Why hyper-inflation? I understand the $MV=PQ$ identity. I also understand the benefit for the government in some cases to having non increasing over time (even if high) values of $\Delta P$ (stealth tax & seigneur-age & real gdp shock,...).

What I do not understand is in what cases would the benefits (for the government) of increasing rates (second derivatives) of inflation out weight the costs?

Because it seems to me many countries (Venezuela, Zimbabwe, apparently Iran right now) could have --yet did not-- hike short term rates (to reduce $\Delta M$ and $\Delta V$) when inflation ($\Delta P$) was going from 'high but stable' to 'out of control' (systematically increasing $\Delta P$).

How central banks raise interest rates

This is a common misunderstanding of central banking.  The popular media portrays things as if the act of increasing interest rates prevents hyperinflation.  In truth, "preventing inflation" and higher interest rates are both side effects of restraining the growth of the money supply.

What central banks set is not generally interest rates but a target for the interest rate.  In the United States, this is called the federal funds rate target.  Let's break this down.  Funds are simply money loaned from one bank to another, generally on an overnight basis.  The funds rate is the (weighted) average interest rate at which they loan to each other.  It's the federal funds rate because it is at the national level, not limited to one state.  That number is calculated based on market actions.  It is not government controlled.

The Federal Reserve sets a target for that statistic.  If the statistic is below the target, the Fed sells bonds.  Too high, the Fed buys treasury bonds.  That puts more money in the system.  With more money, more banks have excess reserves (to loan to other banks) and fewer banks are short on reserves (needing to borrow from other banks).  As a result, the average rate falls.  Inflation increases.  Too low, and the opposite happens:  sell bonds; less money; less excess reserves; more borrowers; higher interest rates; lower inflation.

Alternative tools of monetary policy

Are there other tools that a central bank can use?  Sure.  A central bank can change the reserve fraction, pay interest on excess reserves, or change the rate at which it loans to banks.

In the US, the rate at which the Fed loans to banks is called the discount rate.  It is generally set to around the same value as the federal funds rate target.  Banks hardly ever borrow from the Fed, as the rules around it make it undesirable.  They would generally prefer to pay a premium and borrow from other banks.  But there is a limit to the premium that they will pay.

Changing the reserve fraction means changing the amount of deposits that banks must hold.  The higher it is, the less that banks can lend.  This is hard to change suddenly.  If the reserve fraction was 10% (as it currently is in the US) and is raised to 11%, that means that most every bank will have to borrow and few will have money to lend.  This would force many to borrow money from the Fed or be forced to suddenly sell assets.  They would prefer to have some notice.  Then they can save up to have 11% by the announced date.

Paying interest on excess reserves has a similar effect to changing the reserve fraction, as it increases the amount of money held as reserves.  It is more gradual than changing the reserve fraction.  However, it can also compete with banks that are trying to borrow from other banks.  The other banks may prefer the deposit interest to the loan interest.  Paying interest also costs money.

Monetary policy and hyperinflation

Let's get back to hyperinflation countries.  The governments of those countries don't want to pay interest on excess reserves.  Even if they didn't mind the other side effects, those countries don't have the money to spend.  And they generally aren't constrained by the reserve fraction.  In a hyperinflation, times are uncertain.  Banks don't want to get stuck loaning money at an interest rate that will be lower than tomorrow's market rate.  But rates are always increasing.  So banks often refuse to loan even if they are awash in reserves.

For the same reason, central banks don't increase the reserve fraction.  It won't help.  And in a hyperinflation, the central bank would want to increase the discount rate and loan less.  But as already said, central bank to bank loans are rare.  Even eliminating them will not have much of an effect.

We're back to the one tool remaining, open market operations to try to meet a higher interest rate target.  But for those to work, the central bank has to sell government bonds in competition with the government.  The central bank has a limited amount of assets that it can sell.  Once it is out, it's helpless.  It can set any interest rate target it wants, but it lacks any way to hit it.

Meanwhile, the government retains plenty of tools to cause inflation.  For example, they can print money.  And they are likely to do so, as everything is becoming more expensive.  So they need more money just to offer the same level of services as they did previously.  So they print it.  But then everything is more expensive and they need more money.

Effective policy

The solution of course is to lower the level of government spending instead of trying to maintain it against inflation.  But in the short term, that is extremely painful for all those that suddenly don't get the level of government support to which they are accustomed.

They still need to do that eventually.  Because in a hyperinflation, everything only gets worse.  Because they are trying to trade a permanent increase in inflation for a temporary improvement in the economy.  Obviously trading a permanent harm for a temporary benefit is a bad deal.  But in the short term, it's hard to make that decision and go for the permanent benefit in exchange for the temporary harm.  And the longer this goes, the larger the short term harm is.  And desperate people think more about the immediate term than the long term.  That's part of what it means to be desperate.

TL;DR:  in the short term, benefits of more money seem to outweigh the harm of inflation.  And in a hyperinflation, people tend to think in the short term, trying to survive today while letting tomorrow fend for itself.


Is unemployment benefit part of national income measurement? Since unemployment benefit is a form of government spending, which is a component of aggregate expenditure, then shouldn't unemployment benefits be part of national income measurement?

It is included. It is a transfer payment, so the trick is to ensure that the income flow is only counted one time in the national accounting.

Statistics Canada, National Gross Domestic Product by Income and by Expenditure Accounts link to description

  The Income and Expenditure Accounts record the distribution and use of
  income by the six main institutional sectors: households, non-profit
  institutions serving households, general governments, non-financial
  corporations, financial corporations and non-residents. These accounts
  articulate revenues to the sector (including current transfers from
  other sectors, such as employment insurance received) and current
  expenditures of the sector (including transfers to other sectors, such
  as income taxes paid to government).

("Employment insurance" is the Canadian equivalent of unemployment insurance.)


Were there any contemporary economists (or other observers) who considered the 1920s a "New Era" of greater growth and stability? Eichengreen (2015, Hall of Mirrors):

In the 1920s it was said that the world had entered a “New Era” of economic stability with the establishment of the Federal Reserve System and independent central banks in other countries. ...

The 1920s were known as the “New Era,” capturing the idea that the country had entered a new age not only of faster productivity growth but also of greater economic and financial stability. Shades of the “Great Moderation,” the supposed diminution of the business cycle in the lead-up to the 2008–09 crisis.

I had never heard of this "New Era" notion before. I know of Irving Fisher's infamous "permanently high plateau" quote, but I do not know of anything similar to the triumphalist "Great Moderation" messages sent out by (some) economists before 2008.
(I wonder if Eichengreen is perhaps exaggerating this in order to craft stronger parallels between the Great Depression and Great Recession.)
So, were there any contemporary economists (or other observers) who considered the 1920s a "New Era" of greater growth and stability?

There are several other economists/observers that share view that this was time of growth and relative stability. For example, according to  Smiley writing for the Economic History Association

1920s are a period of vigorous, vital economic growth.

Smiley is drawing upon quite wide list of research on economic history that generally shares that view. I mean in this case you could say that data speak for themselves. As you can see from the figure produced by Smiley, shown below, based on Romer (1988) estimates of the real GNP, we can see that real national output increased significantly during the 20s (excluding Great Depression of course).

More recently, Gordon (2016) in his The Rise and Fall of American Growth considers this period to be a period of significant economic growth and prosperity/stability.
This being said, I think Eichengreen was just referring to how historians call 20's rather than how economists call them. According to The American Yawp
A Massively Collaborative Open U.S. History Textbook by Sandford University Press [emphasis mine]:

The 1920s, of course, would be anything but “normal.” The decade so reshaped American life that it came to be called by many names: the New Era, the Jazz Age, the Age of the Flapper, the Prosperity Decade, and, most commonly, the Roaring Twenties.

The use of quotation marks seems to indicate that Eichengreen was just referring to the commonly used historic term for that period in American history. The way how I read the passage is that Eichengreen is saying this was 'the New Era' - a pun/reference to the historic term, and then just saying that it was a period of economic growth and stability - which is supported by data  and, as mentioned above, also argued by multiple other economists/observers.


Why N>T is required on system GMM estimations? My question is why N>T is required when working with dynamic panel estimations based on system GMM, such as xtdpdsys at stata. Is that based on the potential lost of information due to orthogonal deviations?
Thanks

Asymptotics are done as N goes to infinity and not T. If T is larger than N, this assumption is odd. True for most of what is commonly called panel data estimators.


Ramsey growth model - per capita vs aggregate variables I am reading a paper Okuguchi,1981 with a Ramsey growth model. There is a constant population growth $n$ such as $L(t)=e^{nt}$. The way he writes the Hamiltonian is quite interesting (equation 5 on page 658.)

$$H=e^{-(\rho+n\left(1-\sigma\right))t}\left[\frac{C^{1-\sigma}}{1-\sigma}+\psi AG+\phi\left(Q-C\right)\right]+\lambda R$$

where capital letters represent the aggregate variables and not per capita variables. $\lambda$ is constant in the model.

In fact, the author does not deal with per capita variables but aggregate variables. Of course, he takes into account the population growth since $n$ appears in co-state dynamics. However, I can not understand how he puts the exponential term $e^{-(\rho+n\left(1-\sigma\right))t}$ in the Hamiltonian in this way.
I would really appreciate if you give me some hints or suggestions.

In the same fashion, is it correct to write something like

$$H=e^{-(\rho-n)t}\left[U\left(C\right)+\lambda\left(AK-C\right)+\mu\left(\left(1-S\right)S-\gamma AK\right)\right]$$

For per capita $c=C/L$, I guess it comes from

$$\int_0^{\infty}e^{-\rho t}\frac{c(t)^{1-\sigma}}{1-\sigma} dt = \int_0^{\infty}e^{-\rho t}\frac{(C(t)/L(t))^{1-\sigma}}{1-\sigma} dt$$

$$= \int_0^{\infty}L(t)^{-(1-\sigma)}e^{-\rho t}\frac{C^{1-\sigma}}{1-\sigma} dt$$

Normalizing initial population to $L_0=1$ we have $L(t) = e^{nt}$ so

$$...= \int_0^{\infty}(e^{nt})^{-(1-\sigma)}e^{-\rho t}\frac{C(t)^{1-\sigma}}{1-\sigma} dt =  \int_0^{\infty}e^{-(1-\sigma)nt}e^{-\rho t}\frac{C(t)^{1-\sigma}}{1-\sigma} dt$$

$$=\int_0^{\infty}e^{-[\rho+(1-\sigma)n]t}\frac{C(t)^{1-\sigma}}{1-\sigma} dt$$

Regarding the application of the discounte factor throughout the whole Hamiltonian, it just implies that the multipliers $\lambda$ and $\mu$ are to be treated as present-value multipliers, and not current-value multipliers. In the textbook formulation and in per capita magnitudes, we usually use current (running) value mutipliers and we totally ignore the discount factor. This affects the intertemporal first-order condition. In generic notation, denote $B$ the state variable, $q$ the mutliplier and $r$ the discount factor. Then, with present-value mutlipliers, we have

$$\frac{\partial H}{\partial B} = -\dot q$$

while if we interpret $q$ as a current-value multiplier we have

$$\frac{\partial H}{\partial B} = r q-\dot q
$$


The Sonnenschein-Mantel-Debreu results, what are the implications for macroeconomics? I have always wondered why such a negative result such as Sonnenschein–Mantel–Debreu theorem (SMD) theorem is usually ignored or disregarded by practitioners, specially in macro.

The result roughly says that there is no deep empirical implications of the rationality assumptions when aggregated because the aggregate demand itself only inherits Walras' law and homogeneity of degree zero. The lack of empirical implications can be translated to the lack of identification and so we cannot hope to recover the underlying preferences from an aggregate demand setup as much of the cross-country empirical macro or calibration macro seems to try. For additional discussion of the importance of the SMD theorem, see this piece.

In sum what are the implications of the SMD theorem for macroeconomics.

This is somewhat broad and argumentative question, but I'll try to provide one possible answer (disclaimer: I'm not a GE theorist). SMD theorem states that imposing only the abstract structure of general equilibrium has few empirical implications. Fair enough. But it's incorrect to say that

  we cannot hope to recover the underlying preferences from an aggregate demand setup as much of the cross-country empirical macro or calibration macro seems to try.

because once we impose additional assumptions about preferences, technologies and endowments, we do obtain additional empirical predictions (and of course, different sets of assumptions will lead to different predictions). This is precisely what economists are doing most of the time! It's not 1950's anymore - these days, there is very little research that tries to arrive at general results from abstract axioms. Most theoretical work in macroeconomics will present specific model, derive its qualitative and quantitative implications and often will include an empirical component.

Another often-heard argument, present e.g. in the linked piece, is that SMD theorem shows GE models are in general unstable or indeterminate, and thus we must resort to unrealistic restrictions, such as representative agent assumption, to guarantee that our models are well-behaved. But this confuses necessary and sufficient condition: yes, assuming representative agent is sufficient to obtain well-behaved aggregate demand function, but that doesn't mean all, or even most, models with heterogeneity are somehow automatically unstable. And a logical possibility of unstable equilibrium says nothing about whether such situation is empirically relevant (maybe it is, but critics usually provide no evidence for such claims).


What would monetarists do when interest rates approach zero and become ineffective? So monetarist economists such as Milton Friedman focus on managing the money supply and lower interest rates as a solution to economic downturn. But what would they do in a recession as the interest rate approaches zero and recognises that monetary policy becomes ineffective? Would they turn to fiscal policy even though they highly oppose government intervention?

An excellent overview of not just monetarist view on this issue but all other mainstream views on monetary policy at zero lower bound  (ZLB) can be found in Ullersma (2002) The Zero Lower Bound  on Nominal Interest Rates and Monetary Policy Effectiveness: a Survey - which is an excellent literature review on this subject (albeit slightly dated).
In short, under the monetarists view 'true' liquidity trap should not exist. Even though there is zero lower bound monetarists would argue that central banks can still stimulate economy through other transmission channels. For example,  Meltzer (1963, 1995, 1999) and Brunner and Meltzer (1968), would argue that central bank can still stimulate the economy through asset purchases with implicit assumption that even when the short-run nominal interest rates are at the zero lower bound, yields on non-monetary assets are not necessarily at their lower bounds.
Bernanke and Gertler (1995) would argue that central banks can stimulate economy also through the credit channel that is not affected by ZLB. Basically this amounts to central bank making sure banks are willing to extend credit as much as possible by providing incentives for them to do so. For example, this is actually being tested in the Europe by ECB through long-term repo operations (LTROS). LTROS is a way how central bank should can borrow money to commercial banks and in theory encourage them to lend more. Furthermore, there is no ZLB when it comes to LTROS as long as central bank is just willing to soak up the losses it makes on such lending (at it can always do that by creating more money).  See this article in the Economist that talks about LTROS in greater detail.
Lastly, Milton Friedman  himself would probably just suggest that central bank engages in 'helicopter money', that is directly giving money to households/firms in some form (see Friedman (1969) The Optimum Quantity of Money: And Other Essays). Such a direct transfer should in principle 'circumvent' the liquidity trap. However, for many central bankers this would be quite radical as such practice could lead to politicization of central banks (since once people learn they can get 'free money' from central banks there might be political pressures for such transfers even when they are not needed).
However, it is worth while noting that the monetarist view above, while still technically being part of mainstream, is not the predominant view in the macroeconomic profession. The current predominant view on ZLB would be more close to the New Keynesian (NK) view on liquidity trap which in the paper cited above is denoted as the 'Krugman's View'). Under the NK view the above mentioned prescriptions would still be helpful, but insufficient and government spending would still be necessary.


Does labor negatively affect output (from empirical perspective)? I am estimating Cobb-Douglas production function for US using time series framework:
$$Y_t=A_t \times L_t^\alpha K_t^\beta $$
where $Y$ is output, $A$ is $TFP$, $L$ is labor and $K$ is capital. After log-transformation I estimate the following linear model:
$$lnY_t=\alpha_0+\alpha_1lnL+\alpha_2lnK+\epsilon_t$$
The estimation results are: $\alpha_0=16.8$, $\alpha_1=-0.72$ and $\alpha_2=0.64$. I am concerned about negative coefficient of labor. Is there any theoretical/empirical explanation for it?

The negative coefficient could imply decreasing returns to scale under some specifications. For example, Miller & Upadhyay (2002) show that under certain assumptions (see the paper for details) the Cobb-Douglass functions of a form $Y =AK^{\alpha}L^{\beta}$ could be rewritten as:
$Y =AK^{\alpha}L^{\alpha +\beta-1}\implies \ln Y = \ln A + \alpha \ln K + (\alpha+\beta-1)\ln L$
The authors even find some negative estimates for coefficient on labor, although most of them not statistically significant. However, the study above is done on a panel with short $T$ and might not necessarily be applicable to time series setting that you seem to imply you are having by only including $t$ subscripts.
Moreover, if you just directly applied the specification you are showing above most likely your coefficients are biased so that could be an explanation for this result.
As pointed out by @Bertrand in his +1 comment there is likely to be an endogeneity present, and hence it is common to use methods that address that. For example, the Wooldridge method that relies on GMM is becoming very popular in the literature (see . Wooldridge, 2009), and some older but still popular approaches include the
Levinsohn and Petrin (2003) or Olley and Pakes (1996)  approaches. Also,  Van Beveren, I. (2012). Total factor productivity estimation: A practical review and sources cited therein provide good overview of different estimation methods.
Furthermore, if you are estimating the production function on datasets with long $T$ you need to take into account possible unit-roots (or use some cointegration models) and in addition to the stochastic trends also deterministic ones. If you run this on a time series data I would find it very likely that the result is just due to some stochastic or deterministic trend that was not properly taken into account.


What would happen to the world GDP if we run out of minerals to mine but not of sources of energy? I was thinking about how the depletion of minerals (metals and all material we mine from the Earth) will affect the GDP of the world. I read somewhere that at the current extraction rate, we will run out of easily accessible iron ore in the next 60 years.

I found this Wikipedia article and it seems that about 1% of the world GDP is in extracted or mined minerals.

A simple prediction would be that the world GDP would drop by 1%, but thinking about this more and not I'm not sure if this is correct, I think the lack of mined metals and minerals would have a much bigger effect on the economy than 1% since this 1% actually contributes a huge sum to the rest 99% and without it, we will just end up with whatever we have and will not be able to construct, manufacture or build anything anymore.

So to make this simple and to make sure that energy sources are out of the equation, let's assume that humanity will never run out of sources of energy like oil, gas and coal, but will run out of all metals and minerals that we can mine so that there is no any additional minerals being added to our economic system. How would that affect the GDP of the world?

You're correct that just because 1% of the economy is in mineral extraction, doesn't mean that stopping it would reduce global GDP by 1%. Similarly, roughly 10% of global GDP is on energy expenditure, and about the same on food, but if we stopped either of those, global GDP drops to pretty much zero.

Do bear in mind that we will stop using coal, oil and gas quite shortly - probably within 2-3 decades. Nevertheless, we've got no shortage of energy - we've got far more solar energy available than we can use.

However, minerals aren't going anywhere: very little of them leaves the planet. Which means almost all the elements we've ever mined, are still around.  And retrieving valuable elements from waste is largely just a matter of energy, and a bit of know-how.

Let's think about what might happen if an element starts to become scarce.

First, its price goes up.

As a result, there's investment in substitutes, as they're now more competitive.

There's also investment in recovering the scarce material from waste streams, as this investment now has a higher payback -  a higher financial return on investment.

The short-term effect on global GDP would be a small reduction. The longer-term effect could go in either direction. But, ultimately, the global effect on humanity will be positive: sooner or later, we're going to need to move a circular economy, and this transition will be part of it.


Does stronger regulation of economic activity impede economic growth in constitutional democracies with basic constitutional rights? Given similar democratic constitutions and basic constitutional rights such as property rights, freedom of speech, freedom of press etc and the definition of regulation according to OECD:

Regulation is broadly defined as imposition of rules by government, backed by the use of penalties that are intended specifically to modify the economic behaviour of individuals and firms in the private sector.

Is there a relationsship between regulation of economic activity and economic growth? Are there any reviews / meta analysis?

Broadly speaking regulation can be defined as done by OECD see Khemani and  Shapiro (1993):

Regulation is broadly defined as imposition of rules by government, backed by the use of penalties that are intended specifically to modify the economic behaviour of individuals and firms in the private sector.

This is broad definition that includes anything from price controls via regulations on export/imports to antitrust laws.
Research shows that inclusive institutions which include things that would very well fall under the label regulation like antitrust laws, laws that enforce competition, prevent firms from creating barriers to entry and so forth do promote economic growth (see Acemoglu 2008, Acemoglu & Robinson 2000a, 2000b, 2001, 2006, 2008; Olson 1984, Bates 1981, 1983, 1989 and sources cited therein*).
On the other hand extractive institutions, such as government grants of monopolies, restrictions on who or what classes of people can participate in labor market (e.g. like many apartheid regulations that restricted blacks form some professions) many of which again would fall under the label of regulation, tend to depress growth (see the same sources as above).
So growth is neither retarded or supported by 'heavy regulation' per se, it all depends what sort of regulation we are talking about. Regulation that fixes market failures, promotes openness and competition generally supports growth whereas regulation that creates market failures, closes economy and favors concentration generally retards growth.
An important caveat is that under a more narrow definitions of regulation such as the definition used by  Breyer et al (2016):

governmental actions to control price, sale and production decisions of firms in an avowed effort to prevent private decision-making

which is more narrow the answer might change as literature is less kind to price controls and command and control regulations, but still the answer would depend on what sort of regulation are we talking about.
In literature,  the focus is not really on how 'heavy' regulation is, but on regulation quality. For example, Jalilian et al (2007) show that developing countries with higher regulatory quality grow faster, where regulatory quality is defined as "burden on business via quantitative regulations, price controls, and other interventions in the economy" (where more burden means lower quality - but burden is not the same as having many regulations as one can implement large quantity of well designed regulations that jointly might have smaller burden than few very badly designed regulations).

* Note this research is also summarised in Acemoglu & Robinson: Why Nations Fail? which is book that is written in a way that is more accessible to laymen readers than the papers mentioned above).


Marxist Economics: Understanding Capitalist Exploitation through "speed up" or increase in intensity I have been studying Marxist economics for a little over 3 months now and have been really trying hard to understand the arguments made.

Right now I'm watching Dr. Stephen Resnick's course on Marxist Economics and am having difficulty understanding the Matematics behind the concept of capitalist exploitation through "speed up" or increase in intensity.

He presents the following formula:
$$\text{I}=\frac{SV+V}{lh}$$

where
-$\text{I}$ is intensity
-$SV$ is surplus value
-$V$ is variable capital/ value added by workers
-$l$ is number of laborers
-$h$ is the number of hours each laborer works

he argues that there exists a way to increase surplus if we hold l and h fixed. Mathematically im not sure how this is done.

It cant be from a change in machinery/capital stoc as noted in a previous video of Dr. Resnick since this would decrease the capitalists level of surplus1.

So if there is no change in hours worked, number of workers and machinery- How is this increase in surplus value occurring?

1. this is based on the equation:

$$r=\frac{SV}{V}\left(1-\frac{C}{V+C}\right)$$

where:

-$r$ is the rate of profit

-$\frac{S}{V}$ is the rate of exploitation/rate of surplus value

-$\frac{C}{C+V}$ is the index of mechanization/rate of mechanization, where $C$ is value of machinery.

Though I'm certainly no expert in Marxist economic theory, I think he's presenting it as an "if you can figure out a way of speeding workers up, then you can generate additional surplus from their efforts. Effectively, this is just saying "say you're Ford, and have teams of 7 working on assembly lines that generate 4 cars per hour. Suppose there is some way to increase the intensity of labor, by getting them to work faster and produce 5 cars per hour, and not changing the number of assembly lines or workers per line. Then, if you can hold their wages fixed, you can generate more surplus for the capitalist."

(I'd imagine a historical example might be the semi-mechanized looms of the English Industrial Revolution, where the speed of the mechanisms was dictated by the machinery, and could therefore be bumped up by a factory owner.)

I don't think there's a single coherent mathematical argument explaining why this speed would be increased, just highlighting the different ways the ratio of interest can change.


Is there an alternative metric to GDP that measures a country's assets rather than flows? The Gross domestic product measures the value of all final goods and services produced in a period.

This has some limitations.  For example, if a car hits a lightpost and both need to be repaired and some parts replaced, that contributes to the GDP, even though at the end we're in roughly the same situation as before and the reparation process has used resources (labour and material) that might have been used elsewhere.  From Wikipedia:

  The UK's Natural Capital Committee highlighted the shortcomings of GDP in its advice to the UK Government in 2013, pointing out that GDP "focusses on flows, not stocks. As a result an economy can run down its assets yet, at the same time, record high levels of GDP growth, until a point is reached where the depleted assets act as a check on future growth". They then went on to say that "it is apparent that the recorded GDP growth rate overstates the sustainable growth rate. Broader measures of wellbeing and wealth are needed for this and there is a danger that short-term decisions based solely on what is currently measured by national accounts may prove to be costly in the long-term".

I'm aware of numerous alternative measures focussing well-being, such as the Human Development Index,Gross National Happiness, and the Happy Planet Index.  They don't really measure either wealth or well-being (the latter may be impossible to measure), and their exact definition is somewhat arbitrary.

Is there any reasonably well-established alternative to GDP that measures the total value of the stocks of a country?  Wikipedia has a financial assets list, but it relates only to households (and on this list, Greece is almost twice as wealthy as Norway, which is so vastly against expectations that I wonder if the definition is useful).  I can reasonably well estimate the value of my own assets (except for the value of my education/skills, which is harder to quantify).  Is there any metric that is used to estimate the value of all assets in a nation: infrastructure, real estate, machines, goods, farmland, but also education, human capital, unmined natural resources, etc.?

You are correct in pointing out the flaws of GDP. But honestly, it's the best we have came up with; measuring the stock of wealth is just impossible.

You say you can estimate the value of your own assets, and of course it seems that way. We all feel that way, until we actually try. How much is your cellphone worth? Easy to know, go to ebay and look for similar phones in similar conditions and you get a price for that phone, right? Why don't we do that for all the country's cellphones? Well, because if we put all the cellphones in the market prices would drop down dramatically. Now, we're not obviously sending all the cellphones to the market, we just want to know how much its value is, but the point is that in order to have an estimation, we have to make assumptions regarding the market. The value of one cellphone is easy to calculate because it doesn't really affect the market.

Additionally, who says that you have to sell it in your local market? Why don't you offer it in Australia or Japan or Algeria or wherever the price is higher? And why not sell it another day when the price is up? The same goes for your car, house, computer, clothes, books, etc.

Think now of your financial assets. You can know that, right? All one needs to do is see what the bank says, and it says that my investment portfolio is worth \$100.... at least today... and exactly at the time I checked. But remember: the value of any financial portfolio is going up and down with high volatility even in the same day. Now try to figure out the value of the financial assets of all the country. Well, you can do that. But again: just today, and just at the time you checked.

Even though it doesn't seem like it, the same goes for goods. Take oil. Say that there is an oil reserve with x millions of barrels just waiting to be extracted and sent out to the market. How much is the value of that reserve? It depends on when you ask, whom you ask to, and who else is asking the same question (other sellers). What about gold? silver? copper? coal? wood? You can estimate a value in a given time, but that can be very different in another time.

What about land? Well, if I sow corn it will give me a given amount, but if I sow rice it will yield a different amount. Besides, would it be the same if my neighbor sows corn and I sow rice than if we both sow the same (competition)? What if I want to use it to generate electricity instead of growing food? Or what if I use it for livestock? What if I sell it? What if we all decide to do the same thing at the same time? Or what if we decide to do completely different things? Indeed to value farm land, you have to assume what use will it have.

Infrastructure is even more complicated. How much is a railroad valued? And what does value even mean in this context? There is no secondary market for railroads is there? There are opportunity costs, though. And we have the construction value too. But the we would have to make assumptions on its depreciation, and say how much of the revenue of the country is due to this specific segment of the railroad. To estimate this, one has to say how much the country would loss was the railroad not there (which is a fictitious value). What about parks?

Don't even get me started on natural resources (rivers, lakes, coast, jungles, biodiversity, etc). But even then, those are the easy ones. We still have left education, experience, skills and the like. But this post is long already, the point has been made and now I have a headache for thinking all this.

Indeed, there are many serious and very clever attempts to make estimations of these. But they are used to answer very specific questions. GDP, on the other hand is easy: just sum all the sells of goods and services. We (economists) don't like it very much either, but that's all we have. Just try to accept it, and don't take it to seriously.


Implication of the concavity of u Suppose that we have the following inequality:
$u( y- d) - u(y-d') \ge u(y' - d) - u( y' -d')$.
The concavity of $u$ together with $y\le y'$ then implies that $d \le d'$.

I sometimes come across this problem in economic papers, but I don't understand the logic behind this. Can anyone explain this?

I think you have the inequality sign wrong. The inequality as you write it cannot hold. To see this, consider the numerical case where $y=4 > y'=3, d=1 < d'=2$ and $u= \sqrt(x)$. I let you do the calculation yourself.

I suppose the correct inequality is:
\begin{equation} u(y−d)−u(y−d') \leq u(y'−d)−u(y′−d′) \end{equation}

This equation holds true due to the curvature of a concave function. If you're on the upward sloping part of the function, the closer you are to the summit, the flatter the slope. So the vertical distance between the images of the two points in the domain on the left side of the equation is smaller than the same on the right side if $y>y'$ and $d \leq d'$, and equal if $y=y'$. On the downward sloping part it's the other way around. The set of inequality conditions imply each other. Changing one inequality requires changing the others. You can check this for yourself by picking points on a concave curve that satisfy the inequalities.

It is really just an elaborate version of $|x| \leq |y|$, where if $x \leq 0, y \leq x$ and if $x \geq 0, y \geq x$.
I hope this answers your question.


Does the Tax Allowance Benefit Everyone (Regardless of Their Income)? Note:

This question is in the context of the UK and its policies.

I understand that if your income is below the tax allowance (£11,000), you pay no income tax. However, I think my textbook is also trying to say that if your income is above the allowance, you only pay income tax on whatever is left after the allowance has been deducted from your total income. Is this true?

Oh, and I was also wondering if there was any difference between the 'personal tax allowance' and the 'tax allowance'.

Thanks

Not everyone benefits from the tax allowance, but most people with incomes above £11,000 do.  As you suspect, those with income higher than £11,000 only pay tax on their income above £11,000, and thus gain £11,000*tax rate.  However, the personal tax allowance is clawed back at incomes above £100,000 at a 50% rate, so those with incomes above £122,000 don't benefit from the allowance at all.  The official UK government site is here.

At the margin, this means that those with incomes less than £11,000 pay no tax on each additional dollar of income, while those with incomes between £100,000 and £122,000 pay an extra-high amount in tax for each additional dollar earned (the tax allowance doesn't change the tax rate on each additional dollar of income for incomes between £11,000 and £100,000, or above £122,000).  The entire picture is nicely summarized in this Telegraph article.

Here in Canada, we have an equivalent tax allowance, but without the clawback.


Is my reasoning about economic growth and environmental sustainability correct? Below is my attempt to answer question "Can we have economic growth and environmental sustainability?".

Please correct me if I got something wrong in my reasoning.

Let's see what is counted as economic growth and environmental sustainability:

a)Economic growth happens when we produce more goods and services than our Production Possibility Frontier (aka PPF) previously allowed. In its turn, this usually happens when we have more labor and/or capital and/or land than previously. The exception for this are gains from specializing  in producing and selling goods in which given country holds comparative advantage. If we just started using our labor/capital/land more efficiently we still would be inside our Production Possibility Frontier, although it would lead to increase in production of goods and services.

b)Environmental sustainability happens when following conditions are met, according to Herman Daly:

For renewable resources, the rate of harvest should not exceed the rate of regeneration (sustainable yield);
[For pollution] The rates of waste generation from projects should not exceed the assimilative capacity of the environment (sustainable waste disposal); and
For nonrenewable resources the depletion of the nonrenewable resources should require comparable development of renewable substitutes for that resource.

Considering this, we can have economic growth thanks to producing and selling goods in what we have comparative advantage, while reducing production of goods in what have have comparative disadvantage. To put it simply, more free trade, less subsidies of uncompetitive industries - more growth. And it's compatible with environmental sustainability.
We can increase labor (which is understood as amount of work done) by, say, curing people who are unable to work due to an illness. Or by punishing people who can work, but don't want to work (like USSR did). Or by making people work for longer hours. Or by making sure that workers are really working instead of, say, wasting time playing games and checking their Facebook. Or by training part-time workers to be qualified for a full-time job. Such measures can lead to economic growth and are environmentally sustainable.
As for land, there does NOT seem to be any environmentally sustainable way to get more of it. If only you don't seriously consider scenario when one country gets territory that previously belonged to another country. But it's a zero-sum game.
Capital is understood here as all man-made goods which are used for further production of wealth.
We can increase our capital in an environmentally sustainable way if we will increase capital  thanks to more efficient use of existing labor, land and capital. Again, increased production of capital (and other goods) due to more efficient use of existing labor, land and capital does NOT itself mean economic growth. But after capital will be increased it will be possible to produce outside previously established PPF, meaning that economic growth have occured.

It is on a correct track but I would not say it is 100% correct as you miss the main source of growth which is technological progress. Consider the following Cobb-Douglas production function with all 4 main factors of production, capital ($K$), human capital ($H$), labor ($L$), land ($M$) then the standard production function would look something like:

$$F(K,H,L,M) = A K^{\alpha}H^{\beta}L^{\gamma}M^{1 - \alpha - \beta - \gamma}$$

Well you are correct to say that production $F$ increases if either more capital is put to work, or more labor hours are employed ($K$ or $L$ increases). However, there can be limits to how far you can push capital accumulation and still be sustainable.  Machines after all need some raw materials. Also, people only have 24 hours a day so there is a limit there. Land is for all practical purposes fixed (at least before we can make an effective use of real estate on moon or mars - and I am saying that just half-jokingly). However, you missed one important factor of production: human capital. That is things like better education which as far as I can see can increase almost without bounds.

Second thing you missed is $A$ that is the total factor productivity, which makes any factor more productive and loosely speaking it is the technology avaiable. For example, thanks to our better understanding the computer technology modern computers are more faster than the old ones yet they are produced with the same amount resources (or sometimes even less resources - think of the old bus size punch card computers). This does not apply just to capital, thanks to telecomunication technology you can make better use of your human capital, thanks to better management techniques better use of labor, thanks to agricultural techniques like crop rotation better use of the same land.

Also some things that were not correct in your answer:

Comparative advantage (unless you are considering some dynamic endogenous growth effects) generally does not lead to higher economic growth. It will increase the efficiency so GDP will have one time jump from lets say 100 to 200, and sometimes the jump may take several years so on yearly accounts it looks like growth, but its really just increase in level - that is still good but its not really long run growth.


Why does the Federal Reserve seem to hold so much weight on the unemployment rate, relative to labor participation? The title is the question.

In the press conference held yesterday, the Chair referenced "unemployment" 18 times, relative to "participation" 3 times.  If the labor participation is at the lowest since the 70's, is it delusional to buy into the seemingly marketed strength of the U.S. economy?

As mentioned in the press conference, other data is taken into consideration, such as the Employment - Population Ratio, especially when filtered/line series graphed to detail trends in demographics.

This blog post provides a bit more insight on the data.

It seems relatively, the U.S. is climbing to employment-pop levels we haven't seen since the mid 80's, which seems to suggest the outlook isn't as terrible as initially thought.


Solow Model, Growth rate of K/L and Y/L in steady state I have been given the following setup:

$$ Y=K^\theta (AL)^{1-\theta }$$

Where Y = Output, K = Capital, L = Labour and A = Productivity.

$$ \frac{\dot{L}}{L} = n $$
$$ \frac{\dot{A}}{A} = g $$

The Capital Accumulation Equation has also been given as:

$$ \dot{K} = sY - \delta K $$

Using this, I have calculated expressions for capital per worker and output per worker in the steady state as:

$$ \frac{K}{L} = A(\frac{s}{n+g+\delta })^{\frac{1}{1-\theta }} $$

$$ \frac{Y}{L} = A(\frac{s}{n+g+\delta })^{\frac{\theta }{1-\theta }} $$

The next questions asks: Calculate the growth rates of capital per worker and output per worker that will hold as the economy moves along a steady state growth path.

This is confusing because I was under the impression that growth of capital and labour is the steady state is 0, and then therefore growth of K/L and Y/L should also be 0. However this seems wrong and I believe the answer should be g. Can anyone help please?

If $Y = C \cdot X$ where $C$ is constant and $\frac{\dot{X}}{X} = g$ then we can solve for $\frac{\dot{Y}}{Y}$ as follows:
$$ \frac{d}{dt} Y = \frac{d}{dt} C \cdot X = C \cdot \frac{dX}{dt} = C \cdot \dot{X} \Rightarrow$$
$$ \frac{\dot{Y}}{Y} = \frac{C \cdot \dot{X}}{ C \cdot X} = \frac{\dot{X}}{X} = g$$

Therefore, as you concluded, they both grow at rate $g$.


Why would creditors accept a comfort letter that can't be legally enforced, from borrowers that refuses to guarantee? Prof Richard Taylor, Damian Taylor. Contract Law Directions (6 edn, 2019). p 65. Para 1.

  You might be wondering why businessmen bother to make an agreement
  if it has no legal effect—is it worth the paper it is written on? [mine] In banking practice ‘comfort
  letters’ are used literally; they provide comfort to a lender that he will get his money back. A
  borrower’s parent company (i.e. its main shareholder) will usually be saying something along
  the lines of ‘we will try to be a really good parent company and make sure that our subsidiary
  pays all your money back’. The bank does not expect to be able to sue on this agreement, and
  should insist on receiving a guarantee if it wants to have legally enforceable rights against the
  parent company.

  In Kleinwort Benson Ltd v Malaysia Mining Corp. (1989) the following wording in a comfort
  letter was held not to contain an intention to create legal relations: ‘It is our policy to ensure
  that the business of [our subsidiary] is at all times in a position to meet its liabilities to you under
  the above arrangements.’ Companies must be careful not to promise to ‘guarantee’ or ‘undertake’
  to do anything as this is more likely to be interpreted as a binding agreement.

I'm not swayed by Shawn Goldmintz's (BBA York, JD Windsor) article.

  It may seem as though comfort letters have the potential to be absolutely useless in certain circumstances. The question begs as to why a creditor would accept a comfort letter from a guarantor that refuses to give a guarantee when the letter cannot be enforced at law. The commercial reality is that when a business fails to meet its obligations, whether as a matter of law or honour, it reflects poorly on that business. A bad reputation can be much worse for a company than having to repay a loan. While this may not seem like an appropriately harsh punishment for one who fails to honour a non-contractual version of a guarantee, it should be remembered that, unfailingly, the author of a comfort letter will have been asked for a guarantee first, and the author refused.

  In addition to this weak punishment, it is entirely possible, and in fact probable, that some comfort letters will be interpreted as binding legal obligations in the future. All it would take is for the wording of a letter to indicate promissory intent (perhaps by using the future tense), and the letter would be held to create legally binding obligations.

I'm asking just about comfort letters that aren't legally enforceable. Viz. disregard last para above.

If I were the lending officer at a rich bank, I'd distrust any borrower or its parent company that can't guarantee! I wouldn't lend it. After all, I can just lend to someone else who does guarantee, though I may have to sweeten my offer.

The emboldened quote doesn't feel true to me. Any examples? To the lender, "a bad reputation" won't get her his money back, and feels useless.
Are there other reasons?

It's not as much a matter of reputation as much as it is a question of signaling. not paying one of your debts could signal to other suppliers that you are NOT ABLE to pay which will prompt them to stop any on-going deliveries and require much stricter payables conditions. These kind of things happen more often in companies facing difficulties. The apparent (real or fake) inability to pay could materialize into a number of headaches with other counter-parties and render you unable to follow up with any business if you just used this "excuse" to gain more time. There is an example where we recently saw this with TESLA in their relationship with Panasonic, turned out they just wanted to delay payments as much as possible but it raised a number of questions on their cash flow situation and ended up causing a lot of volatility (albeit temporary) in their listed liabilities (equities and bonds).


Data: Trade imbalance at the County level in the US I am looking for some source of data on trade imbalance at a county level in the US, preferably for years 1990-2007. Does anyone know if such data can be publicly available? Or any paper that have been using data like this?

UPD: to break this down, I am looking for:

international import/export statistics by US county,
intra-state trade statistics, meaning shipments between counties within the US.

Thanks in advance!

The data you're asking for doesn't exist.

Most data on things like this are captured through the taxing authority, which means that there are entities that want to collect taxes and therefore care very much where you do something and where your inputs came from. That's very true when there are international borders involved, and it's true to a much lesser degree when there are state borders involved, but in the US, there's generally no administrative interest in tracking the flow of goods across county borders.

Source: I used to create state and local area GDP estimates for the Bureau of Economic Analysis.


How can I tell if 2 different utility functions represent the same preferences? I need to verify that $u(x,y)=x^{1/3}y^{1/3}$ represents the same preferences as $v(x,y)=x^3y^3$. Obviously these are completely different functions with different derivatives, so what am I comparing? What makes 2 utility functions represent the same preferences?

Recall the definition:

  The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$

We can show that if $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$, then for any strictly increasing function, $f: \mathbb{R} \rightarrow \mathbb{R}$,  the function $v(x) = f(u(x))$ also represents $\succeq$

(Short) Proof: For any $x,y \in X$, if $u$ represents $\succeq$, then $x \succeq y \iff u(x) \geq u(y)$ by definition.

Because $v$ is strictly increasing, $f(u(x)) \geq f(u(y)), \implies v(x) \geq v(y)$.

Thus $x \succeq y \iff v(x) \geq v(y), \implies v$ represents $\succeq$

In your case, you have $u(x,y) = x^\frac{1}{3}y^\frac{1}{3}$, and can set $v(x, y) = (u(x,y))^9 = x^3y^3$, your desired result.


Is inequality a micro or macro economics issue? In a clear and comprehensive blog post, Tejvan Pettinger describes economics problems. He starts by making brief considerations on scarcity, enumerates the fundamental economic problems (what/how much produce, how to produce, whom to produce to) and then goes to specifics, commenting about micro and macro economic problems.
For me, it seems fair to conceptualize "microeconomics" as a bottom up approach for economy, in which the focus is the study of the behaviors and interactions of the economic agents (producers and consumers) facing scarcity, and "macroeconomics" as a top down approach whose goal is the study of the great economic aggregates like unemployment and inflation, for example. As such, it surprised me that Pettinger sees inequality as a microeconomic issue instead of a macroeconomic one: aren't macroeconomic tools intended to stimulate growth or redistribute income by means of monetary/exchange/fiscal policies? Isn't the goal of macroeconomics policies to grant country-wide sustained economic growth to benefit the society as a whole? In a way, isn't microeconomics the "Invisible Hand Science", according to which everyone including you and I seek our own benefit and helping the society by accident in this process?
Hope you guys can shed a light on that doubt of mine.

Inequality and public policy has actually its own field (Public Economics) so it does not narrowly fit in either of those categories, but given that most research and work on inequality is of microeconomic character (see Stiglitz Economics of the Public Sector 3rd ed) I think it is fair to put inequality under microeconomics if you have to make such binary choice as it fits there better.
First let us get the definitions of micro and macroeconomics straight. Following Mankiw Principles of Economics pp 27:

Microeconomics is the study of how households and firms make decisions
and how they interact in specific markets.
Macroeconomics is the study of economy-wide phenomena.

Neither of them is top-down or bottom-up, in fact most macroeconomics models are build upon micro-foundations (in fact even you do research in pure macro you need to learn a lot of microeconomics), so by your descriptions in your question they would both be 'bottom-up' - although I do not think it is sensible to talk about these fields as top-down or bottom-up. The distinction is based rather on the scale at which you are looking.
When it comes to inequality, often the appropriate level of analysis is microeconomic analysis. For example, do you worry about income inequality? Well income inequality arises from the fact that people are heterogeneous in their ability. Hence you need to study how this affects rewards they get for their labor supply on a labor market. Such analysis does not require tracking of inflation and other macro factors that would likely just 'muddy the waters'. To study that you want some micro models not macro models where everyone is aggregated to an 'representative agent' (some sort of average human). Similar arguments would apply to most topics to be considered in the realm of income inequality.
Macroeconomics, also touches bit on inequality as there can be interplays between inequality and economic growth (and growth is definitely macro topic). Or unemployment benefits or redistribution from people who have high marginal propensity to save (usually rich people) to ones with low propensity to save (usually poor) can be considered macroeconomic stabilizers. But macroeconomics does not really concern itself with inequality per se. If it would empirically turn out that poor people have higher propensity to save than rich people from pure macro perspective there should be redistribution from poor to the rich.
This is not to say that there is absolutely no research into inequality where inequality would be in the main focus and the research is just using macro models/tools. But this strand of literature is miniscule compared to microeconomic literature devoted to this topic.
For example, consider this. In Romer Macroeconomics, which is the standard widely used graduate macro textbook the word income tax is printed only 8 times. In the Varian Microeconomic Analysis you have separate sub chapter on optimal taxation, separate chapter for welfare analysis, separate chapter for public goods etc.
This is not because macroeconomists would want to ignore inequality but it is simply not considered to be part of the subject matter. This primarily comes to the fact that macroeconomists often have to aggregate when they try to describe whole economy. In order to aggregate you need to introduce average households, average individuals, average firms etc. Once you aggregate whole population to set of average individuals then income/wealth inequality amongst those averages is zero.
There are macro models that also look heterogeneous agents but even there you will usually have just some small limited number of options. For example, having two households that are different and each is supposed to be an avatar of their own category. It is possible to still introduce more heterogeneity into macro models but the more heterogeneity you include the complexity of the model raises disproportionately. This is less problematic in microeconomic models and thus they are preferred.
Lastly, it is possible that in the future this will change and people studying inequality and policy economics in general will start focusing more on macro models and treating inequality as a macro phenomenon. From my own experience work on inequality from macro side is nowadays more common then when I was undergraduate. Nonetheless, as of now I don't think anyone could reasonably disagree with statement that most inequality work focuses solely on micro  (for example, almost whole literature on optimal income taxes and transfers is microeconomics) or uses micro models. Hence, if you have only two options I think it makes more sense to put it under microeconomics and I think this was why the author did that.


not following a step in a paper on rational expectations Hi: I'm reading a paper by Broze and Szafarz titled "On Econometric Models with Rational Expectations and I don't follow a step in section 3.2 which discusses the method of undetermined coefficients. So,  I was hoping that someone could explain the derivation. I will write down the equations and then explain the step that I don't follow.

1) $p_{t} = a \times E(p_{t+1} | I_{t}) + u_{t} $ This equation  is given. It comes out of the Cagan Hyperinflation model. $u_{t}$ is an error term.

2) $ u_{t} = \epsilon_{t} + \alpha \epsilon_{t-1} + \ldots + \alpha_{k} \epsilon_{t-k} + \ldots = A(B) \epsilon_{t} $

3) $ p_{t} = \psi_{0} \epsilon_{t} + \ldots \psi_{k} \epsilon_{t-k} + \ldots = \Psi(B) \epsilon_{t} $

4) $ E(p_{t+1}| I_{t} ) = \frac{\Psi(B) - \psi_{0}}{B}    \epsilon_{t} $
So, $B$ denotes the lag operator which is often denoted as $L$ in the literature. The explanations of the equations follow.

1) As stated above, this is given. This is clear.

2) This is the standard infinite MA representation that follows from Wold's decomposition. This is clear.

3) This is the guess that is made in the method of undetermined coefficients. This is clear.

4) This is not clear to me. They say "it follows". I see that it's kind of coming by subtracting $\psi_{0} \epsilon_{t}$ from both sides of 3) but $B$ is the infinite lag polynomial and they are dividing by it which confuses me.

Thanks in advance for the help. Some discussion has been omitted so, if anyone needs to see the actual pages, I can scan and send them. My email is my name with a 2 on the end of it and the mail server is that famous one that starts with a g.

                                        Mark

Presumably $I_t$ stands for the information set at time $t$. So note that $E[\epsilon_j|I_t] = \epsilon_j$ $\forall j \leq t$. This will come into play when working out the LHS.

We wish to verify Equation (4), which is

$$
E(p_{t+1}|I_t) = \frac{\Psi(B) - \psi_0}{B} \epsilon_t,
$$

which is equivalent to

$$
$$

$$
\begin{align*}
B(E(p_{t+1}|I_t)) = \Psi(B) \epsilon_t - \psi_0 \epsilon_t. &&(*)
\end{align*}
$$

$$
$$

We are given that

$$
\Psi(B) \epsilon_t = \psi_0 \epsilon_t + \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_k \epsilon_{t-k} + \dots
$$

so the RHS of $(*)$ is

$$
$$

$$
\boldsymbol{\Psi(B) \epsilon_t - \psi_0 \epsilon_t = \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_{k} \epsilon_{t-k} + \dots}.
$$

$$
$$

Now, the LHS. Since $p_t = \psi_0 \epsilon_t + \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_k \epsilon_{t-k} + \dots,$

$$
$$

$$
\begin{align*}
E(p_{t+1}|I_t) &= \psi_0 E(\epsilon_{t+1}|I_t) + \psi_1 E(\epsilon_t|I_t) + \psi_2 E(\epsilon_{t-1}|I_t) + \dots  + \psi_k E(\epsilon_{t-k+1}|I_t) + \dots \\
&= \psi_1 \epsilon_{t} + \psi_2 \epsilon_{t-1} + \dots + \psi_{k} \epsilon_{t-k+1} + \dots
\end{align*}
$$

$$
$$

since $E(\epsilon_{t+1}|I_t) = 0$ because $\epsilon$ is an independent white noise process.

Given that $B(\cdot)$ is the lag operator, the LHS of $(*)$ is thus

$$
$$

$$
\begin{align*}
\boldsymbol{B(E(p_{t+1}|I_t))} &= B(\psi_1 \epsilon_{t} + \psi_2 \epsilon_{t-1} + \dots + \psi_{k} \epsilon_{t-k+1} + \dots) \\
&= \psi_1 B(\epsilon_{t}) + \psi_2 B(\epsilon_{t-1}) + \dots + \psi_{k} B(\epsilon_{t-k+1}) + \dots \\
&= \boldsymbol{\psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_{k} \epsilon_{t-k} + \dots}.
\end{align*}
$$

$$
$$

Thus the LHS = RHS; that is,

$$
\begin{align*}
E(p_{t+1}|I_t) = \frac{\Psi(B) - \psi_0}{B} \epsilon_t. &&\blacksquare
\end{align*}
$$


Why do prices go up when the central bank increases money supply? From my understanding, when the money supply curve shifts to the right, interest rates go down, it follows that the price level decreases and the prices should go down?

Thanks in advance.

Price levels are not necessarily linked to interest rates, especially real rates.

Speaking strictly from an IS-LM standpoint, an expansionary monetary policy will increase the real money supply and depress interest rates, holding output constant. This will cause a rightward shift in the LM curve; holding money demand constant, the relative value of currency must fall.

Speaking practically, lower rates reduce the hurdle rate for investments and thus promote asset inflation. Further, the lower rates will also find their way into consumer credit, increasing purchasing power among consumers and thus driving up prices of consumer goods. Mind that the former precedes the latter by a good couple years.

Hope that answered your question.


$p(0)\geq c'(0)$ in monopoly $p(0)\geq c'(0)$ is assumed so that the inverse demand curve originates above the marginal cost curve.

But what is the intuition behind this assumption? The highest willing-to-pay consumer is more than the marginal cost of producing the first unit? Is this going to be the variable cost of producing the first unit?

You intuition about the assumption is correct.

Given the usual assumptions of downward sloping demand curve ($p'(q)<0$) and non-decreasing supply curve ($c''(q)\ge0$), if $p(0)<c'(0)$, then demand would lie entirely below supply. This means the equilibrium quantity must be zero---as if the market did not exist. Therefore, for the analysis to be non-trivial, the assumption of $p(0)\ge c'(0)$ is maintained.

With the use of calculus, we are assuming implicitly that quantities are infinitely divisible. Thus $c'(0)$ can be roughly interpreted as the cost of producing the "first infinitesimal unit" of the good. Mathematically,
\begin{equation}
c'(0)=\frac{\mathrm d\, TC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,(FC+VC(q))}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,VC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}.
\end{equation}
Thus $c'(0)$ is the derivative of the variable cost function, evaluated at $q=0$.


What is the use of knowing elasticity of average product? Firstly they have given output elasticity of a factor, use of which is clear to me. It says to what degree my total output will vary due to a change in the quantity of a factor.
But what else do we get to know from elasticity of Average product?

If you know Elasticity of average product you can identify how your output per worker or machine will change as $L$ or $K$ increases.

Recall the definition of elasticity is:

$$\mathcal{E}=\frac{\Delta y/y}{\Delta x/x}$$

average product of labor and capital:

$$AP_L=\frac{Q}{L}$$
$$AP_K=\frac{Q}{K}$$

therefore the elasticity of average product of labor and capital is:

$$\Gamma_L=\frac{\Delta AP/AP}{\Delta L/L}$$
$$\Gamma_K=\frac{\Delta AP/AP}{\Delta K/K}$$

This is useful for evaluating how existing worker or capital productivity will be impacted as a result of an increase in capital or labor.


Why it is unit elasticity If I always spend a total of exactly $\\\$10$ per week on coffee, then does my demand function have unit elasticity?
According to me, the change in income doesn't affect my coffee consumption, so it does not seem like unit elasticity. Can someone help me understand this?

Assuming you mean unit elasticity of demand with respect to price the answer is yes.
From the information you have, we can deduce that the demand function is as follows:
$$Q^d(p)=10/p$$ (I always spend $\\\$ 10$ on coffee, which buys me exactly $10/p$ units of coffee, where $p$ is the price of coffee).
Now recall the formula for computing elasticity of demand with respect to price:
$$\varepsilon \equiv \frac{dQ^d}{dp} \frac{p}{Q^d}$$
And the rest I leave to you.


Who is the first one to equate "rational" with "complete and transitive preference"? MWG taught that, suppose that the menu is finite, "rational" is the same as "complete and transitive". But it seems that it does not cite any sources. Who said this first?
vNM said in 1944 that their EU is rational behavior, which is followed by computer scientists, statistician, and Finance literature. Why didn't MWG also follow this?

MWG equate "rationalizable" with "existence of utility function" or "complete and transitive preference". To be clearer about my question, I am most interested in learning the first literature which argue that the "existence of utility function" or "complete and transitive preference" is rational.

As pointed in the comments this was done by  Ragnar  Frisch.  At least Barten and  Böhm. (1982) as well as Johansen (1969) attribute these axioms to one of these two publications:

Frisch, Ragnar (1926). "Sur un problème d'économie pure [On a problem in pure economics]". Norsk Matematisk Forenings Skrifter, Oslo. 1 (16): 1–40
Frisch,(1926). "Kvantitativ formulering av den teoretiske økonomikks lover [Quantitative formulation of the laws of economic theory]". Statsøkonomisk Tidsskrift. 40: 299–334.

Also I think this question has a misconception.
Von Neumann and Morgenstern (1944) indeed in their Theory of Games and Economic Behavior equate maximization of expected utility $E(u)$, with rational behavior.
However, note that the $E(u)$ of vNM is already based on axiom of transitivity and completeness. The issue here is that vNM were not just deriving utility of rational person. They were deriving cardinal utility of rational person.
The assumption of transitivity and completeness is only condition for preferences to be rational in general (see MWG pp 6.).


How does a perpetual futures contract affects the price of the underlying? As long as I understand, a futures contract is kind of a prediction for the underlying's price at the time when it expires. But what happens if this futures contract is perpetual, and doesn't have a date where it'll expire? Like bitcoin's futures.

How does buying/selling this contract affects the price now?

(More of a long comment.)

The "futures" in "perpetual futures" found on cryptocurrency exchanges is a bit of a misnomer, as already pointed out. It's not a futures position, but a leveraged spot position. As a result, crypto "perpetual futures" prices typically track spot prices. (There are also bitcoin futures contracts, in the standard sense--- those traded on CME.)

Such positions are not restricted to crptocurrencies. Leveraged spot FX positions are common. The type of exposure is the same whether it's cyptocurrency or central bank currency. More broadly, it's not restricted to currency. Similar instruments exists for equity, etc (but the somewhat unusual vernacular "perpetual futures" is only used in crypto).

In principle, the relationship between spot order flow (leveraged or not leveraged) and price (exchange rate in the case of currency market) is the same in any market. Price equilibriates supply and demand.
(In comparison, the relationship between futures and spot rices derives from spot-forward parity.)

For example, you might buy USD 100 using CNY at 7 CNY/USD exchange rate but only making a deposit CNY 35C for a position with notional value of CNY 700. That's a spot position with 1:50 leverage. This would be a bet on the spot depreciation of CNY.
If CNY depreciates by 1% against USD, the broker credits your margin account by CNY 70. If CNY appreciates, you might get a margin call.

The broker would then hedge his net exposure from aggregate order flow. In the above example, if you were the only customer, the broker would hedge by exchanging USD for CNY in the inter-dealer market. This in turn would drive up CNY against the USD---until the exchange rate equilibriates to where the long USD-short CNY spot bet is no longer profitable.

Similar mechanism leads to crypto perpetual futures prices tracking spot prices, although price dislocation may occur during episodes of illiquidity.


Why doesn't economy of scale bring down prices of food? I have little to no experience in economics and would enjoy a simple answer, if possible. Additionally, I base this question on subjective data, so I might be completely wrong in my assumptions.

EDIT: The title was deemed confusing. What I mean by "food" is any ready-to- or close to ready-to-eat meal. Restaurant meals definitely qualify, but so do frozen meals that might need a microwave/oven. I'm unsure about instant ramen, since I can hardly imagine enough people consuming enough ramen to create economic pressure on the rest of the food industry. Anything requiring attentive cooking disqualifies, so e.g. pasta is out.

Having some experience in preparing food for groups of people of up to 50, I could easily observe economies of scale working, driving the price per meal down as the number of participants (and thereby meals) increased. Roughly speaking, I'd spend about 3\$ on a meal for myself, 2\$/meal for 2-3 people, 1\$/meal for 10 people, and so on.

EDIT: I should emphasize I'm not asking about a specific price, but rather use them to show the difference between expectation and reality.

So I would expect companies preparing hundreds or thousands of meals per day (i.e. restaurants) to sell meals at comparably low prices. Now most restaurants don't work like that and prepare every meal when it's ordered and I'd assume customers pay because they value the experience more than the food itself.

But why aren't there any places selling decent meals for .5\$? My explanations don't quite satisfy me:

They do exist, it's called McDonalds. The prices aren't that low because there is an organizational overhead, at some point the number of people required rises linearly with the food that can be distributed. I'm not quite satisfied by this, because if it were driven by economics of scale, I'd expect similar concepts for better quality meals.
They do exist, they are called soup kitchens. At some point, the prices are just so low that demanding the actual price causes people to question it's quality. I.e. when selling food for .1\$/meal, people start wondering what's wrong with it. The only way people accept these prices is by acting as a charity, giving the meals away for free and generating money through donations, rather than meals sold. But wouldn't that mean there should be something like for-profit soup kitchens? I know a pay-as-you wish restaurant in my city, they rely heavily on economics of scale, and still invest about 4\$/meal.
Rent is the essential problem. A restaurant can only seat so many people, and prices
would drop drastically enough only at higher numbers. In that case, wouldn't bigger restaurants or restaurants owning their venue offer cheaper food?
In the thought of all of the above, the solution are frozen foods. Prepared in huge quantities at industrial scale, they may be shipped anywhere and are storable, even selling in stores is possible. Then again, the cheapest frozen food I know still costs around 1\$/serving (and is by no means a full meal). There is actually a company sponsored by the red cross delivering frozen meals to elderly people, which has prices starting at 3\$/meal.
People just don't accept cheap food. Though possible, there is a much stronger social component to eating, driving people to avoid low priced food as to avoid a status-penalty. But there are cheaper places (e.g. cafeterias) in existence, already associated with a status-penalty, so why would even cheaper food result in unbearable status penalties?

In the US at least, purchases and wages cost more than rent for a fast-food type business:

Purchases include some other hidden labor costs (drivers for transportation etc.) I'm not sure what's in the "other" category, perhaps insurance, lawyers etc.
And I'm guessing in Europe (where you are) "value menus" might not be as common as in the US. These are loss leaders in the US by the way, i.e. the restaurants lose money on $1 double cheeseburgers (but they are required to sell them in order to keep the franchise.) Or so the story went around 2014. And you're not the only price-sensitive customer. A lot of McDonalds' US customers apparently were:

In January 2018 McDonald’s finally brought back its dollar menu after a five-year hiatus, marking the return of a deal structure that had been displaced by alternative — and less popular — offers, like the McPick $2. The removal of the dollar menu coincided with the company’s initial decline, and its return makes it competitive among other chains with similar value menus, such as Taco Bell and Wendy’s.

If you want even lower prices in the short run, your best bet might be that Amazon buys a fast food chain. They're known to sell entire categories of products at a loss.
And depending whom you choose to believe, McDonalds did or din't envisage labor cost savings when rolling out kiosks to replace cashiers. Their statements on that have been somewhat vague and thus interpretable insofar. I find it a bit hard to believe that a company which pioneered low food costs by cutting on labor didn't consider that as a factor; flashback to the 50s:

The [McDonalds] burgers sold for 15-cents, about half of what a burger cost at regular diners of the time.

And I think you might have some misconception about the level of scale needed to cut costs substantially (say by half). You expect to pay half as much per meal by ordering 10, but you are hardly in a position of bargaining power here given the scale of a large food chain. In contrast, the actual economies of scale achievable have long been realised when nation-wide (and later world-wide) franchise chains appeared:

The predominance of fast food restaurants changed the food supply chain all the way down to the farmer. McDonald's quickly became the single largest buyer of beef, pork, potatoes and apples in the U.S. That gave them tremendous economic clout.
The fast food system is all about standardization, and so when the companies went looking for someone to supply their meat, they choose to deal with their large, corporate counterparts in the packing industry. IBP began to produce "boxed beef," where the final cuts of beef, including hamburger, were produced at the processing plant rather than the local grocery. IBP became the largest supplier of hamburger meat to the fast food industry.
Kentucky Fried Chicken buys all of its chickens from huge suppliers like Perdue, Tyson and Pilgrim's Pride. McDonald's gets its fish products from the giant supplier Gorton's of Gloucester.
Because consumers with busy lifestyles needed food fast, the chains needed raw materials in standardized packages. So, meat packers needed a consistent supply of standardized animals to produce their meat. They couldn't afford to deal with the uncertainty of many, small family farms. So, livestock producers became bigger and bigger. McDonald's and other chains have also been accused of using their huge buying power to keep farm produce prices artificially low.

What you really want is technological progress at any of these levels, because the foreseeable economies of scale are already there.
And (being focused on restaurants), you might not know that

Chicken is America's favorite meat today, surpassing beef and pork. Chicken consumption per capita has nearly doubled in recent decades, whereas consumption of beef and pork has declined or stagnated. [...]
Far from being automatic, these changes are the consequence of technological and organizational innovations that have made chicken products cheaper, abundant, and more diverse over the last century. This transformation is directly related to the rise of new lead firms and governance structures in the chicken value chain. The increased sophistication of processed chicken has given rise to a set of firms known as integrators, which vertically integrate through ownership a wide range of activities from breeding to processing and distribution. The shift to away-from-home consumption favored fast food giants like KFC and McDonald's, whose mass demand for processed chicken required higher levels of explicit coordination by the integrators. [...]
The chicken value chain is characterized by a hierarchical governance structure made up of a few large integrators and tight coordination between them and large food retailers and fast food chains. The integrators have become highly concentrated as well. Two leading integrators, Pilgrim's Pride and Tyson Food, accounted for 48% of the US market in 2006, up from 30% in 1996. They exert great power over the entire value chain by connecting a wide array of activities from inputs to distribution. [...]  Their linkages with thousands of contract growers, mostly small farm owners, can be portrayed as a captive relationship, where the growers heavily depend on the integrators for market access and resources. Meanwhile, the integrators have relational or modular linkages with large food retailers and fast food chains, which are now highly concentrated.

The latter paper goes into some detail into what the captive relationship consists of, but I'd be digressing too much detailing all that here; the high-level definition is that it is "a means to maintain control over the growing process with minimal ownership" (compare this with franchise model). How far vertical integration goes (and thus economies of scope) is limited by some legal issues like antitrust legislation.


Market demand independent of distribution of income If preferences are identical and homogeneous, then show that market demand for any good must
be independent of the distribution of income.

My workings are as follows:
$$q^{d}(p)=\sum_{i=1}^{n}f(p_x,p_{-x},I)$$

$$q^{d}(p)=f(p_{x},p_{-x})\sum_{i=1}^{n}I$$
Here, $x$ is the price of that particular good and $(-x)$ is the price vector of all the rest of the goods except $x$. Market demand is given by the submission of all consumers from $1$ to $n$.
But, I'm still confused how this function is multiplicatively separable in the price vector $P$.Can someone give a proof of it.
Thanks!

I take homogeneity of preferences here to mean that preferences are homothetic; that is, preferences can be represented by a utility function that is homogeneous of degree one.

For an arbitrary good $x$ and a vector of prices $\mathbf{p}$, let individual demand be $x_i(\mathbf{p}, M_i)$ for each consumer $i$ with income $M_i$. Thus market demand is $D = \sum_{i} x_i(\mathbf{p},M_i)$. Since preferences are identical,

$$\sum_{i} x_i(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},M_i).$$

Then, by homogeneity,

$$\sum_{i} x(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},1) M_i = x(\mathbf{p},1) \sum_{i} M_i = x(\mathbf{p},1)M,$$

where $\sum_i M_i = M$ is aggregate consumer income. Thus $D$ is independent of the distribution of income and is solely a function of aggregate income across consumers, $M$.


Kőszegi - Rabin (2006) model problem First of all, that's a homework question, and I will try to make it as useful to future readers as possible.

Problem

So, I was given a problem about the model described by Botond Kőszegi and Matthew Rabin in "A Model of Reference-Dependent Preference" in 2006:

  A person deals with two goods: tea ($c_1 \in \{0, 1\}$), and money ($c_2 \in R$).
  $$
c_1 = \begin{cases}1, \mbox{if he buys tea}\\ 0, \mbox{if he does not buy tea} \end{cases}
$$
  His consumption utility is given by: $$m(c) = v \times c_1 + c_2$$
  His gain-loss utility is given by:
  $$
n(x | r) = \begin{cases} (x - r), \mbox{if } x \ge r\\ \lambda \times (x - r), \mbox{if } x \lt r \end{cases}
$$
  Tea costs $P_H = 30$; also, let $v = 13, \lambda = 4$.

There are multiple tasks, but I'm struggling even with the first one:

  Prove that the reference point $r = (c_1, c_2) = (1, -30)$ with probability 1 is his personal equilibrium.

  Do not forget that in the gain-loss utility in case of deviation from expectations, the lost utility should be considered - as it looks in the consumption utility.

Confusion 1: What does the second sentence even mean and how do I consider the lost utility??

My attempt

First of all, let's define what we're looking for. According to page 1143 of the paper:

  A selection $\{F_l \in D_l\}_{l \in R}$ is a personal equilibrium (PE) if for all $l \in R$ and $F'_l \in D_l$ $U(F_l | \int F_l dQ(l)) \ge U(F'_l  | \int F_l dQ(l))$.

So, provided that the probability is 1, I can get rid of the integrals and prove that $U(r | r) \ge U(x | r) \forall x \ne r$, or, given that there are only two possible choices (right?), that $U((1, -30) | (1, -30)) \ge U((0, 0) | (1, -30))$.

Confusion 2: does he really have only two choices and is the second choice $(0, 0)$?

Next, compute $U(r | r)$. As per p. 1138 of the same work:

  We assume that overall utility has two components: $u(c|r) = m(c) + n(c|r)$, where $m(c)$ is "consumption utility" typically stressed in economics, and $n(c|r)$ is "gain-loss utility."

$$
U(r | r) = U((1, -30) | (1, -30)) = m(1, -30) + n((1, -30) | (1, -30))
$$

As per the same page:

  We also assume that gain-loss utility is separable: $n(c|r) = \sum_{k=1}^K n_k(c_k | r_k)$.

So I can do:

$$
m(1, -30) = v \times 1 - 30 = 13 - 30 = -17\\
n((1, -30) | (1, -30)) = n(1 | 1) + n(-30 | -30) = (1 - 1) + (-30 + 30) = 0\\
\mbox{Thus, } U(r | r) = -17 + 0 = -17
$$

Now for the second option:

$$
U((0, 0) | (1, -30)) = m(0, 0) + n((0, 0) | (1, -30)) = 0 + n(0 | 1) + n(0 | -30) = \lambda \times (0 - 1) + (0 + 30) = 30 - \lambda = 30 - 4 = 26\\
\mbox{Thus, } U(x | r) = 0 + 26 = 26
$$

Confusion 3: So, I got $U(r | r) \lt U(x | r)$, which is quite the opposite of what I need to prove...

UPDATE

I'm being told by a fellow student that one should multiply the second part of the gain-loss utility by $v$ (!) to get:

$$
n(x | r) = \begin{cases} (x - r), \mbox{if } x \ge r\\ v \times \lambda \times (x - r), \mbox{if } x \lt r \end{cases}
$$

So that the computation of $U(x | r)$ I get:

$$
U((0, 0) | (1, -30)) = \lambda \times v \times (0 - 1) + (0 + 30) = -22
$$

...and the same for $U((1, -30) | (1, -30))$.

They also say that this is what the "Do not forget that in the gain-loss utility in case of deviation from expectations, the lost utility should be considered - as it looks in the consumption utility." part of the task tells me to do. Unfortunately, I still have no idea what this part means or why I should multiply only the second part of the function by exactly $v$. I mean, why would you just change the function out of nowhere?

Question

Where am I wrong? And why do I need to multiply by $v$, and do I actually need to do this?

You can interprete $c_1 \in \{0,1\}$ as "I have the tea" and "I don't have the tea", and this tea is valued $v=13$. $c_2$ is just a wealth transfer, i.e., it is $-p$ if the tea is bought and $0$ if the purchase is rejected.
Gain-loss utility is assesed separately in each dimension. Such that
$$n(c|r) = n_1(c_1|r_1) + n_2(c_2|r_2).$$

In your case the reference point is $\widehat r =(1,-30)$. In words, the buyer expects to purchase the good at price 30.
In a PE, it must be that given this reference point
$$U (\widehat r| \widehat r) \geq U ( c | \widehat r) \quad \forall c.$$
That is, the buyer prefers to execute his plan.

This leads to utility
$$U (\widehat r| \widehat r) = v * 1 - 30 + 0 = 13-30=-17.$$
Suppose instead that the buyer does not buy the good. In pure strategies and with a fixed price, there are only two options "buy at price 30", $(1,30)$ and not buying, $(0,0)$.
That is, she does not spend $p$ of her wealth and does not get the good with value $v$,
$$U ((0,0)| \widehat r) = v * 0 - 0 \:+ \quad \lambda(v*0-v*1) \:+ \quad (0-(-30)), $$
where the first part is the consumption utility from not buying, zero.
The second part is the loss-utility from not having the good weighted by the loss-aversion parameter $\lambda$. He does not have the good, so he feels a loss of $v$, the value of the good. The third part is the gain-utility from saving 30 bucks.
Hence,
$$U ((0,0)| \widehat r) =30 - \lambda v = -22$$
with the given parameters. We can verify the given PE.

$v$ is the value of the good. Think of this setting as separable consumption utility
$$m (c) = m_1 (c_1) + m_2(c_2)$$
with $m_1 (c_1) = v c_1$, i.e., if you have the good you garner $v$ and otherwise zero.
The money just enters linearly $m_2 (c_2) = c_2$. Gain-loss utility is always evaluated with respect to consumption utility, see the paper. That is,
actually
$$n_i (c_i|r_i)  = \mu [m_i (c_i) - m_i(r_i)]$$
with the given "universal gain-loss function".

EDIT: To make this useful for all, let me mention the paper corresponding to this model: Heidhues and Kőszegi (TE, 2014). See how they define the product and the money dimension, $k^v =v b$, where $b$ is the $c_1$ of your model (buying or not buying), and $k^p=-bp$. This paper has interesting insights on the impact of consumer loss aversion on pricing and it supersedes their earlier paper.


What are the necessary and sufficient conditions on a utility function for gross substitutes? What are the necessary and sufficient conditions on a utility function for gross substitutes? Gross substitutes is:

$$
  \frac{dx^*_i}{dp_j} \ge 0\ \ \forall_{i,j}\ i \ne j
$$

The paper @Henry referred to in comments pretty much answers this question. For closure, and to summarize the jist, the paper opens:

  As is very well known, the case in which all excess demands have the gross substitute property is one in which very strong results can be obtained concerning the uniqueness and stability of general equilibrium. Considering how long this has been known it is perhaps remarkable that we apparently do not possess a convenient characterization of the class of utility functions which yield individual demand functions with the gross substitute property. ... [This] note provides a characterization of the entire class of utility functions having the gross substitute property. Since the primary interest in that property is in the analysis of general equilibrium, we shall mainly concern ourselves with the demand functions which arise in a condition of exchange rather than with those that arise when income is fixed, independent of prices. The extension of the results to the latter case being given at the end of the paper.

The main result is, in an exchange market, for any initial endowments (quoting directly):

A necessary and sufficient condition for $U(x)$ to have GS is that for every $i, j = 1,...,n, j \ne i$, positive income $Y$, and strictly positive price vector $p$:

$$
  \eta_{ij} > max(e_i, e_j, e_i(1 - \frac{1}{\alpha_j}), e_j(1 - \frac{1}{\alpha_i}))
$$

Where:

$e_i$ is the income elasticity of the i-th commodity, $\frac{\partial x_i}{\partial Y}/\frac{Y}{x_i}$.
$\alpha_i$ is the proportion of expenditure devoted to the i-th commodity, $\alpha_i = \frac{p_ix_i}{Y}$.
$\eta_{ij}$ is the elasticity of substitution i-th to j-th of the so called "Allen-Uzara" kind (what ever that is - see paper).

There are some corollaries.

The main complication with all this is you have to take income or rather income elasticity, into consideration. Some broad classes of utility functions that are sufficient for GS regardless, are only very briefly mentioned in the intro of paper.


Why is the price of capital ‘r’ ? (From Cost function) according to the Cost formula in microeconomics class,

Total Fixed Cost is represented as “rK” (K as in unchanging, fixed K)

now my economics teacher tells me this ‘r’ is the interest rate at which you rent capital, which is pretty vague and ambiguous to me. If r represents interest “rate” and not the “interest-added cost of capital”, then how can this r be the “price” of capital?

So for example, if the amount of capital is 5 (K=5) I understand that you assume you rent it, or borrow the money from a bank to get the capital, incurring interest. What I don’t understand is the formula. If r is the interest rate (aka rental rate)

isn’t the cost formula supposed to be “K * {P(K) + P(K)*r}”, instead of “rK”?
 (P(K) here represents the price of capital per unit)

so if a unit of capital costs $10, and the interest rate (r) is 3%,

in the end you will be paying $ 5 * (10 + 10 * 0.03) in total , not 5 * 0.03 right?

I mean, r is just the percentge, not the ‘amount’ that you have to pay for the capital. So I don’t really understand why the textbooks and the teachers tell me that the “price” of capital is r when it seems like it’s supposed to be ‘P(K) * (1+r)’ instead of just ‘r’ .

I hope I can get a satisfying answer here. Would appreciate detailed explanations!

Firstly, in your example the value of $r$ (as used by economists in this context) would be $1.03$, not $0.03$. Economists call this the "interest rate", but you might prefer to think of it as the "rate of return on capital".

Secondly, what we define as constituting one unit of capital is pretty arbitrary. Is a computer one unit of capital or ten units of capital? It doesn't make any difference, just like it doesn't matter whether you measure distance in meters or feet, so long as everyone knows what units you are using. Since this choice is arbitrary, economists usually make the simplest possible choice and measure units of capital in units of money (e.g., dollars), meaning that one unit of capital has a price of one dollar by definition: $P(K)=1$.

If we return to your formula with $r=1.03$ and $P(K)=1$:

$$K[P(K)+0.03P(K)]=P(K)K(1+0.03)=Kr,$$

we get exactly the cost you saw in your course.


What methods - inspired by Haavelmo’s Structural Econometrics - can show that a partial equilibrium model is unreliable? According to Spanos 2014 Revisiting Haavelmo's Structural econometrics: Bridging the gap between theory and data Dynamic Stochastic General Equilibrium models are statistically inadequate, in such an order of magnitude as to render them useless.

  The most serious problem with DSGE models is that they are often statistically inadequate. Advocates of DSGE modeling, however, have
  put forward a number of arguments to explain away the unreliability of
  their evidence, such as ‘the price one has to pay for rigorous and
  policy-relevant theoretical models’ (Lucas, 1980, p. 696), or simply
  inevitable: “The models constructed within this theoretical framework
  are necessarily highly abstract. Consequently, they are necessarily
  false, and statistical hypothesis testing will reject them. This does
  not imply, however, that nothing can be learned from such quantitative
  theoretical exercises.” (Prescott, 1986, p. 10)

Part of my research involves using a partial equilibrium models to simulate a particular segment of an economy.

Considering the above, what methods would you start with to show that a given equilibrium model is statistically unreliable?

I can think of two big ways.

Formal tests of model fit: If you are using SMM, GMM or indirect influence check out the J-stat. If you are using maximum likelihood you want a likelihood ratio. Bayesian model testing is similar to likelihood ratios, but more complicated. Many of the DSGE models were rejected on these bases (you can see Sargent discussing it briefly in this interview). These tests are all very similar.
Out of sample performance: Estimate the model, then use it to predict out of sample data. This is a tough test, but it is super useful. They aren't so common in economics, but statisticians use them a lot (often times they leave out part of the data and try the model out there). You probably want your model to pass a formal test in sample before you do an out of sample test.

All these tests rely on the same, extremely basic, intuition: If the model is good then it should fit the data well. If the model does not fit the data well it is probably not good.


How to get the effect of one dummy variable against many others? I have the following regression:

wage = constant + (beta1)*michigan+ (beta2)*california+...+(beta49)florida+(beta50)education + u

where michigan is equal to one if the person is from michigan and zero otherwise, and so on for all of the next state variables.
education is equal to the number of years of education of the person.
The base case is South Dakota, which is the state that does not appear in the regression, that is why the coefficient for florida is 49 and not 50.

I want to get the effect of being from south dakota against any other state in wage. My idea is to run this regression, and then add up betas for all other states (beta1 to beta49). If the sum is positive that would mean that people from any other state earn on average more, and the opposite if the sum is negative. Is this correct? if so , why? if not, why not? and what would be the right way to do it?

If you only want to test whether the mean wages of South Dakotans is higher than for residents of other states, run:

Where South Dakota is a dummy variable like the one you've described earlier. I'm assuming you simply have a cross-section of wages at a single point in time, but if you have either repeat observations of individuals or a repeated cross-section, then you can include a t subscript for each of the variables and include time controls.

If  is positive and significant, it would be interpretable as the level or percent (if the model is run in log wages) increase in mean wages for a resident of South Dakota compared to the mean for the pool of residents of all other states.

If instead you wanted to perform pair-wise comparisons between South Dakotans and residents of an individual state, you could include a dummy for that state and then run a command like lincom in Stata to perform hypothesis tests between those states' coefficients.

If you wanted to compare South Dakota against every single other state, then make South Dakota the omitted category in your regression, and you could run the model you described of including dummies for all non-South Dakota states. The big issue with this approach is the massive multiple hypothesis testing issue you will encounter, which could be addressed through Bonferroni correction.


How to interpret income elasticity of demand My question:

  Income elasticity of demand for apple is $\gamma$. Cross elasticity of
  demand between apple quantity and orange price is $\psi$,  What is the
  impact on apple's revenue if consumers' income increase by x% and
  orange price fall by y%?

What I think the answer is?:
Am I right I say that revenue would increase for the first instance (income) and the revenue will fall for the second instance.  However, how would I know the percentage of revenue increase in the first instances and fall in the second instances?

Hints:

Looking at the definitions, if the income elasticity of demand for apples is 2, how much does apple demand increase when consumers' income increase by 10%?
Looking at the definitions, if the cross elasticity of demand between apple quantity and the orange price is 0.5, how much does apple demand decrease when the orange price falls by 15%?
How might you combine the increase with the decrease?

Multiplication might be a useful tool.


Why does supply equal demand? Problem if large increase of price, decreases demand only by a bit I have looked at my coursenotes and this thread here:Why does supply equal demand?, but they did not adress, why I do not understand the law of supply and demand.
The way I understand this law is the following: Supply is the amount of goods that can be produced,given the price of the good. Demand is how much of the good is demanded, given the price of the good. Supply is equal to demand.
The reasoning from my coursenotes is:If supply is higher than demand, sellers will decrease the price, so they can still sell everything they have. If Demand is higher than supply, there is no reason for the sellers to keep the price low, so they increase the price in order to get more profit.
I still dont understand one aspect. If we are in a situation, where a high increase of the price leads to only a small decrease of demand, the cost of potentially not selling a proportion of your goods is outweighed by the gain in profit from each good. So sellers do not have a reason to decrease the price.
I must be missing something, so I am very happy to see some explanation.

"Sellers" as a whole indeed do not have a reason to decrease the price under these conditions. But sellers don't decide together collusively, acting like a monopoly seller. Instead, each of the many individual sellers decides on its own. And each does have an incentive to lower its price, because by posting a lower price it can suddenly sell all of its individual supply (as long as this is below total market demand), which always increases profit. In other words, even if the market demand curve is very inelastic, the demand facing a single seller is highly elastic. It is therefore profit maximizing for an individual seller to slightly lower its price. If all others follow suit, the market price has decreased. This continues until equilibrium is reached.