Are quantitative easing and helicopter cash really different tactics? And how does QE relate to Modern Monetary Theory? I've been trying to get a clear understanding of exactly what economists mean by Quantitative Easing (QE). It seems to me that different people mean different things by it. I find simplistic analogies such as "it's like printing money" unhelpful.
My best guess from what I've read is that QE applies to the action of a national government that has a fiat currency selling one of its own bonds to its central bank, thereby increasing the government's reserve account balance with that bank.Since most governments hold few, if any, of their own bonds, the bond sold to the central bank would generally be newly issued specifically for this purpose. Importantly, under this meaning, QE is purely a way of the government raising funds, via an increase in its central bank account balance, and says nothing about how those funds might be spent.
I further guess that, when a govt does QE, it generally uses the increased account balance to make payments to the private sector, which may be to individuals or companies, and may be gifts (as in a cash handout to individuals) or purchase of goods or services (as in where a govt  undertakes a new infrastructure project, and uses the cash to pay workers and for materials).
This makes perfect sense to me, and seems to align with the 'printing money' analogy. But there are a number of difficulties:

Some sites describe 'helicopter cash' as an alternative to QE. That conflicts with the above understanding of QE as a way of raising funds, not of spending them. Under my understanding, one might do QE in order to finance a helicopter cash strategy.

Some sites say the aim of QE is to lower interest rates, while other sites (and sometimes even the same sites!) say that QE is used when lowering interest rates has ceased to have any effect on economic activity. One of these must be incorrect. Under my understanding, the first is incorrect.

Some sites include as QE the process of the central bank buying bonds from the private sector, which may be government bonds or private sector bonds. That is inconsistent with the above understanding. It does not increase the government's account balance and hence provides no funding for the government to increase spending. Also, it does not differ from what the central bank does as part of business as usual (BAU), which is buying and selling bonds, outright or via repo, in order to manage the cash supply and influence interest rates. The Reserve Bank of Australia calls this activity 'Open Market Operations', and it is a core component of its BAU. So if this is counted as QE it then seems to provide no new information to say that a government has commenced QE.

Some accounts of Modern Monetary Theory (MMT) say that it involves a government creating money without taxing or issuing debt. That would rule out the above notion of QE, which involves the government issuing a bond (debt) to the central bank. Does that mean that QE could not be regarded as a MMT-oriented activity? Also I struggle to work out how such a transaction would be accounted, because it would involve the central bank increasing its liabilities (the government's account balance) but not its assets. Hence it would break the fundamental principle of double-entry book-keeping.

QE as outlined above differs from ordinary deficit spending only by the party to whom the government sells the new bond. In QE it is sold privately to the central bank. In deficit spending it sold to investors, typically through a public auction. In Australia those auctions are conducted by AOFM - the Australian Office of Financial Management. Since the difference is only in who is the first owner of the new bond, I do not see why QE is regarded as radically different from what government's normally do.

I am hoping someone can help me resolve these apparent conflicts and clarify the mechanics of quantitative easing and its relationship, if any, to Modern Monetary Theory.

What is QE:
QE is simply an asset purchase by central bank. As explained by Fed St. Louis QE is defined as:

large-scale asset purchases—in the hundreds of billions of dollars range—of, for example, mortgage-backed securities and Treasury securities.

Furthermore, under QE this is done with newly created reserves. Generally these assets are actually not government assets (i.e. bonds) but rather the assets of banks (although they can be government assets as well).
The idea of QE is to basically make an asset swap - a swap of central bank's  reserves for some assets of banks which make them reluctant to borrow more (for example the first QE Fed did focused heavily on mortgage based securities - which were at the heart of Great Recession - as it was argued that banks were reluctant to borrow more while holding them). Later QE's also included treasury bonds and other more safe long term debt. This is not printing money, but to the extent it encourages new borrowing it increases the money supply so the money printing analogy has some measure of merit in it.

Answer 1

Some sites describe 'helicopter cash' as an alternative to QE. That conflicts with the above understanding of QE as a way of raising funds, not of spending them. Under my understanding, one might do QE in order to finance a helicopter cash strategy.

Yes 'helicopter cash', or more correctly helicopter money is an alternative to QE. QE is an asset purchase program. Helicopter money just literary means giving away cash/deposits to people. The term originated from Milton Friedman's quip that in this case central bank might as well be literary dropping money out of helicopter. Helicopter money does not require any asset purchase from central bank. Central bank can as well just send every household a check or directly deposit the money into their account.

Answer 2

Some sites say the aim of QE is to lower interest rates, while other sites (and sometimes even the same sites!) say that QE is used when lowering interest rates has ceased to have any effect on economic activity. One of these must be incorrect. Under my understanding, the first is incorrect.

No both are correct and you are misunderstanding what they are saying. There are multiple interest rates in the economy. QE is used when Federal Reserve cannot bring down further its federal funds rate - when it hits zero lower bound. However, as explained in the same Fed St. Louis document I linked at the beginning, the aim of QE is to bring down interest rates on long term treasury securities and other financial instruments. Those are completely different interest rates than the one which makes QE necessary. Even if federal funds rate is 0 long term treasury rate or corporate bond rate might be positive.

Answer 3
No, open market operations and QE are different although the difference is subtle.  Investopedia has an excellent article on the difference between the two, here is short summary (but I recommend reading the full Investopedia article):

Open market operations are a tool the Fed can use to influence rate changes in the debt market across specified securities and maturities.
Quantitative easing is a holistic strategy that seeks to ease, or lower, borrowing rates to help stimulate growth in an economy.
Open market operations can be an important tool used in seeking to obtain the goals and objectives of quantitative easing.

Basically, open market operations is much narrower term, while QE is wider term and when central banks execute QE it might be partially done through open market operations.

Answer 4
QE is an asset purchase program, hence it will always involve a debt. However, central banks are part of the government. All profits including the profits from interest on government bonds are sent back to government (after cost of running the central bank is deducted see this previous Q and A about Fed profits - other central bank might have different institutional arrangement but the profits end up at the government).
Also it is technically possible for central bank to just 'scrap' the bonds after purchase, saying government never has to pay it back and giving up any claims to gov. revenue those bonds give central bank. Nonetheless, doing so would represent permanent increase in money supply and most central banks would probably think twice before going this route.

Does that mean that QE could not be regarded as a MMT-oriented activity?

I dont think this can be answered. There is no single formal modern monetary theory (MMT). MMT lives, for most part, in blogosphere not in actual research papers and with every new book on MMT new/different things are claimed about it so there is too much inconsistency to say what is consistent and what isnt with it. As far as I understand from what I have read about MMT central bank purchasing bonds from government would be in view of MMT just an accounting trick and equivalent to directly funding government spending as long as central bank commits not to cash those bonds, but keep all the caveats mentioned in preceding sentences in mind.
Edit in Response to Brian's answer
For example, this blog on MMT says:

It should be noted that QE is conceptually different from MMT, although both involve central bank purchases of government bonds. Under QE, the expectation is that the central bank will sell the government bonds it buys before they mature, so that the government will need to raise money (ultimately through taxes) to pay its debt to private holders of that debt. Effectively, central bank assessments of a temporary need for money-creation to address short-run economic circumstances have driven QE.

Hence the blog claims if assets are purchased without intention of them to be sold later then QE is consistent with MMT as I mentioned above in my answer.
However, an important caveat to the reader here is that does not automatically means that Brian's answer is incorrect (apart from the part arguing this answer made incorrect assertions) - MMT has no definitive agreed upon version of a theory as mentioned before. However, saying that this answer misrepresented view of MMT is itself not correct as this answer both cautioned that people dont even agree on what MMT is and as the quote above shows clearly under some MMT views QE is consistent with MMT as long as central bank does not intend to reverse the bond purchase.

Answer 5
QE and any other monetary policy differs from mere deficit spending (i.e. fiscal policy) tremendously. From macroeconomic perspective the goal of monetary policy is to stimulate economy through changes in the money supply, which in turn affect interest rates and prices in economy. Fiscal policy tries to stimulate economy directly through government spending.
Furthermore, fiscal policy might be conducted even when central bank does nothing and vice versa monetary policy can be conducted even when government does not makes any changes in its fiscal policy or even if government does not have any net debt. In general monetary policy and fiscal policy are macroeconomically different on fundamental level (but this answer is already way too long to go into any further details on these differences but you can consult any standard macro textbook - if you are layman Blanchard et al. Macroeconomics: an European Perspective is excellent source targeted at undergraduates so you wont get yourself lost. If you want something more involved and nuanced then Romer's Advanced Macroeconomics is the book to learn intermediate macro from).


How does reducing government spending increase the money supply? According to the following question, it seems as though cutting back on government spending increases the money supply (the answer is C).

The only possible explanation I can think of is that in the second fiscal year, the government continues to spend, but this time it spends less while not changing tax revenue ("In the absence of offsetting factors"), hence the MS still grows but the following fiscal year's deficit has reduced.

The government cuts public spending but continues to "run a deficit by borrowing from the central bank."
This is the polite way to say "increase yearly the money supply by an amount equal to the government's deficit".

So the money supply will also increase in the second year, albeit by a smaller amount than before. But increase it will.


Monopoly - Where is the Markup formula derived from? Problem:

My lecturer wrote the marginal revenue formula before writing the markup formula, indicating some sort of similarity between the two.

How it was written on the board:

  Market Power being $\text{P>MC}$

  $\text{MR} = \text{P  (1+1/η) = MC}$ (This being profit maximising condition for a monopoly)

  Can be written as:

  $\text{MC/P = 1+1/η}$

  Markup: $\text{(P-MC/P)= -1/η}$

Question:

As much as I can see the similarity, I can't understand how he got to $\text{(P-MC/P)= -1/η}$ from $\text{MC/P = 1+1/η}$?

Index:

P = Price,
MR = Marginal Revenue,
MC = Marginal Cost,
η = Elasticity

It is just a missing parenthesis issue. The expression

$$\text{(P-MC/P)= -1/η}$$  should have been written

$$\text{((P-MC)/P)= -1/η}$$


Compare utility functions I recently joined an econ class. I am so lost on how to prove their equality. As a math standpoint, these are completely different equations. Please help!

u1 (x1, x2) = x1^(2/3) x2^(1/3)
u2 (x1, x2) = 4ln (x1) + 2 ln(x2) +3

According to the logical positivist view of decision theory, utility functions are just descriptions of observable behavior and have no intrinsic meaning absent this vantage. In other words, $u(x) > u(y)$ indicates that $x$ would be chosen in favor of $y$, but the magnitude of the difference carries no addition information. If $u(x) = 1$ and $u(y) = 0$ the decision maker prefers $x$; if $u(x) = 10000$ and $u(y)= 0$ the decision maker prefers $x$. In either case, we can draw the same conclusions.

What does this mean with regard to your question? It tells us what we mean by equality of utility functions: we mean that these two different (so no definitionally equal) functions represent the same observable data. Specifically, exactly when $u_1(x) > u_1(y)$ do we also have $u_2(x) > u_2(y)$. As mentioned in the comments, this is precisely when we can compose $u_1$ with a strictly increasing function and pop out $u_2$. (This is because all we care about is the ordering between objects, and strictly increasing functions preserve order).

So, what might our strictly increasing function look like? Well, we can take the log, so that we get $\frac23 ln(x_1) + \frac13 ln(x_2)$ then multiplying by 6 and adding 3 does the trick. Each of these was strictly increasing so too is the composition:
$$ (x,y) \mapsto 6 (ln(x) + ln(y)) + 3$$
is our desired function.


Why would mortgage rates go up if interest rates go down? What reasons or scenarios could cause a case where mortgage rates would rise if interest rates(federal funds rate) fall? Is that even possible?

Your ceteris paribus logic is correct however that not how the real world works.
And yes this happened in Canada spring 2020, It's because risk premiums went up. We could imagine that if the bank had inflation expectations they would want a real ROI > 0 and thus those could move in opposite directions


Independence and Reduction Axioms I have read that the Independence of Irrelevant Alternatives axiom in expected utility theory implies the fact that compound lotteries are equally preferred to their reduced form simple lotteries. However, I am unable to prove this. I'm sure that the proof should be easy, but I just can't see it.
I read it here: http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html

You can find in many textbooks (for example Mas-Collel, Winston and Green) that the Independence of irrelevant alternatives axiom implies that preferences over lotteries are linear with respect to the probabilities of each event.

Then, a compound lottery is of the form: $\mathcal{L}''=\alpha\mathcal{L}+(1-\alpha)\mathcal{L}'$, If $A$ and $B$ are the outcomes of the simple lottery $\mathcal{L}$ and $C$ and $D$ are the outcomes of the other simple lottery, $\mathcal{L}'$. Then, the compound lottery can be seen as a simple lottery with four outcomes, given the linearity in probabilities: $\mathcal{L}''=\alpha\left(\beta A+(1-\beta)B\right)+(1-\alpha)\left(\gamma C+(1-\gamma)D\right)$.


What is "indirect mechanism" in the context of adverse selectoin? On this slide deck, page 12, it says "then this direct mechanism implements exactly the same allocation as the indirect mechanism we started". What is indirect mechanism in this context and in particular, what is the "indirect mechanism we started" on this page?

In direct mechanism agents directly report their preferences (preferences are observable).

In indirect mechanism agents don’t report their preferences directly. Preferences can be observed only indirectly through signals or behavior.

By Revelation Principle if some outcomes can be implemented in indirect mechanism they must be also implementable in the direct mechanism where everything is observable - that’s what the quote is trying to say.


General Equilibrium Involving Production I need a little conceptual clarification.
For a standard $N*K*M$ general equilibrium model, would an allocation, say, $y^k$ be Pareto Optimal if it does not solve $max(py^k)$?
I understand that the competitive allocations would Pareto Optimal given strictly convex, monotonic utility functions, but does the implication hold reversely as well?

You seem to be looking for the Second Welfare Theorem.

  The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.


When does US Federal Reserve redeem the corporate bonds it bought? The US Federal Reserve did the corporate bond buying program in the 2008 crisis and again now during the 2020 pandemic. What happens to the bonds it is buying? That is, someday it matures and corporates pay the money back to Fed. Where can one get this data? I can see the Fed's asset holding from its weekly report (https://www.federalreserve.gov/monetarypolicy/bst_recenttrends.htm). This one seems to give a picture that bonds bought in 2008 are still held by the Fed. Now in 2020 again there is big spike in holdings due to the purchase again. So, my questions are:

When do the corporates pay back?
What is the impact on economy when they have to payback?
If there is data on such redemptions, where can I find it?
where does the Fed get money from for this corporate bond buying? Tax? But, Would'nt this go cyclical and beat the purpose? That is, government wants to increase money available to public & businesses. So, it buys corporate bonds. If it is done from tax money, it'll spend less on welfare. If it levies more tax, then that will suck money from the hands of public resulting in money crunch again. So, how does the Fed avoid this?

When do the corporates pay back?

These are bonds.  It doesn't matter who owns them.  They get "paid back" -- dividends interest and principal -- according to the terms of the bonds.

What is the impact on economy when they have to payback?

None, because the central bank makes the money disappear.  On the plus side, it slightly reduces the money supply.

where does the Fed get money from for this corporate bond buying?

They "printed money".
A better question (which has also been answered, if you search for it) is why the trillions of printed dollars (aka Quantitative Easing) hasn't produced inflation.


Gold Prices and Terrorism With the Federal Reserve set to review interest rates next month, the price of gold may very well drop if they increase it. But with the current prevalence of terrorism and insecurity, is the price of gold likely to be affected (increase/decrease)?

Very unlikely. Terrorism events are not economically significant enough to move a commodity like gold.  In fact, terrorist attacks only effect markets if they are likely to lead to a major economy entering a full-scale war like it was after 9/11, 2001. The cost of terrorism, while at historical height, is still very small to effect the markets significantly: http://fortune.com/2015/11/17/terrorism-global-economic-cost/. Even economy of a country like France that have been recently hit by a terrorist attack is not likely to be affected by it too much. Politicians may exploit the terrorist attack to push their agenda though, which may have negative effects on the economy, like this articles argues: http://www.marketwatch.com/story/frances-real-challenge-is-growing-stagnant-economy-not-fighting-in-syria-2015-11-19.

Yet, gold is a very volatile market and the price of gold can move a lot in any direction for no obvious reason. Purchases and sales of gold by central banks and institutional investors often go unannounced.

Another thing you should keep in mind if you want to speculate on the rate hike is that some big players may be waiting for that moment to BUY gold in large amounts, because that is when they can expect the most people placing sell orders. In that case the price of gold can jump contrary to expectations.


Could money be tied to a basic good like the gold standard? I don't have a formal education in economics but I had an idea for creating a better and fairer currency. So the gold standard ties gold to the dollar. Gold is usually used in luxury items like jewellery and electronics. So as the economy booms demand for luxury goods increases and thus the demand for dollars. So this would reduce capital investment as dollars would be better to have right, which would reduce growth?

If you tied the dollar to say the Big Mac, then as the economy boomed the dollars relative value would decrease as people are more well off so they would be more likely to invest in businesses. Also the advantage of this system is that in a crash people are poorer so the demand for more basic goods goes up thus the dollar goes up. This would have the effect of making recessions easier on poor people as they mainly have their assets in cash. Also as means of production for the big mac become more refined, inflation will occur so it doesn't have the gold standard problem.

I might be wrong about some points above but in general could making 1 dollar always give you 1 big mac work as a standard for a currency?

The way that the gold standard ties the value of a dollar to gold is usually by making a dollar exchangeable for a fixed amount of gold. For example, from 1934 until 1971 the US government would exchange \$1 for 1/35th of an ounce of gold.

One of the charms of using gold as a currency is that it is durable and extremely valuable for how much room it takes up. How would anyone actually implement monetary conversion for Big Macs? Would there be a Fort Kroc where 3 billion Big Macs (worth the \$11 billion of the current US Gold stockpiles) would be kept? The same  perishability and low value per unit volume of make Big Macs  delicious meals to millions also make them a terrible anchor for a commodity currency.

It is true that consumption of gold services in the form of gold jewelry and the like make demand for gold pro-cyclical. But empirically, prices were stable and inflation low during the gold standard. Therefore it is hard to imagine this was a severe problem. Of course, the gold standard has lots of problems including:

The supply of gold increases slowly over time relative to the economy, leading to gradual deflation which is thought to make recessions worse.
The use of gold as a backing for currency significantly increased demand for gold, leading to tremendous diversion of resources into gold production.
Like any pegged currency, a gold standard depends on reserves which can be exhausted in a panic about the future ability of a government to convert.

Big Macs might not suffer from all these problems but, as mentioned above, suffer from lots of other problems that make it problematic to use them as or a backing for money. And how would you handle counterfeiting?

One reason all major counties have moved to fiat money is that it doesn't suffer from any of the problems of gold listed above. It also doesn't suffer from the additional weaknesses of Big Mac being both perishable and a low value item.

One negative of fiat money is that the monetary authority (e. g. the central bank) can't abuse their discretion to create high inflation. But Big Macs suffer from this problem too. Because Big Macs are easy to create in large quantities and short run demand is relatively inelastic, the government could create lots of inflation by just grilling up their own, just as they could run their printing presses.

Yes, it is true that consumers of Big Macs are probably poorer on average, and so they would benefit relatively more from cheaper burgers if the government tried to inflate away the debt by printing money (grilling up burgers) but even if they were free how many would people eat? That's why their demand is inelastic, people don't want that fourth burger in one day nor do they want to eat them everyday forever. As for the other costs of higher inflation, that's a more complex issue and probably best posed as another question about what are the benefits and costs of higher inflation and who gets what. But original question asks would it be workable. The costs of conversion to a commodity backed money based on Big Macs seem to have little to recommend it and enormous costs both in transition off fiat money, governance (counterfeiting and monetary policy), and ongoing implementation owing to low value per item and perishability.


Understanding Price Elasticity of Supply I understand that the price elasticity of supply is just the percentage change in quantity supplied for a percentage increase in price of an item. However, when I look on Wikipedia, I am struggling to understand why the Supply Elasticity for Oil would be more than 1, which is contrast to what is mentioned in Investopedia. However, I feel like Investopedia is right on this one, in that suppliers of oil will not be able to significantly change supply for an increase in oil price.

To elaborate on 123's comment, crude oil is extracted from the ground and can be manufactured/refined into a variety of produces (vehicle fuel/plastics/kerosene/etc.) If the price of kerosene increases, then it is natural for produces to want to substitute from producing plastic to producing more kerosene. Even if the supply of oil (of all varieties) is inelastic, the supply of individual oil-based products need not be.


Big Mac Index based on other products or services? The Big Mac Index compares the cost of a McDonalds Big Mac in different currencies around the world. In doing so, it provides:

an informal way of measuring the purchasing power parity (PPP) between two currencies and provides a test of the extent to which market exchange rates result in goods costing the same in different countries

Have any academics, organisations, or think tanks made similarly useful indices based on other product/s or service/s?

Yes there are several other such indexes although none as popular as the Big Mac one.
For example the Economist themselves provide not just Big Mac index but also 'Tall Late Index' based on Starbucks' tall late prices.
Another example is the Bloomberg 'Billy bookshelf index' [ based on Ikea product].
Yet another example is one Australian bank for while used iPod index (although this one was criticized on the grounds that it requires transportation and transportation costs are non-trivial).
Probably you could find more such indexes. In principle any homogenous good can serve as a basis for such index.


Comparative dynamics in forward-looking equation I have set up and solved an optimization problem with time $t$ endogenous state variables, $\alpha_t$ and $\beta_t$ and choice variable $s_t$. After some manipulation, the first-order condition for $s_t$ is of the form:

$f(\alpha_t,\beta_t,s_t,s_{t+1})=0$

where $f(\cdot)$ is a non-linear and contains expectations over future realizations of shocks. In some specifications there is no explicit solution for $s_t$.

I want to derive testable implications from the underlying theory. In particular, I am interested in the signs of:

$\dfrac{\partial s_t}{\partial \alpha_t}$ and $\dfrac{\partial s_t}{\partial \beta_t}$.

How should I treat $s_{t+1}$ when I take the total derivative? Should I treat it as a constant, or include it? What's the rationale? If more elaboration is needed, do let me know and I'll be happy to explain in more detail the problem.

Thanks!

Try writing your equation explicitly with a conditional expectation operator:

$$ \mathbb{E}_t f(\alpha_t, \beta_t, s_t, s_{t+1}) = 0$$

Hopefully this clarifies things.  You can take the total derivative as you normally would with any function, however $s_{t+1}$ is a random variable since it is not predetermined.  Ultimately you are considering all cases of $s_{t+1}$ weighted by their probability.

Since $\mathbb{E}_t$ integrates over shocks, you can differentiate under the integral sign by Leibniz's integration rule.


Budget hyperplane in n dimensions Take the set of all vectors $x = (x_1, \cdots, x_n)$ that are
  solutions to $p_1x_1 + \cdots + p_nx_n = I > 0$. Show that this set
  has $n-1$ dimensions.

I have somehow managed to get myself stuck on the last part of this proof it seems. I am not using the fact that this set is a hyperplane and that hyperplanes are $n-1$ dimensions of the space they are in.

It is easy to show that $\{x_1, \cdots x_n\}$ spans the set we are considering, since $\sum p \cdot x$ is a linear combination and all that. However, $x_n$ can be expressed as a linear combination of $\{x_1 \cdots x_{n-1}\}$:

$$x_n = \frac{I - (p_1x_1 + \cdots p_{n-1}x_{n-1})}{p_n}$$

So we can remove $x_n$ from the span and the resulting set still spans. Now we want to show $\{x_1, \cdots n_{n-1}\}$ are linearly independent. That is, if $p_1x_1 + \cdots p_{n-1}x_{n-1} = 0$, all $p_i = 0$. If the set is spanning and linearly independent, then it is a basis. Since it would have $n-1$ vectors, it would be of dimension $n-1$ and we'd be done.

So I note that $p_1x_1 + \cdots + p_{n-1}x_{n-1} = I - p_nx_n$, and that $I > 0$.

So I assume there is a case where $I - p_nx_n = 0$ and where $I - p_nx_n \neq 0$. I am not sure how to finish off this proof, which makes me sad, because I think I'm just missing something obvious. Any assistance would be appreciated.

Let matrix $A = \begin{bmatrix} p_1 & p_2 & \ldots & p_n \end{bmatrix}$. Let $\mathbf{x}^*$ be a fixed solution to $A \mathbf{x} = c$. Then for any vector $\mathbf{u}$ that belongs to the null space of $A$, we have $A \mathbf{u} = 0$ hence $\mathbf{x} = \mathbf{x}^* + \mathbf{u}$ is also a solution (furthermore, all solutions $\mathbf{x}$ can be written this way). The dimension of the set of solutions to $A\mathbf{x} = c$ is thus the dimension of the null space of $A$.

By the rank-nullity theorem, the rank of matrix $A$ plus the nullity (dimensionality of the null space) of matrix $A$ equals the number of columns of $A$:

$$ \operatorname{rank}(A) + \operatorname{nullity}(A) = n$$

Matrix A is rank 1 (assuming not all $p_i$ are zero) and hence the nullity is $n-1$.


Lemma 2 in Tirole and Maskin's Dynamic Oligpoly I (1988) The full paper can be found here: http://www.dklevine.com/archive/refs4397.pdf. I refer here to the Lemma 2 in page 557.

My doubt is regarding the first statement of the proof: "Because reaction functions are nonincreasing and $0$ is a realization of $R^1(q)$, firm 1 must react to any quantity above $q$ by setting $0$ with probability $1$". Why is that so? If $0$ is a possible realization of $R^1(q)$ (if I understood correctly what the authors mean by realization -- since the reaction functions can be understood as random variables --, and maybe that is where I am troubled), what guarantees the fact that any small quantity above $q$ will make firm 1 set its reaction function equal $0$?

The monotonicity result that they prove in their lemma 1 (p. 556) is stronger than you think: it states that whenever $q>q'$, the support of $R_1(q)$ is "uniformly below" the support of $R_1(q')$. In other words, it is impossible to find any realizations $(r,r')$ of $R_1(q)$ and $R_1(q')$, respectively such that $r>r'$.

The proof of this result relies on the idea that firms decisions are strategic substitutes in the Cournot duopoly model: the more firm 2 produces, the less firm 1 wants to produce. Thus, if firm 1 finds it optimal to produce an amount $r'$ with positive probability when firm 2 produces $q'$, it cannot find it optimal to produce a larger amount ($r>r'$) when firm 2 itself produces more ($q>q')$.

Consider therefore a production level $q$ by firm 2 such that $0$ is a realization of $R_1(q)$. Take $\epsilon>0$ and suppose that there exists $r>0$ that is a realization of $R_1(q+\epsilon)$. The condition
\begin{equation*}
r > 0 \text{ and } q<q+\epsilon
\end{equation*}
violates their lemma 1, since $r$ is a realization of $R_1(q+\epsilon)$ and $q$ is a realization of $R_1(q)$. This proves that $R_1(q+\epsilon)$ has all its weight on 0.


Books about chaos theory For someone who already knows micro and macro economics, are there any good books to understand chaos theory in economics and mathematical economics in general?

Some older references, from the first wave of fascination with the subject:

1) Baumol, W. J., & Benhabib, J. (1989). Chaos: significance, mechanism, and economic applications. The journal of economic perspectives, 77-105.

A general introductory overview.

2) Lorenz, H. W. (1993). Nonlinear dynamical economics and chaotic motion (Vol. 334). Berlin: Springer.

Theoretical treatment

3) Brock, W. A., Hsieh, D. A., & LeBaron, B. (1991). Nonlinear dynamics, chaos, and instability.

Applied econometrics for the detection of chaotic behavior in economic time-series, from two of the most known researchers in the field (W. Brock and B. LeBaron)/


Inequality and Growth: the Role of Democracy In the Alesina and Rodrik (1994) study on the impact of inequality on growth, why did they include a dummy for democracies in their empirical estimation? In other words, why did they want to study whether the impact would be different for democracies vis-a-vis non-democracies?

Alesina and Rodrik (1994) wanted to study whether the impact would be different for democracies vis-a-vis non-democracies because they show that

  in a democracy (where the ``median voter theorem'' applies) the rate
  of taxation is higher and the rate of growth lower, the more unequal
  is the distribution of wealth.

Their model implies that democracies with a more unequal distribution of capital ownership grow less rapidly than more egalitarian democracies. They argue this is because
the median voter has a relatively small endowment of capital when wealth is unequally distributed, and thus favors high taxes on capital which keep growth low.

They wanted to estimate this prediction by adding a dummy variable for democracies vis-a-vis non-democracies


What is homebody economy? What is this thing named homebody economy? It seems to be related with the virus
It refers to working at home or maybe at online shopping?

A homebody is someone who doesn't leave their house much. The "homebody economy" refers to economic activities that do or do not do well when people tend to stay at home, which is exactly what is occurring now with the coronavirus pandemic.
Things like online shopping, media streaming services, or cook-at-home meal programs tend to do well when people are stuck at home. Things like movie theaters, restaurants, and cruise vacations require people be physically present, and do not do very well when people are stuck at home. The new "homebody economy" that's taken hold over the past few months is a result of people staying home due to the pandemic, and has had major, industry-specific impacts.


Could a "Japanese-style strike" work as it's supposed to? Under what conditions? This is an urban legend, mostly believed in Spain and some countries in South America.

The legend is based on the premise that their way to make a strike would be to work much more than usual in some goods production, this would cause prices to be lower due to increased supply in the presence of constant demand, so the owners of the business would end up losing money (or making less money) because of that, due to usually planning goods production with the "just in time" method. In addition they would have to pay employees for that day of work and storage costs for excess production, which might be high.

I don't think this would work for every case, but maybe for some it would be plausible.

Would such a type of strike be able to do something similar for at least some cases? If so, in which ones?

We can consider that legally speaking there's no limit to no the number of goods that can be produced.

Without the cooperation of at least mid-level management I dont think that such an approach can work even in theory. Management sets the quantity of ressources and intermediate goods the striking employees have at their disposal to use as inputs.

The theory is inconsistent in the sense that on the one hand it relies on assuming JIT (for its lack of large storage facilities for final products) to obtain the dumping of a large quantity of output on the market. On the other hand "no limit to no the number of goods that can be produced" implies either large storage facilities for inputs or that the company striking is Bertrand's water well.


Additional components in utility functions (behavioral economics) I am looking for a term describing the second part of a utility function in behavioral economics and related disciplines.

For example, Thaler (1983) describes a utility function that could be simplified like

$u_i=\overline{p}-p \pm v(p^*-p)$

with $\overline{p}$ valuation of the good, $p$ price of the good, and $v(p^*-p)$ the so-called transaction utility; the utility gain/loss from making a bargain or being ripped off.
Similarly, a simplified Fehr-Schmidt utility (1999) function consists of

$u_i=(\overline{p}-p)-\alpha (p^*-p)-\beta (p-p^*)$.

with the second and the third term of the utility function describing negative preferences for inequity.

Thaler calls the first term "acquisition utility" and the second term "transactional utility". Is this a consensus in (behavioral) economics, i.e., could we call the second/third part of Fehr-Schmidt "trabsactional utility" or is there another, better term for this second part of utility that is added to the "classical" utility?

Do you know of a paper comparing these terms? I only do so with respect to fairness (Fehr, Schmidt, 2006; Clavien, Chapuisat, 2016).

...

Clavien, Christine; Chapuisat, Michel (2016): The evolution of utility functions and psychological altruism. In: Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 56, S. 24–31. DOI: 10.1016/j.shpsc.2015.10.008.

Fehr, Ernst; Schmidt, Klaus M. (1999): A Theory of Fairness, Competition, and Cooperation. In: Quarterly Journal of Economics 114 (3), S. 817–868. DOI: 10.1162/003355399556151.

Fehr, Ernst; Schmidt, Klaus M. (2006): The Economics of Fairness, Reciprocity and Altruism – Experimental Evidence and New Theories. In: Foundations, Bd. 1: Elsevier (Handbook of the Economics of Giving, Altruism and Reciprocity), S. 615–691.

Thaler, Richard (1983): Transaction Utility Theory. In: Advances in Consumer Research 10 (1), S. 229–32.

Your formulas contain an undefined $p^*$, and the Fehr-Schmidt utility function is wrong. The brackets should be $\max\{p^*-p,0\}$ and $\max\{p-p^*,0\}$, respectively. Apart form that, the two additional terms are usually called the disutility from advantageous inequity and the disutility from disadvantageous inequity, respectively. Calling them "transactional (dis)utility" would not fit in this context.


Understanding what's in the frequently reported and discussed high demand in IT specialists in Germany - is the IT Fachkräftemangel real? So far, I can observe the following contradictory evidences what I perceive from the media.

Statement group #1. In the press, there are repeated reports from different sources about missing specialists in magnitude like many ten thousands - keyword here is the IT Fachkräftemangel. They write about growing numbers of open positions etc.

Statement group #2. In social media, like comments to such articles, especially on heise.de, they say it would be kind of employer propaganda to attract more people to the market (through studying and immigration) to be able to pay less. So, actually, the demand is much more for cheap workforce to save money.

Evidence #1. If I check salary statistics from surveys and also hour rates in Gulp, and what I know from my HR contacts in many companies, payment indeed grows a little bit but high salaries are featured for almost unrealistic profiles, which in my opinion would mostly likely map to two-three people both by skills and responsibilities range (but do not correspond to a two- or threefold salary of course).

So I from the above I could judge like the following. From what I know about the market laws and demand/offer correlation, actually growing real demand in workforce should lead to exploding salaries and hourly rates or at least growth comparable to growth of market leaders , shouldn't it? Which renders the statements group #2 to be closer to the real situation?

Are the above evidences enough to judge like proposed, or have I missed something?

is the IT Fachkräftemangel real?

It is not. Reports fail to filter out the abundance of fake job posts with which intermediaries saturate --and obfuscate-- the labor market. This is the actual reason why intermediaries hardly ever give candidates any concrete information about the alleged job positions. Consequently, reports headlining a "shortage of skilled professionals" are meaningless.

To a great extent this also happens in the U.S., Canada, and perhaps other regions. But the fact that one same position is advertised by several intermediaries (very often their list of requirements being the same, verbatim) obviously does not mean that there are several open positions.

The broad unawareness regarding this issue allows for air castles when it comes to discussing the pseudo "shortage" of labor supply in labor markets.


How are housing costs reflected in factor shares of GDP? The labour share of GDP in the UK is around 55% (Source).
Presumably all labour is supplied by households and all households need a house.
The average percentage of total monthly household income spent on private rent is around 30% (Source). For the sake of simplicity I assume that this percentage is the same for the imputed rent of owner-occupiers.
How do housing costs enter into the factor shares of GDP?
If a household earns 30k/year from labour and pays 10k/year rent, does that result in 30k x 0.7 = 21k towards labour share of GDP and 30k x 0.3 = 9k towards land/capital share of GDP?

They enter through the output. By definition the labor share of GDP is: $w/Y$ where $w$ is factor income that is by statisticians assigned to the labor. Note  this does not necessarily correspond to the economic concept of labor as the statistical labor share of GDP calculation lumps together multiple factors of production and calls them labor - the same way as ‘capital’ share lumps together returns to land, capital, entrepreneurial labor etc.
Now the labor share of income is derived from incomes, but at the macroeconomic aggregate level all spending must be equal all income (although statistically there might be discrepancies due to measurement error). So the numerator can be replaced by total spending in the economy giving us:
$$\frac{w}{Y}=\frac{w}{C+I+G+NX}$$
Where $C$ is consumption spending, $I$ is investment spending, $G$ is government spending and $NX$ is spending on net exports. Now expenditure on rent is included in the $C$ part so that is how these expenditures enter the labor share of income calculation.
However, the back of an envelope calculation at the bottom of your question is not correct for several reasons.
First, If the labor income is 30k then that 30k is already the result of multiplying GDP by labor share i.e. it is $\frac{w}{Y }(Y)=w=30k$
Second if the labor income per year is 30k and rent is 10k and we assume that there is nothing else going on in the economy,  then it is the 10k that goes to the capital income and  calculation of its share. To be more precise here the total output by income approach would be 40k (30k wages 10k rent - this would by the way be also equal to all spending where rent would be part of $C$ and the other income also part of $C$ or other components of aggregate spending) so the labor share would be 30k/40k or $75\%$ while the capital share would be 10k/40k or $25\%$.
Lastly, using that $30\%$ statistics in this case is not really appropriate to begin with. All factors of production are provided by households, and that statistics is an average across both labor and capital providing households. Some capital providing households will just pay that income to another capital providing households (not even mentioning that large share of households empirically provides both). In addition, even though you specify that the 30k is only from a labor income of a households, I don’t think it is reasonable to assume that capital and labor providing households spend the same share of income on housing. The numbers for such household would likely be different from this average.


Aggregating individual demand curves I'm aggregating individual demand curves but the demand curves are inelastic or elastic. Say, 20 people have an elastic demand for a product at price of $2.

Another 40 people have an elastic demand for a product at price of $4. There is a third category who have an inelastic demand of 50 units.

How do I aggregate these demand curves (graphically and in equation form)?

Mathematically

To calculate the total demand curve (aggregate demand has a different meaning), you just need to ask yourself that for each price, how many units will be demanded.

You have three groups:

A. 20 people, perfectly elastic at $p = 2$. This group will buy 20 units as long as $p \leq 2$.
B. 40 people, perfectly elastic at $p = 4$. This group will buy 40 units as long as $p \leq 4$.
C. 50 people, perfectly inelastic. This group will buy 50 units no matter what.

Mathematically,

$$A(p) = \begin{cases} 20 & \text{if $p \leq 2$} \\ 0 & \text{otherwise}\end{cases},\quad
B(p) = \begin{cases} 40 & \text{if $p \leq 4$} \\ 0 & \text{otherwise}\end{cases},\quad
C(p) = 50$$

Total demand is just a function of $p$: $Q(p) = A(p) + B(p) + C(p)$.
If we start at $p = 1$, what's $A(1) + B(1) + C(1)$? Do this for all prices, and you'll get the total demand curve.

Graphically

You could also do this graphically... draw up the demand curve (would look like a rectangle) for each group. The total demand curve will just be the horizontal stack of these rectangles.


Cobb-Douglas and Logarithm Utility Functions Suppose I have a consumer with a utility function $U(x,y) = x^\alpha y ^{1-\alpha} $ where $a \in (0,1)$. Suppose this consumer has wealth $w$ and the prices for $x$ and $y$ are $p_x$ and $p_y$ respectively. I have already set up budget constraints, calculated demand and expenditure functions.

But now I am given another utility function $\alpha \log x + (1-\alpha) \log  y$. Supposedly I can calculate the demand function for this without needing to do further calculation. I don't see how though. What property of logarithms are useful here? I know obviously the definition of a logarithm, but I haven't seen it in this context and am confused what kind of math I should apply to it to find a demand function. Is this just arithmetic? Is it calculus? What is pertinent here to solving this problem?

Utility functions are invariant with respect to positive monotonic transformations (PMT).
Take $U(x,y)=x^\alpha y^{1-\alpha}$, and let $V(x,y)=\log(U(x,y))$ be a PMT of $U$.
Thus $V$ and $U$ both represent the same preference, and thus demand functions for $x$ and $y$ are the same.


Dynamic Optimization : Resource Stock as a Sink I think about a dynamic problem where social planner maximizes the following utility ;

$$\underset{c\left(t\right)}{max}\int_{0}^{\infty}u\left(c\left(t\right)\right)e^{-\rho t}$$

subject to two constraints

$$\dot{K}\left(t\right)=AK\left(t\right)-c\left(t\right)$$

$$\dot{S}\left(t\right)=\left(1-S\left(t\right)\right)S\left(t\right)-\gamma AK\left(t\right)$$

where $S(t)$ and $K(t)$ hold for natural capital (seas, lakes, forests etc.) and physical capital respectively. $c(t)$ represents consumption.

In this $AK$ model, physical capital generates some wastes which are harmful to natural resource stock with a constant parameter $\gamma$.
This feature is close to Wirl (2004) in environmental economics literature.

The Hamiltonian of the problem is

$$\mathcal{H}=u\left(c\left(t\right)\right)+\lambda\left(t\right)\left(AK\left(t\right)-c\left(t\right)\right)+\mu\left(t\right)\left(\left(1-S\left(t\right)\right)S\left(t\right)-\gamma AK\left(t\right)\right)$$

$\lambda$ and $\mu$ are co-state variables for physical and natural capital.

Dynamics of co-state variables are

$$\dot{\lambda}=\rho\lambda+\mu\gamma A-\lambda A$$

$$\dot{\mu}=\rho\mu-\mu\left(1-2S\right)$$

In fact, in this model, it is easy to remark that natural capital don't have any amenity value don't provide any positive utility and it is just like a "sink".

As the social planner just takes into account the negative effect of capital accumulation (it creates wastes), I think natural capital enters in this model as a "cost".

Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?

"Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?"

No. $\mu$ can be thought of as the shadow price of natural resources. Being a "price", it has to be non-negative.
Note that if we set $\mu=0$ the problem reverts back to the standard model, which has an intuitive explanation: if "we don't care" about natural resources, their "price" is zero.

To see this from another route, for the standard CRRA utility function

$$u(c) = \frac {c^{1-\theta}-1}{1-\theta}$$

we would get

$$\frac {\dot c}{c} = \frac{1}{\theta}[A-\rho] - \frac{1}{\theta}\frac{\mu}{\lambda}\gamma A$$

So if $\mu <0$ we would obtain a higher consumption growth rate, which doesn't sound very plausible if by  $\mu <0$ we would want to express that "we care" about natural resources and so we would lower capital accumulation that hurts them, and consequently lower the consumption growth rate.


How could having more US dollars solidified the 1991 Argentine peg to the US dollar? In an economics textbook, Modern Principles of Economics, the authors, Tyler Cowen and Alex Tabarrok, write that "The Argentine government did not have enough U.S. dollars to keep up the value of the peg,"(referring to the 1991 peg) almost suggesting that had they had enough US dollars that the peg would be kept. How does having more US dollars keep the value of the peso up?

If the exchange rate of the Peso was dropping due to decreased demand(people were converting pesos to dollars), shouldn't the government buy a non-USD currency with pesos to reduce the supply and push the price up, keeping the peg intact. How does having "enough" US dollars solve that problem?

Clearly, looking at this from just one perspective, the exchange rate, is limiting in itself, but the sentence seems to suggests it was a possibility. I would appreciate it if anyone could explain the authors rationale.

Simply having more dollars may have served two purposes. Firstly, more dollars would mean that the central bank could have bought more pesos and exerted some upward pressure on the exchange rate. Secondly, fixed exchange rates can fall because of expectations about the central bank's credibility to its peg. If the prevailing expectation is that the central bank does not have the resources (reserves) or the will to defend the peg, pesos will begin to be sold en masse in the expectation of a devaluation.


Interpretation of Interesting Utility Function Solving introductory microeconomics problems I have come across the following type of utility function:
$$ f(K,L) = (\alpha K^{\frac{\sigma - 1}{\sigma}} + (1 - \alpha) L^{\frac{\sigma - 1}{\sigma}})^{\frac{\sigma}{\sigma - 1}} $$

I find it slightly reminiscent of the logarithm version of the Cobb-Douglas function, but clearly the exponents don't fit with that. So the question is: How would you interpret $\alpha$ and $\sigma$ in this case?. Is $\alpha$ still the relative fraction of capital and labour? How can I think about $\sigma$?

This is the CES production function, where CES stands for constant elasticity of substitution.

The parameter $\sigma$ captures the (constant) elasticity of substitution and $\alpha$ is the share parameter.

The Cobb-Douglas production function can be obtained as a special case of the CES class by taking $\sigma\to1$. For proof, I'd refer you to this post.


Elasticity of Supply for a Monopolist Is the supply of output for a monopolist less elastic than that of a perfectly competitive firm with the same production function?

On one hand, it seems impossible to calculate the elasticity of supply for a monopolist since it has no supply curve (i.e. no explicit relationship between quantity supplied and prices since it determines both quantity and therefore indirectly, prices by producing where MR = MC). As a result, it would be impossible to calculate the elasticity of supply for a monopolist and the above statement is false.

However, it would seem to make sense that the elasticity of supply is lower for a monopolist because if for example there is an increase in demand, leading to higher prices, the additional output produced by a competitive firm would be higher than a monopolist as the monopolist would tend to restrict output to keep prices higher. Thus, the monopolist's supply is less elastic and responsive to price changes.

Both sets of reasoning seem to be correct to me. What am I missing?

Elastic with respect to which variable?
Contrary to what is said in some bachelor textbooks, the monopoly has a supply curve, but it cannot directly be seen on the MR=MC Figure.
As the price is endogenous in the case of monopoly, the price elasticity of supply is not defined, unless you have some exogenous variable shifting the price. For instance if the inverse output demand function is $p=P(y,z)$ where $z$ denotes an exogenous variable (population size, aggregate unemployment rate, etc), then it is possible to compute the elasticity, even in the case of a monopoly. Then you can compare the responsiveness to shifts to $z$ in both the competitive and the monopoly cases. At the monopoly optimum, the condition $$P(y,z)+\frac{\partial P}{\partial y}(y,z)y=c'(y)$$ implicitly defines the supply function $y^*(z)$ and it can be shown by the implicite function theorem that $$\frac{\partial y^*}{\partial z}(z)=-\frac{\partial P/\partial z +y^*\partial^2P/\partial y\partial z}{2\partial P/\partial y+y^*\partial^2P/\partial y^2-c''},$$ where for simplicity I omitted the arguments of the different functions. From the second order condition for an optimum, we know that the denominator is negative. It turns out that for demand shifts $z$ which increase $P$ the monopoly output increases (provided the term $y^*\partial^2P/\partial y\partial z$ is well behaved, if $P$ is linear in $z$ for instance).  In the competitive case, the comparable expression is $$\frac{\partial y^c}{\partial z}(z)=\frac{\partial P/\partial z }{c''},$$ which is positive as well. Now compare both expressions, and conclude…. As we are interested in elasticities, we should actually compare $$\frac{\partial y^*}{\partial z}\frac{z}{y^*}(z) \text{ and } \frac{\partial y^c}{\partial z}\frac{z}{y^c}(z),$$ and consider that $y^*>y^c$. If the inverse demand is linear, comparison is easy, but is involved in general as it depends on global properties of all functions involved in the expressions above.


Question on Central bank loss function interpretation How do you interpret this central bank’s loss function:

$$L(\hat x, \pi)=\alpha \hat{x}^2+(\pi-\pi^T)^2$$

where $\alpha >0$ and $\hat x$ is output gap.

Phillips curve is $\pi = \theta \hat{x}+\pi^T $

Sorry I am not economist, I am studying math, and I need its interpretation.

I guess I need to interpret the value of $a$, and I need to look at the geometry of the loss function in the Phillips curve diagram.

What is its interpretation in economic view?

Thank you so much!

The loss function reflects the central bank’s policy objective (i.e. their preferences for stabilizing inflation and some real variable around target levels, in this case output), and it says that the central bank is concerned about inflation fluctuating from their target level ($\pi^{T}$) and their perception of the equilibrium output (here reflected in the output gap).

Now, because the objective of the central bank includes two factors (output and inflation), we can know from the loss function how concerned the central bank is about stabilizing the output gap from the size of coefficient $\alpha$. Indeed, at varying levels of $\alpha$ it is easy to see that the reaction of the central bank to the same output gap will differ. Since $\alpha$ > 0 we know that they are concerned about the welfare loss associated with the output gap (if = 0 they would simply not be concerned about this, and if < 0 this would reflect that bigger output gaps are best since the function is to be minimized).

I can imagine that when looking at the geometry of the loss function in the Phillips curve diagram this may become clearer to you as someone from maths.

And just to note that once you derive the policy rule - the slope of the MR curve - (as in your other question), you will see that this policy rule not only depends on the central bank’s preferences $\alpha$ but also on the behavior of the private sector, the price flexibility as given by the parameter $\theta$.


Multiplant Monopoly With Third Degree Price Discrimination I am trying to do an exercise with a Monopoly Firm with 2 production plants that supplies 3 different markets.

I know how to do every case (Multiplant monopoly or third degree price discrimination), but since this exercise combines both cases, I dont know how to solve it.

Can anyone tell me how to proceed? This are the costs for every plant, and the demand for each market, for clarification.

Hint.
The following are the necessary conditions for profit maximization:
Monopoly with 1 plant and 1 market: $MR = MC$
Monopoly with 1 plant and 2 markets: $MR_1=MR_2=MC$
Monopoly with 2 plants and 1 market: $MR = MC_1=MC_2$
I'll let you figure out the rest.


:History: Industry threatening to leave if social change takes effect With growing inequalities and salary gap, left-wing politicians are asking for social reforms in some countries (I am thinking of France).

Social reform include: equal pay for women, maternity leave, increasing minimum wage, higher taxes on multinationals, lowering retirement age, etc.

In all these countries, people are threatening that if these changes come into effect, "the money will leave".

I seem to recall similar threats: Ford threatening to close its plants in the UK after the 1968 strike if the women pay were increased, some French based companies after the 35h-week reform (all worked hours above 35 in a week are considered extra and payed more), and currently politicians in the US when people speak of holidays or maternity leave.

My questions are:

1: is there any historic precedent of these threats to "leave" being actually carried out and

2: does it even make sens from an economics point of view ? I can understand manufacture going to Asia, but Starbucks, McDonald, and all these companies that have to stay in France to make profit and, with a clever manipulation, do not pay taxes on the companies in France (impôts sur les sociétés), would they really leave ? Starbucks makes 2 billion benefits in France, so if social reform and it paying taxes would mean it would only make 1 billion benefits, wouldn't that be still worth it ? And even if the benefits are less than that, wouldn't McDonald stay in France, if only to maintain the hegemony (leaving would mean taking the risk of someone taking its place, like Burger King or something, and that would mean losing the monopoly, which I would think is a bad thing for such a company). And even if they do leave, couldn't France just introduce a 5 years tax-free period for any new start-up replacing  them ? It's not getting its taxes anyway, better give it up for 5 years than for ever... no ? Am I being naïve ?

What's going on when a for-profit company makes a threat?

The same thing that is happening when does anything. It is trying to increase its profits, relative to what would happen if it didn't do the thing.

In this case, it's trying to influence policy to protect its profits. It has decided that issuing the threat gives their profits a better chance than not issuing the threat. That probably means that they believe that their profits will be lower with the new policy than without it.

The thing it's threatening is, in a sense, neither here nor there: the threat itself is the action.

Should the policy that a company opposes, get implemented, then the company then has a choice, to stay or to go, and the decision criterion will be: will them staying result in higher or in lower profits than them staying. That's a completely different issue to the thing they issued the threat over, and the threat doesn't tell us anything useful about it.

And companies are often wrong about the impacts of policy anyway.  In summary, treat their statements on policy with a lot of scepticism.


Can I Upload my Preprint on Arxiv Before Submitting it to JPE I wrote a paper relating the optimal deterrence strategy for crime to concepts on statistical physics, and I am considering submitting the paper to the Journal of Political Economy. I'm wondering if posting the preprint on Arxiv violates the journal's policy, or lowers the chances of my paper being accepted.

I'm not too familiar with the standard practices in the field of economics. Where do people commonly publish their preprints, if at all?

Most papers I have read in the JPE circulated in form of preprint before (not on arxiv, but this should not be an important point). See for instance https://ideas.repec.org if you want to find examples of working papers later published in JPE:

But it is probably better to read the journal's policy before submission.


Papers that use R vs Stata Is there any way to tell if a paper used R instead of Stata or Matlab for the estimation? I know it doesn't matter for the quality of the paper, but I'd just be interested to see how many papers use one software over the other.

I'm thinking an easy way might be by looking at the figures- ggplot means R, for example.

Again, not really an economics question, but it's about economics papers.

The main economic journals are slowly starting to require authors to make their data and the code of their analysis available as part of the online appendix. When this is the case, it is easy to figure out which software was used.

One example are recent publications in the American Economic Review. For instance,

Calsamiglia, Caterina, Guillaume Haeringer, and Flip Klijn. 2010. "Constrained School Choice: An Experimental Study." American Economic Review, 100(4): 1860-74. https://www.aeaweb.org/articles.php?doi=10.1257/aer.100.4.1860

the online appendix of which contains the complete STATA code of the statistical analysis.

If the code is not available at the journal's webpage, other alternatives involve :

Checking the website of the authors.
Contacting the authors themselves to ask if they could send you their code.

None of these methods are great if you're trying to gather a sizable or representative dataset of software used in econ papers, but for anecdotal data, they should do the trick.


Decomposition of an additive functional into a Martingale part and other This question relates to a theorem about the decomposition of additive functionals---a technique especially useful in macroeconomics and finance. This question has two objective. First, I don't have a reference to the theorem that shows when this is possible, and I would like to find it. Second, I would like to know how to actually find this decomposition.

Consider the following autoregressive model:
$$
X_{t+1} = \alpha_0 + \beta_0 (X_t - \alpha_0) + W_{t+1},
$$
where $-1 < \beta_0 < 1$ and $W_{t+1}$ is distributed as a normal with mean zero and variance one. Now, consider the additive functional
$$
Y_t = \sum_{j=1}^t X_j.
$$
How would I produce the decomposition of the form
$$
Y_t = r_1 t + M_t - r_2 (X_t - X_0)
$$
where $\{M_t \}$ is a Martingale? (How can obtain the values $r_1$, $r_2$, and $M_t - M_{t-1}$?)

Using the given formulas (inserting the expression for the $X$'s in the sum), we arrive at

$$Y_t = (1-\beta_0)\alpha_0\cdot t + \beta_0\sum_{j=0}^{t-1}X_j +\sum_{j=1}^{t}W_j $$

We do not need a theorem to obtain that, given the description of the problem.

Manipulating,

$$Y_t= (1-\beta_0)\alpha_0\cdot t  - \beta_0(X_t-X_0) + \beta_0Y_{t}  +\sum_{j=1}^{t}W_j$$

$$\implies Y_t = \alpha_0t +\frac{1}{1-\beta_0}\sum_{j=1}^{t}W_j-\frac{\beta_0}{1-\beta_0}(X_t-X_0)$$

and we want to match this with

$$Y_t = r_1 t + M_t - r_2 (X_t - X_0)$$

We immediately get that we must have  $r_1 =  \alpha_0$, $r_2 = \beta_0/(1-\beta_0)$, and
$$M_t = \frac{1}{1-\beta_0}\sum_{j=1}^{t}W_j$$

Is $M_t$ as obtained a martingale? We have

$$E[M_t\mid \sigma(t-1,t-2,..)] = \\=\frac{1}{1-\beta_0}E[W_t\mid \sigma(t-1,t-2,..)] + \frac{1}{1-\beta_0}\sum_{j=1}^{t-1}W_j$$

$$= \frac{1}{1-\beta_0}E[W_t\mid \sigma(t-1,t-2,..)] + M_{t-1}$$

So if $E[W_t\mid \sigma(t-1,t-2,..)] = 0$, namely, if the original disturbance is mean-independent from the past (in a more fancy term, if $\{W_t\}$ is a "Martingale difference"), then $M_t$ is a martingale and we have our mapping.


Nominal and real government expenditure I just need to know that how to make a data series of real government expenditures using nominal government expenditure data series.

If you have more detailed information about what kind of expenditures you have then you can use the CPI data. Current price data for government does not exist under the price program for BLS for you to use to convert the nominal to reals.

However, you can use the BEA industry data which does have a government sector price indexes for either valued added output, gross output concept of GDP or you can select the government sectors price indexes for intermediate inputs.  It depends on the topic you are analyzing.

https://apps.bea.gov/iTable/iTable.cfm?ReqID=51&step=1


Utility Representation of Lotteries We went over a lemma in class leading up to a larger theorem. The Lemma states:

  Let $\succeq$ be a rational preference relation on $\mathscr{L}$ and let $\succeq$ admit utility representation under Von-Neumann-Morgenstern expectations. Then

  $U(\sum_{k=1}^{K} \alpha_k L_k) = \sum_{k=1}^{K} \alpha_k \ U(L_k)$
  $\succeq$ satisfies independence
  Every linear representation $V$ of $\succeq$ is a positive affine transformation of $U$. So $V = \beta U + \gamma$ where $\beta > 0$

So in proving this, here is some work so far:

1:

Let $L_k = (\Pi^k_1, \Pi^k_2, ..., \Pi_s^k)$
$$U(\sum_{k=1}^{K} \alpha_k L_k) = \sum^s_{i=1}\sum^K_{k=1}\Pi^k_i \alpha_k u_i$$ by Von-Neumann-Morgenstern, where $U(L) = \sum^s_{i=1}\Pi_i u_i $

$$\sum^s_{i=1}\sum^K_{k=1}\Pi^k_i \alpha_k u_i =  \sum^K_{k=1} \alpha_k \sum^s_{i=1}\Pi^k_i u_i = \sum_{k=1}^{K} \alpha_k \ U(L_k)$$

2:

Consider $L_1, L_2, L_3 \in \mathscr{L}$ and say $L_1 \succeq L_2$.

$L_1 \succeq L_2 \iff U(L_1) \geq U(L_2)$

Take $\alpha \in (0,1)$ and define

$$L_4 = \alpha L_1 + (1-\alpha)L_3$$
$$L_5 = \alpha L_2 + (1-\alpha)L_3$$

Say $L_5 \succ L_4$

$\implies U(L_5) > U(L_4)$
and from 1.)

$$U(L_5) = \alpha U(L_2) + (1-\alpha) U(L_3)$$
$$U(L_4) = \alpha U(L_1) + (1-\alpha) U(L_3)$$

$$\implies \alpha U(L_2) + (1-\alpha) U(L_3) > \alpha U(L_1) + (1-\alpha) U(L_3) \implies U(L_2) > U(L_1)$$

which is a contradiction, so independence must hold.

I assume 3. is some sort of "monotonicity" condition. How would I approach the proof for this condition? Any hints would be appreciated. I also am wondering what this Lemma is leading up to, so I can study for that in advance. Does anyone have any idea?

So I figured I'd post the proof for the third condition now that I know it.

  We want to show that if $U$ and $V$ are linear representations of $\succeq$ on  $\mathscr{L}$, then $\exists \ \beta > 0, \gamma \in \mathbb{R}$ such that $V = \beta U + \gamma$

Case 1: All the lotteries are all equally preferred.

If $\forall \ L_1, L_2 \in \mathscr{L}$ we also have $L_1 \sim L_2$, then $U$ is constant.

$V = \beta U$ holds $\forall \beta > 0, \gamma  > 0$

Case 2: All the lotteries are not equally preferred.

$\forall L \in \mathscr{L}$ where $L \neq L_b, L_w$
$$L_b \succeq L \succeq L_w$$
$$L_b \succ L_w$$

For a lottery $L \in \mathscr{L}$,

$$\lambda_L \equiv \frac{U(L) - U(L_w)}{U(L_b) - U(L_w)}$$

for $\lambda \in [0,1]$

$$1 - \lambda_L \equiv \frac{U(L_b) - U(L)}{U(L_b) - U(L_w)}$$

So now we can write $U(L), \ \forall L \in \mathscr{L}$ as

$$U(L) = \lambda_L U(L_b) + (1-\lambda_L)U(L_w)$$
$$\implies L \sim \lambda_L L_b + (1-\lambda_L)L_w$$

For $V$ to also represent $\succeq$,

$$V(L) = \lambda_L V(L_b) + (1-\lambda_L)V(L_w)$$

$$V(L) = \bigg[\frac{U(L) - U(L_w)}{U(L_b) - U(L_w)}\bigg] V(L_b) + \bigg[\frac{U(L_b) - U(L)}{U(L_b) - U(L_w)}\bigg] V(L_w)$$

$$V(L) = \frac{U(L)V(L_b) - U(L_w)V(L_b)}{U(L_b) - U(L_w)} + \frac{U(L_b)V(L_w) - U(L)V(L_w)}{U(L_b) - U(L_w)}$$

$$V(L) = \frac{V(L_b) - V(L_w)}{U(L_b) - U(L_w)} \cdot U(L) + \frac{U(L_b)V(L_w) - U(L)V(L_b)}{U(L_b) - U(L_w)}$$

Define $\frac{V(L_b) - V(L_w)}{U(L_b) - U(L_w)} \equiv \beta$ and $\frac{U(L_b)V(L_w) - U(L)V(L_b)}{U(L_b) - U(L_w)} \equiv \gamma$

Thus $V(L) = \beta U(L) + \gamma$, $\forall L$

$\square$


What's the difference between the Life Cycle Hypothesis and Permanent Income Hypothesis? Would it be correct to say that the Permanent Income Hypothesis (PIH) stipulates that current consumption decisions are made based on future income projections/expectations, while the Life Cycle Hypothesis (LCH) claims that consumption is constant over the average person's life time, and this is made possible, despite changes in income level throughout his/her lifetime, through borrowing when younger and savings during the elderly years?

The ideas are very similar and functionally virtually equivalent although there arguably is subtle difference. For example, Romer in his Advanced Macroeconomics, which is widely used intermediate macroeconomics handbook, calls both the life cycle hypothesis and permanent-income hypothesis just permanent income hypothesis. To be more specific he says

Thus the individual uses saving and borrowing to smooth the path of consumption. This is the key idea of the permanent-income hypothesis of Modigliani and Brumberg (1954) and Friedman (1957).

Where the Modigliani and Brumberg (1954) refers to the paper where the life-cycle hypothesis originates.
There is very subtle difference though. Friedman's permanent income hypothesis, focuses more narrowly on income. Friedman's argument is also very 'instrumentalist' and it goes as follows. We know from macroeconomics that consumption depends on income so we will have:
$$C_t = a +  bY_t + e_t$$
where the estimated $\hat{b}$ is given as covariance of income and consumption over variance in income $\hat{b} = \sum(C_t-\bar{C})(Y_t-\bar{Y})/\sum(Y_t-\bar{Y})^2$ and $\hat{a}= \bar{C}- \hat{\beta}\bar{Y}$. Now if we assume that $Y$ can be decomposed into permanent $Y^P$ and temporary $Y^T$ income (i.e. $Y=Y^P +  Y^T$) and that the two are uncorrelated with transitory income having mean zero we can following Friemdman show that:
$$b= k \frac{\sum (Y_t^P- \bar{Y_t}^P)^2}{\sum ( Y_t- \bar{Y_t})^2} \text{ and } \hat{a} = \bar{C} -\hat{b}(\bar{Y}^P+\bar{Y}^T)$$
Since transitory income is on average assumed to be zero, and because variation in transitory income which is component of the denominator in calculating $\hat{b}$ is arguably much smaller than the variation in permanent part of the income one can conclude that consumption is determined virtually completely by changes in permanent income.
The life cycle hypothesis of Modigliani and Brumberg is more analytical and pays also attention to person's whealth. Following presentation in Romer's Advanced Macroeconomics,  a person at any time $t$ a person would make expectation of his/hers lifetime income $\sum^T_{t=1} E_1 [Y_t]$ together with any initial assets/wealth $A_0$ and their current consumption at the time $t$ will be just an equal fraction of all income (i.e. permanent income in Friedman's lingo) and assets  for that particular time period. That is the consumption in present period 1 would be given by:
$$C_1 = \frac{1}{T}\left(A_0+ \sum^T_{t=1} E_1 [Y_t]\right) \text{ and }$$.
This implies that people will try to smooth their income over their life as it implies that people will save when the income is higher than average income and dis-save when it is lower than average since:
$$S_t=Y_t-C_t= \left( Y_t - \frac{1}{T} \sum^T_{t=1} E_1 [Y_t]\right) - \frac{1}{T}A_0$$
Thus both PIH and LCH are functionally equivalent and they are both consistent with each other. They both argue that consumption is determined by your permanent income/lifetime income including any inherited assets.  The main difference is in presentation and arguably the mechanisms are subtly different. The Friedman's PIH presentation is more 'empirically oriented' and focuses on income itself (although arguably Friedman implicitely included any assets in $Y$). The LCH is more analytical and pays more attention person's wealth and later versions also to demographic factors.  Since they are so similar as already mentioned before the ideas are usually treated simultaneously in textbooks and some authors such as Romer call both ideas permanent income hypothesis and I recall seeing the reverse as well.


Stimulus Effect of the Minimum Wage In the question about the minimum wage found here, I have heard several times that an increase in the minimum wage can, as one of several competing effects, "boost the economy" by increasing the income of some workers. Under what conditions could this happen, if at all? Are there any papers (peer-reviewed journal articles) out there that make this claim?

From the Chicago Federal Reserve:

  Following a minimum wage hike, household income rises on average by about \$250 per quarter and spending by roughly \$700 per quarter for households with minimum wage workers. Most of the spending response is caused by a small number of households who purchase vehicles

http://www.chicagofed.org/digital_assets/publications/working_papers/2007/wp2007_23.pdf

Aaronson, D., S. Agarwal, and E. French  (2011): "The Spending and Debt Responses to Minimum Wage Increases" Federal Reserve Bank of Chicago Working Paper, WP 2007-23.


Optimal production level for a typical firm in long-run Assuming all firms have identical cost functions. Now suppose there is an increasing shift in the demand curve. As we all know that for increasing costs case, both average costs (AC) and marginal costs (MC) faced by a typical firm would be higher. The question is, whether the optimal production level of a typical firm remains constant or increases (or decreases! is it possible?) as compared to previous long-run equilibrium position?

In other words, as both MC and AC curve will shift upward, can one predict with certainty the direction in which optimal production level moves to ensure MC=AC_min?

It appears we start at long-run equilibrium point.

The fact that all firms operate at the level where $q^*:MC = \min AC$, means that given demand $Q^d$ and the cost structure, what is endogenously determined is the number of firms $m$:

$$Q^d = mq^* \implies m^* = \frac {Q^d}{q^*}$$

Assume that aggregate demand increases. In the microeconomic setting we are examining things here, the cost/technology structure will not be affected by this change (we do not place bounds and the production inputs available). So in the new equilibrium, each individual firm will produce exactly the same as before.

What will change is the number of firms, which will increase, in order to accommodate the new higher required output.


Why is the USA not capable to pay its debt? Why is the USA not capable to pay its public debt on its own?

The country needs to create new debt in order to pay its due debt.

Not only that, but the debt is also only increasing and the country is not capable to reduce the debt. Since 2000 or so, the country is only capable to increase its debt. And the prediction is that the debt will only increase - Link1 - Link2 - Link3

Hungary and the Czech Republic are capable to pay their debt on their own but in the same time USA is not capable to pay its debt on its own. Why is that?

What makes a country not capable to pay its debt on its own?

The US is in a privileged position of producing a currency that people in other countries want to use (as reserve). There's even a French name for that. Anyway, a result of that is cheap credit for the US government:

Many believe that this dollar dominance has allowed the United States to live beyond its means, running sizable current account deficits financed by borrowing from the rest of the world at cheap interest rates. Some other countries have chafed at this “exorbitant privilege” enjoyed by the United States. [...]
There are tangible and intangible benefits to a country whose currency serves as a reserve currency. In addition to the prestige conferred by this status, it also means access to cheap financing in the country’s domestic currency and the benefit of seigniorage revenue—the difference between the purchasing power of money and the cost of producing it—which can be extracted from both domestic and foreign holders of the currency.

In more mundane terms, the Chinese (and others) don't just hold on to their reserve dollars, but they then lend them back to the US gov't.
So the incentive to not be indebted is less than elsewhere.
In particular, the classical idea is that:

Economists say the US need not ever pay its debts back entirely, so long as the economy keeps growing enough to cover the payments.

But there are even more extreme ideas like the MMT, which says the US can pay even less than that.
Of course, you could ask, why don't the Chinese (and others) do something else with their dollars. The catch is that there's nothing better on the horizon, in a certain (risk averse) sense:

Moreover, there are no alternative currencies or investments that provide a similar degree of safety and liquidity in the quantities demanded by investors.
The reason the United States appears so special in global finance is not just because of the size of its economy, but also because of its institutions—democratic government, public institutions, financial markets, and legal framework—which, for all their flaws, still set the standard for the world. For instance, despite the Federal Reserve’s aggressive and protracted use of unconventional monetary policies, investors worldwide still seem to trust that the Fed will not allow inflation to get out of hand and diminish the value of the dollar.

(That was written in 2014, quantitative easing is even less controversial now.)

As for "can not": it is a matter of time frame and it is relative to other objectives. Here are some silly ways to pay the debt (quickly):

print money; it will causes substantial inflation, loss of confidence etc., and not really solve the problem because a lot of costs the government faces will rise with inflation.

mega-austerity, i.e. cut suddenly spending, a lot. The a-word is nearly unheard of in the US, unless it's news about Europe. Largely unacceptable politically. Also a shrinking economy means less tax revenue. Classical example nowadays:

Under the auspices of ‘bailouts’ from the IMF and the EU, Greece cut more than 20 per cent of GDP in spending. It lost nearly 30 per cent in final consumption.

There's of course debate whether a little austerity will do that; it depends what you'd cut.
But here's an example what Greece cut (and didn't):

The International Monetary Fund (IMF), which joined the rescue of Greece in 2010 alongside the EU and the European Central Bank (ECB), never asked for defence cuts, but insisted on reductions to wages. So while the defence equipment budget remained intact, soldiers suffered a near 40% drop in salaries.

And that's actually a little misleading too, because (except for defense) investment tanked as well, in fact more than the GDP did, to some 40% of the pre-crisis value.

Someone with better macroeconomics knowledge may be able to quickly estimate what paying the whole US debt at one would do. But since serious sources dismiss the idea off-hand, I can't find a published calculation.
There's no quick fix for the debt that won't impact some other long-term goal.


Contract curve and Pareto frontier Consider an exchange economy with two agents.
Each agent $i \in \{1,2\}$ derives utility $u^i(x_1,x_2) \in \mathbb R$ by consuming $(x_1,x_2) \in \mathbb R_+^2$.
Let $u_j^i(x_1,x_2) = \partial u^i(x_1,x_2) / \partial x_j$ for $(i,j) \in \{1,2\}^2$.
The set of efficient consumption bundles is given by
\begin{align}
C = \left\{(x_1,x_2) \in \mathbb R_+^2 ~ \bigg| ~ \frac{u_1^1(x_1,x_2)}{u_2^1(x_1,x_2)} = \frac{u_1^2(x_1,x_2)}{u_2^2(x_1,x_2)}\right\}
\end{align}
Suppose the function $x_2^c :\mathbb R_+ \to \mathbb R_+$ solves
\begin{align}
\frac{u_1^1(x_1,x_2^c(x_1))}{u_2^1(x_1,x_2^c(x_1))} = \frac{u_1^2(x_1,x_2^c(x_1))}{u_2^2(x_1,x_2^c(x_1))}
\end{align}
The contract curve is the graph $\mathcal G(x_2^c) = \{(x_1,x_2^c(x_1)) \mid x_1 \in \mathbb R_+\}$.
Let $\overline u^i(x_1) = u^i(x_1,x_2^c(x_1))$.
The Pareto frontier is now given by the following utility allocations
\begin{align}
P = \{(\overline u^1(x_1), \overline u^2(x_1)) \mid x_1 \in \mathbb R_+\} \subset \mathbb R^2.
\end{align}

Suppose that the Parento frontier is downward sloping w.r.t. $x_1$ such that
\begin{align}
\frac{\overline u_1^2(x_1)}{\overline u_1^1(x_1)} < 0.
\end{align}

  Q: What conditions must be met such that the Pareto frontier is concave?

I'm especially wondering if the following condition is sufficient:
\begin{align}
\frac{\overline u_{11}^2(x_1)}{\overline u_{11}^1(x_1)} < 0.
\end{align}

Let us assume all these functions exist and are differentiable. Furthermore let us denote the inverse of the function $\overline u^1(x_1)$ by $x_1\left(\overline u^1\right)$.

Note that for any invertible function $f$ we have
$$
\left.\frac{\text{d} f(x)}{\text{d} x}\right|_{x=x_0} =
\frac{1}{\left.\frac{\text{d} f^{-1}(y)}{\text{d} y}\right|_{y=f(x_0)}}.
$$
In our case this means (with slightly simplified notation)
$$
\frac{\text{d} x_1(\overline u^1)}{\text{d} \overline u^1} =
\frac{1}{\frac{\text{d} \overline u^1(x_1)}{\text{d} x_1}}.
$$
We will further simplify notation by writing
$$
\overline u{^i}' =  \frac{\text{d} \overline u^i(x_1)}{\text{d} x_1}.
$$
We will denote second derivatives similarly by affixing $''$.

The question is under what conditions
$$
\overline u^2\left(x_1\left(\overline u^1\right)\right)
$$
is concave in $\overline u^1$. This of course, depends on the sign of
$$
\frac{\text{d}^2 \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}{^1}^2}.
$$
Differentiating
\begin{align*}
\frac{\text{d} \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}^1} & =
\frac{\text{d} x_1(\overline u^1)}{\text{d} \overline u^1}
\frac{\text{d} \overline u^2(x_1)}{\text{d} x_1}
=
\frac{1}{\overline u{^1}' }\overline u{^2}'
=
\frac{\overline u{^2}'}{\overline u{^1}' }.
\end{align*}
Going one step further
\begin{align*}
\frac{\text{d}^2 \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}{^1}^2} & =
\frac{
\frac{1}{\overline u{^1}' }\overline u{^2}^{''} \overline u{^1}'
-
\overline u{^2}'\frac{1}{\overline u{^1}' }\overline u{^1}^{''}
}
{\left(\overline u{^1}'\right)^2}.
\end{align*}
This implies that the Pareto-frontier is concave when
\begin{align*}
\frac{\text{d}^2 \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}{^1}^2} & \leq 0 \\
\\
\frac{
\frac{1}{\overline u{^1}' }\overline u{^2}^{''} \overline u{^1}'
-
\overline u{^2}'\frac{1}{\overline u{^1}' }\overline u{^1}^{''}
}
{\left(\overline u{^1}'\right)^2} & \leq 0 \\
\\
\overline u{^2}^{''}
 & \leq \frac{\overline u{^2}'}{\overline u{^1}' }\overline u{^1}^{''}
\end{align*}
for all $x_1$.


Can You Help Me Understand this Definition of Economic Efficiency Economic Efficiency:

"A broad term that implies an economic state in which every resource is optimally allocated to serve each person in the best way while minimizing waste and inefficiency. "

And here's the part I don't understand:

"When an economy is economically efficient, any changes made to assist one person would harm another."

Can someone please unpack that last part for me?  I'm not sure I understand it.

Suppose the economy is in a state where one person could be made better off without making anyone worse off. Most people would agree this is not a good state to be in, i.e. we can do better.
That means a "pareto improvement" is possible.

So what we would want is to in fact make that person better off (as no one else gets harmed and one person benefits). In trying to improve upon that state, we change it. We keep doing this until it is no longer possible to make anyone better off without making someone worse off. I.e. we stop improving when it's no longer possible to improve.

So a state where pareto improvements are possible is not efficient since we can do better. Hence, states where no pareto improvements are possible (we can't do better) are the best possible ones. The best possible states are called efficient.

Hence, efficiency means a pareto improvement is no longer possible. In other words: it is not possible to make someome better off without making someone else worse off.


What do banks do with "Repaid Principal"? When a bank receives payments towards repayment of the principal of a loan, it is called "Repaid Principal". What does a bank do with this amount?

Does "Repaid Principal" add up to their pool of funds or reserves which they lend again in form of new loans, or do they utilize it in some other ways?

Where exactly does the "Repaid Principal" end up?

When you repay principal of 90 USD, the bank makes two accounting entries in its computerized books.

One is called a debit and it goes like this.  If you pay by check drawn on another bank, their reserves at the Fed are increased by 90 USD. If you paid in currency, their vault cash asset account, which is also an asset of the bank, increases by 90 USD and the cash is dumped into the vault with all the other cash.  If you pay out of your checking account at the same bank, their "deposits" book entry, which is a liability of the bank, is decreased by 90 USD.

The other entry is called a credit.  Your loan balance, which is an asset to the bank, is decreased by the 90 USD.

You asked what they do with "it", meaning the principal paid.  As you can see above, when the currency was dumped into the vault and mixed with all the other dollar bills, or a computerized balance was increased or decreased by 90 USD, "it" ceased to exist.  The 90 USD are no longer distinguishable from any other dollars.  So it is a meaningless question. A lot of people don't understand this!

But if you are asking what the bank does with its reserves (vault cash plus reserve account at the Fed), in general: in normal times, they keep them as low as allowed by the Fed.  As their reserves increase higher than required, they reduce them by lending money to customers and other banks, purchasing securities, or paying their expenses.  Transactions that make their reserves increase are things like (a) loan principal is repaid, (b) currency or checks drawn on other banks is deposited, (c) paying their expenses, (d) loan interest is paid by their debtors.

At present, banks are holding on to more than they are required to, because they are relatively afraid to lend.  The extra is called "excess reserves".


What stops a government in partnership with its central bank from lowering interest rates and then borrowing at this lower rate to fund investments? For example, in the UK, after the Brexit vote, the Bank of England decided to lower reserve interest rates to 0.25%. This, as I understand it, has the knock on effect of lowering the interest rates of loans. If a government wants to borrow to fund particular investments, such as renewable energy, a new bridge, trains, etc, it would cost them less under the new, lower interest rate. So my question is, why aren't low interest rates the standard strategy for a government? Surely they would always want to borrow at cheaper prices?

I can think of a few reasons but I'm not really sure if they're correct.

Savers would get annoyed and move their money to another currency.
Ideological positions (neoliberalism/austerity instead of Keynesian economics)

Seems like I'm missing something obvious here..

Cheers!

Bank of England decided to lower reserve interest rates to 0.25%.

This means the central bank wants to reduce the interest paid on reserved stash, discourage bankers parking too much money in central bank.

With smaller return from central bank interest rate, this is just a policies that central bank hope the bank will use the money to invest or buy bonds.

In short, this has nothing to do with borrowing rates.

Long story
Most country central bank enact rules that require banker to deposit some reserved, which the stash can be used when one of the banker face a sudden surge of withdrawal,etc. It is just a measurement to prevent banker gone rogue, loan too much money to borrower than a safety threshold.  But money in this reserved are "unproductive", thus most central bank will give some interest return.

However, in high risk condition or common rent seeking situation , bankers might do the opposite, they park more money in central bank to earn some dime, when the bond or other money making tools just too expensive or risky to them.

The speculation follow by the question are mostly flaw, perhaps IMHO, those are fallacy impost by media.

In real life.
 1. i. Moving money is costly  ii. it is difficult to find a safe haven that is cheap.
 2. In fact, the move is just pragmatic (Beware that pragmatic doesn't mean it is logical or good decision)

And to OP question title : What stops a government in partnership with its central bank from lowering interest rates and then borrowing at this lower rate to fund investments?

Interest rates DOESN'T SOLVE the problem of money supplies.
Central bank only control money supplies, it doesn't do the "borrowing"
Government borrow money by printing bonds. HOWEVER, if too much bonds are printed and too much money are used to buy the bonds, it will affect liquadation of money. So there is always some "money printing" (AKA inflation) action follows.


How are robocallers able to make international calls profitable? I have read that robocallers are untraceable because they are usually companies in India or the Phillipines.

I don't understand how this is possible. If I call India or the Phillipines, it is very expensive, a dollar or two per call. If the robocallers are paying this kind of money for each call, I don't see how that could be profitable, considering how many hangups they get.

Are foreign companies getting some cheap way to call into the US so that the pricing of international calls is highly asymmetric?

The short answer is yes

In general, technology today allows to transfer calls between telephone switches, over the internet, at zero cost (with only the fixed broadband cost), using SIP protocol

So one of the main reasons for international calls to cost differently between countries is the internal market structure of each country

Internal market structure includes in many cases operators interconnect costs or wholesale costs, regulated or not. It can also include taxes.

Calls to the US are extremely cheap
So in theory, one can download a SIP program and connect to any other person with a SIP program. All that is left is connecting to an international calls provider, and make calls...