash equilibrium,” where both countries’ payoffs are worse than if they were to simply cooperate with each other. Specifically, assuming that each country continues to choose strategies that protect its best interests in a time of uncertainty, it’s likely that there will be 1-3% reduction in global output in the next several years.

  In its entirety, Davies’ article demonstrates that the game theory logic underlying the prisoner’s dilemma is omnipresent. Almost every economist and investor examining the impacts of the budding trade war has assumed that the two countries will continue to retaliate against one another without cooperation. Their models paint a dismal (albeit realistic) picture of this trade war’s global repercussions. Still, economists and investors might be better off employing the same game theory logic to illustrate the mutually beneficial benefits of cooperation. Perhaps the prisoner’s dilemma will help to show how cooperation, not just among inmates, but among countries can lead to better payoffs than self-preserving strategies.

https://www.bloombergquint.com/global-economics/game-theory-explains-why-the-u-s-china-trade-spat-is-worsening

  Experts surveyed by Bloomberg News lay out four scenarios for how the U.S. trade conflict with China could end: Both sides back down, which now seems unlikely in the short term; China blinks; the U.S. blinks; or both sides keep escalating.

  The “China blinks” scenario assumes President Xi Jinping won’t want to endure a downturn in the Chinese economy, which showed signs it underperformed in May. The “U.S. blinks” scenario assumes China calls Trump’s bluff, knowing how much he enjoys a strong economy, rising stock market, and the support of voters in farm states that China could target.

  The fourth scenario, in which neither blinks, is the most damaging. “We’re not there yet, but it’s scary, because it seems like we’re on a path toward major conflict, and it’s hard to see the offramp,” says Michael Smart, managing director at Rock Creek Global Advisors LLC in Washington and a former international trade director on the National Security Council.

  The World Trade Organization was created to prevent exactly this kind of bluffing and brinkmanship. “Trade agreements are a way of escaping from a prisoner’s dilemma in which each country acting rationally is stuck doing something that is bad for it individually, but they can’t get out of it without a collective agreement,” says Dartmouth College economist Robert Staiger.

  The Bagwell-Staiger Theory of Trade Agreements and the "Reciprocity Approach"

The WTO literature seldom fails to mention prisoner's dilemma (avoidance), e.g.:

  At the latest since the writings of David Ricardo and John Stuart Mill the welfare-enhancing qualities
  of international trade have been common wisdom in the realm of economics (Irwin, 1996). Reality,
  however, paints a different picture: Countries are often reluctant to open up their borders to foreign
  goods and services and unwilling to liberalize trade unilaterally. Yet, there are plenty of examples that
  countries have overcome what appears to be a mercantilist and protectionist stance and engage in
  bilateral or multilateral trade agreements. What is the rationale for such cooperation? Why do
  sovereign countries contract over trade issues?

  Starting with the seminal work of Harry Johnson (1953), economists have strived to address this
  question. The basic intuition that has guided much of the literature is that countries cooperate in an
  effort to constrain unilateral “beggar-thy-neighbour” policies. Trade protection by one country may
  reduce the welfare of trading partners; it provokes negative externalities, or spill-overs.
   The strategic
  set-up of a prisoners’ dilemma emerges. Excessive trade protection, albeit inefficient, becomes the
  dominant strategy for importing countries. The purpose of a trade agreement is to eliminate the
  inefficient restrictions on trade volumes that arise when policies are set unilaterally, and thereby offer
  governments a means to escape the prisoners’ dilemma.

  Theories of trade agreements based on externalities come in two variations.
   The best-known and
  most elaborate theory of trade cooperation is based on a terms-of-trade driven prisoners’ dilemma that
  (large) countries overcome by means of concluding an international trade contract. The terms-of-trade
  school contends that it is solely the ability of economically large countries to influence world prices
  for goods and services – and their potential to actively deteriorate other states’ terms-of-trade - that
  motivates governments to conclude trade agreements. Large countries realize that the unilateral
  setting of import tariffs creates inefficiencies that can be avoided by mutual trade cooperation. This
  theory is represented most prominently by Kyle Bagwell and Robert Staiger. Their research
  programme is the only one that formally integrates an explanation of why countries cooperate on trade
  matters with an explanation of how this can be done. In other words, it currently is the only approach
  in economics that can explain both the existence of the multilateral trading system and its architecture.

  The political externalities strand of the literature, which gained prominence through the writings of
  Wilfred Ethier (2004; 2004a; 2006), is critical of the terms-of-trade approach. It alleges that “a trade
  agreement serves governments to get credit for the reduction in foreign trade barriers. International
  cooperation is not about the elimination of economic (world-price) externalities, but about political
  externalities. The latter arise when politicians in one country believe that their political status is
  directly affected by actions of politicians of another country” (Hauser and Roitinger, 2004: 652).

  Different as these rationales for trade agreements may be concerning possible policy implications,
  their basic intuition is identical: Both schools rely on a simple game set-up, where two or more
  rational players are faced with a prisoners’ dilemma situation.

(Avoiding) prisoner's dilemma also applies to subsidies, not just tariffs:

  We begin our investigation of the role for retaliation by looking at export subsidies. Bagwell and Staiger
  (2001b) show how a subsidy agreement that limits government payments and avoids subsidy escalation
  can be of value to governments of exporting firms. Without such an agreement, each government is
  tempted to subsidize its exporters so as to create a competitive advantage in a third market – i.e., a
  prisoners' dilemma problem.

It's actually not clear to me if a more formal turn-based "beggar thy neighbor" rather than the simpler simultaneous version (prisoner's dilemma) has received much attention game-theory wise.

Since trade wars are back in fashion, perhaps we'll see more theoretical literature dealing with how they happen rather than how they are avoided...

The tit for tat strategy and the associated iterated prisoner's dilemma have been studied theoretically quite a bit; perhaps there's more literature relating them to trade wars...

Actually there is some literature from three decades ago how trade wars start and iterate, but it seems mired in controversy as far as modelling goes; the hypothesis  of rent-seeking by internal pressure groups adds another dimension to simpler model of the (external) trade war; this conflicting issue of modelling hasn't been fully resolved as of then:

  In Trade Wars, Professor John A.C. Conybeare, a political scientist,
  employs a game-theory approach in an attempt to formalize a predictive
  theory for when trade wars will occur.  While Conybeare praises the
  rent-seeking literature for rendering trade wars "more theoretically and
  empirically understandable," he finds the interest group explanation
  conveyed in that literature to be incomplete because it "has largely been
  confined to examining the interindustry structure of protection within
  countries and, with a few exceptions, has not made international comparisons." In other words, Conybeare takes the position that, while interest
  groups may set the internal political agenda within a particular country,
  no interest group explanation of trade wars can be complete without an
  understanding ofthe constraints facing that country due to its bargaining
  power vis-a-vis other nations. [...]

  In Chapter Five, he looks at the Anglo-Hanse trade wars to test his
  assertion that bilateral trade wars between large trading units will result
  in a prisoners' dilemma that will evolve into a cooperative equilibrium as
  the two sides learn to employ contingent retaliation. Unfortunately, four
  hundred years worth of trade-a sufficient time span to produce iterative
  opportunities for even the most exacting game theoretician-did not
  produce cooperation. By the author's own admission this was because
  "domestic rent-seeking in conjunction with a severe public good problem in the use of the tit-for-tat strategy of retaliation" caused "outbreaks
  of conflict to escalate."

  In Chapter Six, Professor Conybeare describes the Anglo-French
  trade wars beginning in 1664. These trade wars are used to illustrate the
  author's hypothesis that trade wars between two large countries should
  evolve into a cooperative interaction after some initial defection because
  such wars resemble iterative prisoners' dilemmas.3o Contrary to Professor Conybeare's general hypothesis, the trade wars lasted for over two
  hundred years and did not evolve into a cooperative equilibrium at the
  pace his model predicts. As with the Anglo-Hanse trade wars, the author
  is forced to admit that his own example "illustrates the ease with which
  the effects of iteration may be' disrupted." Once again, the cause of a
  protracted trade war can be traced to domestic rent seeking. [...]

  The Chicken Wars began because of Regulation 22, promulgated in
  July, 1962, which curbed the export of frozen chickens into EEC countries. The Chicken Wars brought the powerful agricultural lobby in
  France into direct conflict with the powerful agricultural lobby in the
  United States. Not surprisingly, France, the largest poultry producer in
  the EEC, was unwilling to embrace a free trade policy whose benefits
  would be spread among consumers throughout Europe. The benefits of
  protectionism were concentrated in France and the costs were borne by
  consumers throughout Europe. Indeed, the stakes were sufficiently high
  that France might have quit the EEC had not French agricultural interests been appeased.

So a fuller (game theory) model of trade wars seem to need to account for internal groups as different players rather than treat countries as single entities. I don't know if more progress on this has been made since that book.

A somewhat more recent paper (only two decades old) on the US-Canada wars from more than a century ago:

  The classic analysis of retaliation with optimal tariffs by Johnson (1954) is shown to
  have empirical validity. The pattern of US–Canadian tariff interactions in the period
  1868–70 shows evidence of a retaliatory adjustment process of the kind described by
  Johnson. Both countries perceive they have market power and therefore think they
  can gain from applying retaliatory optimal tariffs. In the short run, changes in the tariff
  level of each country are seen to be Granger-caused by changes in the tariff of the
  other. An interpretation of that result is that a tariff innovation in one country due to
  exogenous political economic forces in one country, say a highly influential protection-
  ist lobby, is met with retaliation by the partner country which seeks to reverse the
  initial increase. This leads to a tariff war. The Granger causality tests capture the short-
  run reactions that give credence to this story.

  What is more interesting, however, are the questions about the long-run steady-state
  solution after the retaliation war ends. In the long run, the USA maintains and even
  increases the tariff shock that it initiates, while the Canadian policy tends to back down
  from the tariff shock that it initiates. This is consistent with the Kennan–Reizman
  hypothesis that big countries win tariff wars, no matter who starts it. A second long-run
  result is that the Canadian tariff demonstrates a permanent change following a US
  tariff shock; that is, the cross-country effect is positive. On the other hand, in the long
  run, the US tariff returns to its own internal equilibrium (retaliation component is
  significantly no different from zero), unaffected by innovations in the Canadian tariff.
  This result is consistent with Chan’s (1991) model of bargaining which emphasizes
  the availability of outside options. Since the USA has greater outside options, it
  chooses not to participate in a costly retaliatory war. However, since Canada is highly
  trade-dependent on the USA, its outside options are limited and the best it can do is to
  retaliate in an attempt to bring down the US tariff increase.

  Interestingly, an additional inference that may be made from the study is not one
  about retaliation, but about internal political economic factors. While Canadian retali-
  ation does not succeed in reversing a US tariff increase due to political economic
  factors, US retaliation is shown to succeed in at least partially reversing a tariff
  shock caused by political economic factors. Hence, not only is the size and lower
  trade dependency of the USA vis-à-vis Canada beneficial in non-cooperative or bar-
  gaining games, but the internal political economic factors in the USA seem to be more
  stubborn in the face of retaliation than are political economic factors in Canada. It was
  perhaps in the best interest of Canada to establish the Canada–US Free Trade Agreement in which the political importance of the issue was far greater in Canada than in
  the USA.

I have to say however that Chan's model (although based on game theory) seems pretty obscure; the paper only has 10 citation in Google scholar. The Kennan-Riezman hypothesis (that big countries win trade wars) seems a lot more discussed, with a few hundred citations; but it doesn't seem primarily based on based on game theory; it uses an Edgeworth box
 analysis to draw its main conclusion as to the bargaining power in a trade war.

A highly-cited 1993 paper by Grossman and Helpman does attempt to model national-level interest groups: it includes not only the global country utility but also (domestic) political contributions in the government's utility function when doing trade negotiations.

I think this is still pretty limited, since it only considered campaign finance contributions as part of the latter. Potential for votes is not distributed among groups, but only modelled via the general country welfare utility function. But even this results in a pretty complicated analysis. Only by the 34th page in their paper they get to introduce the competitive model including foreign governments, which is not developed a lot. The paper notes/conclues the obvious trade-off that a national government has to face between industry lobby groups and general population welfare with respect to tariffs, and also (obviously)

  Tariff rates are highest in industries with the largest stakes in trade policy (i.e., greatest output) and in industries where foreign export supplies and mohme import demands are most inelastic. [...]

  Industry groups that are politically stronger than their counterparts abroad will secure exceptions from across-the board tariff cuts or from reductions in the levels of export promotion.

But practically all of this involves equilibrium analyses rather than much game theory; there are no (game) iterations involved in this paper. These two authors have an even more highly cited paper title "protection for sale" which elaborates on the costs borne by lobby/industry groups.


What precisely is scarcity? I'm aware that certain terms in economics have precise definitions that don't necessarily line up with our intuition about what they mean. I'm currently writing a paper that heavily relies on the concept of scarcity, and I'd like to make sure I am using the term properly. Currently I have this sentence written:

  Suppose widgets are a scarce resource that are distributed among
  agents

Then I rewrite this mathematically as $\mu = \sum\limits_{i=0}^n x_i$ where $x_i$ is the number of widgets agent $i$ has. I don't see any problems with this myself, but I'd like to make sure that economists view this as orthodox. Are there any references that I could use to verify this? Is this accurate? What is the precise definition of scarcity?

Edit

A Dictionary of Economics states scarcity is: "The property of being in excess demand at a zero price. This means that in equilibrium the price of a scarce good or factor must be positive" This is the most authoritative source I've found so far, but this doesn't seem like a rigorous definition to me - it clearly isn't mathematically defined. Am I wrong about this?

Your formula $\mu = \sum\limits_{i=0}^n x_i$ just denotes the total amount of goods, so this just means that $\mu$ units were distributed among $n$ agents. Scarcity (in this context) means that no matter how you distribute these $\mu$ units, as long as everything else is unchanged (ceteris paribus) at least one agent would like to get more, i.e. $\forall (x_i)_{i = 0}^n$ where $\mu = \sum\limits_{i=0}^n x_i$ there is an agent $i$ and a positive $\epsilon$ such that
$$
x_i \prec_i x_i + \epsilon.
$$
This is context dependent, there may be other definitions as well (e.g. ones that allow for production).


Should we get rid of the Penny? Jeff Gore has suggested scrapping the 1 cent coins a few years ago. Why doesn't the US cancel the pennies? Is it that terrible to the economics to keep the copper coin?
As mentioned in the comments, Greg Mankiw has also discussed it here:

Per year, the average citizen wastes 12 minutes paying with pennies.
That’s like half of an episode of The Office. Twelve minutes is also
the length of the first human spacewalk by Soviet cosmonaut Alexey
Arkhipovich in 1965.
Mankiw also estimated this 12 minutes of wasted time costs the U.S.
economy approximately $1 billion every year.

Should we get rid of the Penny?

No. That would be as foolish as suppressing a symptom instead of addressing the illness.

The reason why "a penny does not buy a penny" --or being more accurate, why it no longer buys a penny-- is that currency depreciation has been policymakers' answer for too long and for too many issues.

The permanent, excessive printing of money inevitably weakens the purchase power of the USD. Metal prices, like everything else, go up where economic problems are "addressed" by issuing more bonds, giving more subsidies, ignoring the rapidly changing demographics, and pretending that welfare state still is sustainable.

The weakening of a currency by inflationary measures is not limited to rendering coin metals out of reach. At some point, even the cost of paper on which money is printed can also exceed the purchase power of the denominations printed on that paper. History has many examples of this.


How to calculate the minimun of Average Fixed Cost knowing the Average Fixed Cost Given the Average Fixed Cost, how does one calculate its minimum?

As the quantity of output of a good increases, fixed cost (by definition) remains the same and therefore average fixed cost per unit of output continuously decreases.  Therefore the average fixed cost curve does not have a minimum, except in the sense that it is asymptotic to zero.
A much more important concept than average fixed cost is average total cost, that is, the average of the sum of fixed and variable costs.  This often has a minimum, but can be continuously decreasing when there are significant economies of scale.