What are the Impacts on an economy when a large mining venture begins to reduce production and lay off workers in a remote region? I am currently unsure about how such a question above should be answered. To clarify , I am confused as  of which topics relating to the question i should discussion in order to answer it. (For example) Fewer jobs in the region means fewer small business due to poor economic conditions which impacts the region's long term trade.

Does any with experience in the economics have an idea of how such a question is to be answered?

It depends on what they mine there and how big are the layoffs. Boomtowns are usually formed when they set up mines in remote regions. After the mines are depleted or become economically non-viable, workers leave and the boomtown becomes a ghost town. Since the economy of such places heavily relies on income from mining, it cannot survive once the money stop flowing. You can read up on the history of boomtowns and ghost towns on wikipedia to see if that is what you are looking for.

Boomtowns:
https://en.wikipedia.org/wiki/Boomtown

Ghost towns:
https://en.wikipedia.org/wiki/Ghost_town


Did/does a decline in crime rates encourage materialistic values? Robert Shiller writes (2000 Irrational Exuberance, 2015 3rd edition):

  A decline in crime rates also, on the flip side, encouraged materialistic
  values by making people feel more secure, less worried that they would be
  robbed or physically harmed. In the United States, the rate of property crimes
  per 1,000 people fell 49% between 1993 and 2003, and the rate of violent
  crimes per 1,000 people fell 55%. One could then more comfortably flaunt
  wealth, and so wealth became more attractive. Living in an ostentatious
  home was now more appealing. Fear of terrorism increased, but terrorists
  did not seem to strike wealthy people preferentially, and generally not in
  their homes. The declining crime rates in the U.S. made a capitalist lifestyle
  seem a better model for the entire world.

These are interesting claims but Shiller does not seem to offer anything to back them.

I wonder if there has been any actual research showing that "a decline in crime rates ... encourage[s] materialistic values by making people feel more secure"?

Did/does a decline in crime rates encourage materialistic values?

  These are interesting claims but Shiller does not seem to offer
  anything to back them.

Much of it sounds quite logical, whence I would not expect the author to present evidence on that. But, despite my consistent preference for neoliberalism, I think the author's conclusion is unverifiable and is rather an opinion that leaves several gaps unaddressed.

People seek and compete to impress and to be admired. That is part of our nature. For that purpose, some people develop their talent & skills, others --regrettably-- get tattoos, and others who can afford it show off goods and services that they think will make them stand out. Unlike a talent or a tattoo, objects such as jewelry, cars, or the latest iPhone can get stolen or at least vandalized. Thus, it is entirely reasonable that a generalized perception of safety will make the materialistic sector uninhibited to go out and engage in a materialistic competition.

Likewise, the author's remark on terrorism follows almost from definition. A terrorist's targets are those he views as strategic to his political motive: typically public figures, and/or masses whose support of a regime or institution would weaken upon terrorist acts. By contrast, targeting a merely wealthy person or household would not significantly impact the public's mindset in a way that advances the terrorist's purpose.

But, as I mentioned above, the author's conclusion that

  [t]he declining crime rates in the U.S. made a capitalist lifestyle seem
  a better model for the entire world

is hardly susceptible to verification, research, or consensus. That conclusion entails multiple facets and subjectivity gaps that the author apparently did not address.

For instance, Marxists will disagree with the author's conclusion by portraying the state as the mechanism specifically devised to preserve the elite's capitalist lifestyle. In short, Marxist will refute any link between "declining crime rates" and "capitalism as a better model".

Likewise, an analyst will dispute the adequacy of extrapolating a trend of "declining crime rates in the U.S." for reaching conclusions about what exactly would be "a better [economic] model for the entire world" (brackets and emphasis added). Much of the capitalistic lifestyle in the U.S. relies on its indebtedness, whereas already in the 1920s Hjalmar Schacht was concerned about how the capitalistic lifestyle (perhaps frenzy) in Germany was undermining the German economy.

An environmentalist, a Catholic priest, or a Buddhist monk will refute the notion that materialistic values are any beneficial for a single country --let alone for the entire world--, no matter how the idea of "declining crime rates" is slid in the argument.


Substitutive or supportive factors of production Given the function:
$y=A K^\alpha L^\beta, A,\alpha,\beta >0$

$y=aK+bL^{0.5}, a,b>0$

Decide whether capital and labour are substitute or supportive factors of production. How to solve this question? To answer your possible questions i want to highlight the fact that this is neither my homework assignment nor anything i need to submit, furthermore i have simply no idea how to answer this question that is why i am not able to show any attempt to solve this question.

EDIT: My ideas where to somehow look for the slope of MRTS curve if negative this will imply perfect substitutes if not existing then theoretically it should imply supportive factors, however the answer key states that:

if $\frac{\Delta MP_L}{\Delta K}>0 \rightarrow$ supportive

if $\frac{\Delta MP_L}{\Delta K}<0 \rightarrow$ substitues

if $\frac{A\beta K^\alpha L^{\beta-1}}{A\beta \alpha K^{\alpha -1} L^{\beta-1}}>0 \rightarrow $ subsitutes.

And here i am utterly lost.

Let's find the Marginal Product of Labor, $MP_L$:
$$
\begin{align}
MP_L \equiv \frac{\partial Y(K,L)}{\partial L} &=\frac{\partial }{\partial L}\left( aK + b L^{0.5} \right) \\
&= 0.5\cdot b \cdot L^{-0.5}
\end{align}
$$
Now finding $\frac{dMP_L}{dK}$ is trivial:
$$
\begin{align}
\frac{dMP_L}{dK} &= \frac{d}{dK} \left( 0.5\cdot b \cdot L^{-0.5}\right) = 0
\end{align}
$$
We see that it does not depend on the capital. So a change in $MP_L$


UK GDP accounts for profits on sale of home A house is built and sold to a family as their primary residence for £1000 in 2000. The purchase price is included in GDP for 2000. In 2001 the house is sold for £2500. The seller pays solicitors' and agents' fees of £200. This is included in services part of GDP for 2001. He pockets £300 profit on the sale. Is this profit part of GDP?

I think not, but can find no authoritative answer.

The $£2000$ appears in GDP in 2000, as production (less intermediate goods consumed) and as value added (the wages of the builders and profit of the developer) and as expenditure (capital formation or investment by the original buyer).

The $£2500$ actually does appear twice in 2001, once positively (by the new buyer) and once negatively (by the seller), as fixed capital formation in the expenditure measure of GDP.

So in aggregate this cancels out and does not add to GDP in 2001 (correctly so, as no value has been added in 2001 - equivalently there has been no production in 2001), but might appear in more detailed elements of the national accounts, for example if the new buyer was an individual while the seller was part of government.


US trade outpaces GDP I notice that US exports doubled between 2000 and 2011, imports also increasing greatly, but GDP only grew by 20%. Thus, trade in the United States seems to have grown substantially faster than production. Why is this?

The fact that "trade in the United States seems to have grown substantially faster than production" could be explained by reduction in trade costs. For instance, a decrease in importing partners' tariffs tends to favor US exports. So, for the same level of production or GDP, US sales will be more export oriented. It works similarly for US imports following a reduction of US tariffs.

A first explanation to the reduction in trade costs is the reduction of tariffs and non-tariff barriers (such as administrative fees or quotas), that is changes in trade policy. Trade liberalization policies could be unilateral, multilateral, regional or bilateral. Regional trade agreements for instance have become increasingly prevalent since the early 1990s, according to the WTO.

A second explanation to the reduction in trade costs is related to technological progress such as faster communications.

Note that it is common to compute a country's trade openness index as the ratio of a country's trade (exports plus imports) to its GDP. Controlling for GDP allows economists to evaluate the outward or inward orientation of a given country's economy. Moreover, this ratio could be computed over time to measure the extent of globalization. For instance, according to the World Bank data Chinese trade openness ratio raised from 2.6% in 1970 to 39% in 2006. The US ratio increased from 9.7% in 2001 to 13.5% in 2011, which is consistent with the facts you mentioned.


Existence of optimal strategy in a choice problem with uncertainty and information structure Consider a decision maker choosing an action, $y$, from the finiteset $\mathcal{Y}$, possibly without having complete information about the state of the world.

More precisely, let $V$ be a continuously distributed random variable (or vector) representing the state of the world, with support, $\mathcal{V}$. Let $P_{V}\in \Delta(\mathcal{V})$ be the probability distribution of $V$.

Nature draws a realisation, $v$, of $V$ from $P_{V}$. The decision maker is not aware of $v$. However, she can refine her prior, $P_{V}$, upon reception of a private signal which may be informative about the state of the world. In particular, let $T$ be a random variable (or vector) representing the private signal received the decision maker, with support $\mathcal{T}$. Let $P_{T|v}\in \Delta(\mathcal{T})$ be the probability distribution of $T$ conditional on $v$.
Nature draws a realisation, $t$, of $T$ from $P_{T|v}$. The decision maker observes $t$, uses the Bayes rule to update $P_{V}$ with the posterior $P_{V|t}\in \Delta(\mathcal{V})$, and chooses an action, $y\in \mathcal{Y}$. Finally, the decision maker receives a payoff, $u(y, v)$.

Let us now define an optimal strategy of the decision maker.

A (mixed) strategy is $P_{Y|T}\equiv \{P_{Y|t}\in \Delta(\mathcal{Y}): t\in \mathcal{T}\}$,  collecting probability distributions of  $Y$ conditional on every realisations $t$ of $T$.

A  strategy $P_{Y|T}$ is optimal if it allows the decision maker to maximise her expected payoff:  $\forall t\in \mathcal{T}$,  $\forall y\in \mathcal{Y}$ such that $P_{Y|t}(y)>0$, and  $\forall \tilde{y}\in \mathcal{Y}\setminus \{y\}$
$$
\int_{ \mathcal{V}} u(y,v) P_{T|v}(t)  P_{V}(\text{d}v) \geq \int_{ \mathcal{V}} u(\tilde{y}, v) P_{T|v}(t) P_{V}(\text{d} v;\theta_v). $$

Question: does an optimal strategy always exists for any information structure $S\equiv (\mathcal{T}, P_{T|V})$? Could you make an example on non-existence, if any?

Yes, an optimal strategy always exists for any information structure.

Proof: 1st- the set $\mathcal{Y}$ is finite (thus compact). 2nd- for each state of the world, $v$, $u(y,v)$ is continuous in $y$ using the discrete metric. 3rd- Given the linearity of integrals, for each $t\in\mathcal{T}$ we have that $\int_{\mathcal{V}}u(y,u)P_{T|v}(v)P_V(dv)$ is also continuous in $y$. Therefore Weierstrass theorem ensures the existence of a minimum and a maximum.

Let $y^*_t$ be one such maximizer when the signal is $t$ (note it can be different for each signal realization, and that the maximizer might not be unique. In the latter case, I arbitrarily choose one for simplicity). Finally, define $P_{Y|T}(y|t)=\mathbb{1}\{y=y^*_t\}$. I.e. to play $y^*_t$ with probability one when the signal is $t$.

In this setup, existence might only fail if $\mathcal{Y}$ was not compact, or if $u(y,v)$ was not continuous with respect to $y$. Note I said might, even if these two conditions are not present, an optimal might still exist.


In what kind of economic environment would the cliche of "keeping money under the mattress" be warranted? I'm trying to apply (ideally empirical-backed) economic theory to an age-old cliche.

We must be very articulate about the assumptions, because if it's a total collapse of the dollar, then that money would just be as useful as tinder. Maybe it would be warranted if there was a run on the banks but the dollar still retained value. It's a difficult call though.

Question

What must we assume in terms of underlying economic conditions for someone with a giant pile of cash under their mattress to be favored over someone with their savings in a bank?

A fractional-reserve bank typically creates a lot of value by lending out your money to people while you are not using it. If you decide to take out your money on any given day, your dollar might not be there, but if not, someone else’s is! The bank will give you their dollar and wait for yours to come back from a loan.

However, during a bank run, many loans are going to default. So your dollar might not make it back, and if you’re not quick, you can’t take out someone else’s dollar cause they already got it back. Therefore your money could disappear.

No one can come into your house and take your dollar out from under your mattress though...


Why should small open economies avoid interest rate policies? I've been taught that small economies open to capital flows should avoid manipulating interest rates because increasing interest rates will mean capital inflow which causes currency appreciation. Overall, interest rate policy causes exchange rate volatility. But why is the case for small economies? Why is the US or China less susceptible to this theoretical argument?

The trilemma or impossible trinity is said to be that it is impossible to have all three of the following at the same time:

a fixed foreign exchange rate
free capital movement (absence of capital controls)
an independent monetary policy

The United States prioritises the second and third, and only worries slightly about the first and its impact on trade.  Being an extremely large economy, capital inflows can only make a small difference to domestic market conditions, though during the world financial crisis some moments of fear did tend to push the relative price of the dollar up as international money sought a safe haven (abbreviated to risk-off sentiments). Currently it seems that the US can make moderate increases to interest rates in contrast to Europe or Japan, without a major impact on US exchange rates

China prioritises the third and also the first in a managed exchange rate sense.  There are significant capital controls in China (more now than a few years ago), and they make it difficult for short-term capital flows to cross the border in either direction.  If somebody abroad wants to buy yuan-denominated shares or bonds, they will find it much easier in the rather peculiar markets in Hong Kong than in the domestic markets in Shanghai and the two often disconnect with significantly different prices for similar investments. This enables the Chinese government to pursue a domestic monetary policy without too many international concerns

A small open economy cannot follow the US example (being too small) or the Chinese example (being too open).  And in many cases smaller countries are more affected by international trade than similarly structured larger ones. So the exchange rate implications of changes in domestic monetary policy are more important to small open economies


Would a society of cannibals be self-sustaining? Bit of a weird one, but here goes. Say there is a human society whose diet is 100% the human flesh of their own members. Would this society be able to maintain replacement population, or even shudder grow? I suppose figuring this out would require daily caloric requirements, caloric value of human flesh, length of gestation, average age and a host of other variables, but is there some economic principle to short-cut to a solution here? Apologies if you've been sick.

No, an animal population that subsist solely by consuming its own members is not sustainable.  This is easy to see because there's no external energy input into the population (all energy comes from other members of the same population), yet the individuals will necessarily expend some energy in their metabolism (turning it into heat).  Eventually, if the population cannot find any other food sources, the individuals will die (and be consumed) one by one, until the last one will run out of food and starve.

That said, there are animal populations in nature where some of the individuals may subsist almost entirely on cannibalism.  This is typical of e.g. fish, which are very tiny when born, grow gradually over their lifetime, and tend to consume prey smaller than themselves.  Thus, in small lakes that happen to host only one species of fish, it's not uncommon for the large adult fish to mostly eat smaller juvenile fish of the same species (which in turn eat plankton and invertebrates, providing an external source of energy to the system).


Are there data for the evolution of global economic inequality Please excuse my ignorance of economics. I recently learned about the Gini-Coefficient as a measure of the economic inequality in a country. I have since seen several plots of this coefficient over time, each for an individual country. The following is from Equality trust and concerns the UK.

Another plot, concerning various countries, is found here:

At first sight, it appears impossible, to construct such a curve for the entire world. Yet, in discussing extreme poverty,  I recently learned that incomes can be adjusted to equal purchasing power. Naively it seems, one could measure economic inequality globally. So, are there such data concerning the world as a whole? I know there are maps with countries colour coded by Gini, but this is not what I mean.

My interest is not limited to Gini, if there is another measure, by which the global temporal evolution of inequality can be captured, please tell me!

It is possible, but it involves consiberable work as you need to collect underlying income distributions across the world or to be able to estimate them.

This has been done, for example in a World Bank policy research working paper 6719, Global Income Distribution From the Fall of the Berlin Wall to the Great Recession by Christoph Lakner and Branko Milanovic in December 2013. It included this chart of global income distributions:

In fact the real story here is the reduction in the number of very-low-income people in absolute terms, especially in countries such as China over 20 years, plus the rise of middle incomes (in a global sense).

It is possible to use this to estimate Gini indices and the paper gives the following estimates:

1988: 72.2
1993: 71.9
1998: 71.5
2003: 71.9
2008: 70.5

Points to take from these numbers are

global inequality is wide with a Gini index higher than almost all countries or regions
the numbers suggest a global reduction in inequality even when inequality had been increasing within most major countries (especially China)
the reduction has been small and can easily reverse (either in the future or on re-estimation of the past)

Regional trends and other inequality measures are in Table 3 of the paper, while other people's estimates of a global Gini index are in the Appendix Table 1.


Applying modern economics to Edgeworth's Hedonical Calculus Preface
The late 1800s was a genesis for many economic theories. As math spilled over into the social sciences, one of the more obscure applications of measurement was: Hedonical Calculus, developed by Edgeworth. Of course, centuries later, some of these Victorian Age studies aged better among modern academic communities than others. Hedonical Calculus gained little traction and faded to the periphery as modern economic thinkers steered the field away from quantifying everything. On the notion that if one cannot measure something one does not have a satisfactory understanding of the matter, Frank Knight sarcastically interjects:

If you cannot measure a thing, go ahead and measure it anyway.

 Hedonic Calculus
The work tackles several broad areas of study, and to keep my question within a reasonable scope, I will limit the focus to capital formation / efficiency. With that in mind, I will try to unpack the basics of Hedonical Calculus:
Objective: To find  the distribution of means ($α$) and labor ($β$), the quality ($γ$) and
number of population (δ), so that there may be the greatest possible happiness

Pleasure: The term includes absence of pain. Greatest possible happiness is the greatest possible, integral of the differential: Number of "enjoyers" × duration of enjoyment × degree thereof
Means: are the distributable proximate means of pleasure, chiefly wealth as destined for consumption
An individual has a greater capacity for happiness than another, when
for the same amount whatsoever of means he obtains a greater amount
of pleasure; and also for the same increment (to the same, amount)
whatsoever of means a greater increment of
An individual has more capacity for work than another, when
for the same amount whatsoever of work done he incurs a less amount
of fatigue, and also for the same increment (to the same amount)
whatsoever of work done a less increment of fatigue.

With this framework, Edgeworth proposes we may allocate human capital in such a way that we minimize pains of production.  Pains of production may include blood and sweat, but could be negative if we suppose the thrill one may get from being a pilot (especially during the first few flights). Formally, by finding the largest possible value of:
$$\int_{x0}^{x1} n\left [ F(xy)-p \right ]dx$$

n = number of each section
F(xy) = a unit's pleasure of consumption (x=capacity for pleasure, y=means for living up to his capacity)
p = pains of production

Fully aware of the black box limitations and pitfalls of attempting to quantify the sentiment and decisions of a human being capable of exerting his/her own free will, I am still very curious about the impact on long-term capital formation and efficiency of such a system
Question
Assuming complete omniscience on measurements such as "units of pleasure" and "pains of production," would modern economic theory expect differences in long-term capital formation from a pleasure maximizing economy than that of a profit/utility-maximizing economy of today (or whatever you want to call our economy)? If so how, if not why?
Reference:
https://babel.hathitrust.org/cgi/pt?id=njp.32101063551350&view=1up&seq=414

Hedonics comes from Benthams utilitarianism. The main problem with reifying this into mathematics is that these are not measurable. Instead, one develops proxies for such, which I'd suggest the hedonics is and it suffers from the same problem.
Whilst it's easy to say that six apples are six apples and there are no two ways about it, it is much more difficult to measure 'a happiness index' in eating apples: someone may have an allergy to them, or perhaps they had a bad experience with them.
It reminds me of the Social Realism of Stalins Soviet Union. Official pronouncements of the happiness of the people whilst the people were suffering. Of course, Capitalism has its own Capitialist Realism - just look at adverts - they are the analogue to this art: happy consumers being 'overjoyed' in recieving new products. There's a good reason when many happiness surveys show that the people aren't as happy as one might think ...


All-Pay Auction Mixed Strategy Equilibrium I am currently struggling with this exercise.
Professor Nash announces that he will auction off a 20 dollars bill in a competition between two students chosen at random. Each student is to privately submit a bid on a piece of paper; whoever places the highest bid wins the 20 dollars bill. In the event of a tie, each student gets 10 dollars. The auction is all-pay: each student must pay his/her bid regardless of who wins the auction.

HERE is the question

Suppose each student could borrow money from the other students in the class, so that each of them had a total of $11 to bid. Find a Nash equilibrium in this case?
I know I have to find a mixed-strategy  nash equilibrium but I don't understand how I can't compute it.

Thanks

In response to our policy on homework questions, I will work on the following abstract version of your question:

A single prize is to be allocated by an all-pay auction. Assume the value of the prize is 1 to both bidders, and this is common knowledge. Both bidders have a budget constraint $m$. If there is a tie, each bidder receives the prize with $\frac{1}{2}$ probability. Find a Nash Equilibrium in this game.

We shall focus on the symmetric equilibrium in this game, that is, both bidders are using the same equilibrium strategy. Before we get started, let's make two observations:

Claim 1: Whenever there is a mass point in the equilibrium strategy (i.e. there is positive probability to submit one bid), this mass should be put on the upper bound $m$. For if not, one bidder can always deviate to a slightly higher bid and obtain higher surplus.

Claim 2: Whenever a bidder randomizes, the bid should be uniformly drawn from $(0,t)$, and the probability that this happens should be $t$. This is because only in this case, a bidder is indifferent between any bid $b\in(0,t)$:
$$\text{Expected payoffs}=t\cdot \frac{b}{t}\cdot 1-b=0.$$

Therefore, we can guess that the equilibrium strategy takes the form
$$t\cdot \text{Uniform}(0,t)+(1-t)\cdot\delta_m.$$
That is, with probability $t$, a bidder submits a bid uniformly drawn from $(0,t)$, and with complementary probability she submits $m$. The only thing left is to pin down $t$. Note that if this is an equilibrium strategy, a bidder should be indifferent between any bid $b\in(0,t)\cup\{m\}$. Therefore, the payoffs from submitting $m$ should also be 0, i.e.
$$t\cdot 1+(1-t)\cdot\frac{1}{2}-m=0,$$
which implies $t=2m-1$. (Notice that for the existence of such an equilibrium, we must have $2m>1$.)

Therefore, a Nash Equilibrium in this game is that both bidders play the strategy
$$(2m-1)\cdot \text{Uniform}(0,2m-1)+(2-2m)\cdot\delta_m.$$

How does it map back to the original problem? I believe the OP can take it from here.


Find all of the Pure and Mixed Strategy Nash Equilibria When I do the basic calculations for mixed probability, I get that the Column player always plays B. However, I am getting a negative probability for the row. Any help is appreciated.

I assume that you are calculating a mixed strategy for the row player so that the column player is indifferent between choosing $A$ and $B$.  And the reason why you get a negative probability is that the row player cannot make the column player indifferent by choosing, say strategy $C$, with a positive probability $\in [0,1]$.  This is because the column player (always) strictly prefers $B$.

  Side-note: To avoid such confusion in the future, try to see whether you can apply math to calculate a mixed strategy to a game.  Here you clearly cannot (because of the above argument), and that is why math is giving you a ''weird'' answer.  Namely, math yields an answer to a question: What would be a hypothetical probability with which the row player chooses, say, strategy $C$ to make the column player indifferent?


Sequential Game Rollback Equilibria Question For this question, I believe Albus has 3 strategies: N, E, and S, and Minerva has 8: aaa, aab, aba, abb, baa, bab, bba, and bbb.
I've also found the rollback equilibrium to be the case in which Albus chooses N and Minerva chooses b. However, I'm not sure how to represent this equilibrium (equilibria?).

Do we say that the unique equilibrium is simply (N, b), since those are the choices made by the players, or do we say there are 4 equilibria: (N, baa), (N, bab), (N, bba), (N, bbb) since Minerva's choices are in reality a set of strategies, all of which result in her choosing b in response to Albus playing N?

You're correct on the definition of the strategies: Albus' strategy set is $S_{A} = \big\{N,E,S\big\}$, and Minerva's is $S_{M} = \big\{a, b\big\} \times \big\{a, b\big\} \times \big\{a, b\big\}$.

Remember, a strategy is a contingent plan that prescribes an action at every possible node at which a player plays.

Then, the unique equilibrium (via Backward Induction) is $(N, b, a, b)$. The others you mention are not equilibria, because they involve incredible threats, e.g. if Minerva found herself at the decision node following a choice of $E$ by Albus, she would never choose $b$.


Post Brexit Referendum employment theory According to Chuka Umunna, Labour MP for Streatham and prominent pro-EU campaigner:

  People were told: "if immigration levels fall, it will be easier to
  get a job, and easier to get a job that's paid well." Since 2016, net
  immigration from the EU has fallen. Yet this indicator has gotten
  worse.

He did not cite which study the numbers came from, but let's just assume it's true. From a purely supply / demand perspective it does seem counter-intuitive, but it could be more nuanced and complicated than that.

My personal best guess was: this result could be from a lagged effect of all the previous immigration up until then. With the red tape and time needed to immigrate, settle down, get a sponsored visa, find a job, it may take months or years for immigration levels to have a measurable influence on jobs. I first envisioned conducting a lag / lead regression analysis to test my hypothesis, but have since redirected my interest in theory.

Question: Is there anything in existing economic theory that could explain a fall in immigration correlating and/or causing higher difficulty in finding employment? (note I use and/or for causality) And related, what other empirical findings outside the UK support Umunna's claim?

Further Clarification:

"easier to get a job": I am not quite sure how to quantify this section of Umunna's claim, I'm guessing he meant: average number of months unemployed before finding a job, or perhaps he meant: average availability of jobs in said sector. In your answer, explain which interpretation you use.

Is there anything in existing economic theory that could explain a
  fall in immigration correlating and/or causing higher difficulty in
  finding employment?

In addition to the lagged effect you outline, there are at least two possible causes for the paradox you describe. These theoretical possibilities are independent of which interpretation of unemployment is adopted. I'm unaware of (and not even searching for) statistical evidence of whether this actually occurs.

A reduction of immigration could also entail a reduction of demand of goods and services. If there's no demand, some businesses are likely to reduce their staff, leading to an oversupply of labor (i.e., unemployment).
High-skilled immigrants might boost the industry in a different country to an extent that renders that foreign economy more competitive than the domestic economy. A significant trend in that direction could prompt domestic companies to slash jobs or even to relocate to one such country where supply of high-skilled labor is stronger. That would lead to the unemployment of low-skilled workers in the domestic labor market.

As for 2., it is reasonable to say that --almost by definition-- the low-skilled workers outnumber high-skilled ones.


How to calculate payback period of the project? In order to finish a technological product, it was spent 50000 dollars (for materials, salary). 500 dollars per month is needed for support of the project (will be spend for servers). This product automated some process that required manual work for 10000 dollars per month. So, how to calculate payback period of the project?

My solution would be 50000:10000 = 5 month, but I don't know what to do with the cost of support per month.

This is more finance than economics but here it is:

The payback period is pretty much how much time you need to recover the initial investment.

Each month, you are saving 10000 - 500 = 9500.  Then you simply just divide 50000 by 9500. This comes out to be around 5.

I made some simplifying assumptions: Discount rate is 0. This is usually not true, so you would probably have to discount the cash flows. Note: This will probably involve monthly compounding. Check out the formula for annuity.

Also, just FYI: payback method often results in wrong decisions. When you have a choice, always use NPV.


Weak axiom of revealed preference and choice coherence - how to show they are equivalent $B$ and $B'$ are elements of the family of subsets of $X$

WARP
For every pair $x,y \in B \cap B' $ and if $x \in c(B)$ , then if $y \in c(B'), x$ must $\in c(B').$

Choice Coherence
For very pair $x,y \in B \cap B'$ and if $x \in c(B)$ and $y  \notin c(B)$, then $y$ must $\notin c(B').$

Are these two equivalent. If yes, how can we prove it?

Yes, they are equivalent. Here is a formal proof by contradiction.

WARP $\Rightarrow$ Choice coherence

Suppose that WARP holds but that choice coherence is not true. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B)$, $y \notin c(B)$ and $y \in c(B')$.

But WARP applied to the conditions $y \in c(B')$ and $x \in c(B)$ implies $y \in c(B)$. This is a contradiction.

Choice coherence $\Rightarrow$ WARP

Suppose now that choice coherence is true but that WARP is falsified. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B), y \in c(B')$ and $x \notin c(B')$.

But choice coherence applied to the conditions $y \in c(B')$, $x \notin c(B')$ yields $x \notin c(B)$. This is a contradiction.


Policy to maintain cleanliness In countries where littering is a problem and the regulations and rules alone cannot give the desired effect due to vary high population,large surface area,etc.. ,will it be effective if the government starts buying garbage on a price per kg basis?
(This money spent can be compensated by reducing the number of sanitary workers and other departments if this method proves effective)

How effective it will be in the end depends on how the policy is shaped and enforced exactly.

Buying garbage is in fact the same as subsidizing people to pick-up the garbage. In theory, by setting the subsidy at the appropriate level (where the marginal damage of an additional kg of litter on the street equals the marginal cost of picking up that garbarge and handing it in) one would achieve the "optimal amount of litering".

I do see a problem though,especially for the longer term. If people get paid for handing in their garbage, there are no incentives to reduce the overall amount of garbage, in fact people might be tempted to buy products that come with extra packaging so they can cash in on the garbage.

I think a better solution in the long term would be a tax on material that is likely to end-up on the streets.


Is the minimum of short run average cost equal to the minimum of long run average cost? I understand that for perfect competition, the price is equal to minimum short and long run average cost in the long run as there cannot be any supernormal profits.

Does this mean that the short run and long run average cost at their minimum are equal for perfect competition in the long run? My textbook states that they are "tangential" to one another, but shouldn't they be equal if there are no supernormal profits?

A wikiedpia article states one STC curve is tangent to LRTC at the long–run cost–minimizing level of production. At the point of tangency LRTC = STC. At all other levels of production STC will exceed LRTC. Why is STC equal to LRTC at this point and not at others? And does this also refer to average costs?

My textbook states that they are "tangential" to one another, but shouldn't they be equal

The cost curves are tangential (at least the ones the unnamed book seems to be talking about), and cost levels are equal given the optimal production level.


First Order Stochastic Domination and lottery preferences Let $L$,$L'$ be two lotteries over the real numbers. Let $u$ be an increasing Bernoulli utility function. Let $F_L$, $F_{L'}$ be the CDFs of the two lotteries. We wish to show that
$$L \succ_{FOSD} L' \iff \mathbb E(u(L))\geq \mathbb E(u(L')).$$
I am able to prove the necessity. I am not able to prove sufficiency. I get an expression like $\int u'(x) f_L dx \leq \int u'(x) f_{L'} dx$ after integrating by parts, but am unable to proceed after that.
This is not a homework problem.

Suppose $L \succ_{FOSD} L'$ and let $L$ have CDF $F$ and $L'$ have CDF $G$.
That is, let
$$F(x) \leq G(x) \quad \forall x \quad \mbox{or} \quad 1-F(x) \geq 1-G(x) \quad \forall x. $$
Now consider two lotteries, one is $L'$ and the other one is $L'$ but instead of paying out $x$ it pays out
$$y(x) = F^{-1}(G(x)) \geq x,$$
where the inequality follows by FOSD. That is, the lottery always pays out more. So everyone with an increasing utility function (more is better) prefers the second lottery. How are the payoffs from this lottery distributed?
$$Pr(y(X) \leq y) = Pr ( F^{-1}(G(X)) \leq y ) = Pr ( X \leq G^{-1}(F(y)))=
G(G^{-1}(F(y)))= F(y).$$
Thus,
$$\mathbb E[u(L)]= \int u (y(x)) d F(y(x)) = \int u (y(x)) d G(x)
\geq \int u (x) d G(x) = \mathbb E[u(L')]$$
for any increasing $u$ by the $y(x)\geq x$.

The other direction would be that $\mathbb E[u(L)]\geq \mathbb E[u(L')]$ for all increasing $u$ implies
$L \succ_{FOSD} L'$. Let's show the equivalent statement that if $L$ does not FOSD $L'$, then there is at least one increasing $\widetilde u$ such that  $\mathbb E[\widetilde u(L)]< \mathbb E[\widetilde u(L')]$.
If there exists some $z$ such that $1-F(z) < 1-G(z)$, then there is the increasing utility function
$$ \widetilde u (x) =\begin{cases}
1 \mbox{ if } x> z\\
0 \mbox{ if } x \leq z \end{cases}$$
such that
$$\mathbb E[\widetilde u(L)]= 1-F(z)  < 1-G(z) = \mathbb E[\widetilde u(L')].$$
That's it.


International Capital Asset Pricing Model (CAPM) I wanted to value a High-tech start-up of which I have the cash flows of the coming 6 years. I decided to use the Discounted Cash Flow Method. To do so, I calculated the Discount rate.
For a High-tech start-up the discount rate  = Rate of equity (which can be determined by the Capital Asset Pricing Model).
So I calculated my Discount rate, used it in the DCF model And found an approximation of the Value of the High-tech Firm.

But I didn't took in consideration that the company sells their products globally. This fact has an impact on its beta Factor! and consequently on the discount rate and finally on the Value of the firm!

One option is to use the International CAPM model, but 'my' company is going to sell globally, so how do I know the impact of that to the Discount rate?

When using DCF you would discount using the weighted average cost of capital; and yes you can then find the expected return on equity using CAPM - if you have beta. But how do you know what beta is in your case?

That being said it seems like you are just using DCF for the 6 years where you know the cash flow? Will the company seize to exist after that? If not, you need to discount all future cash flows as well making assumptions on how income/profit will continue in the future (usually a constant growth rate from e.g. Year 10-15 and into all eternity).


Game theory with rational and irrational players Are there game theoretic models where one player is rational and the other is irrational (i.e., plays with behavioral limitations)?

The motivation for this question is that behavioral economics has documented how people often make choices that don't maximize expected utility, but those irrational people might be interacting with near-rational computers or organizations. I've never seen a game theory model that has both an expected utility-maximizing player and a behavioral player.

Yes, a whole book has been written on Behavioral Game Theory.

More specifically, standard solution concept such as Nash equilibrium requires that players best respond to a correct belief about other players' moves. The following are examples that relax one of these cognitive restrictions:

Quantal response equilibrium allows for the possibility that players do not always choose a best response. Instead, they "quantal respond" (or best respond with mistakes) to a correct belief about others' quantal response.
Level-k or Cognitive Hierarchy Theory relaxes the correctness of beliefs assumption, and allows players to best respond to a set of mutually inconsistent beliefs.
Psychological games allow players to derive utility directly from beliefs (and belief-dependent emotions). Relevant solution concept include sequential reciprocity equilibrium and Rabin's fairness equilibrium.

For a complete review of models of behavioral game theory, see Chapter 13 of Dhami (2016).

For a specific example of a game with mixed players, see this paper on robust protection of fisheries. A more comprehensive treatment of mixed games is available in Rani Spiegler's Bounded Rationality and Industrial Organization per Michael Greinecker's comment on the original question.


How much interest rate would make his earnings the same A rural landowner can deposit his savings in a commercial bank and receive an annual interest rate of 8%. Alternatively he can lend to
villagers who need credit. If all loans are of the same size and only sixty per
cent of them are repaid, the how much interest rate would make his earnings the same as from depositing his savings in a bank?

I took the case that suppose landowner has 100 dollars and he is landing to one villager for one year, then if he deposits in bank for a year he gets 108 after one year, but if he gives it to villager, he gets 60 dollars back as given and to compensate he must have charged 48%, so that total is 108 dollars. But answer given for this question is 80%? Why is that?

I don't know how you got 48% (and it would have been appropriate to include your work in your question), but think about it intuitively. Take your second to last step. If the landowner lends to the villagers and on average gets 60 dollars plus interest back, then if the interest rate was 80%, you'd get 60 + 48 = 108 dollars, the same as depositing to the commercial bank.

Algebraically, here's how you'd set up solving for $r$:

$$(100 \cdot (1+r))0.6 + 0(0.4) = 108$$

Where you have a 60% chance of being paid back 100 with interest. Then you just solve.


reference request - dynamic discrete time optimization methods Next semester I am taking business cycle theory course. I emailed my teacher and he replied that I need firm understanding of dynamic discrete time optimization methods. I am gonna study mainly from Sydsaeter / Hammond Further math. Yet I even need to catch up with static optimization, but I have Winter break.
My question is, could you please recommend me some good sources (net / textbooks) for dynamic discrete time optimization methods? Rather introductory. Thank you

As suggested by 1muflon1:
I don't really use dynamic discrete optimization (yet) in my work, but I did explore several books once. Here are some suggestions:

Economic Dynamics: Theory and Computation by John Stachurski: It's very easy to fall in love with this guy. He writes casually yet authoritatively. The book covers required real analysis and includes python codes and exercises.

If you like the above book do explore https://quantecon.org/lectures/

Recursive Methods in Economic Dynamics by Stokey, Lucas and Prescott


Finding a nash equilibrium in pure or mixed strategies The question I'm dealing with:

  Let $x=3$, find any Nash equilibrium in pure or mixed strategies.

I have observed that the pure strategy Nash equilibrium is $(u,r)=(3,3)$.

The answer to this question is:

$(\pi^*,\rho^*)=(1,0)$

Am I correct in interpreting the above as:

Given the PSNE of $(u,r)$, the row player will play $u$ with probability $1$ and the column player will play $r$ with probability $1-0=1$?

Thanks

Once again, thanks to denesp (and Herr K.) for the help. I thought that I'd try to answer the question just for reference purposes, and I guess that it's also a step toward improving the "answers per question" metric on Economics SE.

As aforementioned, the PSNE is $(u,r)=(3,3)$.

Player 2

  If $P1$ plays $u$, $P2$ can play either $l$ or $r$.
  $l$ would give $P2$ a payoff of $2$ whilst $r$ would give $P2$ a    payoff of $3$.
  Therefore, $P2$ will play $r$ with probability $\pi=1$

Player 1

  If $P2$ plays $r$, $P1$ can play either $u$ or $d$.
  $u$ would give $P1$ a payoff of $3$ whilst $d$ would give $P1$ a    payoff of $2$
  Therefore, $P1$ will play $u$ with probability $1-\rho=1$, $\rho=0$.

The result

$(\pi^*,\rho^*)=(1,0)$


Dynamic Optimization: What if the second order condition does not hold? Consider the following dynamic optimization problem
\begin{align}
&\max_u \int^T_0{F(x,u)dt}\\
\text{s.t.}~& \dot{x} = f(x,u)
\end{align}

FOCs

The Hamiltonian is given by
\begin{align}
H(x,u,\lambda) = F(x,u) + \lambda f(x,u)
\end{align}
The necessary condtions for optimality are given by the maximum principle
\begin{align}
\frac{\partial H}{\partial u} &= 0\\[2mm]
\frac{\partial H}{\partial x} &= -\dot{\lambda}
\end{align}

Suppose $u^*=\arg\max_u H(x,u,\lambda)$ is a maximizer, i.e. $H_{uu} < 0$.

SOC

The Arrow Sufficient Theorem states, that the necessary condtions are sufficient if the maximized Hamiltonian
\begin{align}
H^0(x,\lambda) = \max_u H(x,u,\lambda)
\end{align}
is concave in $x$, i.e. if $H_{xx} < 0$.

Problem

Suppose the FOCs hold, but the SOC fails to hold.

What can be said about the optimality of the solution?

There is not a single answer, it will depend on the particulars of each problem. Let's look at a standard example.

Consider the benchmark intertemporal optimization problem for the Ramsey model

$$\begin{align}
&\max_u \int^{\infty}_0{e^{-\rho t}u(c)dt}\\
\\
& \text{s.t.}\;\; \dot{k} = i-\delta k\\
& \text{s.t.}\;\; y = f(k)=c+i
\end{align}$$

The current value Hamiltonian is

$$\tilde H = u(c) +\lambda [f(k)-c-\delta k]$$

Maximizing over $c$ alone we have

$$\frac {\partial \tilde H}{\partial c} = u'(c) - \lambda =0 \implies u'(c^*) = \lambda \implies c^* = (u')^{-1}(\lambda)$$

and the 2nd-order condition will hold if the utility function is concave,
$$\frac {\partial^2 H}{\partial c^2} = u''(c^*) < 0$$

Moreover, from the first-order condition with respect to consumption, $\lambda >0$ if local non-satiation holds. Assume that we do have such "usual" preferences.

The maximized over consumption Hamiltonian is

$$\tilde H^0 = u[(u')^{-1}(\lambda)]+\lambda [f(k)-(u')^{-1}(\lambda)-\delta k]$$

The partial derivatives with respect to the state variable, $k$ are

$$\frac {\partial \tilde H^0}{\partial k} = \lambda[f'(k) - \delta],  \;\;\;\; \frac {\partial^2 \tilde H^0}{\partial k^2} = \lambda f''(k)$$

So here, the Arrow-Kurz sufficiency condition boils down to whether the marginal product of capital is decreasing, constant, or increasing (which will depend on the sign of the second derivative of the production function). In the standard case $f''(k) < 0$ and we have the sufficient condition.

In the most famous case of deviation, Romer's $AK$ model that initiated the Endogenous Growth literature, $f''(k) =0$, and the marginal product of capital is a positive constant.

So what can we say in this case?

Here,
Seierstad, A., & Sydsaeter, K. (1977). Sufficient conditions in optimal control theory. International Economic Review, 367-391. provide various results that can help us.

In particular, they prove that if the Hamiltonian is jointly concave in $c$ and $k$, it is a sufficient condition for a maximum.
The Hessian of the Hamiltonian is

(we can ignore the discount term)

$${\rm He}_H = \left [ \begin{matrix} u''(c) & 0\\
0 & \lambda f''(k)\\
\end{matrix} \right]$$

In the standard case with $u''(c) <0, \; f''(k) <0$ this is a negative definite matrix and so the Hamiltonian is jointly strictly concave in $c$ and $k$.

When $f''(k) =0$, checking that the matrix is negative-semidefinite is straightforward using the definition. Consider a vector $\mathbf z = (z_1, z_2)^T \in \mathbb R^2$ and the product

$$\mathbf z^T{\rm He}_H\mathbf z = z_1^2u''(c) \leq 0$$

this weak inequality holds $\forall \mathbf z \in \mathbb R^2$, and so the Hessian is jointly concave in $c$ and $k$.

So in the $AK$ model of endogenous growth, the solution is indeed a maximum (subject to the parameter constraints needed for the problem to be well-defined of course).


Are there fundamental reasons why (exponential) economic growth is highly desirable? One of the most widely published measures of the economy is the economic growth as a % of the GDP; i.e. the degree to which an economy grows exponentially.  In my understanding, when the rate of economic growth is declining or close to 0%, this is considered undesirable and acts are undertaken to avert recession.

In a situation where population is not growing, why do we want the economy to grow?  In other words, is there something inherently wrong with a steady state economy?

See also: Why is economic growth measured exponentially rather than linearly?

Because according to utility theory: more is better- or at least not worse (nonsatiation- or free disposal).

Your question seems to actually be: why is growth good?

Firstly, It's axiomatic. Utility theory very formally assumes the law of nonsatiation/free disposal.  Things that result in the death of the cosmos are bad not because we are dead, but because we cannot consume and improve our happiness afterwards. Evidince: We observe that consumption improves happiness.

Second, in cases of limited resources, we have situations where optional growth paths are negative-But this is only because it allows for increased overall consumption.


Can there be a sample such that SRF and PRF do not intersect? I am confused if it is possible for to randomly select a sample such that SRF and PRF do not intersect. I have thought of the concept of parallel lines, which allows for such a case but I am unsure if concepts of probability and statistics allow for it.

It is theoretically possible, but extremely unlikely if the random sample is large.

To see that it is possible, consider the very simple case shown below in which the population consists of just four points in the X-Y plane which happen to lie at the corners of a parallelogram.

Assuming linear regression lines, the population regression function (PRF) will be as represented by the continuous line.  If a sample regression function (SRF) is fitted to a sample of two points and the points happen to be A and B, then the function will be as represented by the dashed line which is parallel to the PRF.

Note however that even in this case, the probability that a random sample of two points will produce a SRF parallel to the PRF is only one third.  There are six possible such samples, two of which (AB,CD) yield SRF's parallel to the PRF, and four of which (AC,AD,BC,BD) do not.  In a realistic case in which both the population and the sample are much larger, and the points do not lie in such a simple geometrical pattern, the probability of a SRF parallel to the PRF will be very small indeed.


Canonical set of books covering Bachelors and Masters in Economics? I am a Physics student and would like to learn some Economics in my spare time. Preferably, I would chronologically go through textbooks which cover entire Bachelors and Masters degrees. For instance, in Physics there is a fixed set of topics that is covered within a Bachelor and Master degree. If you happen to know German, you could take the book series by Wolfgang Nolting (seven or eight books, numbered consecutively) and read them. When you're through, you will have covered an entire Bachelors and Masters in Physics (and more, actually). So I wonder if there is such a canonical series of books (maybe from different authors) in Economics that you could recommend to me? Thanks for any suggestion.

PS:
It should be textbooks and not courses or videos, since textbooks usually have a higher information density and transmission rate. Also, preferably it should be a canonical series, to ensure seamless transition from one book to another (no information gaps).

Econ isn't as sequential as physics. The two core undergrad classes that everyone takes are Intermediate Micro and Intermediate Macro. Varian is the standard for Intermediate Micro.

Undergrads also take some sort of econometrics class, which can widely vary depending on the target audience (i.e., business majors vs double-majors). It's not uncommon for there to be an Econ degree geared towards double majoring.

After these courses, students generally pick about 5 electives. The classic books largely depend on the subjects at the undergrad levels. For Game Theory, Osborne is a good text as is Fudenberg and Tirole. The latter is a solid grad textbook as well. A good Auction Theory text is Krishna's.

At the grad level, folks generally go for PhDs over Masters degrees. While Masters programs exist, they aren't as common as in math, physics, or CS. A good math for economists text is Simon and Blume. Mas-Collel is the classic for the grad micro sequence, which highlights game theory, IO, and general equilibrium. Jehl and Reyni is another good reference for micro, and it's a friendlier read.

Folks at the grad level also take sequences in macro and metrics. You may wish to check out Aris Spanos' text for metrics.


Solow model: Partial derivative of y with respect to n I am trying to find the derivative of $y$ with respect to $n$. All I need to do is take the partial derivative $\frac{\partial y}{\partial n}$ of the function: $sf(k) = (n+g+\delta)k$  Here is what I get using the chain rule, but it's wrong:

$$sf'(k)  \frac{\partial k}{\partial n} = (n+g+\delta)\frac{\partial k}{\partial n}$$

To solve $\frac{\partial y}{\partial n}$ some more steps are required, but as far as I know the equation is already wrong. From what I have seen the right side should be augmented by k and then it would be right. Any ideas where the derivation went wrong?

You need to use the implicit function theorem. The long-run relationship implies

$$
0 = F(n,k) = s f(k) - (n+g+\delta)k,
$$

and this implicitly defines function $k(n)$. The derivative is then

$$
\frac{d k(n)}{d n} = -\frac{\partial F / \partial n}{\partial F / \partial k} = \frac{k}{s f'(k) - (n+g+\delta)},
$$

which can be evaluated once one has solved for particular long-run (n,k) combination. To finally obtain derivative of output $y = f(k)$ wrt. $n$, simply use chain rule:

$$
\frac{d y}{d n} = f'(k(n)) \frac{d k(n)}{d n}.
$$


Pricing a European call option while absence of arbitrage is violated Assume that we have a general one-period market model consisting of d+1 assets and N states.

Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with payoff $X$,  on the asset $S_1^2$ with strike price $K = 1$ given that

$$S_0 =\begin{bmatrix}
2 \\
3\\
1
\end{bmatrix}, S_1 = \begin{bmatrix}
S_1^0\\
S_1^1\\
S_1^2
\end{bmatrix}, D = \begin{bmatrix}
1 & 2 & 3\\
2 & 2 & 4\\
0.8 & 1.2 & 1.6
\end{bmatrix}$$

where the columns of D represent the states for each asset and the rows of D represent the assets for each state

What I tried:

We compute that:

$$X = \begin{bmatrix}
0\\
0.2\\
0.6
\end{bmatrix}$$

If we solve $D'\phi = X$, we get:

$$\phi = \begin{bmatrix}
0.6\\
0.1\\
-1
\end{bmatrix}$$

It would seem that the price of the European call option $\Pi(0;X)$ is given by the value of the replicating portfolio

$$S_0'\phi = 0.5$$

On one hand, if we were to try to see if there is arbitrage in this market by seeing if a state price vector $\psi$ exists by solving $S_0 = D \psi$, we get

$$\psi = \begin{bmatrix}
0\\
-0.5\\
1
\end{bmatrix}$$

Hence there is no strictly positive state price vector $\psi$ s.t. $S_0 = D \psi$. By 'the fundamental theorem of asset pricing' (or 'the fundamental theorem of finance' or '1.3.1' here), there exists arbitrage in this market.

On the other hand the price of 0.5 seems to be confirmed by:

$$\Pi(0;X) = \beta E^{\mathbb Q}[X]$$

where $\beta = \sum_{i=1}^{3} \psi_i = 0.5$ (sum of elements of $\psi$) and $\mathbb Q$ is supposed to be the equivalent martingale measure given by $q_i = \frac{\psi_i}{\beta}$.

Thus we have

$$E^{\mathbb Q}[X] = q_1X(\omega_1) + q_2X(\omega_2) + q_3X(\omega_3)$$

$$ = 0 + \color{red}{-1} \times 0.2 + 2 \times 0.6 = 1$$

$$\to \Pi(0;X) = 0.5$$

I guess $\therefore$ that we cannot determine the price of the European call using $\Pi(0;X) = \beta E^{Q}[X]$ because there is no equivalent martingale measure $\mathbb Q$

So what's the verdict? Can we say the price is 0.5? How can we price even if there is arbitrage?

Edit: I noticed that one of the probabilities, in what was attempted to be the equivalent martingale measure, is negative. I remember reading about negative probabilities, but these links 1 2 mentioned by Wiki seem to assume absence of arbitrage so I think they are not applicable. Or are they?

Is it perhaps that this market can be considered to be arbitrage-free under some quasiprobability measure that allows negative probabilities?

Edit (to address a deleted answer):

Thanks BKay.

1 So you mean there is no unique price for $X$ but we can find upper bounds? Like in your example the least upper bound so far is 0.3 then we can continue to find lower upper bounds $u_1, u_2, ...$ (or even higher lower bounds $l_1, l_2, ...$) to say the price of $X$ is in $[0,\inf_n u_n]$ (or $[\sup_n l_n,\inf_n u_n]$)?

2 Re stochastic domination, I haven't heard that term in classes, but I think I read about that before. Might that depend on the (quasi)probability measure? Under this probability measure $0.5 S_1^2$ dominates $X$ but what about under some quasiprobability measure?

3 the $q_i$'s, not the $\psi_i$'s are the probabilities

I believe there is not a unique price. Say, instead of buying the option  you spent 0.5 on a half a unit of the asset $S^2_1$ This asset pays out $[0.4, 0.6, 0.8]$ which first order stochastically dominates the option. So, no matter your probability beliefs about the states,  in that setting you'd never pay $0.5$ for the option which pays less in every state. This suggests the right price is less than $0.5$. Similarly, buying $0.25$ units of the $S^0_1$ asset or $0.167$ units of the $S^1_1$ asset would likewise stochastically dominate the option. In fact, because for $0.375$ units of asset $S^1_2$, which costs on $0.375$, you can still have an asset that pays out $[0.3, 0.45, 0.6]$, it seems unlikely that the price could even be as high as $0.375$. Asset 0 implies a price below $0.4$ and asset 1 below $0.45$

Some python code to solve:

import numpy as np
S0 = np.array([[2],[3],[1]])
D = np.array([[1,2,3], [2,2,4], [0.8, 1.2, 1.6]])
X = np.array([[0.0],[0.2],[0.6]])
phi = np.dot(np.linalg.inv(D.transpose()), X)
print('The weights of the portfolio that replicates payoff X are: \n', phi)
P_X = np.dot(S0.transpose(), phi)
print('With a price: ', P_X)
print('Normalizing to pay a fixed price P_X for each of the three assets, what payoffs can you get?')
D_norm = D/(2*S0)
print(D_norm)
print('Notice that all three first order stochastically dominate the option for a price of: ', P_X)
print(D_norm - X.transpose())
print('Using each of the base assets, what\'s the minimum quantity that dominates?')
D_relative = X.transpose() / D
print(D_relative)
Min_dominating_fraction = np.max(D_relative,axis=1)
print('Minimum fraction of each of the assets that dominates X\n', Min_dominating_fraction)
P_Min_dominating_fraction = S0.transpose() * Min_dominating_fraction
print('At prices of: ', P_Min_dominating_fraction)
print('Therefore the option price should be less than: ', np.min(P_Min_dominating_fraction))

This code doesn't spell out the price of the option, it just show my calculations for the paragraph above. I believe the real price of this option would actually be zero if shorting is permitted. If you buy three units of asset $S^2_0$  and short one unit of $S^1_0$ you get an asset with payouts $[ 0.4,  1.6,  0.8]$. This position costs nothing to take, has positive payouts for all states, and first order stochastically dominates the option itself. Since it is possible to make a better than replicating portfolio at zero cost the price should be zero. Oh the insanity at work when arbitrages are present!


Wealth - Bond Market and Liquidity Preference Framework I've been told that an increase in wealth will lead to an increase in both the demand for Bonds determined by the Theory of Asset Demand. This should shift the bond curve to the right, increasing Price and decreasing interest rate.

But for the liquidity preference framework, it says that an increase in income leads to an increase in wealth thus shifting the demand curve for money to the right and increasing interest rate.

How can wealth increase decrease the interest rate in the bonds graph but increase it in the quantity of money supply graph. I'm sure I'm missing out on something obvious.

These are theories, so need empirical justification. They may not be true, and even if they appear to be true they may still be wrong about the cause and effect.

If you regard your first statement as saying that people with increased monetary wealth will want to see a income increase from this wealth then they will buy bonds and other less liquid investments pushing interest rates down, and your second statement as saying that people with increased non-monetary wealth will want to see an increase in liquid assets such as money so they can spend some of their wealth then they will sell bonds and other less liquid investments pushing interest rates up, you will find there is no intrinsic contradiction. Instead you get an explanation of the existence of secondary markets in bonds and other investments to reconcile the needs and desires of different people


Are there western governments that don't have to borrow in their own currency? Are there any western and/or democratic countries where the central bank buys all government debt, interest free, directly from the treasury?  I.e. are there any such countries where government spending is effectively financed by money printing without the intermediary of a private market for government bonds?  I don't now for sure, but I suspect this is what happens in autocratic states like China and I wondered if such a regime exists elsewhere.
Edit:  @H2ONaCl is correct that I mean to ask about a case where the central bank holds the interest free debt in perpetuity. Alternatively, and in effect identically, the government just prints money when it wants to spend more than it has.

You have different questions in your title and body I will try to answer them separately:

Are there western governments that don't have to borrow in their own currency?

Yes, all western governments that I know of are allowed to borrow in any currency they want. They usually borrow in their own currency because it is easier for them to repay the debt if they borrow in currency they can control.

Are there any western and/or democratic countries where the central bank buys all government debt, interest free, directly from the treasury?

I do not know of any western government that would get sell this debt interest free  de jure (whether on primary market or not). However, de facto all interest payments that are collected by central banks are send straight back to the government treasury,  so any interest payments that government pays on government bonds are de facto just free and just money juggling - like when government employees have to pay taxes (government first spends money on paying their salaries and then takes some of that money back in taxes instead of paying the post tax salary).

are there any such countries where government spending is effectively financed by money printing without the intermediary of a private market for government bonds?

There are some countries that are not autocratic and do that but are not 'western' countries either. For example, Indonesia is democratic country and their central bank recently started to purchase bonds on primary market.
However, some European central banks are considering it. For example, Bank of England (BOE) recently stated it will:

"keep under review the case for participating in the primary market”, suggesting it could eventually buy bonds from the government or companies directly rather than on secondary markets as it does now.

Nonetheless, this has nothing to do with the interest payments themselves, they will likely not be eliminated (and why would it even matter given that the government essentially pays that interest to itself?).
Fed now has a program to buy bonds on primary market from companies, but not from government.
Czech republic recently tried to fast-track a bill that would allow their central bank to do that, but only during financial crises, but the motion failed. However, it is possible the bill will pass in the future (this was only fast-track process that would skip debate with experts/scientists and public debate).


Is a production function bilinear? I believe the following is the multiplicative property of bilinearity:

$$ Y=F(K,AL) $$
$$ c_1 F(K,AL) = F(c_1 K, AL) $$
$$ c_2 F(K,AL) = F(K, c_2 AL) $$

But when we have multiplied through the production function with a constant we have done so through each term as below

$$ c_3 F(K,AL) = F(c_3 K, c_3 AL) $$

e.g.

$$ \frac{1}{AL} F(K,AL) = F\left( \frac{K}{AL} , \frac{AL}{AL} \right) = F(k,1) = f(k) $$

What is the name of this property?

Such a function is called homogeneous of degree 1.


When solving game theory problems, how can you "assume" that the equilibrium strategy will lead to something happening? This is something that confuses me a bit when I see my book solve game theory problems.

When solving a model's equilibrium (like a subgame perfect nash equilibrium), the book will assume that a certain situation occurs in that equilibrium. For example, in a model with some firms setting prices and some consumers who have to make a decision to buy a product, it might assume that in equilibrium, all consumers buy.

Then it proceeds to solve for the actual equilibrium, while it always uses the assumption that they all buy.

When having solved for the equilibrium, it checks that it indeed holds in this equilibrium that all consumers buy, by e.g. showing that no consumers have an incentive to not buy.

Why is this acceptable? Isn't the derivation of the equilibrium flawed because you made an unwarranted assumption?

This is a concept that I've seen confuses a lot of people when they first encounter it. However, it is completely perfect to assume that something will happen in equilibrium, use that fact to characterize equilibrium, and then check your assumption indeed holds. This is sometimes called guess-and-verify. Remember that an equilibrium has a very precise definition, it is a set of strategies that satisfy some conditions. If you find them, and they satisfy the conditions it doesn't really matter how you found them.

It might sound silly to just be guessing properties of an equilibrium and be hoping for the best. But actually, doing good guesses is hard; so there is merit to it. So long you check that your equilibrium satisfies the assumed (or guessed) property, then your original assumption is verified.

The alternative would be, in principle, to solve a game by computing for each player their best response as a function of all possible actions of every other player. And then find the intersection of these best responses. This can be an extremely hard thing to do, even for simple games like a Hotelling model where N firms set locations and prices and then consumers choose from where to buy. So making informed guesses can be a great shortcut to find an equilibrium.


Why is modest inflation a good thing? I have been reading a BBC news article about inflation in the UK, which is saying that inflation has recently become negative (http://www.bbc.co.uk/news/business-33147660). The article suggests that this is a bad thing. However, I would have thought that positive inflation is generally a bad thing, because it creates instability, resulting in people spending all their money rather than saving it. Certainly, I know that hyperinflation can cause a huge deal of trouble. Can somebody explain why positive inflation (in modest quantities) is necessarily a good thing?

I am copying and pasting from an answer I wrote at From an economics perspective, what are the ramifications of a currency with fixed money supply?

  A moderate degree of currency inflation serves a number of useful functions in the economy. The most obvious are:

  It induces people to spend their money before it loses its value. In
  a deflationary environment there is an incentive to put money under
  your mattress and spend it in a year when it has greater purchasing
  power. If everbody does this then the lack of demand will lead to a
  decrease in overall economic activity (i.e. a recession).
  It provides a weapon against downward nominal rigidities. For example, workers are generally reluctant to accept a nominal pay cut, even if market conditions are such that the current wage is above the equilibrium level. Inflation means that their employer can simply increase wages at less than the inflation rate so that the real wage is decreasing.
  It erodes the real value of nominally denominated debt. Now, this is obviously only a pseudo-advantage because (whilst it benefits debtors) it harms creditors. However, this kind of erosion of debt may be desirable if national economic stability is threatened by high debt levels. Also, since debtors are usually poorer on average than creditors, it can reduce inequality, which may be a normative objective for the government.

  Without inflation you miss out on these benefits. The first benefit might not seem like a big deal if you think you can simply set the rate of inflation at zero percent. But it is very hard to hold inflation constant at some target level so attempting to hit zero inflation will almost certainly result in occasional lapses into deflation, with the attendant negative economic consequences.


Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8 Letting $f_i(\mathbf{x})=\partial f(\mathbf{x})/\partial x_i$, ($\mathbf{x}$ is a vector, a commodity bundle, and $x_i$ is a scalar, commodity $i$ in the bundle) show that,

$\sigma_{ij}(\mathbf{x})\equiv -\frac{x_if_i(\mathbf{x})+x_jf_j(\mathbf{x})}{f^2_j(\mathbf{x})f_{ii}(\mathbf{x})+2f_i(\mathbf{x})f_j(\mathbf{x})f_{ij}(\mathbf{x})+f_i^2(\mathbf{x})f_{jj}(\mathbf{x})}\frac{f_i(\mathbf{x})f_j(\mathbf{x})}{x_ix_j}$

Then using the above formula, show that $\sigma_{ij}(\mathbf{x})\geq0$ whenever $f$ is increasing and concave.

PS: I know $\sigma_{ij}$ can be written as

$\sigma_{ij}=-\frac{d (x_i/x_j)}{x_i/x_j} \frac{f_i(\mathbf{x})/f_j(\mathbf{x})}{d (f_i(\mathbf{x})/f_j(\mathbf{x}))}$

How can we derive the result from this known fact? Thank you.

I emailed the question to one of the authors. He said the solution manual will come soon!


Derive marginal productivity conditions in DW Jorgenson paper I'm reading Jorgenson, Dale W. (1963), "Capital Theory and Investment Behavior", American Economic Review. Vol 53, No. 2, pp. 247-259. http://www.jstor.org/stable/pdfplus/1823868.pdf

I would like to know how to derive the marginal productivity conditions given in pg 4.

This is a deterministic Optimal Control problem in continuous time. Jorgenson uses nominal terms, so prices are explicitly present. To copy the original

and we want to arrive at

A standard neoclassical production function, is a constant-returns to scale one over Labor $L$ and capital $K$, $Q = F(K,L), F_L, F_K >0, F_{LL}, F_{KK} <0$, satisfying also the Inada limit conditions, while the law of motion of capital is described verbally and it is the standard one, $\dot K = I-\delta K$.

To solve this we can set up the current value Hamiltonian (analogous to the Lagrangean)

$$H = R(t) - D(t) + \lambda \cdot [I-\delta K]$$

where $\lambda$ is a multiplier, and we maximize it over the decision variables $L$ and $I$, and the state variable $K$.
The first order conditions are

$$\frac {\partial H}{\partial L} = 0 \Rightarrow \frac {\partial R(t)}{\partial L}   -\frac {\partial D(t)}{\partial L}=0$$

$$\Rightarrow \frac {\partial }{\partial L} \left(pQ-sL\right)  -u\frac {\partial }{\partial L}\left(pQ-sL\right)=0 $$

$$\implies (1-u)\left(p\frac {\partial Q}{\partial L} - s\right) = 0 \implies \frac {\partial Q}{\partial L} = s/p  \tag{1}$$

Also

$$\frac {\partial H}{\partial I} = 0 \implies -q +\lambda =0 \implies q = \lambda \tag{2}$$

from which we also obtain (to be needed next)

$$\dot \lambda = \dot q \tag{3}$$

The first-order condition related to the state variable $K$, when we use the current value Hamiltonian is

$$\frac {\partial H}{\partial K} = r\lambda - \dot \lambda$$

$$\implies \frac {\partial R(t)}{\partial K}   -\frac {\partial D(t)}{\partial K} = r\lambda - \dot \lambda$$

$$\implies p\frac {\partial Q }{\partial K} -up\frac {\partial Q }{\partial K} + (v\delta q + wrq - x\dot q) -\lambda \delta = r\lambda - \dot \lambda  \tag {4}$$

The rest are algebraic manipulations (are they correct?)

A good educational presentation on the workings of Optimal Control in Economics is to be found here, mathematical appendix A.3, p. 604. Dorfman's 1969 paper remains a classic in mapping the mathematical structure of Optimal Control to economic concepts and intuition, Dorfman, R. (1969). An economic interpretation of optimal control theory. The American Economic Review, 817-831.


How does a national budget differ from a household budget? We are often told by pundits that "Country X cannot spend more than they take in in revenue, because a house that makes A and spends B will always collapse if B>A". Essentially the siren calls of a balanced budget as promulgated by the talking heads.

Sadly, despite even conservative economists like Friedman citing a k-percent rule and the necessity of inflation (since the number of people demanding money seems to continue rising), many entry level students get hung up on this concept.

Essentially, what I'm asking is:

  What are the key functional differences between a household budget (or other simple toy model) and a national budget and what's the best way to communicate this to people with only a passing familiarity with economics?

I'm looking for a way to effectively communicate to group of first year students that beyond just the ability to control the supply of money (monetary policy), and control income and spending more readily than an individual or household (fiscal policy), economic policy really has nothing in common with the ability to balance a checkbook, and the dramatic oversimplification is a disservice to the dismal science.

If you think about it, there is really not much difference between a government budget constraint and a household budget constraint.

Both have uncertain income streams, which are labor and capital income for the household, and mostly tax revenue for the government. On the other side of the constraint, you will find consumption and saving for the household. The government will spend most of its income on (public) goods and transfers to households (social system).

Both can move income inter temporally using borrowing. However, neither can run a permanent deficit where it is "not expected to ever pay back", something that is typically referred to as a No-Ponzi-Game condition. As an implication, the "life-time budget constraint" will have to hold for both:

sum of all expected future income = sum of all expected future expenditures

I'm pretty informal here, you would typically rather derive "present value" of these constraints, but latex is not yet allowed here, and I'm already writing much.

What should government debt be used for?

We typically assume that households and governments have concave utility, they prefer consumption processes that are second-order dominating. Hence, they want to engage in smoothing of consumption over fluctuations.

Since all debt has to be paid back eventually, there is limited use of government borrowing. Most typically, one wants to stabilize consumption and output over business cycles, by financing consumption in busts and paying back in booms. Also, given an interest rate r, investments that promise a higher return than r can be financed through debt, as the government will not violate consumption smoothing.

Potential Difference

One potential difference is the commitment. Developed governments are typically more
trustworthy, they are more likely to pay back there debt. Hence, they can borrow at cheaper interest than most households. As a conclusion, they can use debt more aggressively than households for the two purposes consumption smoothing and investment, as emphasized above.

tldr: Besides that, they're very much alike. Note also the Ricardian Equivalence, which is not at the heart of this question, but very much related.


Showing that a transformation is measure preserving Note: This question is related to this question about the construction of stochastic processes. Specifically, it relates to the transformation $\mathbb S: \Omega \rightarrow \Omega$ that is mentioned. The following is an example to help understand such transformations. If such transformation are measure preserving, then the distribution function of $X_t$ is identical for all $t \geq 0$.

If we suppose that $\Omega = [0,1)$ and that $Pr$ is the uniform measure and that
$$
\mathbb S(\omega) =
\begin{cases}
2 \omega & \omega \in [0, 1/2) \\
2 \omega - 1 & \omega \in [1/2, 1),
\end{cases}
$$
how would we show that $\mathbb S$ is measure-preserving?

It definitely suffices to verify this property for just the open intervals in $\Omega$. But trying to so explicitly for an arbitrary interval $(a,b) \subset \Omega$ is complicated by the fact that it's hard to say more than $\mathbb S((a,b)) \subset [0,1)$ when $(a,b) \subset [0,1/2)$ and that again
$\mathbb S((a,b)) \subset [0,1)$ when $(a,b) \subset [1/2)$. Given this, what is an easy way to go about arguing that
$\mathbb S$ is measure-preserving?

To show that transformation is measure preserving you need to show that full preimage $S^{-1}(A)$ of any set A in the Borel $\sigma$-field on [0,1) is again in the same $\sigma$-field, i.e. it is measurable, and that $Pr\{S^{-1}(A)\} = Pr\{A\}.$

As you have correctly pointed out, it will suffice to show it for any base of usual topology, namely, all open intervals.

It is evident that preimage of $(a,b)$ is $(\frac{a}{2},\frac{b}{2})\cup (\frac{1}{2}+\frac{a}{2},\frac{1}{2}+\frac{b}{2})$. It is, evidently, in Borel $\sigma$-field, as it is a union of two open intervals, and $Pr\{(\frac{a}{2},\frac{b}{2})\cup (\frac{1}{2}+\frac{a}{2},\frac{1}{2}+\frac{b}{2})\} = Pr\{(a,b)\} = b-a$

We proved that preimage of any open interval is measurable and has the same measure as the interval. As open intervals form a base of the usual topology we can extend this result to any measurable set and it completes the proof.


What happens to effective consumption and capital in the Ramsey/Cass-Koopmans(RCK) model when technological growth decreases? Is the conclusion that both decline correct? Both long and short run.

Let $g_x$ denote growth rate of $x$, $g_c$ the growth rate of $\hat c$. In the model there is no depreciation so,

$$ g_c = \frac 1{\theta}[r - \rho - \theta g_x] $$

$$\dot {\hat k} = f(\hat k) - \hat c - (n+g_x)\hat k$$

At steady state growth of $\hat k$ and $\hat c$ are both $0$. When that is the case:

$$\hat c^* = f(\hat k^*) - (n+g_x)\hat k^*$$

Also since $r = f'(k)$

$$f'(\hat k^*) = \rho + \theta g_x$$

Somehow based on these equations I reach the opposite conclusion, i.e. that $\hat c^*$ and $\hat k^*$ will increase rather than decrease if $g_x$ is reduced.

When the (exogenous) rate of technological change/efficiency is smaller, the corresponding steady-state levels of consumption and capital per unit of effective labor, increase.

For capital, we have

$$f'(\hat k^*) = \rho + \theta g_x \implies \frac {\partial}{\partial g_x}f'(\hat k^*) = \theta >0$$

So if $g_x \downarrow \implies f'(\hat k^*) \downarrow \implies \hat k^* \uparrow$ due to decreasing marginal product of capital.

For steady-state consumption per unit of effective labor we have

$$\hat c^* = f(\hat k^*) - (n+g_x)\hat k^*$$

$$\implies \frac {\partial \hat c^*}{\partial g_x}= f'(\hat k^*) \frac {\partial \hat k^*}{\partial g_x} - \hat k^* - (n+g_x) \frac {\partial \hat k^*}{\partial g_x}$$

$$=[f'(\hat k^*) - n - g_x]\cdot \frac {\partial \hat k^*}{\partial g_x} - \hat k^* $$

The term in brackets is assumed positive, i.e. we have already assumed

$$f'(\hat k^*) = \rho + \theta g_x > n + g_x \implies \rho > n + (1-\theta)g_x$$

in order to exclude infinite utility.

Moreover, evidently $\frac {\partial \hat k^*}{\partial g_x} <0$ so in all

$$\frac {\partial \hat c^*}{\partial g_x} <0$$

Therefore, if $g_x \downarrow \implies \hat c^* \uparrow $.

See Barro & Martin, ch. 2 page 102, where they discuss the  $g_x \uparrow$ case (in p. 101 they discuss the constraint on the parameters).

Comment: This result may appear counter-intuitive, but a deeper examination of the model shows that if $g_x$ is lower, utility per capita is lower. Using "consumption per unit of effective labor" is a modeling tactic, what interests us is what happens per individual. So no, the model does not argue in favor of lower productivity/technology.


Showing the Utility Possibilities Frontier is downward sloping The slope of the UPF must be equal to the slope of the MRT and the MRS. Is it therefore enough to show that the slope of the MRS is negative?

The first statement does not seem to be true. The UPF exists in the space of utilities whereas the other curves (production, indifference) exist in the product space. As a result, the slope of the UPF may change if a monoton transformation is applied to one of the utility functions whereas the other curves and their slope will definitely not change.


Discounted (present value) calculation I had a homework problem regarding calculating a discounted present value. My solution is:

15000 * (1+.08)^1 + 15000 * (1+.08)^2 = $33 696

The question is this:

  Laura has an undergraduate degree in economics and has been working as
  a utility rate analyst at the local electricity utility. If she
  continues on her present career path, the present value of her
  lifetime earnings is \$250,000. If she takes two years off and gets an
  MA degree in economics, the present value of her lifetime earnings is
  \$275,000. The annual cost of an MA degree in economics is \$15,000
  and the interest rate is 8%. Assume that school fees are paid at the
  beginning of the year.

  a.) Calculate the discounted (present value) cost of Laura's graduate
  degree in economics. (2 points)

The given solution is this:

Answer: $28, 889

I don't understand the logic of this question. To me, it seems the present value cost would be the cost of what I would pay if I was accounting for the interest of the tuition.

The calculation that you gave doesn't make sense. To answer this question, you should revise your definition of "present value." The definition you gave isn't specific enough to be operational:

  To me, it seems the present value cost would be the cost of what I would pay if I was accounting for the interest of the tuition.

What does present value mean? Think about it like this. How much money do you need today to pay the future costs. Well you need $\$15,000$ to to pay for the first year and $\$ X$ to pay for the second year. Why did I write $X$? Because we want to know how much money we need today to pay for the second year and that amount is less than $\$15,000$. This is because we can invest $X$ amount of dollars at $8\%$ interest so that we have $\$15,000$ in a year. So, we want to
solve
$$
X * 1.08 = 15000.
$$
Thus, the proper present value calculation is
$$
28,889 \approx 15,000 + \frac{15,000}{1.08}.
$$


What will happen to the world economy if the price of oil/Coal/natural gas/electricity and all energy sources drops to lowest? Imagine if some country identifies a large resource of oil/coal/natural gas or other sources of energy that can be supplied to the entire world for the cheapest rate ever.(say for electricity 1 Cent/unit,petrol 1 cent/litre).How will it affect the world economy ?

There would be a short-term boost to profits, as a significant cost driver has dropped. There will be some substitution, as energy replaces other factor inputs (probably human labour in particular). In theory, overall, everyone collectively is better off, but redistribution effects will be significant and unpredictable.

There would be a massive shift in the geopolitics of energy: countries that are currently powerful primarily due to their energy supplies (e.g. Saudi Arabia) could become insignificant on the international scene.

If it's a fossil fuel, then there would be a catastrophic medium and long-term reduction in total global wealth owing to accelerated anthropogenic climate change. A significant chunk of global economic activity would have to be diverted to adaptation to a rapidly changing, highly random, highly variable, highly unpredictable climatic period. International refugee migrations would be of the order of a billion people or more. There would be a non-trivial probability that human civilisation would essentially cease to exist in any recognisable form.

If it's not a fossil fuel then, well, we're close to that point now: PV is around 2 (\$/€)cent/kWh for a big proportion of the world, and is expected to halve again over the next decade or so. So all being well, we'll live to see this. Note that the benefits will not be confined to one country, though: the resource is globally distributed.


Can infrastructure costs be covered by taxes on flat sales/rents? Instead of describing the general question, let me give a concrete example.
Suppose in a city they want to extend a metro line but they don't have enough budget for it. Could the city designate a locality which would mostly profit from the extension and say that in the next 10 years any flat sold or rented in the area would have an additional 10% sales tax which would go to the metro building funds? I'm thinking that this way the residents would also profit financially (even those who don't use the metro), as due to the metro extension they could sell their flats for, say, 20% more, so about half of this increase would go to the city, the rest to the residents. What would be some issues with such a tax?

Ps. Of course there are several variants, like whether the tax starts only after the extension is completed (from a bank loan), how long the tax needs to be paid (fix years or until enough money is collected), should it be also for rents or only for sales, should only the first sale of each flat be taxed or also subsequent sales, how should the boundary of the tax be determined (possibly with several zones, like some paying just half the tax), which politician should steal the money etc. I'm mainly interested in why the main idea has never been implemented, AFAIK.

Yes,

What you are describing is called value capture. There are a variety of forms of the tax, each with their own pros and cons. The general idea is that infrastructure, particularly transport infrastructure, disproportionately benefits parties closest to the line.

A good example is happening in Western Sydney in what was semi-rural land. An international airport and substantial extra development. Large land-owners found that their property is just down the road from a planned town-centre and they can build apartments on their paddocks. There was massive value uplift which was not captured by the government, but could have been.

I'm not aware of a reason why the policy shouldn't work, but it is politically tricky.


Principal component analysis interpretation Suppose a wealth index is computed using information on a set of 14 assets that a household possesses.
 The index is generated using principal components, as the 14 individual asset variables are highly
collinear. A OLS regression of education expenditures (in Rupees per household) on the wealth
 index yields a coefficient of 0.4.

It is harder to interpret coefficients principal components in terms of original regressors. But can I interpret wealth index as its own ie can I say - increase in wealth index by a score of 1 will increase education expenditure by 0.4?

Is there any alternative
 formulation of doing this exercise that may be preferred?

It is harder to interpret coefficients principal components in terms of original regressors.

Indeed, since you are now dealing with components, you can't. That being said, if your original regressors are highly colinear and all capturing wealth-related phenomena, a possibility is that your first component is still somehow strongly related to wealth, and other components only controling for phenomena that are not related to wealth at all. Put differently, you could reduce the dimensionality of your problem to this unique first component, and maintain your interpretation.

But do not take what i am saying for granted: if you really want to know how abusive maintaining your economic interpretations would be, it's all a matter of visualizing/diagnozing the relative weights of your components, e.g. you need to check the eigen decomposition of your correlation matrix.

I recommend you to read Michael Friendly and Ernest Kwan (2009) "Where's Waldo? Visualizing Collinearity Diagnostics" The American Statistician, Vol. 63, No. 1 (Feb., 2009), pp. 56-65.

Also, Yoel Haitovsky (1966) "A Note on Regression on Principal Components" is very clear and concise.


Understanding subscripts in first order conditions of dynamic optimization problems Suppose we have a simple maximization problem as described in Equation 1.1 here or here. This leads us to the Lagrangian Equation 1.3: $$\begin{align*}\mathcal{L} &= \sum_{t=1}^\infty \beta^{t-1}\left\{u(c_t) + \lambda_t \left[ f(k_t) + (1 - \delta)k_t - c_t - k_{t+1}\right]\right\} \\ &= \sum_{t=1}^\infty \left[\beta^{t-1} u(c_t) - \beta^{t-1}\lambda_t c_t + \beta^{t-1} \lambda_t f(\mathbf{k_t}) + \beta^{t-1}\lambda_t(1-\delta)\mathbf{k_t}  - \beta^{t-1}\lambda_t \mathbf{k_{t+1}}\right]  \end{align*}
$$

When we derive the first order condition with respect to $k_{t+1}$, which is:
$$\frac{\partial \mathcal{L} (\cdot)}{\partial k_{t+1}} = 0 :   \beta \lambda_{t+1} \frac{\partial f(k_{t+1})}{\partial k_{t+1}} +  \beta \lambda_{t+1} (1 - \delta) -\lambda_t=0$$

why do we use the subscript $\phantom{.}_{t+1}$ in  $\lambda_{t+1}$ and why does $\beta^{t-1}$ becomes $\beta$? I cannot understand how the first two terms are combined with the last one ($-\lambda_t$).

The relevant terms (with $k$) of the Lagrangian in period $\phantom{.}_{t+1}$ are: $$ \beta^{(t+1)-1} \lambda_{t+1} f(k_{t+1}) +  \beta^{(t+1)-1} \lambda_{t+1} k_{t+1} (1 - \delta) - \beta^{(t+1)-1} \lambda_{t+1} k_{(t+1)+1}$$ so for this part of the sum we do not "care" about the last term when we take the derivative with respect to $k_{t+1}$. So for this period this part of the sum is $$\frac{\partial \mathcal{L}_{t+1}}{\partial k_{t+1}} = \beta^{t} \lambda_{t+1} \frac{\partial f(k_{t+1})}{\partial k_{t+1}} +  \beta^{t} \lambda_{t+1}  (1 - \delta)$$

The relevant terms (with $k$)  of the Lagrangian in period $\phantom{.}_{t}$ are:
$$\beta^{t-1} \lambda_t f({k_t}) + \beta^{t-1}\lambda_t(1-\delta){k_t}  - \beta^{t-1}\lambda_t k_{t+1}$$
so for this period the part of the sum is $$\frac{\partial \mathcal{L}_t}{\partial k_{t+1}} =  - \beta^{t-1}\lambda_t$$
Now the First Order Condition with respect to $k_{t+1}$ should be:  $$\frac{\partial \mathcal{L}_{t+1}}{\partial k_{t+1}} +  \frac{\partial \mathcal{L}_t}{\partial k_{t+1}} = \beta^{t} \lambda_{t+1} \frac{\partial f(k_{t+1})}{\partial k_{t+1}} +  \beta^{t} \lambda_{t+1}  (1 - \delta) - \beta^{t-1}\lambda_t = 0$$right?

In an intertemporal maximization problem, we seek to find the optimal sequence  of the control and the state variables. It is the recursive nature of the problem that permits us to consider a "typical" point in time and just one condition per variable.

For each such problem, we need to find out (carefully) in how many distinct periods a specific realization of a variable appears. To do this properly we should distinguish between the "absolute" index, and a "running" index. In the formulation of the Lagrangean as appears in the question, this is not done (and it is usual practice not to, but it may become confusing).

So I would use the $t$ symbol as the absolute index (to arrive at same-looking first-order conditions), and some other symbol for the running index, say

$$\mathcal{L_t} = \sum_{j=0}^\infty \beta^{j}\left\{u(c_{t+j}) + \lambda_{t+j} \left[ f(k_{t+j}) + (1 - \delta)k_{t+j} - c_{t+j} - k_{t+j+1}\right]\right\} $$

Note that $t$ no longer affects  the discount factor $\beta$, and this is because the discount factor has to do with looking at the future, which is represented by the index $j$. Also, note that $j$ starts at zero, indicating that the first period is the $t$ period.

Written this way, the Lagrangean says "we are at some point in time indicated by $t$ (that can take the value zero or whatever positive value), and we are looking forward period by period counted by index $j$".

For any given $j$ we have

$$\mathcal{L}_t =...+ \beta^{j}\Big\{u(c_{t+j}) + \lambda_{t+j} \left[ f(k_{t+j}) + (1 - \delta)k_{t+j} - c_{t+j} - k_{t+j+1}\right]\Big\} + \beta^{j+1}\Big\{u(c_{t+j+1}) + \lambda_{t+j+1} \left[ f(k_{t+j+1}) + (1 - \delta)k_{t+j+1} - c_{t+j+1} - k_{t+j+2}\right]\Big\} + ...$$

Pondering this, we realize that the variable $k_{t+j+1}$ will appear in only two consecutive periods, and so the first order condition for a "typical" element of the sequence $\{k_{t+j}\}_{j=0}^{\infty}$ can be exrpessed by differentiating only these two periods with respect to $k_{t+j+1}$. Doing so we get

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+j+1}} =  -\beta^{j} \lambda_{t+j}  + \beta^{j+1}\Big\{ \lambda_{t+j+1} \left[ f'(k_{t+j+1}) + (1 - \delta)\right]\Big\} $$

Take common factors (which will simplify the discount factor) and set equal to zero

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+j+1}} = \beta^{j} \Big[-\lambda_{t+j}  + \beta\Big\{ \lambda_{t+j+1} \left[ f'(k_{t+j+1}) + (1 - \delta)\right]\Big\}\Big] = 0$$

To lighten the indexing burden, we can express this for $j=0$, to obtain

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+1}} = 0 \implies -\lambda_{t}  + \beta\Big\{ \lambda_{t+1} \left[ f'(k_{t+1}) + (1 - \delta)\right]\Big\} = 0$$


How to determine the riskparamter PD, EaD, LGD for the IRB-Ansatz | Basel II A company borrows $100 million from Bank A. According to Basel II, the bank must have a certain amount of equity for this balance sheet item in order to grant the loan.
The exact amount of equity can be calculated using the IRB approach. The IRB formula depends on the parameters

probability of default (PD)
exposure at default (EaD)
lose given default (LGD)
maturity of the exposure (M)

Banks in Europe can use their intern models to compute this quantities. Does someone know what models banks use to compute this quantities? I am especially interested in the computation of PD.

Exact models are proprietary but they usually do that using statistical analysis. For example, in this paper the authors estimate the probability of default of various banks using the probit model. Or another example is in this paper.
Models with a limited dependent variable like probit or logit and others are quite common because these models let you estimate the probability of some event occurring. However, these models can be set up in various ways depending on data.