Pareto optimality when the allocations are restricted This question is about Pareto optimality (PO) in cake-cutting.

The basic definition is: an allocation is PO if there does not exist another allocation in which all players are weakly better off and at least one player is strictly better off. For example, consider the following cake (where "C" means Chocolate and "V" means Vanilla):

  CVCV

Suppose there are two players: Alice wants only chocolate and Bob wants only vanilla. Then, there is only one PO allocation, and that is to give slices 1 and 3 to Alice and pieces 2 and 4 to Bob.

Now, suppose we restrict the cake-cutting such that the pieces must be connected. With this restriction, there are two possible definitions of PO:

If we remain with the previous definition, then there exits no PO allocations with connected pieces, since the only PO allocation requires disconnected pieces.
On the other hand, we can change the definition and say that an allocation is PO if there does not exist another allocation with connected pieces in which all players are weakly better off and at least one player is strictly better off. Under this definition, of course there exist PO allocations, such as, giving slice 1 to Alice and slices 2-4 to Bob.

Both definitions are reasonable. My question is:

which definition is more common in the economics literature?

Let me elaborate on the comments quickly - maybe only for the sake of this question not being "unanswered".
Pareto optimality only makes sense when defined with respect to feasible allocations. That is, your basic definition above lacks an important word:

  an allocation is PO if there does not exist another feasible
  allocation in which all players are weakly better off and at least one
  player is strictly better off.

Without this word, you can easily find an allocation that dominates "A gets slices 1&3; B gets slices 2&4". For instance, you could give Alice eight additional full cakes of chocolate. However, there are no other cakes and, hence, this allocation is not feasible and therefore does not Pareto dominate what you suggest.

As a result of your feasibility restriction, any allocation in which one agent gets one slice of the preferred kind and the other one gets the other three is PO.
If A gets the entire cake, she can make B better off by giving him a slice of V which she does not care about; analogously if B has the whole cake. If both agents get two connected slices, a similar improvement exists.


Derive the demand functions: Hotelling-style Model So I have this economics question that I have been trying for a while now and I can't seem to get the answer correctly. Below is the question and after I will show what I have so far. An explanation would help so much.

  Consider the following Hotelling-style model. There are 3 Arms, each offering a single type of milk (always in 1 quart containers) that is horizontally differentiated along a single dimension, the percent of fat.

  Firm-1 offers non-fat milk.

  Firm-2 offers low fat milk, with 10% fat.

  Firm-3 offers high fat milk, with 50% fat.

  There are 1,000 potential consumers. Some consumers hate fat and some love it. But the most fat any individual consumer would ideally have in their milk is 50%. In particular, assume that people's tastes for the ideal percent of fat is uniformly distributed from 0% to 50%. That is, the line starts at zero and goes to 50 (instead of 1 as we have done in the past).

  The utility that individual i obtains from purchasing a quart of milk from seller j is given by

  Uij = 5 - pj - 1/10*|Xi - Xj|

  Where Xi € [0,50] is individual i's ideal fat content and Xj € [0,50] is the fat content offered by firm j.

  For example, if a consumer is located at Xi = 2 were to purchase from firm 2, and firm 2 happened to set a price P2 = 3, they would obtain utility:

  Uij =5 - 3 - 1/10*|2 - 10| = 1.2

  Derive the demand functions for each of the three firms.

So far I have been able to derive the demand function for firm 1 by first solving for Xi for the indifferent consumer between Firm 1 and Firm 2.

The line looks like this

  |--x--|-------------|

with the first line representing 0% fat, second 10%, and third 50%. The x represents the indifferent consumer between firm 1 and firm 2.

The math for the first firm is done by taking the Utility of the Indifferent consumer for firm 1 and setting it equal to the utility of the indifferent consumer for firm 2.

  5-P1-1/10*|Xi-0|=5-P2-1/10*|Xi-10|

by doing this, you end up with

  Xi=5P2-5P1+5 with the Q1=(Xi-0)*1000/50

Therefore the demand curve for firm 1 is

  Q1=100P2-100P1-100

The next step would be to be to solve for the demand curve for firm 3. My problem is when I set U2=U3 the Xis cancel. The math is below.

  5-P2-1/10*|Xi-10|=5-P3-1/10*|Xi-50|

This reduces to:

  -P2-1+Xi/10 =-P3-5+Xi/10

Finally this reduces to:

  -P2+P3+4=0

The "Xi/10" on both sides cancel which does not make sense. Because of this result, I have no idea how to solve for the demand curve of Firm 3.

You are treating the absolute value wrong. You seem to have done it right the first time around, do it like that again. (Make a little drawing, think about which side of 10 and 50 the indifferent $X_i$ will be on, etc.)

Another thing: The demand function you derived for $Q_1$ only holds for certain ranges of $P_1$, $P_2$ and $P_3$, because if the prices of firms 1 and 2 are very high but $P_3$ is low, a customer indiffirent between firms 1 and 2 would go to firm 3.


Can an appreciating currency lead to inflation? If a currency, e.g. the rupee (INR), is appreciating against, say, the USD, it possibly means that there is a high demand for the former. India could have net exports, high remittances, increasing FIIs all leading to the appreciation of the INR. Does that not mean a lot of rupees flowing into circulation in the country? Should that not decrease the value of the currency and lead to inflation? Where exactly am I wrong?

You are wrong/vague here:

the rupee (INR), is appreciating against, say, the USD, it possibly
means that there is a high demand for the former.

If the rupee becomes more expensive w.r.t. the dollar, demand for the rupee usually decreases, as Indian exports are now more expensive to pay for when measured in dollars, and hence fewer people seek to convert their dollars to rupees to pay for these exports.


Have the Greek bailouts been included in state budgets Eurozone members? A lot of people (mainly journalists and politicians) say that Eurozone taxpayers pay for the Greek bailouts. On the other hand, a lot of economists say that those money have not been added in the yearly state budgets of the Eurozone members because they are not physical money coming from the tax payers.

Contrary, most of the Greek bailout funds are given in the form of intergovernmental guarantees for which Greece pays actual real interests. Additionally, the bailouts not being bond-like give the creditors as the Eurozone countries the right to ask back the value of these guarantees whenever they like.

Can someone objectively clarify the situation? Specifically, can you give some details on the form of these intergovernmental guarantees?

I don't know of a full accounting in the Eurozone but Measuring the True Cost of Government Bailout (Block (2010)) provides a look at the on- and off-budget costs of the bailouts in the United States and finds they were substantially off-budget.

  Government intervention to assist individual businesses and industries
  during the 2008–2009 economic crisis was extraordinary in variety and
  scope. Despite official protestations of "no more bailout" in the
  Dodd- Frank Wall Street Reform and Consumer Protection Act of 2010,
  future government interventions are inevitable, should economic
  circumstances become sufficiently dire. Moreover, even if Congress
  eliminates overt bailout-type interventions, indirect forms of public
  bailout are likely to continue. Understandably, taxpayers have been
  concerned about the cost. A simple tally of dollars authorized or
  disbursed is wholly inadequate to accurately assess the costs of
  various interventions. This Article addresses the challenges of
  providing reasonable budgetary information with respect to different
  types of bailout expenditures. In addition to looking at costs for the
  more obvious bailout programs, the analysis explores the special cost
  estimation challenges for other more covert actions, such as special
  tax breaks or relief from burdensome regulation, that serve a
  "bailout" function. The Article also takes issue with the
  fragmentation of intervention efforts among different "on-budget" and
  "off-budget" entities and with some of the methodologies used by the
  government to value assets obtained in its bailout efforts, arguing
  that decision making about the appropriate allocation of aggregate
  resources is hampered when some expenditures are "off-budget"
  altogether and when even "on-budget" agencies use different accounting
  methods. Finally, the Article calls for transparency and budget
  accounting for public bailouts accomplished more indirectly through
  the tax system and other regulatory regimes. Adequate and transparent
  budget accounting for bailout costs requires greater consistency in
  valuation and accounting methods, and a more unified presentation of
  aggregate information in the budget with respect to all government
  bailout-type activities.

Pages 23-25 explain what she means by off-budget and it takes her 3 pages to say exactly what she means, but approximately "on-budget" means upfront costs that appear on the government's budget while "off-budget" is everything else, including the value of contingent liabilities and expenditures by entities in the control of the US government but for whatever reason do not have accounts fully consolidated into the budget.


REE with asymmetric information and Bayesian Nash equilibrium? What is the difference between a Rational Expectations Equilibrium (REE) with asymmetric information and Bayesian Nash Equilibrium  (BNE)? Since agents in both cases play some game or have a strategic behavior what is the difference? I would like an extended answer with some examples to understand it because I have not found any specific details until now (maybe I am searching in the wrong direction). Thank you in advance!

The idea of a rational expectations equilibrium is more general than BNE. It simply means that the belief system of agents is consistent with the model and incorporates all available information. This abstract idea can be applied for games, markets or other types of interactions. BNE is a solution concept for non-cooperative games. The expectations being rational also poses no restriction on how agents in a model update their beliefs, while BNE assumes beliefs are updated according to Bayes rule.

Let me be more concrete. If you are modeling a competitive market, the REE is not a Nash equilibrium. Players are simply maximizing giving prices and information (instead of given the strategies of other players). In contrast, if you are modeling a market where the buyers or sellers are strategic (for example a Cournot model with incomplete information), the typical solution concept is BNE and it will also be an REE.

To sum up, any BNE is an REE, but the converse is not true.


Euclidean spaces,vector spaces,metric spaces,commodity spaces,Cartesian spaces What exactly is the difference between Euclidean space,vector space ,metric space ,commodity space,Cartesian space?

I am reading about consumer choice and I came across this line:

"A commodity vector (or commodity bundle ) is a list of amounts of the different commodities and can be viewed as a point in Rl,the commodity space.
We can use commodity vectors to represent an individual's consumption levels.The lth entry of the commodity vector stands for the amount of commodity l consumed."

When I googled the meaning of commodity space ,it says - commodity space is Euclidean space and here,as we have taken l entries ,it is l-dimensional Euclidean space.

By l dimensions ,do we mean l directions?

Because when we consider only two commodities in a simple Cartesian plane,we can show them as perpendicular distance from two axis (moving in two perpendicular directions).

Is an l dimensional Cartesian plane similar to l dimensional Euclidean plane?

Since we are taking consumption bundles as commodity vectors,commodity space should be same as vector space.

We are now left with metric space ,the Google search engine defines it as " a set for which distances between all members of the set are defined"

We read about various consumption sets defined as a subset of the commodity space Rl.These consumption sets can be open as well as closed..

This shows that all the spaces listed above are interconnected..or similar to each other.

I will try to do this at an intuitive level without being technical. If you want rigorous definitions of any of this I can edit my post.

We need a way to mathematically model the consumption choices that an agent faces. To accomplish this, we use $\mathbb{R}^L_+$ to denote the agent's level of consumption of each of 1,2,..,L goods.

Vectors tie into this because we want to think about how much of each good an agent chooses when optimizing over her preferences. So we don't just think about how much of good $i$ the agent wants. We want to think about how much of good 1 the agent wants, good $2$ he wants, etc. simultaneously. So, there are tons of different "mixes" of these goods that an agent can afford (that belong to this agent's budget set given some prices and wealth) and one of those "mixes" (vectors) represents how much of each of the L goods an agent chooses to consume when maximizing his or her representative utility.

Now - Cartesian space. You should actually discuss the Cartesian coordinate system on Euclidean space. We use Cartesian coordinates to denote a point in Euclidean space. We could, for example, instead use polar coordinates to denote points in Euclidean space.

A metric space is a bit more complicated  but you can think of it (this is very simplified) as a space where the distance between any two points is well defined. The distances themselves are called a metric. Having a metric space allows us to use topological methods (for example, the concept of open balls when discussing the local non-satiation of preferences). The most common example of a metric space is three-dimensional Euclidean space.


How does profit correlate with the duration of options? To wit, do  you profit more off options, the shorter-term they are like 0 DTEs? I know that  option premium correlates negatively with duration.
Pre-suppose you think some biotech will discover a medical cure and skyrocket, but you don't know when. So on a Thurs, you buy one each of a whole option chain for a company. So you have 1 option each of a 0 DTE, 7 DTE, 21 DTE, ..., 365 DTE, ..., to a 2-year LEAP.
The next day, the cure is discovered and the company's share price rockets! Do the shorter-term options necessarily outperform the longer-term options? Or does this depend on other variables?
I ask because gain-flaired posts appear to feature shorter-term options, like a \$600K+ gain on SLV in 3 days, $69,420 in a month, \$35K to \$1.25M in approx. 4 months, \$800 to \$40K in 2 months. I'm not wealthy enough to, and don't, gamble on options, and I know there are many posts flaired Loss too.

You may profit more off of short-term options in terms of returns. The reason is simple: with less time to expiry, the time value of an out-of-the-money option is near 0. However, this is only true if the options expire in-the-money -- a huge "if."
There are a few caveats that are crucial:

Options impound the entire future expected distribution of returns, so you are unlikely to make easy money.
Any post on Reddit or other social media on money made trading or trading opportunities is likely to be worthless and could be trying to lure you into a pump-and-dump or other manipulation. I would stop reading such posts on Reddit.
You mention options maturing in 7 and 21 days as well as LEAPS. Weekly expirations tend to be very illiquid as do LEAPs. Quarterlies are your most liquid expiries; I'd stick to those.
The bid-ask spread can be quite wide for weeklies, LEAPS, and out-of-the-money options near expiry. You could easily pay a bid-ask spread that is nearly the quoted price of the option -- so a 100% bid-ask spread.

Glad to hear you aren't planning to trade this on your own. I've traded options for years professionally and even I will not trade options in my personal account.


How is the utility function with constant relative risk-aversion obtained? In this slide, it says that constant relative risk-Aversion utility function have this form.

$u(x) = \frac{1}{1-b} x^{1-b}$ for $b≠1$

$u(x) = In(x)$ for $b=1$

When I tried to derive the utility function from $b = -x \frac{u''(x)}{u'(x)}$ (the representation of CRRA), I got $u(x) = x^{1-b}$.

While I understand that preference will not be changed by linear transformation of the utility function, I wonder why is it necessary to make the utility function be $u(x) = \frac{1}{1-b} x^{1-b}$ for $b≠1$.

Also how is the "$u(x) = In(x)$ for $b=1$" obtained?

In the slide, we're given the marginal utility (or the derivative of the utility function) as $m(x) = x^{-b}$. The utility function whose derivative is $m(x)$ is

\begin{eqnarray*} u(x) =  \int m(x) dx = \int x^{-b} dx = \begin{cases} \frac{1}{1- b} x^{1-b} & \text{when } b \neq 1 \\  \ln x & \text{when } b = 1 \end{cases} \end{eqnarray*}

Verify that $u'(x) = m(x)$.


Fractional reserve banking and bank insolvency I am trying to understand the fractional reserve banking system (with say 10 % reserve ratio). If a bank becomes insolvent because 100 % of the loans of a bank go bust, then who really becomes liable for all that money lent out in excess of the bank's value ? Let us say, the depositors are all FDIC-insured and the stockholders of the bank lose all their money, but isn't the loaned amount still larger than the (stock value + deposit amount) ? So perhaps all that happens is that this money simply gets written off (and adds to inflation) ?

Let's assume (as you do if I understand well) that the bank's only liabilities are deposits, so that the bank's equity valuation is simply the value of assets minus deposits. In order to provide loans the bank will have to hold 10% of those funds as reserves in its central bank account (the account it holds at the central bank, written in central bank money or base money), the the maximum amount the bank could lend out would be 90% of its liabilities + equity.

Then if 100% of the loans defaulted at 100% loss severity, 10% of assets would remain, to be shared between the depositors and the equityholders. By the traditional laws of bankruptcy and claims priority, the equityholders are the first to incur losses, so they would lose all their money (assuming the depositors represent at least 10% of the bank's balance sheet). Depositors would incur the remaining loss. At that point, if the FDIC had previously insured all depositors, it would have to repay them what they had lent to the bank, and the FDIC Deposit Insurance Fund would suffer from the loans' defaults. If you assume no external debt for the bank, then by accounting equality the total loan amount cannot be higher than the equity + deposits. (Assets = Liabilities + Owners' Equity)


For the case of two goods, give an example of preferences that are represnted by a continuous utility function that allows for fat indifference curves The question in the title sounds like a trick question, due to the monotonicity property that indifference curves have, such that for two goods x and y, strong monotonicity implies y > x.

Possible that you could answer the question by stipulating the area under a curve?

$$u(x, y) = 2$$
is a simple example of a continuous utility function with thick indifference curve.

\begin{eqnarray*} u(x, y)= \begin{cases} x & x < 1 \\ 1  & 1 \leq x < 2 \\ x - 1 & x \geq 2 \end{cases} \end{eqnarray*}

is another continuous utility function with thick indifference curve for $u=1$.


Listed property price v.s. transaction price A house may be listed at a certain price (e.g. £200,000), but sells for a different price (e.g. £210,000).

In terms of economic theory, how does the transaction price tell us more information than the listed price?

Most papers use listed price because of data availability, and then state that it is more preferable to have transaction price. Why is this the case?

The listed price tells you something about the expectations of the vendor and their estate agent. It's a leading indicator, hinting at where the market may be going next.

The transaction price tells you something about the actual market during the period of the sale, in the period between the property going on the market and the date of exchange. It represents a real meeting of supply and demand - it is a market price. It only becomes available in the weeks or months after completion of the transaction, so the market will already have changed by the time one can get all the data.


How does Ricardo justify this math? In Ch 5, Of Wages, Ricardo tosses this gem:

  if the labour of ten men will, on land of a certain quality, obtain 180 quarters of wheat, and its value be £4 per quarter, or £720; and if the labour of ten additional men, will on the same or any other land produce only 170 quarters in addition, wheat would rise from £4 to £4 4s. 8d.; for 170:180::£4:£4 4s. 8d.

That seems like a bold proposition to me, and I'm not sure where to look for the justification of that math.

Is it earlier in Ricardo and I just missed it?  Is it intuitively obvious given some fact I just don't know yet?

Because in Ricardo's view, the money is neutral. The price reflects only the quantity of labor which is incorporated in the product. If it takes you 2 hours to hunt a deer and only 1 hour to hunt a rabbit then the deer will be two times more expensive.

That being reminded, one can go back to Ricardo's example. First, we have 180 quarters of wheat that needed 10 men (which gave an output of 720£, at 4£ per quarter). But afterwards you have 20 men producing 350 quarters of wheat, which obviously takes into account the less productive land. Let's say we wanted to double up our first production with the new land: then we would in fact need about 21 men to meet our target of 360 quarters (2 x 180 quarters). But as you see this can't be worth 1 440£ , because there is mork work incorporated ! So there must be a new price for the quarter of wheat ...

Therefore the following question is: what shall be the new price ? One can turn that around into asking: how much more work is incorporated in the new production ? Well there is about 5% more work (1 man = 5% of the 20 that should be normally if there weren't any diminishing returns), which you can also find out by putting: 180/170 = 1,05 or 1+5%. If it is still not clear: this fraction represents the production advantage of the first land in comparaison with the second one. It is 5% more productive. Which means that you have to incoporate 5% more work to get the same production, which in returns, means the quarter of wheat has to be about 5% more expensive...


Converting a GDP series to constant international dollars I want to convert a GDP series (I have it in current euros) of a country to constant 2005 international dollars. My idea is to divide the series by year 2005's $PPP$ conversion factor (the exchange rate) and adjust for inflation by dividing the whole series by the country's GDP deflator $(base \; year=2005)$.

Is this the right way to go? Do I need to adjust for inflation?

You are correct. That is the right way to go.

Take your starting nominal GDP series in local currency units (e.g. Euros), and use the PPP rates to transform local currency units into nominal US dollars. You can find PPP rates here or here.
Use the US GDP deflator (or a price index) to transform nominal GDP in USD into real GDP in USD. The base year is given by the chosen price index.

You can find GDP in constant US dollars for 2011 form almost every country in the world here.

See this document or this website for further references.


Omited Variable Bias and descriptive statements Let's say $y=c+ax+by+error$
 (where the error term fulfills all the assumptions) descibes reality. If we have $z=vx$ than $y=h+max+s$ will also describe reality. When reading about OVB I have seen statemets like "if the true modell is $y=c+ax+by+error$
 and x and z are correlated than regressing only on x will overestimate(or underestimate) the true coefficient of .” But this seems to be trying to establish causality? If we are only talking about descriptive models than there could be many different correct coefficients for
depending on the variables included in the model couldn't there?

This is not about causality. Causality is a statement that $x$ causes $y$ but it is not the value of the coefficient ($a$ in your model). Even if you care just about coefficient estimates you will get wrong answer. This is because the formula for estimate of the coefficient $a$, that is $\hat{a}$, in simple OLS can be expressed as:

$$\hat{a} = a + \beta \frac{\text{COV}(x,z) }{VAR(x)} $$

That is the estimate of $a$ you get from OLS is the true estimate $a$ + the true value of $\beta$ which would be the coefficient of $z$ if you would actually include it in a model by estimating $y=c+ax+\beta z + error$ multiplied by the covariance of $x$ and $z$ and divided by variance of $x$. Now variance of $x$ must be non-zero just to be able to estimate OLS in the first place and variance is always positive so the over or underestimation will be given by the sign of the covariance and the $\beta$.

Now whether you care about causality or no this bias is still important consider the following example. You want to estimate the effect of education of parents on number of children they have. You dont care about causality you just want to calculate whats the coefficient inside your sample i.e. you only care about whats the average number of children for a couple with average years of education, and in addition suppose that education is also correlated with income. Also to make calculations easy lets assume that VAR$(x)=0.5$ and COV$(x,z)=0.5$ and that average years of education of parents are 10 years. Now lets suppose the true model looks like this (I already replaced $c,a,\beta$ with actual numbers):

$$y_i= 10 -0.3x_i-0.2z_i + u_i$$

In this true model a family with average education of 10 would have on average 3 less kids compared to family with no education. Note I am not saying anything about causation this is just purely caring about descriptively describing the household with average education.

However, note what happens if we now omit $z$ from the regression model and calculate $a$ using the formula above:

$$y_i= 5 - 0.42x_i+ u_i$$

now the estimate suggest that family with average education has 4 less kids which is plain wrong - this estimate underestimates the number of kids for average family by $25\%$. In this case the sign stayed the same but thats just due to the numbers I chose in real life bias could easily change the sign of the coefficient altogether.

Hence, even if you dont care about causality and you just want to use regression for some description of dataset you care about omitted variable bias.

PS: Also in my answer I left out $by$ from your example: $ y=c+ax+by+error$ since there must be some typo you cant regress $y$ on itself, yet your formula includes $y$ on both sides. Or to be specific you could do that but mathematically coefficient is always set to $1$ when you regress dep variable on itself so there is never a reason to do that.


GDP per industry and per region in Europe I'm looking for a breakdown of GDP per region and industry in Europe. Do such data exist, ideally in the public domain? I'm looking for a breakdown of GDP per region and industry in Europe. Do such data exist, ideally in the public domain? If so, please identify and link to such sources.

Eurostat is the way to go.

Website: http://ec.europa.eu/eurostat (advised that you start from here to get where you want to be because then you see what is actually available)

GDP specifically (I think this is not what you need exactly - GDP per country over time - but its a starting point.): http://ec.europa.eu/eurostat/web/national-accounts/data/main-tables

Keep it in mind. Need Europe+numbers? Eurostat.


What will happen to bank loans in the event of hyper inflation? Say for example a person gets a home loan of $100,000 in a certain country at an interest rate of 10%. Inflation is normally around 5% in this country.

What could happen to that person's loan if the country's inflation rises to say 1000%?

If the loan is indexed to inflation, "nothing" would happen. The remaining loan payments would simply increase.

If the loan is not indexed to inflation, when inflation occurs, the debtor's wage would most likely increase with the inflation rate. Hence, he would very quickly have the money (in nominal terms) required to pay back the debt (which is now less worth, in real terms).

Consider the loan $L$ in monetary terms, and denote the price level by $P$. The lender was expecting $L/P$ when he formulated $L$. Now, instead, he is getting $L/P_2$ in real terms, which is (for the much larger price level $P_2$) a much smaller real value.

Which leads us to the conclusion that in this case (and very often), higher inflation is a redistribution from lenders to debtors.


Why is oil price increasing in May 2016 Why is the oil price and correspondingly the down and midstream company stock prices are increasing in May 2016, when

The economic data across the world in weak
China's economy is
sluggish. They had another round of devaluation in early May 2016.
The
OPEC meeting in Doha in late April did not yield any meaningful
resolution in stemming the production of oil output.

This may be rather late, but it seems from an article at the time that this was due to supply disruptions in Canada, Nigeria, and Venezuela. Canada was experiencing wildfires which shut down oil production, Nigeria suffered from attacks in the Niger Delta (the main source of Nigerian oil), and Venezuela was experiencing severe political crisis. With a supply decrease, oil prices were bid up.


Why's Bernie Sanders wrong to allege that JP Morgan received more than $390 billion in financial assistance from the Fed? This r/badeconomics post avouches:

    [JP Morgan] received more than $390 billion in financial assistance from the Fed.

  Sanders has repeated this lie for several years. He gets the \$390bn number from Table 8 of this report [p. 131 on the printed page, or 144/266 of the PDF] but forgets to adjust for the length of the loans. Table 9 [p. 132 on the printed page, or 145/266 of the PDF] adjusts for the term of the loan and finds that JP Morgan received about $31 billion in assistance, one-tenth of Sanders' amount. So he's established that he can't read a GAO report.

Why ought Sanders use the term-adjusted borrowing? What's wrong with using the not term-adjusted total transaction amount?

Example 1. If I repeatedly make 365 one-day \$1 loans to you over the course of 365 days, then the unadjusted loans (or "financial assistance") I make to you are \$365.

However, in effect I've simply lent you \$1 for one year. So, the term-adjusted (or we could say year-adjusted) loans I've made to you are only \$1.

Example 2. Loan A: I make you a one-day \$365 loan.

Loan B: I make you a one-year \$1 loan.

Both Loans A and B are \$1 term-adjusted loans.

But note that they are not obviously exactly equivalent. Most would consider Loan A a bigger favor than Loan B. This though is debatable. But in any case, the point here is that nobody would consider these two Loans to be exactly equivalent.

This is explained in the cited document, paragraph above Table 9 and again in the Note to Table 9. An example is also given on pp. 130–1:

  Table 8 aggregates total dollar transaction amounts by adding the total dollar amount of all loans but does not adjust these amounts to reflect differences across programs in the term over which loans were outstanding. For example, an overnight PDCF loan of \$10 billion that was renewed daily at the same level for 30 business days would result in an aggregate amount borrowed of \$300 billion although the institution, in effect, borrowed only \$10 billion over 30 days. In contrast, a TAF loan of \$10 billion extended over a 1-month period would appear as \$10 billion. As a result, the total transaction amounts shown in table 8 for PDCF are not directly comparable to the total transaction amounts shown for TAF and other programs that made loans for periods longer than overnight.

The cited Sanders quote is not obviously "stupid" though, as the Reddit commenter suggests. Example:

Say in an emergency, the Fed has to lend \$10T to Bank A for just one day. By the above procedure, we'd translate into to a term- or year-adjusted loan of \$10T ÷ 365 ≈ \$27.4B.

Say the Fed also lends \$27.4B to Bank B for one year.

It's not clear that the degree of assistance the Fed provided to both banks is equivalent. Most would say that the Fed did Bank A a bigger favor.


Is there a financial hypertrophy ? Could you make a case for finance? I've the impression that finance is in hypertrophy. But I'm not an expert on it, so I'd like those who are more knowledgeable than I am to do a [steelman][1] of it.
The things that make me thinking this are :

It uses many of the most talented people to lead activities that seem sometimes positive, but often neutral or negative from a global point of view (i.e either it redistributes money between ~equally rich people or it takes money from poorer people to richer ones ; the opposite seems unlikely given that financial services are more used by richer people).
The reasons for this (for picking among most talented people) is that finance, for whatever reasons, seems to have a rent (https://www.nber.org/system/files/working_papers/w14644/w14644.pdf), partly linked to past deregulation (http://www.parisschoolofeconomics.com/reshef-ariell/papers/BGR_finwage_RoF_final.pdf) and gives much higher salaries than other sectors, making it more attractive than other sectors for many people.

My non-expert view on different parts of finance is the following :

Actions and obligations are useful because they're directly linked with the real economy.
Some other tools are useful in some peculiar situations (futures, options and some derivatives to hedge, reduce some risks or whatever).
But most of it is useless (most of the flows of derivatives especially, which represents a huge amount of transaction (https://www.investopedia.com/ask/answers/052715/how-big-derivatives-market.asp#:~:text=The%20derivatives%20market%20is%2C%20in,quadrillion%20on%20the%20high%20end.) and thus a huge amount of human capital dedicated to it ; it looks like forex is more and more used for speculation aswell, given that the flows more than doubled since 2 decades).

If we were total utilitarianists (i.e we would like to improve global welfare) and if we were social planners (i.e we could do whatever we want without cost), should we delete a part of finance for the sake of humanity ?
If there is any literature on the utility of finance, I'd appreciate to read it. And I guess that for any asset, we can find a useful goal so please, talk about the representative use of a given tool. I'd especially appreciate if you were able to grossly quantify the share of useful resources / tools and useless ones (but no problem if it's just impossible). Feel free to take any coherent definition of finance.
[1]: https://en.wiktionary.org/wiki/steelman

Financial markets can look unimportant to layperson but empirical research shows that well functioning financial markets have direct impact on economic growth (have a look at Levine,
1997 and Levine 2005), which in turn benefits society greatly. For example, well functioning financial sector is crucial for economic development which helps to rise living standards of the poor (see Beck et al., 2007, Honohan, 2004). In addition, medium and small firms get large
share of positive benefits brought about by financial development (see Beck and
Demirguc-Kunt, 2006).
As a matter of fact non-trivial portion of development economics is dedicated to figuring out how to bring well functioning capital/financial markets you can see in the west to the poor countries (see Banerjee and Duflo Poor Economics: A Radical Rethinking of the Way to Fight Global Poverty).
The studies above look at financial sector very broadly and consequently we can conclude that actually most of the financial sector is useful and beneficial for the society. This does not mean that every single activity that occurs on financial markets is but on the net empirical literature shows large benefits of financial markets for the economy.
For example, derivatives you mentioned in your question help price discovery and are often used for hedging (which you actually put as an example of beneficial activity).
Unfortunately, it is not easy to just list useful and harmful activities. The same activity like mortgage backed security can be excellent tool for financial development depending on the characteristics of the market and appropriate regulation, or it can be harmful to the economy if market has some serious imperfections or is improperly regulated.
In fact the same holds even about less exotic types of financial activity, such as simple insurance. For example, adverse selection (an example of market failure) can in principle lead to undesirable outcomes in the insurance market (see some examples in Freixas Microeconomics of Banking) if it is not corrected by some regulation or private mechanism.
Consequently, one can't really provide any simple list of socially 'bad' and 'good' assets. It would be possible to discuss examples when different financial activities cause problems and when they are beneficial, but providing literature review on all financial instruments would go well beyond the scope of one SE answer (heck it would likely take more than one volume as a book). If you are interested in this sort of discussion you might refer to books such as:

Faia & Ester, Financial Regulation: A Transatlantic Perspective
Armour et al. Principles of Financial Regulation

However, as noted previously if you just want big picture the literature shows that on the net financial markets are quite beneficial for the economy.


Existence of a subgame perfect Nash-equilibrium Existence of a subgame perfect Nash-equilibrium

Given is the following game

The game is repeated finitely many times and the total payoff is the sum of the payoff from each repetition.

If we assume that $T=2$, is it then possible to find a subgame perfect Nash-equilibrium such that the strategy $(B,L)$ is played in the first stage (first repetition)?

So far as I can count, it shouldn't be possible to construct any threats for $T=2$ such that the $(B,L)$ strategy is played in the first stage. This is due to the Nash-equilibrium laying in $(T,R)$ and it takes at least $T\geq 3$ to ensure that $(B,L)$ is played in the first stage.

Though, I'm not quite sure how to prove my statement. Can any of you confirm/deny my thoughts and help me with some arguments? Thanks.

The following proposition is well known:

  If a stage game $G$ has a unique Nash equilibrium, then for any finite $T$, the repeated game $G(T)$ has a unique subgame perfect equilibrium outcome in which the Nash equilibrium of $G$ is played in every stage.

Since your stage game has a unique NE of $(T,R)$, this must be the outcome of the SPE of any finitely repeated game based on this stage game. Specifically regarding your question, it is impossible for $(B,L)$ to be played (in any stage including the first) in any SPE of the $T$-times repeated game.


Is there contradiction between automation created resilience and the full employment (with high wages)? From the one side - full employment with the high wages (but still under the level of creating inflation threat) is nice thing and politicians and central banks are highly motivated to achieve this.
From the other side - automation created technological unemployment or underemployment (assuming that the population still maintains skills) is beneficial for the resilience - if the economy can tolerate and can function with reduced human participation (e.g. by employing automation and software) then economy can direct idle resources towards demand that arises from the needs of crisis management. E.g. today economy can afford to shift workforce and resources to the research and production and delivery of vaccines and personal protection equipment and so many adaptations. If the current economy had required the full participation of the human workforce then there would be loss of production due to lockdowns, rising inflation etc.
So - this sketch/vignette provided the contradiction between the full employment from the one side and the resilience from the other side. Is such contradiction really and issue that can attract the attention of economist and is there literature on it? Maybe there is specific term for this contradiction? This vignette also assumes that there is un-used possibilities for introduction of technology. Such possibilities always exist when human beings are forced to compete against machines to do the same job with less wages/resources. Low social support makes this human/machine competition more fierce and the under-use of technology more pronounced/

There is not necessarily any contradiction. Full employment does not mean every single person in economy is employed. Full employment is (Worswick; 1987):

Full employment does not mean zero unemployment. There can be dislocations where large
numbers of workers are displaced from their present employment, and time is needed before new
workplaces can be created. This can happen at the
end of a war, or following some major technological change. Apart from such special cases, regular
allowance must be made for frictional and seasonal unemployment. Policy would not aim,
therefore, at zero but at the elimination of unemployment attributable to demand deficiency.

So generally speaking the technological change per se might only reduce the rate at which economy will attain its full employment. In addition, the concept of full employment is somewhat vague. In order to make it operational it will often be measured by the non-accelerating inflation rate of unemployment. It is completely possible for NAIRU to become higher as a result of technological change.
Next, automation does not necessarily create unemployment (see Nakamura et al 2018). Automation, can as well increase the number of jobs avaiable through effects on increasing aggregate demand. Of course, it is possible that it would create more unemployment, but empirically that did not happen in economies such as US till now (excluding current pandemic where it is hard to say what unemployment is due to automation and which due to other issues).
There is research that examines impact of automation on resilience of supply chains (see for example Hosseini et al 2020), but there does not seem to be a paper that would look at a tradeoff between resilience and full employment (or at least google scholar search did not show anything up).


Consumption Externalities in partial equilibrium analysis I've been working on a standard partial equilibrium with externalities problem and have been having some difficulty with these (relatively) basic concepts. My issue is primarly with solving for Pareto optimality and comparing it to a competitive outcome.
$$u^A(y_A, x_A, x_B)=y_A+\ln x_A+\frac{x_B^2}{2}$$
$$u^B(y_B,x_A,x_B)=y_B+\ln x_B-x_A $$
Solving for CE is fairly straight forward using the price normalization method of $(p_x,p_y)=(p,1)$
$$\max_{x_A,y_A} \mathcal{L}=y_A+\ln x_A+\frac{x_B^2}{2}+\lambda_A(m-px_A-y_A)$$
$$\max_{x_B,y_B} \mathcal{L}=y_B+\ln x_B-x_A+\lambda_B(m-px_B-y_B)$$
The allocation of goods in CE amounts of $x_A, x_B$ in this context is:
$$\{x_A^*,x_B^*\}=\left\{\frac{1}{p},\frac{1}{p}\right\}$$
Further for the pareto optimal allocation we solve:
$$\max_{x_A,x_B,y_A,y_B} u^A(y_A, x_A, x_B)+u^B(y_B,x_A,x_B)$$
$$\max_{x_A,x_B,y_A,y_B} \{y_A+\ln x_A+\frac{x_B^2}{2}+y_B+\ln x_B-x_A\}$$
Solving for $x_A$ is fairly straight forward via the FOC, we have:
$$x_A^{**}=1$$
for dealing with $x_B$ there is difficulty as:
$$x_B+\frac{1}{x_B}=0$$
where solving gives us
$$x_B=\sqrt{-1}$$
which makes no sense.
Im definately doing something wrong but cant figure it out.

Without any constraint the social planer would set $x_B \rightarrow \infty$, and not solve the "first order condition" given by $x_B+1/x_B=0$ only valid for an inner solution. The centralized problem is not comparable with the budget constrained decentralized problems you solved. (By the way why should $m$ be the same for both consumers?)


Need a math help for the Cagan's model in macroeconomics From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.

  To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is

  $m_t − p_t = −\gamma( p_{t+1} − p_t)$,

  where $m_t$ is the log of the quantity of money at time t, $p_t$ is the log of the price level at time t, and $\gamma$ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, $m_t − p_t$ is the log of real money balances, and $p_{t+1} − p_t$ is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by $\gamma$ percent.

Shouldn't $(p_{t+1} - p_t)$ be the log of inflation rate? Why it says just "the inflation rate"?

  This equation states that if inflation goes up by 1 percentage point, real money balances fall by $\gamma$ percent.

My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all.

$\ln \frac{M}{P} = \ln (\frac{p_{t+1}}{p_t})^{-\gamma} \rightarrow \frac{M}{P} = (\frac{p_{t+1}}{p_t})^{-\gamma}$

So, if the $(p_{t+1} - p_t)$ is just the log of inflation rate, then $\frac{p_{t+1}}{p_t}$ is the inflation rate and,

  inflation goes up by 1 percentage point

would mean $\frac{p_{t+1}}{p_t}$ is going to get +1, right? But I couldn't possibly think it would result the fall of $\frac{M}{P}$ by the $\gamma$ point. What am I missing?

And secondly, if the $(p_{t+1} - p_t)$ is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "$−\gamma(1 + p_{t+1} − p_t)$" at the right side, right? How could this be the case?

The answer to both your questions is that for small $x$ values
$$
\ln(1+x) \approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)

So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.

This should also answer your second question, as the approximation
$$
\gamma x \approx \ln(1+ \gamma x),
$$
works as well.

It may also be worthwhile to look into elasticity.


Facebook's Libra - Why would it vary in value Facebook has come up Libra its version of crypto-currency.

The unique feature of Libra is the "reserve" which backs Libra. The reserve is supposed to be held in stable global currencies (USD, GBP, JPY and EUR).

For a Libra coin to be created and equivalent amount of local currency (based on exchange rate) needs to be transferred to Libra reserve. I am assuming that during this transfer the local currency is converted to USD/GBP/JPY/EUR and then transferred to the reserve.

Essentially for every Libra that is in circulation its equivalent amount is held in its reserve.

Given this, I am assuming that:

Libra would behave like Hong Kong dollar against USD and it would remain stable against USD, GBP, JPY and EUR
There would be volatility in the price of Libra against the local currency (except for USD, GBP, JPY and EUR).

Are these assumptions right?

It will vary in value because trust in it will fluctuate just as trust in facebook fluctuates.

There will also be volatility as trust varies in confidence in the payment system; its integrity and security; how exchangeable it is; its exposure to fraud,  to tax, to money-laundering investigations

For holders, there is also significant counter-party risk in whether Facebook chooses to honour their holding. And the majority voting rights in Facebook are vested in a single person. So their counterparty is, in some regards, essentially one man and his whims.


Ranking products according to Price Elasticity Of Demand Assuming ceteris paribus conditions, rank the following products in order from the most price elastic to the least price elastic. Explain why.

Ice Cream
Peter’s Chocolate Ice Cream
Chocolate Ice Cream
Frozen Desserts
Desserts

For some reason I find this question very subjective.

The question is not subjective if you accept the assumption that similar products are likely to be substitutes, i.e. if you wanted chocolate ice cream but none was available you are more likely to get vanilla ice cream than pork chops.

The trick is that the groups can be ordered from smaller to larger, e.g. all ice creams are frozen desserts but not all frozen desserts are ice creams. If the price of a whole product group increases it is more difficult to find a cheap substitute than if only one product's price had increased. In this case all other products in the product group would be available as substitutes at their original prices.

Using this assumption you should be able to finish the assignment.


Unemployment: Human Capital Deterioration I am looking for Macro-Labor models that include deterioration of human capital. The mechanism is simple: The longer you were unemployed, the more human capital you are losing.

However, this is backward-looking: Your human capital today depends on the length of unemployment - i.e. the periods $t-1$, $t-2$, .... Hence, keeping track of this issue in a macroeconomic model appears to be nontrivial, and I am wondering how and whether the literature was dealing with this fact so far.

I know nothing about this. But I entered "unemployment human capital deterioration mortensen pissarides" into Google Scholar and clicked the first link (Two Questions about European Unemployment, Ljungqvist and Sargent, Econometrica 2008) and Ljungqvist and Sargent just do the obvious thing.

  Employed and unemployed workers experience stochastic accumulation or deterioration of skills, respectively. There is a finite number of skill levels with transition probabilities from skill level $h$ to $h'$ denoted by $\mu_u(h, h')$ and $\mu_e(h, h')$ for an unemployed and an employed worker, respectively. That is, an unemployed worker with skill level $h$ faces a probability $\mu_u(h, h')$ that his skill level at the beginning of the next period is $h'$, contingent on not retiring. Similarly, $\mu_e(h, h')$ is the probability that an employed worker with skill level $h$ sees his skill level change to $h'$ at the beginning of the next period, contingent on not being exogenously laid off. In the event of an exogenous layoff, the transition probability is $\mu_l(h, h')$. After the initial period coinciding with an exogenous layoff, the stochastic skill level of an unemployed worker is again governed by the transition probability $\mu_u(h, h')$. All newborn workers begin with the lowest skill level.

So this is at least one approach. Do you have a particular reason not to use a Markov chain?


Subsidies as incentives The given explanation about how subsidies unintentionally can result in resources being wasted does make sense to me. However, I just wanted a second opinion on my take on this.
To me, it seems like a very trivial claim with no little relevance in the real world. Can't this be easily avoided if, let's say, we let the said farmers consume the resources they want initially. Then, the government only releases subsidies if they deem the amount of resources used as appropriate and necessary.
All in all, please inform me of any aspects I might be missing in interpreting the given picture or any shortcomings in my "solution" to this.

The claim is definitely not trivial, otherwise we would not observe frequent draughts due to government subsidies (like drought in California that is at least partially caused by bad water and farming subsidies), environmental degradation and overuse due to subsidies to coal mining and oil production (see this IMF explainer), not even mentioning that often these subsidies perversely redistribute resources from poor to rich. Overfishing, is not just result of tragedy of commons but also prevalent bad fishing subsidies (see this article in The Economist), and so on.
Furthermore, your proposed solution of:

Can't this be easily avoided if, let's say, we let the said farmers consume the resources they want initially. Then, the government only releases subsidies if they deem the amount of resources used as appropriate and necessary.

would not work, because in repeated interactions people will expect the subsidy. In economics, subsidies are typically considered to be tied to some specific economic activity, more specifically some production (i.e. it is not the same as redistribution where you just transfer resources). Hence, farmers would essentially figure out how the subsidy varies with economic activity (unless you want to propose completely random normally distributed subsidy with mean zero and what would be the point of that?), and eventually realize that if they produce x the subsidy will be y. Furthermore, such arrangement would defeat the purpose of subsidy. As mentioned above subsidy is used to subsidize some economic activity, if you just want to redistribute resources you can give them regular transfers. If you would first let people make their choices and then subsidize the activity, assuming people would be completely myopic and not realize post hoc subsidy was connected to the economic activity, you just created convoluted welfare transfer, for which we already have better systems.
This is not to say that subsidies can't be useful tool for government in some situations. However, in most of the cases those situations occur when the price mechanism does not work properly (e.g. there are some positive externalities meaning the price does not reflect all important information). However, if price system is working properly distorting it often leads to either shortages or inefficient overuse.
Of course, if government would only use subsidies when appropriate there would not be any issue. I mean this is obvious, that is like saying homelessness can be solved by people getting houses, or opioid pandemic by doctors stopping to overprescribe opioids and patients taking them responsibly, or solving police shootings by having them just shoot when arresting dangerous criminals with guns. But neither getting right subsidies or solving those other issues is as simple as that, otherwise our societies would not have the issue.
When it comes to subsidies, the issue is that it is very hard to determine what optimal subsidy is in a field. Next, even once you manage to identify problems which require subsidies and what the value of subsidy is there are still political problems as alluded to by Dayne's +1 comment.
Moreover, the point of the box in your question, which is often mentioned in one way or another in almost every single textbook is to explain how price system functions to distribute information and give incentives to people to act on that information. This is something that most non-economists do not understand.
Consequently, the information in the box is both non-trivial and very relevant. It is relevant and non-trivial because most people do not understand how price system works and what happens when it is distorted. Moreover, we can empirically often see subsidies being misused and often even in areas connected to climate change (as mentioned above), and given that climate change is likely one of the defining issues of our 21st century, I would say this makes this issue as relevant as ever.


Why GDP positive growth does not decrease inflation? My understanding is : If production increases, it would imply the amount of goods increases as well for the same amount of money available in the market. Because of this, we should see prices of goods decrease as the offer got bigger.

Since we say inflation increases with GDP growth, where am I wrong thinking it should be the other way around?

GDP growth would lead to deflation if the money supply remained unchanged. The government usually increases money supply as the economy grows to avoid deflation. Controlling money supply is not an easy task though because the economy responds with lag to every policy decision. But ideally, the government aims at keeping inflation low (1-4%) and avoiding deflation. Zero inflation is also undesirable because if you know your money will be worth just as much a year from now, you have less incentive to invest.


M1 Money supply data? Why do these M1 Money supply charts show data only up to 7th May, when the Fed data is released weekly?

https://fred.stlouisfed.org/series/M1

http://schrts.co/zgaAPi

Thanks

That’s what the source data goes up to at present: Fed H.6

The fact that it is weekly does not mean that it will be close to available in real time, they presumably need time to get the data for some components of the aggregate.


How would privately minted coins enter circulation? I read a great article about private minting in early America that talks about some of the private minters whose copper, silver, and gold coins were widely circulated throughout the country without the federal government having anything to do with the process. I had some questions about how exactly that would work, so in the absence of a historian, I thought I would ask here for some answers that would have to make sense economically.

These coins would have to be considered commodity money rather than fiat money since the government wasn't involved. In order for this to work, would the face value of the coins have to be really close to the intrinsic value?

If the face value and intrinsic value were the same, how would the minters profit without any seigniorage?

Without a central bank, how would money minted by private minters enter circulation?

I am unsure about the American situation, but I learned a bit of history about token issuance in pre-Confederation Canada. These comments are based on the notes in a token collector guide.
Since they were privately issued, they would be called “tokens” and not “coins,” as that was reserved for governmental issuance.
In the Canadian colonies, there was a deficiency of low denomination copper coinage, which were officially minted in Britain (and British governors did not really  care). So the locals took matters into their own hands.
The banks were serious about coinage, and the metal content was close to what was implied by face value. Non-banks also issued tokens, but the metal content was quite often below face value. The banks did not like these tokens, and they typically did not accept them at face value, and melted them down. (This means that the tokens with the least copper are the rarest, and worth the most now.)
The tokens entered circulation via being handed out by the firms that minted them. Even if the tokens were too dodgy to be used elsewhere, people could return them to the same shop.


Gapminder's Dollar Street and the role of self-supply I find it quite hard to get a clear picture of what the income numbers in Gapminder's Dollar Street tell. How to compare \$27 in Burundi with \$10,098 in China? What would it mean that the family in Burundi is 500 times "poorer" than the family in China?

There are two aspects that are unclear to me:

Wouldn't it be a better measure (than income is) to quantify how much of the money one needs to lead a "normal healthy" life one has at ones disposal (in percent)? How is this measure called?
What about self-suppliers (what many families in poorer countries are)? They own and need less money (income) than others do, but how is the value of their self-supplying work measured?

In the context of self-supply: Wouldn't "belongings" be a better measure than "income"?

On Gapminder's website they write

  If you have a question, maybe it's already answered in our Q&A.
   Q&A

The underlined text links to a Google document, which answers several questions. Under one of these answers

  The households’ door numbers represent the consumption values (US dollars) that each adult in the household has per month. This figure comes from a combination of the household’s self-reported consumption and income levels. We then checked official cost of living data in each country to adjust the values for purchasing power parity (PPP) and converted the value into US dollars.

there is a link to a further Google document titled Detailed income calculations for Dollar Street. This second one seems to answer your question, but it is several pages long, so I will not copy paste its full body of text here.


Dixit-Stiglitz Pricing Index I have recently encountered (in a few applications), the Dixit-Stiglitz Pricing index. Consider a Constant elasticity of substitution form of utility $$U=\left(\sum_{i=1}^{N}c_{i}^{1-\frac{1}{\sigma}}\right)$$

This index basically aggregates prices to give a general term as a function of prices that yields a unit cost of utility. How is the index derived?

This index is derived solving the cost minimization problem
$$\min_x \{ p^\top x \lvert \bar u(x) \geq \bar u\},$$
where $p=(p_1,...,p_J)$ is price vector and $x=(x_1,...,x_J)$ is the consumption bundle. The problem can therefore also be written as
$$\min_{x_1,...,x_J} \sum_{j=1}^J p_j x_j \\[8pt]
s.t. \ \ u(x_1,...,x_J)\geq \bar u.$$
One interpretation of this problem is that the underlying question asked and answered by solving the problem is:

How much money is the consumer required to spend at the price $p$ in order to achieve the utility $\bar u$?

Solving the problem you get the Hicks demand functions
$$x^\star(p,\bar u),$$
and inserting these in the budget equation $p^\top x$ you get the expenditure function
$$E(p,\bar u) = p^\top x^\star(p,u) = \sum_j p_j x_j^\star(p,u)$$

It is worth noting that the problem are analogous to the cost minimization problem for firms

Firms minimize cost of producing output $y$ given a production technology captured by production function $f(x)$, hence the firm solve the problem
$$\min_x \{p^\top x \lvert f(x) \geq y\},$$
here prices are now factor prices. The solution here is conditional demand functions $x^\star(p,y)$ and the cost function $C(p,y) = p^\top x^\star(p,y) = \sum_j p_j x^\star_j(p,y)$.
Deriving the unit cost of utility with CES preferences
I will now show how to derive the unit cost of utility given CES preferences. As noted about this is the same as deriving the cost function of firm using CES production technology. I will not show that this is a price index, which would require setting up the formal properties of what a price index is, which is not trivial.
So the problem to be solved is
$$\min_x \sum_j p_j x_j \lvert u(x) \geq u$$
where
$$(1) \ \ u(x) = \left(\sum_j x_j^\alpha\right)^{1/\alpha}$$
To solve the problem I set up Lagrangian function and find FOC thereby arriving at the result that relative prices equal MRS. The Lagrangian is
$$\mathcal L(x,\lambda) = \sum_j p_j x_j - \lambda(\bar u - u(x)),$$
with FOC
$$\frac{\partial \mathcal L(x,\lambda)}{\partial x_k} = p_k + \lambda \frac{\partial u(x)}{\partial x_k} = p_k - \lambda \left(\sum_j x_j^\alpha\right)^{\frac{1}{\alpha}-1} x^{\alpha - 1} =  0,$$
this implies - using that constraint is binding so $\lambda > 0$ - that
$$\frac{p_k}{p_j} = \frac{x_k^{\alpha - 1}}{x_j^{\alpha-1}}\phantom{xxx} \Leftrightarrow \phantom{xxx} x_k = \left(\frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{1}{\alpha - 1}},$$
which is inserted in the constraint $u(x) = \bar u$ to get
$$\bar u = \left(\sum_k \left( \left(\frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{1}{\alpha - 1}}\right)^\alpha\right)^{1/\alpha} = \left(\sum_k \left( \frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha},$$
the sum index is running over $k$ so not binding $x_j$ and $p_j$ in the expression and these can therefore freely be moved outside the sum to get
$$\bar u = \left(\sum_k \left( \frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} = \frac{x_j}{p_j^{\frac{1}{\alpha-1}}}\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha},$$
which gives us the result that
$$x_j^\star(p,\bar u) = \frac{p_j^{\frac{1}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u,$$
which is Hicks demand. To get expenditure function use
$$E(p, u) = \sum_j p_j x_j^\star(p,u) =\sum_j p_j  \frac{p_j^{\frac{1}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u =  \frac{\sum_j p_j^{\frac{\alpha}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u = \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}\cdot \bar u,$$
where it is easy to see that $\bar p:= \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}$ is the cost of each utility point since the derivative of $E(p,u)$ with respect to $u$ does not depend on the level of utility.
It is perhaps worth adding a final property, namely that the Hicks demand can be reexpressed in terms of the price index:
$$x^\star_j(p,u) = \frac{p_j^{\frac{1}{\alpha-1}}}{\sum_j p_j^{\frac{\alpha}{\alpha - 1}}}u \bar p = s_j(p) u \bar p = s_j(p)E(p,u),$$
where $s_j(p):=\frac{p_j^{\frac{1}{\alpha-1}}}{\sum_j p_j^{\frac{\alpha}{\alpha - 1}}}$ is denoted with "s" to indicate that it is a share in the sense that it is a positive numver in (0,1) and $\sum_j s_j(p) = 1$.


What textbook can I use to really understand microeconomics, macroeconomics and mathematical economics What textbook can I use to really understand microeconomics, macroeconomics and mathematical economics. It can be different textbooks. I'm in first year college taking introductory courses in microeconomics, macroeconomics and mathematical economics. I need a really good textbook or online site to understand what I'm studying well. I like the way people on this site answer and explain questions with application. I want to be like that too. I have Greg mankiw books by the way, I have schaum's introduction to mathematical economics. I need to read more quality books to understand.

Thanks

Your question is quite broad, but here are a couple of (open) book sites you might get started with:
https://open.umn.edu/opentextbooks/SearchResults.aspx?subjectAreaId=4
http://www.freebookcentre.net/Business/Economics-Books.html

When you're familiar with the basics, you might ask the question again with a more specific topic you'd like to study.

Expertise on economics (or pretty much any academic subject) is gained through years of studying and thought. It's, at least to me, a fascinating way of thinking about the world.


Why Is the West providing technology to India? Previously, we have seen that Europeans and Americans provided various technologies (high-speed rail[1], aerospace[2][3], microprocessor[4], etc.) to China to earn instantaneous money. Later what happened was, China did its own research on those techs, improved them and became a giant vendor.

Apparently, the same is being repeated in case of India. The USA and EU is giving India missile tech[5], nuclear tech[6], aircraft carrier tech[7], fighter jet tech[8], and so on. India is also making 100% sure that they obtain those techs by enforcing a so-called policy named: "Make in India"[9].

My question is, Why Is the West providing technology to India even though there is an imminent danger to the West of India becoming a 2nd giant vendor?

References

https://en.wikipedia.org/wiki/High-speed_rail_in_China#Acquisition_of_foreign_technology
https://www.ainonline.com/aviation-news/defense/2017-06-20/new-chinese-fighters-tap-russian-us-technology
https://www.cnbc.com/2017/09/18/rolls-royce-chairman-ian-davis-on-chinese-jet-engines-singapore-summit.html
https://en.wikipedia.org/wiki/Loongson#MIPS_patent_issues
http://uk.reuters.com/article/uk-l-t-mbda-idUKKBN15S0XG
http://www.firstpost.com/india/india-us-nuclear-deal-all-you-need-to-know-about-the-landmark-agreement-3596209.html
http://www.news18.com/news/world/us-to-provide-emals-technology-to-india-for-aircraft-carriers-1550359.html
https://economictimes.indiatimes.com/news/defence/us-updates-jet-engine-technology-transfer-policy-with-india/articleshow/50131574.cms
https://en.wikipedia.org/wiki/Make_in_India

I may hazard a small guess to this. India has a very young workforce in comparison to its Asian counterparts (China and Japan). We especially see this in the field of software design where it has an even lower median age of labour than USA. As we know the West gains a lot from brain drain and stands to gain even more from young innovative minds that India produces, we can assume this access to new technology will let India produce even more bright minds.

As for India becoming a large vendor. We can see China will be facing extreme labour shortages soon due to it's high aged labour force and extreme brain drain. This in turn has China to be predicted to fall as a large vendor in the global market (Or at least possible). If it does India can take its place, easing the instability of the market, and if it doesn't we have a new youth force with amazing work ethic gain from.

Also from an Economic view more vendors in the market is decentralizing, which can be seen as good or bad depending.

Hope this had something to gain.


What would happen if tomorrow all private and public debt was erased? The question is what would happen if tomorrow all private and public (government) debt was erased? I am especially curious if normal daily economic activities would go on, what would the role of central banks be and how would the economy come into some equilibrium again.

The question is, indeed, a bit vague but I would like to know some opinions from various aspects. I think it is a very nice gedanken experiment.

It would depend a lot on how the debt was "erased." If you are simply suggesting a transfer payment from creditor to debtor, what would force such an event? Or would it be voluntary? If it were forced, then you would have a threat to private property rights and therefore, a disincentive for further lending. Security of capital would be diminished and all investment might stop overnight.

This question is very broad. But the answer of what might happen would depend very much on the details of how all this might come about.


Building a time-series demand forecasting model I am forecasting demand for certain types of goods and services, which I expect to be correlated to a sub-set of a basket of macroeconomic indicators (considering 15-20 indicators)

I do not know which indicators influence the demand more, whether they have a simple correlation influence or whether a derivative of change influences the demand (i.e. GDP or GDP change for example) or whether there’s a delayed effect on demand (e.g. increased government spend in last year may better predict this year’s demand?). Some macro indicators may be correlated to each other.

I have some basic hypotheses on likely indicators - that may be right or wrong.

Questions 1. What are good time-series forecasting models? What can be considered, apart from just a multivariate regression? 2. Is there a tool whereby I can input the historical demand, historical macro indicators, which will then output which set of indicators best predict the demand and which model works best?

I know how to do regressions in excel, but that’s just one set of indicators at a time. 20 indicators (plus derivatives, plus lag) throws up so many possibilities I cannot manually simulate.

Any help appreciated.

If it's just knowing about the models, I'd suggest starting with Hamilton's "Time Series Analysis" but any book like it will do. You can also explore Google Scholar for inspiration (people likely had built similar models). It seems you have a VAR model, check this wikipedia link to see if it's what you have in mind: https://en.wikipedia.org/wiki/Vector_autoregression
If you are looking for tools to obtain these forecasts, I'd suggest EViews or R. They are not useful on their own: you must have to have a model to do any kind of forecasting.


What would be the defining characteristic of New-Keynsian thought? If you had to summarize New-Keynsian thought into one paragraph, or a set of equations, what would it be?

Gali's Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, provides an advanced undergraduate / first-year graduate student introduction to these models, and I'd recommend it for self studies. According to chapter 1, available online, which offers an overview of the  New Keynesian Model, the key elements are:

Structural characteristics of Real Business Cycle (RBC) Dynamic Stochastic General Equilibrium (DSGE) models
Monopolistic competition
Nominal rigidities
Short run non-neutrality of monetary policy


How to derive utility possibility frontier? I'm trying to derive the utility possibility frontier of a economy whose consumption contract curve is $$y_A = \frac {y} {x} x_A$$ and $$y_B = \frac {y} {x} x_B$$where $x_A + x_B = x$ and $y_A + y_B= y$.

$$U_i = \sqrt {x_iy_i}$$

While deriving the UPF. they have added up $U_A$ and $U_B$, and I don't understand why. Why do we add up utilities while deriving UPF?

(I gave all the details of the question because I'm not sure if they added the utilities only because of the peculiarities of the question or if its an algorithm. )

Given that $u_A(x_A, y_A) = \sqrt{x_Ay_A}$, $u_B(x_B, y_B) = \sqrt{x_By_B}$ are the utility functions of A and B, and total endowment of X and Y in this pure exchange economy is $\omega_X$ and $\omega_Y$, utility possibility frontier (UPF) is the set of all utility pairs $(\mu_A, \mu_B)$ such that the there exist a Pareto optimal allocation $((x_A, y_A), (x_B, y_B))$ with the property: $\mu_A = u_A(x_A, y_A)$ and $\mu_B = u_B(x_B, y_B)$. So to find UPF, we will use the following conditions:

Optimality:

$\displaystyle \frac{y_A}{x_A} =\frac{y_B}{x_B}$

Feasibility:

$x_A + x_B = \omega_X$ and $y_A + y_B = \omega_Y$

Utility:

$\mu_A = \sqrt{x_Ay_A}$ and $\mu_B = \sqrt{x_By_B}$

Solving the above system of equations we get the UPF as:
$\mu_A + \mu_B = \sqrt{\omega_X\omega_Y}$.


Do we have some evidence of overshooting? I was going through overshooting of exchange rates (which I probably haven't seen since undergrad) and it seemed to me that the story is a bit of a stretch. I wonder if there is some empirical test that would convince me otherwise.

It depends on what you mean by overshooting.

The original Dornbusch (1976) story is that following a surprise permanent increase in the money supply, the nominal exchange rate "overshoots" its long-run level (the long-run level increases proportionately with the money supply). This is not a directly relevant story today (and I'm not sure that it ever was), because few changes in monetary policy take the form of surprise permanent changes in the money supply.

The mechanism for Dornbusch's "overshooting" is that since the price level takes time to adjust, in the short term there is an increase in the real money supply, leading to a decrease in nominal interest rates. This decrease in nominal interest rates leads the currency to weaken.

If we cut out the first part about the shock to the money supply and just think about the mechanism itself, where (all else equal) lower nominal interest rates lead a currency to weaken relative to its long-term level, then we have a much more relevant and plausible story. Whenever central banks are looser or tighter than expected in policy, you usually see the exchange rate weaken or strengthen, respectively, as a result. This is related to the notion of uncovered interest rate parity, but it's a weaker version: it doesn't say that UIP always holds (which it definitely doesn't), just that it conditionally holds in response to an interest rate shock.

Surprisingly, the international finance literature has been mixed on whether this is true or not, but Michael Kiley has recently shown that conditional UIP seems to hold in response to monetary policy surprises, as identified by monetary policy statement days. I suspect that the previous literature, which was based mainly on VARs without a clear identification strategy, struggled to find a consistent effect due to bad research design.


Derivation of the elasticity of substitution of a general production function with labor-augmenting technological progress I am following and trying to fully understand a famous and interesting work of Bentolila and Saint-paul (2003). They try to explain movements of the factor's share in terms of a relationship between the labor share ($LS$) and the capital-output ratio ($k$). To summarize as much as possible they start from a general production function with labor-augmenting technological progress and constant returns to scale:

$$Y_{i} = F(K_{i},B_{i}L_{i})=K_{i}f(l_{i}),$$

where $l_{i}=\frac{B_{i}L_{i}}{K_{i}}.$ And show that under the usual microeconomic assumptions of equilibrium (i.e. labor its paid its marginal product), there exists a unique function $g(\cdot)$ such that:

$$S_{Li} = g(k_{i}).$$

where $S_{Li}=\frac{w_iL_i}{p_iY_i}$ the labor share in the industries revenues, with $w_i$ denoting the wage, $p_i$ the product's price and $k_i=\frac{K_i}{Y_i}$ is the capital-output ratio.

Then, at page 6 of the paper, they employ the standard definition of elasticity of substitution to production function above, i.e. $\sigma_{i}=\frac{d(K_i/L_i)}{d(r/w)}\cdot \frac{r/w}{K_i/L_i}$ to obtain the following result:

$$\sigma_{i}=\frac{f'(l_{i})}{l_{i}f''(l_{i})}\left [1-\frac{l_{i}f'(l_{i})}{f(l_{i})} \right ]$$

I am trying to do all the derivations, but I can't find a way to derive this last formula with the same result. Is there someone who could kindly help me and show me the steps?.

You have that in  equilibrium each factor is paid its marginal product, so
$$\tag1\frac{w_i}{p_i}=B_if'(l_i)$$ and $$\tag2\frac{r_i}{p_i}=f(l_i)-K_if'(l_i)\frac{B_iL_i}{K_i^2}=f(l_i)-l_if'(l_i)$$ so deviding (2) by (1) we have:

$$\tag3\frac{r_i}{w_i}=\frac{f(l_i)-l_if'(l_i)}{B_if'(l_i)}.$$

Take the derivative of $r_i/w_i$ with respect to $l_i$:
$$\tag4\frac{d(r_i/w_i)}{d(l_i)}=\frac{(f'(l_i)-(f'(l_i)+l_if''(l_i)))B_if'(l_i)-B_if''(l_i)(f(l_i)-l_if'(l_i))}{(B_if'(l_i))^2}=-\frac{f''(l_i)f(l_i)}{B_i(f'(l_i))^2}$$

Note that $l_i=\frac{B_i}{(K_i/L_i)}$ so
$$\tag5\frac{d(l_i)}{d(K_i/L_i)}=-\frac{B_i}{(K_i/L_i)^2}=\frac{l_i}{(K_i/L_i)}.$$ Using the chain rule ans substituting (4) and (5) we have:
$$\frac{d(r_i/w_i)}{d(K_i/L_i)}=\frac{d(r_i/w_i)}{d(l_i)}\frac{d(l_i)}{d(K_i/L_i)}=\frac{l_if''(l_i)f(l_i)}{(K_i/L_i)B_i(f'(l_i))^2}$$ and by the inverse function theorem we get:
$$\tag 6\frac{d(K_i/L_i)}{d(r_i/w_i)}=\frac{(K_i/L_i)B_i(f'(l_i))^2}{l_if''(l_i)f(l_i)}$$

Finally by multiplying (3) and (6) and dividing  by  $(K_i/L_i)$ we conclude that

$$\sigma_i=\frac{d(K_i/L_i)}{d(r_i/w_i)}\frac{r_i/w_i}{K_i/L_i}=\frac{(K_i/L_i)B_i(f'(l_i))^2}{l_if''(l_i)f(l_i)}\cdot\frac{f(l_i)-l_if'(l_i)}{(K_i/L_i)B_if'(l_i)}=\frac{f'(l_i)(f(l_i)-l_if'(l_i))}{l_if''(l_i)f(l_i)}=\frac{f(l_i)}{l_if''(l_i)}\left[1-\frac{l_if'(l_i)}{f(l_i)}\right]$$

as desired. $Q.E.D.$


Deriving the translog production function Ive been having difficulty deriving the translog production function defined as:

$$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i \ln x_i+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\ \beta_{ij}\ln x_i\ln x_j $$

I know we start with a log-log production function.
$$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i\ln x_i$$

the next step from what i recall is to take the taylor series of this function around the point $x_i=0$. the reason why this is an issue is because $\ln(0)$ is undefined.

How exactly is this function derived?

The idea is indeed to Taylor expand the production function. To justify it, you can start with the constant elasticity of substitution function, which in the two-factor case can be written as

$$
Y = A[\alpha K^\gamma + (1 - \alpha)L^\gamma]^{1/\gamma} \tag{1}
$$

in this case $X_1 = K$, $X_2 = L$. Now we expand $\ln Y$ around $\gamma = 0$ (recall the CES approximates to a cobb-douglas production function when $\gamma \approx0).$

  $\gamma^0$ term

$$
\lim_{\gamma \to 0} \ln Y = \ln (A K^\alpha L^{1 - \alpha}) \tag{2}
$$

  $\gamma^1$ term

\begin{eqnarray}
\lim_{\gamma \to 0} \frac{\partial \ln Y}{\partial \gamma} &=& \lim_{\gamma \to 0}\frac{\alpha  K^{\gamma } \ln (L)+(1-\alpha ) l^{\gamma } \log (L)}{\gamma
   \left(\alpha  K^{\gamma }+(1-\alpha ) L^{\gamma }\right)}-\frac{\ln \left(\alpha
   K^{\gamma }+(1-\alpha ) L^{\gamma }\right)}{\gamma ^2}\\
&=& \frac{1}{2} (1 - \alpha) \alpha  (\ln (K)-\ln (L))^2 \tag{3}
\end{eqnarray}

Up to first order we have then

\begin{eqnarray}
\ln Y &\approx& \color{blue}{(\ln Y)_{\gamma = 0}} + \color{red}{\left(\frac{\partial \ln Y}{\partial \gamma}\right)_{\gamma = 0} \gamma} \\
&\stackrel{(2),(3)}{=}& \color{blue}{\ln A + \alpha \ln K + (1 - \alpha) \ln L} + \color{red}{\frac{1}{2}\alpha(1 - \alpha)\gamma [\ln K - \ln L]^2} \\
&=& \ln A + \alpha \ln X_1 + (1 - \alpha) \ln X_2 + \frac{\alpha \gamma (1 -\alpha)}{2}\left[\ln^2 X_1 -2\ln X_1\ln X_2 + \ln^2 X_2 \right] \\
&=& \alpha_0 + \sum_{i = 1}^2 \alpha_i \ln X_i + \frac{1}{2}\sum_{i, j = 1}^2 \beta_{ij}\ln X_i \ln X_j \tag{3}
\end{eqnarray}

This is naturally extended to $n > 2$ as

$$
\ln Y = \alpha_0 + \sum_{i = 1}^n \alpha_i \ln X_i + \frac{1}{2}\sum_{i, j = 1}^n \beta_{ij}\ln X_i \ln X_j
$$


Calculating standard error Consider a regression model on the form:

$y_{i} = \alpha + \beta_{1}X_{i} + u_{i}$ (1)

I am given $var(u_{i}|X_{i}) = 10$ and I know the OLS estimates for $\alpha$ and $\beta$ and using this I have to calculate the standard errors of the parameter estimates.

I am following Hayashi's book and I know that

$se(b_{k}) = \sqrt(s^{2}(X'X)^{-1}_{kk})$

Where $b_{k}$ is the parameter estimate of interest. I know $(X'X)^{-1}$ but what is confusing me is going from the given model to expressing it in matrix notation and that this does not use the variance of the error term.

I can express (1) as $Y = X\beta + u$ and since $var(u_{i}|X_{i}) = var(y_{i} - \alpha - \beta_{2}X_{i}|X_{i})$ but is this equal to $var(Y - X\beta|X)$?

If so, I can express $var(u_{i}|X_{i}) = 10$ as $\sigma^{2}I_{n}$ where $\sigma^{2} = 10$. And then could I use the fact that

$var(b_{k}|X) = (X'X)^{-1}X' \sigma^{2} I_{n} X(X'X)^{-1}$

To find the standard errors of interest.

Your notation is a bit all over the place, so I'm going to try and standardize it for a general case.

Let $X$ be the regression matrix (which includes a column of 1s for the intercept term) and $\beta$ be the vector of coefficients to be estimated via OLS (which includes the intercept term). $Var(u_i|X) = 10 = \sigma^2$, for all $i$. Thus the errors are homoscedastic (constant variance) and the variance-covariance matrix of the errors $\Omega$ has diagonal entries that are all equal to $\sigma^2$. Furthermore, if you assume that the error terms are not serially correlated, then the off-diagonal covariance terms $Cov(u_i, u_j|X)$ for all $i \neq j$ in the variance-covariance matrix $\Omega$ are zero.

Those are two of the standard Gauss-Markov assumptions used to establish the BLUEness of the OLS estimator. Under these assumptions, $\Omega$ is

$$
E[uu'|X] =
    \left[
    \begin{array}{ccccc}
    \sigma^2                                    \\
      & \sigma^2    &   & \huge0        \\
      &               &   \ddots                  \\
      & \huge0 &   & \sigma^2        \\
      &               &   &   & \sigma^2
    \end{array}
    \right] = \sigma^2I.
$$

To get the variance of the OLS estimates $b = \hat{\beta}$, first note that

$$
\begin{align}
b &= (X'X)^{-1}X'y \\
&= (X'X)^{-1}X'(X\beta + u) \\
&= \beta + (X'X)^{-1}X'u \\
\implies b - \beta &= (X'X)^{-1}X'u,
\end{align}
$$

using $y = X\beta + u$. Then,

$$
\begin{align}
Var(b|X) &= E[(b-\beta)(b-\beta)'|X] \\
&= E[(X'X)^{-1}X'u((X'X)^{-1}X'u)'|X] \\
&= E[(X'X)^{-1}X'uu'X(X'X)^{-1}|X] \\
&= (X'X)^{-1}X'E[uu'|X]X(X'X)^{-1}.
\end{align}
$$

But recall that we derived $E[uu'|X] = \sigma^2 I$ via the Gauss-Markov assumption of spherical errors. Thus, by substitution,

$$
\begin{align}
Var(b|X) &= (X'X)^{-1}X'E[uu'|X]X(X'X)^{-1} \\
&= (X'X)^{-1}X'(\sigma^2 I)X(X'X)^{-1} \\
&= \sigma^2 (X'X)^{-1} X'X(X'X)^{-1} \\
\implies Var(b) &= \sigma^2 (X'X)^{-1}.
\end{align}
$$

The standard deviation of $b$ is just the square root of the variance, or

$$
sd(b) = \sqrt{\sigma^2 (X'X)^{-1}}.
$$

To find the standard deviation of the $k^{th}$ estimated coefficient in $b$, $b_{k}$, we simply extract the $k^{th}$ diagonal element of $(X'X)^{-1}$, denoted as $(X'X)_{kk}^{-1}$:

$$
sd(b_k) = \sqrt{\sigma^2 (X'X)_{kk}^{-1}}.
$$

$\sigma^2$ is the (common) variance of the errors, and unfortunately, this value is unobserved in our sample. In your question, however, it appears that this value is just given to you directly — it's $10$. In general, however, $\sigma^2$ must be estimated using the data. It turns out that given homoscedastic errors, an unbiased estimator of $\sigma^2$ is

$$
s^2 = \frac{e'e}{n-P},
$$

where $n$ is the number of observations, $P$ is the number of columns in $X$, and $e = \hat{u}$; that is, the vector of residuals $y - Xb$.

Thus, the standard error of $\mathbf{b_k}$ is estimated as

$$
\boldsymbol{se(b_k) = \sqrt{s^2 (X'X)_{kk}^{-1}}},
$$

which is Hayashi's result.


Solving Leeper (1991) model I am attempting to solve a variation of Leeper's (1991) model, which deals with the FTPL. This is what I have done so far:

The utility function is $\log(c_t)+\delta\log(M_t/p_t)$.

I obtain two first order conditions:

(1) $\frac{1}{R_t}=β E_t \left[\frac{1}{\pi_{t+1}} \right]$ and

(2) $m_t = \delta c \left[ \frac{R_t}{R_t - 1} \right]$.

Where $R_t$ is the gross nominal interest rate, $\pi_t$ is the gross inflation rate and $c$ the deterministic steady state value of consumption.

Suppose the monetary authority follows the following rule:

$R_t = α0+ απ_t+ θ_t R_t$, where $θ_t = ρ_1 θ_{t−1} + ϵ_{1t}, |ρ_1|<1$.

The fiscal authority follows:

$τ_t = γ_0 + γ b_{t−1} + ψ_t$, where $ψ_t = ρ_2 ψ_{t−1} + ϵ_{2t}$.

I am having trouble seeing how Leeper was able to take all the given information so far and produce the following two equations:

(3) $E_t \hat{π}_{t+1} = αβ\hat{\pi}_t + βθ_t$

(4) $φ_1 \hat{π}_t + \hat{b}_t + φ_2 \hat{π}_{t−1} - (β^{−1} − γ)\hat{b}_{t−1} + φ_3 θ_t + ψ_t + φ_4 θ_{t−1}=0$

where the varphi's are the steady state constants:

$\varphi_1 = \frac{\delta c}{\bar{R} - 1} \left[ \frac{1}{\beta \bar{\pi}} - \frac{\alpha}{\bar{R} - 1} \right] + \frac{\bar{b}}{\beta \bar{R}}$

$\varphi_2 = - \frac{\alpha}{\bar{\pi}} \left[ \frac{\delta c}{\left(\bar{R} - 1\right)^{2}} - \bar{b} \right] $

$\varphi_3 = \frac{\delta c}{\left(\bar{R} - 1\right)^{2}}$

$\varphi_4 = - \frac{1}{\bar{\pi}} \left[ \frac{\delta c}{\left(\bar{R} - 1\right)^{2}} - \bar{b} \right] $

Note: I was able to solve for (3); however, I am still unsure why $\theta_t$ does not have a hat on top of it. If someone can walk me through how to reach (4), I would very much appreciate it!

You have the government's flow budget constraint (re-written in real terms):

$b_{t} + m_{t} + \tau_{t} = g + \frac{m_{t-1}}{\pi_{t}} + R_{t-1}\frac{b_{t-1}}{\pi_{t}}$ (1)

Now all you need to do is substitute (2) and the policy rules (also, I don't think Leeper's utility function had a $\delta$ but that's not important) and linearise. I.e linearise:

$b_{t} +δc[\frac{R_{t}}{R_{t}-1}] + (\gamma_{0} + \gamma_{1}b_{t-1} + \psi_{t}) = g +\frac{δc[\frac{R_{t-1}}{R_{t-1}-1}]}{\pi_{t}} + (\alpha_{0} + \alpha_{1} \pi_{t-1} + \beta \theta_{t-1}) \frac{b_{t-1}}{\pi_{t}}$ (2)

Note: I left some $R_{t}$ and $R_{t-1}$ in there but you will have to substitute the monetary policy rule there as well. Then all you do is a Taylor approximation around the deterministic steady state. Just a warning, the algebra will be very hairy, it took me several tries to get it right.

OP requested some more reading on this topic so I'll add a list here:

[1] Loyo (1999), Tight Money Paradox on the Loose: A Fiscalist Hyperinflation

[2] Woodford (1998), Public Debt and the Price Level

[3] Woodford (2000), Fiscal Requirements for Price Stability

[4] Leeper & Leith (2016), Understanding Inflation as a Joint
Monetary-Fiscal Phenomenon

I have many more but start with these.


What accounts for the high GDP of the United States? Wikipedia defines GDP as:

  Measure of the market value of all final goods and services produced in a period (quarterly or yearly) of time.

So, I take it that the total value of all goods (and services, whatever "Services" means) manufactured within the country per year is the country's GDP. This YouTube video does a nice job of explaining it too:

What is Gross Domestic Product (GDP)?

The United States GDP, also from Wikipedia, is around $19 trillion!

According to the definition, US should manufacture USD 19 trillion worth of goods (finished products) and services within the US, but again, according to this Wikipedia article, US manufacturing industry outputs only about $2.2 trillion worth of goods.

So, what accounts for the remaining $17 trillion? Is it the "Services"? If so, what are these services and how are they measured?

Looking at gross output (which includes using the outputs of other industries) and value-added (largely wages and profits) by industry you get numbers like this for 2017 in USD trillion.

Adding up the value-added gives total GDP, and you can see that there is a lot more to the economy than manufacturing

                                                         Gross output  Value-added
Agriculture, forestry, fishing, and hunting                       0.4  0.2
Mining                                                            0.5  0.3
Utilities                                                         0.4  0.3
Construction                                                      1.5  0.8
Manufacturing                                                     6.0  2.2
Wholesale trade                                                   1.7  1.2
Retail trade                                                      1.8  1.1
Transportation and warehousing                                    1.1  0.6
Information                                                       1.7  0.9
Finance, insurance, real estate, rental, and leasing              6.2  4.1
Professional and business services                                3.8  2.4
Educational services, health care, and social assistance          2.8  1.6
Arts, entertainment, recreation, accommodation, and food services 1.4  0.8
Other services, except government                                 0.7  0.4
Government                                                        3.7  2.5

Total                                                            33.7 19.4


Is Buy Nothing Day a classist protest of Black Friday? I hope this isn't off topic, it seems specifically economic to me even though it probably intersects with other fields of study.

Black Friday is seen as an iconic hallmark of consumerism, and Buy Nothing Day is a reaction to it that's been around since the 90s. It's a protest against both Black Friday as well as consumerism in general, and a temporary boycott to draw attention to the issues. I get that the consumerist mindset being protested during this time is buying a bunch of unnecessary stuff just because it's on sale versus getting that one thing you couldn't afford but now you can. And on a wider scale, I get BND is protesting buying a bunch of unnecessary stuff period (consumerism).

But on a functional economic level, in relation to things people would be buying anyway, doesn't boycotting Black Friday just amount to paying full price (or finding other sales) at some other time of the year versus taking advantage of these sales? And thus wouldn't that be something that people of lower economic status would be excluded from participating in, as they wouldn't necessarily have the luxury of choosing to pay more just to make a statement?

in relation to things people would be buying anyway, doesn't boycotting Black Friday just amount to paying full price (or finding other sales) at some other time of the year versus taking advantage of these sales?

Yes.  Regarding necessities or scheduled expenditures, participating in BND requires a time-shift in consumption that may carry a higher price for the same goods or services, in addition to other potential non-pecuniary opportunity costs.  I suppose in the extreme this could lead to hoarding of perishables, fuel, etc. that might be measurable.

  And thus wouldn't that be something that people of lower economic status would be excluded from participating in, as they wouldn't necessarily have the luxury of choosing to pay more just to make a statement?

Yes.  If BND were enshrined in policy, it would be considered regressive in the sense that its negative welfare impacts are likely to be higher for those of lesser means.  It may also constitute a redistribution of wealth from BND adherents to business owners (although this would be complicated to show empirically).


Estimate unobservable $X_t$ with observable estimation {$Y_t$}, where $Y_t= X_t+\delta e_t$ Suppose there are measurement values {$Y_t, Y_{t-1},..., Y_0$} which come from the relationship $Y_t= X_t+\delta e_t$, where $\delta$ is a known constant, $e_t\sim N(0,\sigma^2_e)$ is a Gaussian distribution with known moments while $X_t\sim N(\mu_X, \sigma^2_x)$ is unobservable whose mean needs to be estimated ($\sigma^2_X$ is known).
Empirically, how does one estimate $\mu_X$ with the measurement value {$Y_t,...$}? Can the estimation be performed if both $\mu_X$ and $\sigma_X^2$ are unknown?
Btw please also tell me if there's any literature I can refer to (or start with). I've come across bunches of papers about Bayesian estimation in macro but can't figure out where to start.
Thanks a lot in advance!

It seems all you need to do is calculate the mean of Y (take the average of observations), which gives you the mean of X, since the expected value of epsilon is zero.
I would approach this problem as follows:
$ X_t = Y_t - \delta e_t$
$\mu_X   = E(X_t)= E(Y_t - \delta e_t) = \mu_y - \delta E(e_t) = \mu_y$


How to derive an aggregate demand function (and optimal uniform price) from two demand functions? If I have two demand functions

$P_1=100-10X_1$
and
$P_2=50-10X_2$

and $X_1=40$ and $X_2=60$ with $MC=10$

We have two consumer groups: 1 and 2, where 1 has a higher willingness to pay, and we have one single good. $X_1=40$ refers to number of customers.

How might I derive the optimal uniform price and its aggregate demand function from this?

Let the market price for the good be $P$.

It is easy to see that a consumer from group $1$'s demand for the good is $$ X_1 = \frac{100-P}{10} $$

Similarly, a consumer from group $2$'s demand for the good is $$ X_2 = \frac{50-P}{10} $$

Total demand $Q$ is then given by $$ Q = 40X_1 + 60X_2 = 700-10P $$

This means that the market inverse demand curve (i.e. aggregate demand) is $$ P(Q) = 70 - \frac{Q}{10}$$

Suppose a single monopolist were serving this market. The monopolist's profit is $$ \Pi(Q) =P(Q) \cdot Q -MC \cdot Q $$

If we take the first-order condition (i.e. take the first derivative of $\Pi$ and set it equal to $0$), we know that the optimal quantity sold satisfies $$ 70 - \frac{Q^*}{5} = 10 $$

This means that $Q^* = 300$. We can plug this back into the market demand function to find that $P^* = 40$, which is the optimal uniform price.


Does brain drain really affect a nation's growth trajectory? Take India as an example. India has been losing most of its bright minds to the west. India also ranks low in development. I would like to get a definitive answer to this cause-effect type question.

If India had somehow ensured that its talented people could not leave the country or If India had prevented its bright and ambitious minds from getting enticed and seduced by the west, would those bright minds have been forced to figure out how to improve India's infrastructure thereby resulting in India's development?

Or is it the other way, i.e., a nation's development is mostly (>80%) decided by the quality of its political leadership? The quality of Indian polity has been really bad since 1947. This is not a judgment but a plain fact. So is it that even if we had retained the bright ones they would have made no difference to India's future?

Could you give a data-intensive answer with stats? Do give intuitive arguments also to back it up.

My thesis is that a nation's polity decides the economic fate of the nation, not its talent pool because talent can be trained by the government of the day whenever it wants to. In India, people are not talented because they are born so talented but because competition forces that in the over-populated society. So my claim is that the fact that Indians have become so successful in the USA where they hold key policy-making positions should not be a cause of alarm. Had they been in India also, they could not have made any difference. And whenever the nation's polity improves, the nation will create any amount of required talent it requires just like monkeys can be trained any time you want them to perform in a circus.

Is my thesis wrong? I hear strong views on both sides and i cannot just make up my mind as to which side to follow. Please don't just give subjective answers/opinions (i know i have!). Provide some rigor and facts.

You have asked a very relevant question, and you must be new to economics or have not got a clue about economics of growth/development. And, you will not be able to get a 'definitive' answer to your question - it is not easy, and there is probably none! Should there be a definitive answer, we would not have to spend years doing PhD in economics.

There are plenty of resources which could give you a better grasp of the issue.
For a start, read https://www.amazon.co.uk/Why-Nations-Fail-Origins-Prosperity/dp/1846684307