Hint 1: Suppose a mixed strategy NE $(\sigma_1,\sigma_2)$ exists, where $\sigma_i$ is a distribution over $[0,1]$. By the property of MSNE,
$$u_i(x_i,\sigma_j)=u_i(x_i',\sigma_j)$$
for all pure strategies $x_i,x_i'$ in the support of $\sigma_i$.

Hint 2: Note that a segment of $u_i$ is linear in strategies, and thus
$$
x_i+\frac12(1-x_i-\mathbb E[x_j])=x_i'+\frac12(1-x_i'-\mathbb E[x_j])
$$


System of Differential Equations- Asymmetric First-Price Auction

I followed Oliv's suggestion, which was quite fruitful. So we have the differential equations:

$$g_{1}^{\prime}(b) = \dfrac{1}{2} \cdot \dfrac{g_{1}(b) - 1}{g_{2}(b) - b}$$

And:

$$g_{2}^{\prime}(b) = \dfrac{3}{2} \cdot \dfrac{1}{g_{1}(b) - b}$$

With the boundary conditions $g_{2}(0) = 0$ and $g_{1}(\overline{b}) = g_{2}(\overline{b}) = 3$, where $\overline{b}$ is the maximum bid.

Now we guess that $g_{1}(b) = \alpha b + \gamma$ and $g_{2}(b) = \delta b + \lambda$. Applying $g_{2}(b) = 0$ yields that $\lambda = 0$.

Next, I substitute $g_{1}(b)$ into $g_{2}^{\prime}(b)$ to obtain:

$$g_{2}^{\prime}(b) = \dfrac{3}{2} \cdot \dfrac{1}{(\alpha - 1)b + \gamma}$$

Integrating $g_{2}^{\prime}(b)$ yields

$$g_{2}(b) = \dfrac{3}{2(\alpha - 1)} ln( (\alpha - 1)b + \gamma)$$

We note there is no constant when integrating, as $g_{2}(b) = \delta b$. We now apply $g_{2}(0) = 0$ again, concluding that $ln( \gamma) = 0$. And so $\gamma = 1$. Thus, $g_{1}(b) = \alpha b + 1$.

We now solve:

$$g_{2}(\overline{b}) = 3 = \dfrac{3}{2(\alpha - 1)} ln( (\alpha - 1)\overline{b} + \gamma)$$

From this and noting that $g_{1}(\overline{b}) = 3 = \alpha \overline{b} + 1$, we obtain:

$$e^{2(\alpha - 1)} = 3 - \overline{b} \implies \overline{b} = 3 - e^{2(\alpha - 1)}$$

Plugging this into $g_{1}(b)$ yields:

$$g_{1}(\overline{b}) = \alpha(3 - e^{2(\alpha - 1)}) + 1 = 3$$

Which implies that the solution $\alpha = 1$. Thus, $\overline{b} = 2$.

So $\delta = \dfrac{3}{2}$.

Thus, $g_{1}(b) = b + 1 \implies \beta_{1}(v) = v - 1$; and $g_{2}(b) = \dfrac{3}{2}b$ implies $\beta_{2}(v) = \dfrac{2}{3}v$ as desired.


The notions of Bayes Correlated Equilibrium and Bayesian Nash Equilibrium in a game with one player

The terms experiment and information structure are used interchangeably by the literature, so both terms are valid regardless of $N$.
I agree if $N=1$ there is really no game, and it is rather a single-agent decision problem (or decision problem with uncertainty), and yes, you don't have a player, rather a decision maker.
For the same token, you don't want to say the solution concept if BNE, instead you want to say that the agent maximizes his expected utility given the information she received.
I was not aware of the term "one-player BCE" but Bergemann and Morris coined the term BCE, so if that is how they call it, it is correct.
In terms of other references, this is an earlier version of that same paper. However, I think that the paper you linked is the right paper for citing the concept of BCE.

In this later paper, the same two authors made the remarkable connection that the problem of Bayesian Persuasion popularized by Kamenica and Gentzkow could be interpreted in light of the concept of BCE when $N=1$. So perhaps, this later reference of BM can be useful if you want to explore more the case of $N=1$.


Is an alternative other than negative interest rate possible?

Negative interest rates means that if you take a 1.000 loan, you will pay 900 to the lender. At the same time, if you deposit this same 1.000 in a savings accounts, you will withdraw 900. This is an expansionary monetary policy. In other words, low or negative interest rates estimulates the consumption and destimulates the savings.

It also weaken a country’s currency, which boost the exports and pushing up import costs. Like Japan and USA, Denmark is a big exporter. One of Trump strategies - along with direct taxations into imported goods - is to decrease interest rates to gain advantage against China.

The stimulus for comsumption is to boost the economy and increase the aggregate demand. An investor would prefer to invest in a company or enterprise rather than government bonds, and a family would rather buy a new house today than save for 5 years.

Other monetary policies could be the reduction of the reserve ratio or quantitative easing, but all expansionary policies can lead to bad consequences in the future, like high inflation rates.


Can increase of NEER cause higher inflation?

An increase in the index from one point in time to a later point indicates a nominal appreciation over that period

Standard economic theory might suggest that would lead to lower prices than without such an appreciation, since imports would be cheaper and demand for exports could reduce

The Czech Krona was devalued at the end of 2013.  An article published a year later by the Czech National Bank suggests it thought it achieved the aims, saying

  Without having depreciated the exchange rate, deflation would have been significant, at least -1%, meaning that the Czech economy would already have entered deflation for an extended period. However, thanks to the exchange rate depreciation, the threat of deflation caused by a lack of demand and a downturn in the economy passed. Despite a significantly greater anti-inflationary effect abroad than was expected at the end of 2013, core inflation returned to positive values after nearly five years.

and this is consistent with standard economic theory

But if you want a rationalisation of why a currency appreciation may not have happened during a financial crisis, try thinking of a world where interest rates are tumbling worldwide and where carry-trade investors looking for nominal yield invest in countries where nominal interest rates are higher, possibly caused by higher inflation.  Such a country might see higher inflation than other countries and at the same time an appreciating currency.  The appreciating currency may then attract momentum investors, with the flood of money pushing interest rates lower than would be suitable to control inflation (especially if the country wanted to keep its exchange rates more stable), and the combination of more money and reducing interest rates could lead to higher inflation

To a small degree this may be what has happened in the Czech Republic more recently: it had slightly positive interest rates (until summer 2017, when they started to be increased) while the Euro area has had zero or negative short-term interest rates, and has seen an appreciation in the Krona against the Euro.  There could be other explanations


Why are banks allowed to resell mortgages?

A bank selling mortgages does not, by itself, increase the money supply.

To see this, work through the balance sheet implications step-by-step:

A bank makes a mortgage loan, swapping cash assets on its balance sheet for a mortgage asset. The customer receives cash. At this stage, the money supply has been increased.

2a. A bank sells a mortgage to another bank, swapping cash and mortgage assets between the two, but without increasing the money supply.

2b. A bank sells a mortgage to something that is not a bank, reversing what it did when it made the mortgage loan: the bank ends up with a cash asset and loses the mortgage asset, while the not-a-bank entity loses cash, decreasing the money supply.

They key here is that expansions or contractions of the money supply arising out of banking have everything to do with the the extent to which banks in the aggregate are lending cash that has been deposited with them (i.e., deposit liabilities). It’s a stock, not a flow, so you have to analyze the balance sheet impacts of what happens when banks sell an asset, paying attention to how the buyer is financing the purchase, to understand what the effect on the money supply is.

The (glaring) exception to this is that many entities that hold mortgages act effectively like banks, but aren’t banks. They’re often referred to as “shadow banks.” There are a lot of different types of entities that do this, but a simple, clean example is a mortgage REIT. If a REIT buys a mortgage security, it will likely fund it by doing a repo (basically borrowing money but with collateral) with a money market mutual fund, which is lending out its depositors’ money, just like a bank.

One note: In responding, I’m deliberately ignoring the part about liquidity requirements, because money creation by banks can be (and has historically been) bound at different times and for different entities by liquidity, capital, or reserve requirements, but liquidity requirements in the US are not generally intended to constrain money creation per se, but rather as a prudential measure. More specifically, GSE MBS is a Level 1 “high-quality liquid asset” under the liquidity coverage ratio, so swapping mortgages once they’ve received a GSE wrap doesn’t actually do anything to improve a bank’s liquidity ratio.


They don't include anything. It is the pure price for 1 MWh.

Edit.

The producers have been paid the tariffs, which lowers their short run marginal costs quite a lot. There has not been added any taxes to the consumers at this point, however.


If porn is free on the Internet, then how do porn stars actually make money?

Using Google is also free, yet the firm is quite profitable, so a possible solution could be ad revenue.

Another solution could be that your sampling is off. E.g. most urban Democrats do not know many rural Republicans and vice-versa. It is similarly possible that people who do pay for porn do not move in your social circle.

There is also an incentive issue to your sampling. This is a subject most people probably do not wish to discuss. As such you can reasonably doubt if you are getting honest answers. If one were to answer your question in the affirmitive, he would incur not only the stigma of watching porn, but also the stigma of having paid for it, which as you say is kind of strange given its free availability.


Dot over a variable, Neoclassical growth model?

Maybe an example will help you. Imagine you start working at a company today and they offer you a salary of $50,000$ USD with a promise of raising it to $55,000$ USD next year. At the same date your boss is also hired with a salary of $100,000$ USD which will be raised to $110,000$ next year. Surely you will think it is unfair your boss will get a lot more next year than you will. But let's look at it from another perspective

  For you

$$
\frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{55,000 -50,000}{50,000} = 0.1$$

  For your boss

$$
\frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{110,000 -100,000}{100,000} = 0.1$$

Which is exactly the same number, people usually call this a relative increment, which in this case is $10\%$. So from this point of view it is the pretty fair. To make this a bit more technical call $S$ the salary,  $t$ today, and $\Delta t$ make it 1 year, so that the equations above becomes

$$
\frac{S(t + \Delta t) - S(t)}{S(t)} = r \Delta t
$$

where $r$ is the example above just means an increment of $10\%/{\rm yr}$. Now let's organize this a bit

$$
\frac{1}{S(t)} \frac{S(t + \Delta t) - S(t)}{\Delta t} = r
$$

And from this, you probably will recognize the notion of derivative. If you take $\Delta t$ a very small number you will get

$$
\frac{1}{S}\frac{{\rm d}S}{{\rm d}t} = r \equiv \frac{\dot{S}}{S}
$$

Note that this is just fancy notation to represent a fractional change of the quantity $S$. In other words: how much does the quantity $S$ change in a moment, compared with this current value.

In most cases the quantity $r$ is not a constant, but depends on time. Such in your second expression, but do not loose track of the meaning there: It tells you how the relative change of quantity $c$ is affected by other factors. For example $r$ will make it grow, but $n$ will make it decrease (because it is substracting)


Definition of a 'Competitive Firm' and a 'Perfectly Competitive Firm'

Without seeing a particular context, it is hard to be 100% sure. But typically the term "competitive" is used as shorthand for "perfectly competitive".

  A market is perfectly competitive if everyone in that market takes the price as given. That means a seller in a perfectly competitive market can sell as much or as little as he likes (and a buyer can purchase as much or as little as he likes) at the prevailing market price. Nobody has any power to influence the market price.

Contrast this with something like a monopolist, which can set whatever price it likes and therefore obviously does not take the price as given.

A firm in a perfectly competitive market would be said to be a "competitive firm".


Yes it is subjective. But subjective valuations are where demand curves come from (at least for final goods and services). Consumer surplus is simply a measurement of aggregate willingness to pay so it is subjective (like utility) and measured in currency units (unlike utility).

For example, if a movie ticket is for sale for ten dollars and you'd pay up to 15 dollars to see it then your consumer surplus is five dollars. This valuation is purely subjective. Another consumer could have a different maximum willingness to pay for the ticket. If you thought the movie was going to be less entertaining or had something else fun to do the maximum you'd pay would be lower and so would the consumer surplus.

One area of this that once confused me is the units. Utility functions typically are discussed as though they were in units of utils but utils cannot be compared across people. However, willingness to pay, which is an implication of the budget constraint and preferences implied  by household utility functions, is measured in currency and can be compared across people. That is, it is sensible to say "you would pay more to see Iron Man IV than I would" but not "you would get more utility from seeing Iron Man IV than I would".

So it isn't exactly as @Kitsune says, that consumer surplus is utility minus price. Rather, consumer surplus is willingness to pay minus price, which are both in currency units and can be aggregated over and compared across consumers.


What metrics can be used to distinguish between capitalistic and socialistic economies?

My opinion is that “capitalism” and “socialism” in this context is in the eye of te beholder. The European countries are mixed economies.

One standard metric is the size of government expenditures as a percentage of GDP. One issue is that different countries have different ways of measuring national accounts data.

The IMF World Economic World Economic Outlook (WEO) has a database that is standardised across countries. For any one country, the national accounts data would probably be a better source, but the key to the WEO is that they worked on comparability for you already.

Link to the data: link to IMF WEO data page

One series to look for is “general government expenditures as a percentage of GDP.” You want “general government” (and not something just like the central government) as countries like Canada have large sub-sovereign governments.

Expenditures gives us a rough idea of the footprint of the government in the economy. However, it will miss things like the effect of regulations. Instead of the government directly taking over some activity, it could force the private sector to follow a very specific set of rules. Whether that is “socialism” or a “properly regulated economy” turns into a political debate.

If you are only interested in European data, Eurostat would likely be your source. Eurostat has worked on getting European countries’ data to follow similar standards, so there is good cross-country comparability for now. I do not know what series to look for there, so you would have to dig. Link to Eurostate website.

The only other sort of data  to work with are various “ease of doing business” indices. The quality of those data series can be debated; there was some recent controversy with some World Bank data of this type in the news. The concern is that it is unclear what to include in these indices, and many of the inputs are qualitative. As may be obvious, I am not a fan of these series, so I do not have any recommendations (you can either search or someone else can give an answer).


When can one read the general theory of employment,interest and money?

I don't think you should read it at any point in your carrer, honestly. It is severily outdated, ambiguous and overall not worth the time and effort put into reading it.

Try Screpanti & Zamagni's An Outline of the History of Economic Thought. Undergraduates do better when studying the works of economists who have been dead for decades as what they are, historical curiosities.

If you are really bent on reading the General Theory and/ or pretend to research history of economic thought, then I'd say after you go through an intermediate macroeconomic theory course you're good to go.


What are the alternative uses of unlimited wants and scarce resource?

Scarce resource : is a (physical) limitation which forces us to make choices accordingly. It is what is available under current technology and cost of extaction. We could say its a supply side perspective and generally refers to material.

Unilimited desires: Is a demand side perspective. Individuals or organizations usually want more tomorrow than today. Even if you get satiated of one good you will want more. For instance if you have enough foood you can't eat more but maybe you would wish a special meal in a fancy restaurant. So even if cooking home was sufficient to narch your need, after an increase of income you will buy more services... bigger house... There are diminishing returns to consume more of the same good but imagination makes it possible to always find a new object of desire. Thus unlimited wants.

By combining these two phenomenons, a scarcity situation arises and humanity must make choices hopefully to fullfill the most pressing needs.


An obligatory draft is a tax collected in kind - productive time. Instead of producing in the private sector, citizens of the economy offer their services to the army.

Now, we should acknowledge that the existence of an army offers some desired public good (or at least that the society implementing the draft thinks so). Is this a utility-enhancing, or a production-enhancing public good?

Thompson, E. A. (1974). Taxation and national defense. Journal of political economy, 82(4), 755-782.

argued that it is safeguarding property rights of the local population against foreign aggressors, and so it is production-enhancing. Moreover it is subject to congestion: the larger the output/capital base of the economy, the larger military spending has to be to provide the same degree of security (one can find a mention of the above as well as a related model in Barro's book on growth but only on the 1st edition, p. 160).

Here, the tax to fund military expenses is collected in kind, so we could model this situation with the following tweaked Cobb-Douglas production function

$$Y = AK^a(N -L_a)^{1-a}\cdot (BL_a/Y)^\gamma \tag{1}$$

where $N$ is population/total available productive time, there is no labor-leisure choice, $L_a$ is the time spent in producing security services through the draft. The last term is the indirect contribution of military spending in output (by safeguarding property rights), and it is divided by output in order to reflect the congestion characteristic. It is treated as an externality from the private sector.

But the government/social planner can maximize output for any choices of the private sector regarding investment in physical capital, by maximizing the above production function, after taking into account the externality, i.e. by maximizing

$$Y = A^{1/(1+\gamma)}K^{a/(1+\gamma)}(N -L_a)^{(1-a)/(1+\gamma)}\cdot (BL_a)^{\gamma/(1+\gamma)} \tag{2}$$

with respect to $L_a$. We obtain

$$L_a^* = \frac {\gamma}{1-a+\gamma}N$$

namely, that the "size of the draft" in terms of productive time should be a constant proportion of the population.

The problem here is that the private sector looks at $(1)$ and "ignores" the last term. So it behaves as though the capital elasticity of output is $a$ while in reality is smaller, $a/(1+\gamma)$ as we see from $(2)$ (this is due to the existence of congestion in the public good).

This means that the private sector will tend to over-invest. In such cases a proportional tax on output is optimal from a welfare point of view because it would internalize the externality...

...and this means that it would be better, aside from possible non-economic issues, if instead of an obligatory draft the government applied a proportional tax on output in order to fund military expenses (i.e. have a professional, paid army). In this way it would both provide this desired public good, and internalize the externality, reducing the rate of capital investment down to the optimal level.

(Note: obviously the above is a very abstract approach. One could point out that armies need much more than just the labor of their soldiers, in terms of equipment, but even in terms of the subsistence expenses of the soldiers themselves. Then one could model the existence of a tax to fund these, and examine whether it can sustain the output maximizing level of Labor drafted and be at the optimal level as regards the target of internalizing the externality).


Are strict dominance solvable games weakly dominance solvable?

You're right. Let $s^*=(s_1^*,\dots,s_N^*)$ be the equilibrium of a strictly dominance solvable game. Then by definition,
$$
u_i(s_i^*,s_{-i})>u_i(s_i,s_{-i})
$$
for all $i$, all $s_i\ne s_i^*$ and all $s_{-i}$. This implies that
$$
u_i(s_i^*,s_{-i})\ge u_i(s_i,s_{-i})
$$
for all $i$, all $s_i\ne s_i^*$, all $s_{-i}$ and with strict inequality for at least some $s_i$ (in fact, for all $s_i\ne s_i^*$). This makes $s^*$ an equilibrium satisfying the weak dominance solvability criterion.

Your quoted definition of strict/weak dominance solvability makes me a little uncomfortable though. I'd say

  A game is strict (or weak) dominance solvable if the process of iteratively removing strictly (or weakly) dominated strategies leads to a unique outcome (i.e. only one strategy for each player survives).


I don't have a copy of Gibbons handy, so I cannot speak to the specific model presented there, but only generally. The intuition of the conclusion is based on the combination of the following factors:

Whenever the firm can tell the high and low types apart, it's willing to pay a high wage to high type and a low wage to low type
If the firm cannot tell the two types apart, it will pay both types the same wage equal to their average productivity
Low type will try to pretend to be high type whenever possible to get a higher wage (note that even the average wage is higher than the low wage)
High type, in order to get the high wage as opposed to the average wage, will try to distinguish themselves from the low type whenever possible

Hence, if $e^*(H)$ is a sufficiently small cost for the low type to pay to pretend to be high type, then the high type choosing $e^*(H)$ will only result in them getting the average wage instead of the high wage.

As a consequence, high type would be willing to pay a bigger cost $e_s>e^*(H)$ --- a cost so big for the low type that they won't find it profitable to pay to get the average wage, but at the same time small enough for the high type so that their gains from getting the high wage instead of the average wage outweigh such a cost --- to distinguish themselves from the low type.

The key here is to take into account the presence of the low type (and asymmetric info). They are the ones who cause the reduction of the high type's wage to the average level at cost $e^*(H)$.


Effects of Eliminating Government Controls on Supply (organ donation)

The Iranian organ market
It might be worth looking into the organ market in Iran. Unlike in other countries, it is legal to sell your kidney to patients who need a new one. The process is regulated and charities actively help pay for patients who cannot afford to pay the donor themselves.
The result is that a kidney, which would cost more than $100.000 on the global black market can be bought for a few thousands of dollars. This also highlights an overlooked problem with the concern that a free market would put rich people in front of the line. The fact is that very rich people can always find a way to acquire the goods they want. If they do it illegally, they can hire expert lawyers to get them out of trouble.
The example of Iran shows that allowing people to sell their organs for profit will lower the barrier to entry to the point where middle class people could conceivably afford one. It also becomes possible for charities to afford to buy organs for the poor. This would not be possible if they were to pay the black market price or wait indefinitely to win the waiting-list lottery.
https://en.m.wikipedia.org/wiki/Kidney_trade_in_Iran


1) "To first order" appears to be a shorthand for "to a first-order approximation", meaning in turn, "to a linear approximation".

2) It is not that "the one implies the other", it is that the one is consistent with the other.

In attempting to arrive at a functional specification for the "tax distortion costs" function, say $C_D=h(T)$, first we reason that if tax revenue is zero (not due to inefficiency of tax collection and/or tax evasion, but because the tax rate is set to zero), then distortions due to tax will also be zero. This implies that $h(T)$ should not contain a constant term, so that $h(0) = 0$.

Moreover if we reason (as D. Romer does) that "distortions by taxes are likely to increase more than proportionally with the amount of revenue raised", this implies that $h(T)$ should not be linear in $T$, but it should be convex (this is what it means to say "more than proportionally").

The natural step then is to consider a quadratic distortion cost function, without a constant term:

$$C_D =  aT^2,\;\; a>0$$

which reflects both our theoretical arguments (and in particular, $h(0) =0$).

But this also implies that "distortion costs are zero to a first-order approximation", because we have

$$h'(T) = 2aT \implies h'(0)=0$$

so a first-order approximation (Taylor expansion) around zero would give

$$h(T) \approx h(0) + h'(0)\cdot T = 0+0\cdot T =0$$


Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why?

A clarification: The values $w,r$ are not independent of input combinations: you have already solved for $K/L$.

1) Unless you assume that the price of the output is equal to 1, there is a minor mistake, as output prices affect factor prices.

2) If there is a profit maximizing pair $(K,L)$, then for all $\alpha \in \mathbb{R}_+$ $(\alpha K,\alpha L)$ will also be profit maximizing. This is because the production function you are looking at has constant returns to scale (CRS).

A consequence of CRS is that if the function is differentiable then both marginal products will be homogeneous of degree zero, hence $r$ and $w$ will only determine the ratios of $K$ and $L$, not their level.


Have UK exports (goods and services) to Ireland declined in value since 2011?

The ONS Balance of Payments Pink Book gives the following figures for UK exports to Ireland (£million).  The figures in Euros would be different, as would Irish reporting of the numbers, but there seems to have been a sharp drop in services exports from 2012 to 2013.

2016 data is due to be published in October 2017, though trade in goods numbers may already be available

 £m     Goods Services  Both
CDID    QDJJ    FYVJ    LGIA
2011    18223   11475   29698
2012    17234   11615   28849
2013    18592    8780   27372
2014    18104    9288   27392
2015    16764    9321   26085


Subgame Perfect Nash equilibrium: two stage game

In part a, if $B = D_1 + D_2$, then the SGPE should be

$\left\lbrace D_1 = \frac{W}{3},\ D_2 = \frac{W}{3}, \left\lbrace  P = \alpha (D_1 + D_2), \ S = (1 - \alpha) (D_1 + D_2) \right\rbrace \right\rbrace$

Don't say $P = \alpha \frac{2W}{3}$. That's an action, and the second stage best respond should a strategy (function) to make the equilibrium subgame perfect.


What is the difference between "Social Choice Theory", "Public Choice Theory", and "Collective decision-making?

Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a collective decision or social welfare in some sense.1

A social choice is related to a collective decision. Many different opinions, preferences and interests are taken into account to form a decision which is acceptable by everyone who is involved. Opinions, preferences and interests don't just have a personal background, but a social one, too. The society around you influences your thinking and doing by socializing. Hence the name: Social choice theory.

Group decision-making (also known as collaborative decision-making) is a situation faced when individuals collectively make a choice from the alternatives before them. The decision is then no longer attributable to any single individual who is a member of the group.2

Collective decision-making refers to a group of people who are committed to make a decision. Collective means that the decision isn't made upon the ideas of one person but of several. Responsibility is shared among these people. One alone can't be held responsible for the decision as several people agreed to it.

A. What is Public Choice? Public choice is the study of political decision making.
i. It attempts to understand how public policies come to be adopted using economic models of the impacts of policies on individuals,
ii. and models of individual political behavior under a variety of political institutions.
iii. It tends to use rational choice models to do so, which is largely what distinguishes it from mainstream and classical political economy.3
[...] iii. (Roughly speaking, public choice, and/or rational politics, is the application of
economic models of human action to politics.)

The theory of Public choice believes that every politician is rational and economically (self-)interested. The goal of Public choice is to analyze and predict the behaviour of politicians and bureaucrats - analyzing their public choice.
Social choice theory and group decision-making are somewhat related to each other. However, a social choice doesn't need to be made by several people, it just has to represent several people. By considering a lot of interests and preferences, one or more make a decision to achieve the best.
Public choice is more like a field of research. Hence, it's related to both Social choice and group decision-making. As it's the application of economic models of human action to politics, you research the advantages and disadvantages of doing public decisions.
Long story short: Social choices and group decision-making are actions, Public choice is a field of study which researches political behaviour out of a economic view. All three are related to each other but still are independent approaches for making a decision.

1 Wikipedia: Social choice theory
2 Wikipedia: Group decision-making
3 http://rdc1.net/class/BayreuthU/PUBCL1.pdf


As I said in the comments, without additional details proving anything is either impossible or trivial. Impossible if I do not make any assumptions, trivial if I am allowed to make any assumption I like.

Let me give you a hint on how to solve this in the framework how it is usually taught.

The assumptions made are:

There are decreasing returns to scale to any single production factor
There are constant returns to scale to all production factors together
Technology and the labour force grow at an exogenous rate (which may be 0)
The production function is cobb-douglas form

Then, denoting national income at time $t$ as $Y_{t}$, capital as $K_{t}$, labour as $L_{t}$, technology as $A_{t}$, a fixed factor such as land as $X$ (which is independent of time) and a non-renewable resource extraction as $E_{t}$ we can formulate the production function as:

$Y_{t}=A_{t}K_{t}^{\alpha}X^{\beta}E_{t}^{\eta}L_{t}^{1-(\alpha+\beta+\eta)}$

We can determine the income per capita (or more precisely the income per labourer) by dividing both sides by $L_{t}$. Using small letters to denote a variable in per labourer terms we get:

$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$

Note that whereas $X$ is independent of time, $x$ is not because the labour force grows over time so the amount of land per labourer changes over time. You can now rewrite this equation in terms of growth rates to determine how it is (im)possible to continue to grow income per capita even in the presence of a non-renewable and fixed production factor.


The demand functions you have derived are correct.
The Marshall demands are given as
$$ x^\star(p_x,p_y,M) = \sqrt{\frac{M}{p_y}} \\[8pt]
y^\star(p_x,p_y,M)  = \frac{M - p_x \sqrt{\frac{M}{p_y}}}{p_y},$$
where $M$ is income and $p_x$ is pizza price and $p_y$ is the apple price.
To answer the question of how the agent responds to change in pizza price it would be standard to find derivatives and/or elasticities of demand with respect to the price $p_x$ for pizza.
The derivative of pizza demand with respect to own-price is given as
$$ \frac{\partial x^\star(p_x,p_y,M)} {\partial p_x} = 0, $$
which implies that the elasticity is
$$El_{p_x}x^\star(p_x,p_y,M) = \frac{p_x}{x^\star(p_x,p_y,M)}\frac{\partial x^\star(p_x,p_y,M)} {\partial p_x}  = 0,$$
so you can conclude that demand is perfectly inelastic. When the price increase the consumer simply keeps demand constant and substitutes away from apples. This is manifest from the fact that
$$ \frac{\partial y^\star(p_x,p_y,M)} {\partial p_x} = - \sqrt{\frac{M}{p_y^3}}, $$
which is clearly negative and by implication the cross-price elasticity $El_{p_x}y^\star(p_x,p_y,M) <0$.
This is "unusual" if you expected apples and pizza's to be substitutes with the demand of one increasing when the cross-price increase.


Derive marginal productivity conditions in DW Jorgenson paper

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.


Will SRAS curve definitely shift if LRAS curve shifts?

The SRAS also shifts.

SRAS is normally used in models where the supply side does not adjust immediately to the new conditions in the market, i.e. models with nominal rigidities. Examples are: sticky wages, sticky prices.

Other instances where adjustments in prices or nominal money supply are not immediate and in which money has a real effect are the Lucas 'misperceptions' or 'islands' model or Cash in advance models.

However, and to the point of answering your question, all these models coincide that in the long term and when the agents in the economy have had the time to adjust their choice variables and/or information symmetry is restored, the AS is vertical again (LRAS). So, if there has been sufficient time for the long run AS to adjust, by definition of short run, i.e. shorter than the long run, there has been enough time for the SRAS to adjust as well.


What does an individual's demand function describe?

The demand function does not say how much he's willing to pay for an additional amount dq. The demand function says: given the price of good is $p$, you will purchase $Q(p)$ of them. I think your confusion lies in the fact that it is possible to charge different prices for each unit of good. Notice that our theory is assuming one price, not multiple.

Moreover, the notion that consumer surplus measure the difference between what you paid and what you are willing to pay is completely false. Suppose the demand function is $Q(p)=-5p+100$. Suppose further that the price is $5$, so you would be purchasing at a quantity $75$. The consumer surplus is: $$CS=\int_0^5Q(p)dp=\int_0^5 -5p+100dp=562.5.$$ If interpreting it as the difference between what you paid and what you are willing to pay, we know that you are willing to pay $562.5+5\times75=937.5$ for $75$ units of them. In other words, you are willing to accept a price $937.5/75=12.5$. However, plugging this number into our demand function, we find that $Q(12.5)=37.5$, i.e. you are only willing to buy 37.5 units of the good, which clearly contradicts to our interpretation of consumer surplus.

For a better treatment of consumer surplus, I would recommend you to consult advanced texts on microeconomics theory such as Mas-Colell's or Varian's more advanced book.


Is there a serious forum/chat (not Q&A like StackExchange) on Economics?

This could be useful: https://groups.google.com/forum/#!forum/decision_theory_forum

Also, in econjobrumors sometimes there are interesting discussions...


Solving rational expectations model - Sims form

I think I have managed to solve it. However, not the way I was initially hoping. I simplified the stacked matrices using the given conditions and some assumptions. Here is my solution:

Eq. (3) I write as $\pi_{t+1} = \alpha \beta \pi_{t} + \beta \theta_{t} + \eta_{t+1}$

Forwarding equation (4) one period and arranging it in terms on $b_{t+1} = -\varphi_{1} {\pi_{t+1}} - \varphi_{2}{\pi_{t}}  + (\beta^{-1}- \gamma)\widetilde{b_{t}} - \varphi_{3} \theta_{t+1} \ - \psi_{t+1} - \varphi_{4} \theta_{t} = 0$

$\pi_{t+1}$ simplified by eq. (3)

$\theta_{t+1} = \rho_{1} \theta_{t} + \epsilon_{1,t+1}$

$\psi_{t+1}$ can be written as its AR(1) formulation as well. Putting the four equations in matrix form results in:

Which is what I was after. However, this made me wonder what was wrong with the method I was trying above. Looking at the top corner of the coefficient matrix (i.e the 4x4 matrix in the top left corner) it is clearly $\Gamma_{0}^{-1} \Gamma_{1}$. But the rest of the matrix I am unsure of.

And question (2) remains, once I have solved for the variables of interest, will I be ready to make IRF's?


Raising your wage means that you can consume more. It also means (if you want to) that you can afford to consume more leisure. However, because you have to forgo more consumption per hour of leisure than you did at the old wage, this causes you to substitute away from leisure and towards consumption (more consumption, less leisure) which causes you to supply more labor.

However, with a high enough wage, the marginal utility of consumption is very low while labor supply has gone so high that the marginal utility of leisure is very high. Eventually the wage is so high that the same wealth effect that made you work more makes you consume so much that consuming more looks less attractive than working less. This causes labor supply to drop and the supply curve to bend back.

All of this depends on the household utility function. There are some utility functions where the backward bending never occurs. For example, linear utility.


Textbook on the mathematics of RBC/DSGE models?

The mathematical theory behind DSGE models can be found in any textbook on stochastic dynamic optimisation.

One common reference that economists use for this is Stokey, Lucas and Prescott. Of course, they focus exclusively on recursive methods, but (perhaps) the lion’s share of dynamic problems in economics are solved in this way.

There is also a treatment of dynamic optimisation (including non-recursive and/or continuous-time methods) in Acemoglu.


One-shot deviation principle for infinite repeated games and dynamic programming

A deviation (one-shot or not) can certainly generate a sequence that differs from the optimal one for an arbitrary number of periods.

You could treat a dynamic programming problem as a repeated game between one player and chance. The one-shot deviation principle should then carry over from repeated games to dynamic programming.


Rising Marginal Costs: where is the empirical evidence?

Based on data/work with consultants/business school discussions with executives: A good example of a set of industries where marginal costs are rising is commodity production (metals, agricultural products, and energy such as oil). In such industries, companies own many plants (Glencore, BHP Billiton) and they tend to switch on production in a plant when the price is above the marginal cost of that plant. So it makes sense to order plants in increasing order of their marginal cost, and assume that the lowest-marginal-cost plant will be the last one to be shut down. Of course if the firm acquires a plant that has a lower marginal cost than the lowest-marginal-cost plant, it will shift that firm's marginal cost curve. Look for Carmin Nappi's slides online on the aluminum industry, plenty of that analysis going on.

Some of the slides are in here, see slide 11. http://www.riotinto.com/documents/Media-Speeches/Rio_Tinto_Alcan_Carmine_Nappi_11_June_2008.pdf

Now this is a similar framework adopted by Pindyck, for instance in http://www.jstor.org/stable/41322920?seq=1#page_scan_tab_contents.


Applications/generalizations of a theorem of Debreu

This result is indeed a version of Berge's maximum theorem. If there is a continuous function $u:M\times H\to\mathbb{R}$ such that $x\preceq_e z$ if and only if $u(e,x)\leq u(e,z)$, one can derive the result directly from Berge's maximum theorem. If $H$ is locally compact, as it is the case if $H=\mathbb{R}^n$, then such a function can always be found, this follows from Theorem 1 in Mas-Colell's On the Continuous Representation of Preorders (at least if $M$ is metrizable, I'm not sure on that point). More on such "jointly continuous utility functions" can be found in chapter 8 of Representations of preference orderings, 1995, by Bridges & Mehta.

Now Debreu did not have such a result available, so he worked with preference relations and essentially reproved Berge's maximum theorem (the generalization is mathematically straightforward). Why did he do so? To understand that, one needs to understand the point of Debreu's paper, which is finding a topology on preference relations that has nioce properties and makes economic behavior continuous. The need for such a result comes from the literature on economies with a continuum of agents.

What does it mean that a continuum of agents economy is the limit of a sequence of finite eonomies? One answer is that the distribution on characteristics of agents converges to the distribution of characteristics in the continuum economy, so the notion of convergence is convergence in distribution. To make this idea operational, one needs to topologize the characteristics of agents. Now an agent is characterized by her endowment and by her preferences (and in more general models by her consumption set). There is a natural topology on endowments, the Euclidean topology, but it is less straightforward to topologize preferences, and that is what Debreu did in his paper. An exposition of this distributional approach can be found in Hildenbrand 1974, Core and equilibria of a large economy.

Now, there are cases where one would like to apply Berge's theorem for non-compact sets of choices. This can be important when studying economies with infinite dimensional commodity spaces, in which being closed and bounded does not imply compactness. One way to deal with this problem is to find a compact set so that the correspondence is compact-valued and nonempty-valued when restricted to this set. There is a large, very technical, literature on "generalized games" or "abstract economies" (basically normalform games in which strategy spaces depend on the actions of others), and they implicitely often contain non-compact generalizations of Berge's theorem. If you can get your hands on the book, check chapter 4 of Xian-Zhi Yuan 1999, KKM Theory and Applications in Nonlinear Analysis. My impression, however, is that these results proved to be not that useful in economic applications. To prove existence of Walrasian equilibria in models with infinite dimensional commodity spaces, one usually uses different methods.


Of course markets can be shortsighted. The same problem can apply to any institution though. In order to correct a market failure using government it's necessary to appeal to the self interest of various interest groups, which may end up putting you back in square one.


Finding Cobb-Douglas Hicksian Demand using Duality

You have found the value function $V(p,I)$ which in your case is given as
$$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}},$$
where $\alpha_2 = 1- \alpha_1$ and $I$ is income.
The Hicks demand is a function of the utility level denoted $u$. An agent facing prices $p$ and income $I$ gets utility $V(p,I) = u$. Therefore you can set
$$u = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}},$$
and invert to solve for $I$ in order to get
$$\frac{p_1^{\alpha_1}p_2^{\alpha_2}}{\alpha_1^{\alpha_1}\alpha_2^{\alpha_2}}u = I,$$
which is the expenditure function. You can insert this in your Marshall demand
$$\frac{\alpha_j I}{p_j} = \left(\frac{\alpha_j}{p_j}\right)\frac{p_1^{\alpha_1}p_2^{\alpha_2}}  {\alpha_1^{\alpha_1}\alpha_2^{\alpha_2}}u = h^\star(p,u).$$
Or in the case of $J$ goods: You have CD preferences with well known Marshall demand
$$(1)\ \  x^\star_j(p,I) = \frac{\alpha_j I}{p_j} \Leftrightarrow \frac{x_j}{\alpha_j} = \frac{I}{p_j},$$
and the utility function is
$$(2) \ \ U(x) = \prod_j x_j^{\alpha_j} = A \prod_j \left(\frac{x_j}{\alpha_j} \right)^{\alpha_j},$$
where $A := \prod_j \alpha_j^{\alpha_j}$ the constant $A$ serves the purpose of making the algebra somewhat easier. Inserting from (1) in (2) gives you value function
$$V(p,I) = A \prod_j \left(\frac{I}{p_j} \right)^{\alpha_j} = \frac{AI}{\bar p},$$
where $\bar p:= \prod_j p_j^{\alpha_j}$ and $I$ is constant across goods hence not bound by index so when $\sum_j \alpha_j = 1$ it $\prod_j I^{\alpha_j} = I$. From here you isolate income to get
$$\frac{\bar p}{A}V(p,I) = I$$
you then insert this in Marshall demand to get
$$x^\star(p,I) = x^\star\left(p,\frac{\bar p V(p,I)}{A}\right) = h^\star(p,V(p,I)).$$
Knowing the Marshall demand $\alpha_j I/p_j$ insert $\bar p V(p,I)/A$ with $V(p,I)$to get
$$h^\star(p,V(p,I)) = \frac{\alpha_j}{p_j} \frac{\bar p V(p,I)}{A} ,$$
let $V(p,I) = u$ to get
$$h^\star(p,u) = \frac{\alpha_j}{p_j} \frac{\bar p u}{A} = \frac{\alpha_j}{p_j} E(p,u),$$
where $E(p,u)= \frac{\bar p u}{A}$ is expenditure function (this follows from the fact that we arrived at $\bar p u/A$ as an expression for income by inverting the value function).


Verification of understanding of the substitution and income effect of normal, inferior and Giffen goods?

[Edited: I think that Theoretical Economist is right in highlighting that I'm perpetuating the confusion between the mathematical formalism, and one way of getting to the intuitive understanding. It also comes down to language referring to the slope of the curve versus effects that are positive or negative depending on moving up or down that negatively-sloped curve. So I clarified the table and text consistently against a price rise and Slutsky.]

I think your intuition is correct, but the derivatives have slopes in the other direction in Slutsky's equation. And it's Slutsky's equation that's ultimately given us this theoretical framework, so let's stick with that here ;)

To be clearer, let's discuss for a Price RISE:

Normal Good:   Demand- = Substitution-- Income-  [Same direction]

Inferior Good: Demand- = Substitution-- Income+  [S > I]

Giffen Good:   Demand+ = Substitution-- Income++ [I > S, weirdly]

I start my understanding of this with the demand curve. Under the “usual” laws of supply and demand, the demand-curve is backward-sloping, so as the price rises, we demand less of it. This is the Ecos101 curve we're familiar with; and it applies both to normal and inferior goods. But for Giffen goods, the demand curve has a positive slope... As the price rises, we actually demand more of it. Usually, examples of luxury goods are given as Giffen goods. The argument runs that as price is an indicator of quality and wealth of the purchaser, people actually by more of these goods as the price rises. The actual empirical evidence for the existence of Giffen goods in the real world, though, is slim and hotly debated... Sorry Apple!

So we understand the demand curves are backwards for normal/inferior, but upwards for Giffens. Why?

It was Slutsky who best disentangled demand curves as comprising an income effect and a substitution effect through a synthesis Marshall and Hicks' respective demand curves. (These are the uncompensated and compensated curves per the comments below.)

Per Slutsky's equation, Theoretical Economist is correct: The substitution effect can only ever be negative / downward-sloping. That leaves only the income effect to explain the differences between the three types of goods.

The link is that as the price of a good rises, in effect income falls. For normal goods, as effective income falls, we demand less of the good. For inferior goods, as income falls, we demand more of the good. Typical examples of inferior goods include staple foods. The idea is that as we get poorer, we cannot afford “rich” foods like meats, exotic fruits, chocolate, and so on; but must "downgrade" to cheaper foods.

Giffen is where things can get counter-intuitive. As the price of a Giffen good rises, even though effectively our income falls, we demand the good so much more that it outweighs the (always-negatively-sloped) substitution effect to yield the total positive demand.

I hope that's both more formal and clearer to understand.

Regarding your final question, the intuitive understanding around the increase/decrease in substitutions based on price rises/falls can be confusing against the fact that the slope of the substitution term in Slutsky is always negative.

From a mathematical perspective, the slope of substitution is always negative, but we can move "up" or "down" that curve to produce "positive" or "negative" changes... More or less substitution of a good.

For the second part of your question about the constancy of these effects; goods can apparently change between being normal/inferior. So if we do as we typically do in economics classes and draw out demand curves and indifference curves as straight lines, it holds.

But in reality, we know that demand curves are not immutable, straight lines, but can (a) curve and (b) change under exogenous variables. Think of the example of an asset bubble. For a while, the asset may actually look like a Giffen good: The price rises, people demand more as they jump on the bandwagon. Suddenly, the bubble bursts, and that asset is “demoted” to being a normal good (just as everyone’s income is decreasing... oops!)


How exactly does money supply causes inflation?

The view that money growth causes inflation may have been a consensus view during the (old) Monetarist era, but it is unclear whether that still holds.

(One somewhat related question here discusses whether inflation can occur without money supply growth - link)

As seen in the data around Quantitative Easing, we certainly see deviations between money supply growth and inflation "in the short term."

In the "long run" (which is what you asked about), it is unclear what causes what. Other than in Monetarist models (which are arguably no longer consensus), the money supply is determined by a portfolio allocation by the household sector. That is, people hold money instead of interest-bearing bonds for whatever reasons. What policymakers control is the policy rate of interest. Therefore, money growth is an outcome of the policy environment, like the inflation rate. In other words, faster nominal GDP growth (and hence inflation) tends to cause faster money supply growth.

However, if we want to assume that policy makers are controlling the money supply, and that money growth indeed causes inflation, modern theories would probably lean on the following two mechanisms.

Money does not just appear; it is a government liability, and the growth of government liabilities implies fiscal deficits. That is, loose fiscal policy raises demand, and causes inflation.
Expectations. If people are convinced that money supply growth causes inflation, they will watch the published money supply numbers. (In the United States, money numbers are published weekly, and were followed closely during the early 1980s. Currently, I doubt that 10% of bond traders under the age of 30 know what day of the week money numbers are published on.) They will raise prices based on what they think the money supply is doing now, and what they forecast it will be doing. I am not expert on Market Monetarism (the modern offshoot of "old Monetarism") but I believe that they emphasise this effect; although they might phrase things differently.


I've changed order slightly, for ease of explanation.

  I think I'm confused because I don't quite understand the specifics of
  bond trading. So the bonds that I own in period t−1, is their yield
  flexible?

In the real world (and more complex models) certainly - otherwise bond trading would be rather boring.

During the day, bond traders exchange bonds at various prices, and the yield is essentially the implied internal rate of return (modulo quote convention effects) corresponding to those prices if the bond is held to maturity. That is, if you hold to maturity, you are guaranteed that internal rate of return (IRR)
 - absent default - although total return depends upon coupon reinvestment. Otherwise, your profit/loss depends upon what price you sell at (and coupon income during your holding period).

Dynamic economic models quite often diverge from this reality in two ways.

All bond trading during a period occurs at one price. Therefore, you cannot have gains/losses during a single accounting period.
In many models, "bonds" mature in the next period. ("Bills" would be a better label.) Since there are no intermediate time periods, you cannot suffer a capital loss on bonds in such a period. The only way to have capital gains/losses on bonds is to allow for multi-period maturity bonds (or perpetuals), which complicates the model mathematics.

  I.e, once the CB performs the OMO and raises yield, will the bonds I
  already own also receive this new, higher yield?

If you are investing in one period bills, your existing bill matured, and you would invest in a new bill at a higher yield.

If you hold multi-period bonds, their projected internal rate of return would rise. However, the effect on your wealth is mixed. The way that bond yield conventions work is that you would receive the running yield of the bond (+), but you would get a capital loss (-) which roughly equals the modified duration of the bond in the next period, times the yield change. For long maturity bonds, the capital losses are typically much larger than the running yield. In other words, you would likely experience a short term capital loss, but that is made up by higher projected returns. After all, if you hold to maturity, you get the initial yield as an IRR.

  Say in period t-1 I own some government bonds. In period t the
  central bank performs the contractionary OMO. So they lower the price
  of government bonds but raise its yield. This will increase the debt
  service cost for the government and if they decide to fund it by
  selling more bonds, I will experience a positive wealth effect.

Leeper's text is phrased in a somewhat confusing way. It refers to a model of a representative household, who effectively owns all government liabilities. In a world with multiple households, if you bought a long-term bond at t-1, you could easily have less wealth in period t; the increased interest could be received by other households who held 1-period bills that matured, and did not have a capital loss.

That is, he is really writing about the household sector, and not individual bond owners.

  Also, in period t−1, when I bought the bonds, I paid a certain
  price. Then the CB starts selling bonds, thus lowering its price while
  raising yield. The higher debt service cost results in the government
  issuing more bonds, surely this lowers the bond prices further [1]? So
  when I eventually sell the bond, I should have lost money [2]?

Sub-question [2] is covered by the above; your gain/loss depends upon the trade-off between the running yield and the capital loss.

As for [1], the effects of supply on bond yields is controversial. In a simplified model where there is no term premium, bond yields depend only upon the expected path of interest rates. If the increased interest payments do not cause the central bank to raise interest rates (over the lifetime of the bond), bond yields are unaffected.

In the real world, there presumably is a term premium, which might be affected by the amount of bond issuance. Even so, it is clear that the effect is not that large: we have had plenty of rate hike cycles, and bond yields did not enter a positive feedback loop of ever-rising yields.


how is a trade imbalance related to domestic savings?

When Krugman talks about savings, he is presumably talking about the difference between production and consumption (personal consumption and government consumption) in a country as a whole. Although measured in monetary terms, this is really about stuff, i.e. goods and services. It is not about personal savings of money in a bank.

Some of the difference will be used to make things producing stuff for the future, i.e. as investment. In a closed economy, in this sense savings must equal investment, since building up stocks of stuff for future use would also count as investment, while simply wasting the excess stuff would make it not count as valuable production.

In an open economy, it is possible for production and use (consumption and investment) to differ, and so it is possible for savings and investment to differ.   If use exceeds production, i.e. investment exceeds saving, the extra stuff has to come from somewhere, namely as net imports of stuff.  If production exceeds use, i.e. saving exceeds investment, then the extra stuff has to go somewhere, namely as net exports.

So what Krugman is saying is that having a trade deficit is associated with production being less than use, and so with savings in this sense being less than investment.  So he is suggesting that the US trade deficit is associated with low savings.  Using his definitions, that is almost tautologically true as a matter of arithmetic. It is also more coherent than Trump's simple exports good, imports bad, which is little more than saying send stuff abroad but do not get stuff back.

Krugman's next step is to say that the US trade deficit is a consequence of low savings. That is putting a suggestion of causality into the relationships, something much less obvious. It also raises the question of how to change the position; elsewhere it appears that Krugman would not support efforts to reduce consumption or investment. He would welcome increased production; so too would Trump.


I am having trouble understanding what $C(\{x, y, z\}) = \{x, y\}$ means.

MWG already explain this on p.10:

When [$C(B)$ contains more than one element], the elements of $C(B)$ are the alternatives in $B$ that the decision maker might choose; that is, they are her acceptable alternatives in $B$. In this case, the set $C(B)$  can be thought of as containing those alternatives that we would actually see chosen if the decision maker were repeatedly to face the problem of choosing an alternative from set $B$.

[H]ow can we even have a situation where if the consumer is choosing between $x$ and $y$, he'll always choose $x$, but then when he is faced between choosing $x$, $y$, and $z$, he will choose either $x$ or $y$? Can someone give me a real life example of how this would be possible?

Consider a voting situation as follows. Individuals $x$, $y$, and $z$ are potential candidates for the position of a department chair. A five-member committee is to select the chair by voting via simple majority rule. Let's label the committee members $1$ to $5$ and suppose that they each have a private ranking of the three candidates as follows ($\succ_i$ is member $i$'s ranking):
\begin{align}
x\succ_1 y\succ_1 z \\
y\succ_2 x\succ_2 z \\
z\succ_3 x\succ_3 y \\
y\succ_4 z\succ_4 x \\
x\succ_5 z\succ_5 y
\end{align}
Suppose that each member will vote for his/her top-ranked candidate. Then when only two candidates $x$ and $y$ are under consideration, $x$ will get three votes (from members $1$, $3$ and $5$) and thus always be chosen. But when all three candidates are being considered, $x$ and $y$ will each get two votes ($x$ from $1$ and $5$, $y$ from $2$ and $4$) and thus both be "acceptable" candidates.


Zero transaction cost:

It is correct that the resulting herd will be $\$7,500$, but the transfer from Smith to Jones is not necessarily $\$3,000.$  For instance, the negotiation may result in the transfer amount of $\$4,000$, in which case Smith will be indifferent between producing the ``full-size'' herd and cutting it down by half as $\$7.500 - \$4,000 = \$3,500$. The resulting transfer depends on the negotiation powers of the parties and can be anything between $\$3,000$ and $\$4,000$.

  High transaction cost:

Notice that your decision as a judge should depend on the value of the transaction cost.  Suppose the transaction cost is $\$4,000$, then what would be your decision? Can (or should) you make Smith indifferent in this case? Hope this will help you with the solution.


Instead of calling the three strategies you named the "best possible replies", I would say that they are the most salient replies. As you mentioned, there are many other potential responses to a tit-for-tat strategy. It turns out that what best responds to a tit-for-tat strategy is also a tit-for-tat strategy. Every other strategy will produce a lower payoff than using the tit-for-tat.

Consider the sequence of outcomes generated by each of the three strategies you mentioned (played by $i$ against the other player's tit-for-tat):

Tit-for-tat ($s_i^T$): $(Y,Y)$, $(Y,Y)$, $(Y,Y)$,  $\dots$
Alternating ($s_i^A$): $(Y,N)$, $(N,Y)$, $(Y,N)$, $\dots$
Always $N$ ($s_i^N$): $(Y,N)$, $(N,N)$, $(N,N)$, $\dots$

Note that all four possible stage-game outcomes are observed. As a result, if a player $i$ is playing any other strategy $s_i^O$, her payoff $u_i(s_i^O,s_j^T)$ can be expressed as some convex combination of the payoffs from playing either of the above three strategies. That is, there exists $\alpha,\beta\in[0,1]$ with $\alpha+\beta\le1$ such that, for any $s_i^O\in S_i$,
\begin{equation}
u_i(s_i^O,s_j^T)=\alpha u_i(s_i^T,s_j^T) +\beta u_i(s_i^A,s_j^T) +(1-\alpha-\beta)u_i(s_i^N,s_j^T).
\end{equation}

Now if you established that $u_i(s_i^T,s_j^T)\ge u_i(s_i^A,s_j^T)$ and $u_i(s_i^T,s_j^T)\ge u_i(s_i^N,s_j^T)$, which I assume you have done, it must follow that $u_i(s_i^T,s_j^T)\ge u_i(s_i^O,s_j^T)$ for any other strategy $s_i^O$.


We cannot obtain "demand" in the usual sense, because demand is a random variable. The "best" we can do is first, to obtain the Conditional Expectation of individual demand (conditional on the variables that determine the probability of whether the consumer will demand/buy).
Let a situation where consumers decide to buy or not to buy a single (for simplicity) item of a good $A$. We can model this underlying utility framework by

$$u_i(A) = \alpha p_i+\mathbf x_i'\beta +e_i$$

with $i$ denoting the consumer, $p_i$ is the good's price, faced by consumer $i$, $\mathbf x_i$ containing various other variables pertinent to the case, and $e_i$ representing a "random" preference shock, assumed, say, to follow the standard logistic distribution, conditional on the $x$'s.
The consumer will demand/buy, $q^d_i=1$, if (by convention) $u_i(A)>0$, so we can model the conditional probability of buying by

$$E(q_i^d =1\mid p, \mathbf x_i) = \Lambda(\alpha p_i+\mathbf x_i'\beta)$$

Then the conditional expectation of Market demand is

$$Q_d = \sum_{i=1}^nq^d_i = \sum_{i=1}^n\Lambda(\alpha p_i+\mathbf x_i'\beta)$$

This is still a random variable, and with unknown parameters. To obtain something one dimensional, given a sample of size $n$ on $\mathbf x_i$ and on relevant transactions (and so on $p_i$), one can estimate $\hat \alpha, \; \hat \beta $, say by maximum likelihood. Then we can obtain some concept of "average market demand curve" by using the sample means of the $x$'s as

$$\hat Q_d(p\,; \mathbf {\bar x}) = \sum_{i=1}^n\Lambda(\hat \alpha p+ \mathbf {\bar x'}\hat \beta) = n\cdot \Lambda(\hat \alpha p+ \mathbf {\bar x'}\hat \beta)$$

where here the price $p$ is not indexed anymore, and we vary it to obtain a curve.

As a toy numerical example, assume there are $100$ consumers, that ${\bar x'}\hat \beta =1$, and  $\hat \alpha = -2$. Then $\hat Q_d(p\,; \mathbf {\bar x_i})$ will look like

MULTINOMIAL LOGIT

In a multinomial logit where we examine the choice to buy "this or this or this" (mutually exclusive choices $k=1,..,K$), we end up with (after the usual normalization)

$$\hat E(q^d_{ik} \mid p_{ik}, \mathbf x_{ik}) = \frac {\exp\{\hat \alpha p_{ik} + \mathbf x'_{ik}\hat \beta\}}{1+\sum_{k=2}^K\exp\{\hat \alpha p_{ik} + \mathbf x'_{ik}\hat \beta\}}$$

Then we can obtain $K$ demands (each conditional on the rest) using (say, for good $1$)

$$\hat Q^{(1)}_d(p_1) = \frac {n\cdot \exp\{\hat \alpha p_1 + \mathbf {\bar x}_{1}'\hat \beta\}}{1 + \sum_{k=2}^{K}\exp\{\hat \alpha \bar p_{k} + \mathbf {\bar x}_k'\hat \beta\}} $$

Note that here we have to average the prices of the other goods also.


Theoretical/Empirical Arguments for Migrant Workers in the US

There is a broad literature that attempts to address the very question you pose- what is the impact of immigration (legal and illegal) on native wages? However, if you're interested in the 'real world' evidence, I'd suggest focusing on empirical labor economics as opposed to more theoretical labor models.

The reason for my suggestion is that theoretical economics makes no clear prediction without a lot more information. There are a variety of factors, such as the production function being considered, the labor market conditions observed, the relative substitutability of immigrant and native workers, etc. which tends to make answering the question more of an empirical exercise.

As far as the findings of the empirical papers, they actually tend to find relatively little impact of immigrant labor on native wages, with a few notable exceptions.

For some examples:

This survey paper, though a bit out of date as it's from 2005 outlines the results of 18 papers and finds wide discrepancies depending on modeling techniques and underlying assumptions used.

Borjas, 2003 is one of the better known works, and finds what one might expect based on Econ101 models- that an increase in immigrant labor supply leads to a decrease in wages for native workers.

A couple very well known articles have tried to tease out the impact of immigration on wages by focusing on particular events (called supply shocks) where the number of immigrants increased "suddenly" for reasons having nothing to do with the economic climate at the time. A pair of papers examined the impact of the Mariel Boatlift, which led to a spike in Cuban immigrants in Miami: The first, a paper by Card 1990 found no evidence of a negative impact of immigration on wages, while Borjas 2015 finds a substantial decrease in the wages for the least educated Miami residents.

Several other papers have found either no negative effect, or even positive effects, of immigrants on various worker wages. For example, this paper by Peri in 2011 finds no negative impact of immigrants on wages, but a strongly positive impact of immigrants on productivity. This paper from 2013 finds significant positive impacts of undocumented immigrants on the wages of native workers in Georgia.

Ultimately, the exercise of determining the impact of immigrants on wages really depends on the method of analysis and a number of underlying parameters, and applying "standard" (i.e. Principles of Micro) models can lead to erroneous results and a false sense of certainty.

If there's anything else in particular you're interested in, let me know and I'll try to improve the answer!

EDIT: Sorry, I forgot to include a link to this additional survey paper in the original answer!


Are unit root tests necessary or useful on small samples of time series data?

Cochrane (1991) says that unit root tests have low power, so you need lots of data (at best) to distinguish among non-unit root alternatives:

  This paper exploits the fact that any time series with a unit root can
  decomposed into a stationary series and a random walk. Since the
  random walk component can have arbitrarily small variance, tests for
  unit roots or trend stationarity have arbitrarily low power in finite
  samples. Furthermore, there are unit root processes whose likelihood
  functions and autocorrelation functions are arbitrarily close to
  those of any given stationary processes and vice versa, so there are
  stationary and unit root processes for which the result of any
  inference is arbitrarily close in finite samples.

A critique of the application of unit root tests

However, depending on the test and the realistic alternative, this may be fine. David N. DeJong, John C. Nankervis, N.E.Savin, Charles H. Whiteman (1992):

  Monte Carlo methods are used to study the size and
  power of serial-correlation-corrected versions of the Dickey-Fuller
  (1979,1981) unit root tests appropriate when the time series has
  unknown mean. The modifications do not cause serious size distortions
  or power deterioration in the white noise case. While studies in the
  literature have investigated the operating characteristics of these
  tests in the presence of moving average errors, of particular concern
  in this paper is the performance of these procedures in the presence
  of autoregressive errors. The Philips and Perron (1988) and Choi and
  Philips (1991) procedures are found to suffer from serious size
  distortions and have very low power when errors are autoregressively
  correlated. We conclude that even in the most favorable cases, these
  tests perform poorly against trend-stationary alternatives which are
  plausible for annual, quarterly, and monthly macroeconomic time
  series. The augmented Dickey-Fuller procedure, on the other hand, is
  reasonably well-behaved.

The power problems of unit root test in time series with autoregressive errors

My impression is things have improved a bit in the power of more cutting edge estimators. See Haldrup and Jansson (2005)

  A frequent criticism of unit root tests concerns the poor power and
  size properties that many of such tests exhibit. However, the past
  decade or so intensive research has been conducted to alleviate these
  problems and great advances have been made. The present paper provides
  a selective survey of recent contributions to improve upon both size
  and power of unit root tests and in so doing the approach of using
  rigorous statistical optimality criteria in the development of such
  tests is stressed. In addition to presenting tests where improved size
  can be achieved by modifying the standard Dickey-Fuller class of
  tests, the paper presents theory of optimal testing and the
  construction of power envelopes for unit root tests under different
  conditions allowing for serial correlation, deterministic components,
  assumptions regarding the initial condition, non-Gaussian errors, and
  the use of covariates.

Improving Size and Power in the Unit Root Testing

Nevertheless, I can't help thinking that 16 annual observations is not much data to discipline time series estimation, and there will likely be lots of somewhat plausible specifications you will not be able to reject. Instead, you could use economy theory to justify the functional form and the statistical tests you use. For example, in efficient markets (crudely), price levels are highly serially correlated (random walks) but returns are independent. So if you knew your data were in returns, you might just argue rather than test that they are not a unit root.


Comparative dynamics in forward-looking equation

Try writing your equation explicitly with a conditional expectation operator:

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

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

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


Why hasn't massive derivatives exposures at banks already led to disaster?

While gross notional exposures are huge, net exposures at the banks are much smaller, on the order of 0.1 percent of gross exposures. Since most financial risk (but perhaps not operational risk) is proportional to net exposures this is a more sensible measure of total derivative risk at the banks. Since net exposures at the 6 banks with the largest derivative exposures circa 2014 were on the order of \$300 billion while their total assets were more than \$10 trillion, we may not have seen a disaster because the actual risk is appropriate to the scale of their broader operations.

Here is an example of how this works. At time 0 a bank does derivative trade with notional of 100 with A and collects the offer (or ask) price. That is, the bank sells the derivative to someone who wants to buy it. At time 1 a bank does derivative trade with B with notional of 100 and collects the bid price. That is, they buy the derivative from someone who wants to sell it. Total (gross) derivative exposure is 200, two derivative contracts each with an exposure of 100. However, net exposure is zero, and the bank keeps the spread as profit for its trouble. There is likely some remaining risk in the form of operations and credit risk, but this is what the spread and initial and variation margin are for.


So basically the question is:
If I know the average ($\hat{\mu}$) of the daily temperatures ($y_i$) of last year, does that tell me anything about how many people were born ($x_i$) each day?

Unsurprisingly the answer is no.

The most you can get is the average of the $x_i$ series if you have the parameters of an unbiased regression between $x_i$ and $y_i$.


If I want to create my own currency model what would be the best basic options / things / ideas I need to look into?

In fractional reserve banking the money supply goes up and down according to the rate of new money being created as new loans are made and the rate at which existing money is destroyed as loan (principal) is repaid. So your model needs to include some estimate of the mutual enthusiasm, between banks and their customers, for making new loans. This in turn will be a function of the interest rate amongst other things (IMHO most importantly, people's perception of the future trajectory of house prices).

BTW, I hope you're not assuming that the money supply is capped according to the money multiplier story. That's all completely wrong as the Bank of England confirm => here.

An alternative is full reserve banking where the money supply does not go up and down with loans/repayments. In this case new money can only be created by the government/central bank. This would be far easier to model. When looking at full reserve banking - don't be misled into believing that banks can not lend in such a system (as some textbooks may imply). Peer to peer lenders (e.g. zopa) are essentially full reserve banks and they exist already.


Effect of Strengthening US Dollar on Commodities

Most global commodities are quoted (priced) in US dollars. The more valuable a dollar becomes relative to other currencies, all other things equal, the more commodity prices have to fall to equilibrate. Imagine everything was priced in corn and people start to value corn more or the amount of corn falls. Either way, all the prices in terms of corn (e.g. ears per barrel of oil) would have to fall as a result.


They are not the same.

Basic accounting equation:

Assets = Liability + Shareholder Equity

Assets refers to what the company actually owns: cash, property, inventory, etc.

Assets are paid for in two major ways: debt (liability) and stock (equity). Essentially, everything a company owns is paid for by a combination of (1) getting loans from other entities or issuing bonds and (2) having other entities own a part of the company.

Equity includes a number of things, but the most common form is stock, or ownership of a company.

An example of this: if you own a pen company, assets might include pens, and equity would be your ownership of the company.


The $\sqrt{N}$ convergence of semiparametric estimators of Newey 94, why does it converges to a normal distribution?

The usual proof of the classical Central Limit Theorem (CLT), I believe provides the most intuition there is about this phenomenon. And it is not much of an intuition anyway.

This "usual" proof is through characteristic functions.

Consider a random variable $X$ with characteristic function

$$\phi_X(t) = E(e^{i tX}), \;\;\;i^2 = -1$$

Now consider its centered and scaled version

$$Y = \frac {X-\mu}{\sigma} = \frac 1{\sigma}X - \frac {\mu}{\sigma}$$.

with $E(Y) = 0 , {\rm Var}(Y) E(Y^2)= 1$. Moreover, $Y$ is the sum of two independent random variables, the second degenerate (being a constant, and so also independent of everything). So, by the properties of the characteristic function for the sum of two independent random variables

$$\phi_Y(t) = \phi_Y\left(\frac 1{\sigma}t\right)\cdot e^{-i\frac {\mu}{\sigma}t} = E\left[\exp{\left\{it\frac 1{\sigma}X - i\frac {\mu}{\sigma}t\right\}}\right] = E(e^{i tY})$$

Now take the second-order Taylor expansion of $\phi_Y(t)$, with respect to $Y$ and with center of expansion $E(Y) =0$:
$$\phi_Y(t) = E\left[e^{it\frac 1{\sigma}\cdot 0} + ite^{it\cdot 0}Y + \frac 12 i^2t^2e^{it\cdot 0}Y^2 + o(t^2)= \right]$$

The first term is zero, the second term vanishes because $E(Y) =0$, and for the third term we use $i^2=-1, E(Y^2) =1$ to arrive at

$$\phi_Y(t) = 1  - \frac {t^2}2 + o(t^2)$$

The "phenomenon" is already here because

$$ \frac {t^2}2 = \ln MGF_{Z}(t),\;\; Z\sim {\rm N}(0,1)$$

In words: The characteristic function of any random variable with finite mean and variance that is centered and scaled accordingly, has a strong connection with the moment generating function of the Standard Normal distribution ($\ln MGF_{Z}(t)$ is actually the cumulant generating function).

How can this happen? Have we uncovered a "law of nature" here, this fundamental connection of different "types of uncertain behavior" to the specific type labelled "standard normal distribution",  or it is just our mathematical system, through which we have modeled this thing called "uncertainty", producing some artificial connection that perhaps reveals some aspect of its own internal structure, but it has nothing to do with the real world?

Let's first complete the proof of the CLT: consider the random variable
$$Z_n = \frac 1{\sqrt{n}}\sum_{i=1}^nY_i$$

where the $Y_i$'s are independently and identically distributed. Again by the properties of the characteristic function we have

$$\phi_{Z_n}(t) = \prod_{i=1}^n\phi_{Y}(t\sqrt{n}) = \Big[1  - \frac {t^2}{2n} + o(\frac {t^2}{n})\Big]^n \xrightarrow{n\rightarrow \infty} e^{-t^2/2} = \phi_{Z}(t),\;\; Z\sim {\rm N}(0,1)$$

and so

$$Z_n \xrightarrow{d} {\rm N}(0,1)$$

...and now we have arrived at the Standard Normal distribution proper. And this is a law of nature: mathematics aside, computer simulations aside, real world data consistently validate this result. So there is no further "why?" here -as it cannot be with any law of nature. We just discover them -and we feel like gaining added intuition when we discover the interconnections between these laws (but this is not really insightful, it is just more discoveries).

Of course this is the first and most simple case, in a long line of Central Limit Theorems that increasingly deal with more complicated, interdependent, functions of random variables, stochastic processes etc as the one mentioned in the OP. And there are also the generalizations of CLT to stable distributions and not just the normal, and there is extreme value theory... all these are different aspects of the same conclusion: that collective behavior (even in the simple sense of pooled behavior) is much more homogeneous than individual behaviors, even though it is just the pool of the latter -and this should be one of the most counter-intuitive results ever encountered.

P.S. An inspired attempt at bottom-up intuition is provided by @whuber in this Cross Validated thread: https://stats.stackexchange.com/questions/3734/what-intuitive-explanation-is-there-for-the-central-limit-theorem


GDP Identity:Why does $\text{C+S+T = C+I+G+(X-M)}$?

A concrete example makes it easier to think about:

Let's say during the week you receive \$1000 income. Only 3 things can happen to your $1000:

You could be taxed some/all of it (T: Tax)
You could spend some/all of it (C: Consumption)
You could save some/all of it (S: Savings)

In this example we've used your $1000 income. The distribution of income (Y = C + S + T) is exactly the same but not just one week of your income, but 52 weeks of everyone's income. So it's a measure of how much income was received across everyone in the economy for the whole year.

Now, all this income had to originate from somewhere, and there are only four possibilities:

someone's spending (C: Consumption) - this goes to businesses and is paid out as wages, as interest on business loans, or as share dividends or profits to owners
government spending (G: Government Spending) - payments to government workers and for anything government buys (e.g. materials to build roads, defence spending etc etc)
banks making loans (I: Investment) - loans made to businesses or individuals
sales to foreign economies less what is paid to foreign economies for their goods and services (X - M: exports minus imports)

Adding these components together gives aggregate demand: C + I + G + (X - M); it's the $ amount of all the goods and services purchased in an economy over one year.

GDP is not a perfect way to measure the size of an economy, but it's probably the most common way. So when we talk about GDP = Y = C + S + T = C + I + G + (X - M), what we know is that GDP can be measured in a few different ways, and that more spending will mean more income (and vice-versa).


What would be the disadvantages of a TTIP?

TTIP is a combination of trade agreements and property rights / legal changes.

Trade Liberalization

Most of the current debate in European countries is not so much about trade - we all know trade liberalization yields welfare gains. In the first-order, these are split between consumers and firms: Consumer get a larger variety / cheaper goods, and (surviving) firms can reach more customers.

These are efficiency gains, and could be split up anyhow between these two groups. To the extent that surviving firms get some of these welfare gains, workers of these firms might get some of the benefits. I'm trying to be very cautious here, because on this last bit, I don't think we have sufficient empirical evidence - and theory will support different results based on the wage setting mechanism.

Of course, there will be firms that lose, and workers that will be fired - which is why the US has set up (I think under WIA) Trade Adjustments Assistance for workers who lost their jobs due to trade. This is one way of sharing the welfare gains of trade with those who lose.

Copyright and Regulation

The issue with TTIP is mostly the legal power that firms get. US american firms could sue France for french regulation at US courts. This downside is off topic here. However, the angst here is that is not welfare gains for each single country. Countries with stronger regulations will lose. Even for the world as a whole, it is unclear whether this yields efficiency gains, or whether firms just reap surplus from countries/governments.


Difference between chained CPI and normal CPI in simple terms

The consumer price index is used to measure the rate at which prices increase. It is calculated by taking a basked of common goods that people buy all the time (groceries, appliances,  etc.) and looking at how their average prices change over time.

Suppose we are calculating the CPI using a basket of goods that contains only cinema tickets. Last year a cinema ticket cost \$15, but now it costs \$30. The CPI would then have doubled to reflect that fact that the prices in the basket of goods have doubled.

Some argue that this isn't a very good way to measure consumer price inflation. Here's why: if the price of cinema tickets doubles then people will buy fewer cinema tickets and buy, for example, more Netflix subscriptions instead. So although the price of cinema tickets doubled, people aren't spending twice as much on cinema tickets (because they aren't buying cinema tickets any more!) So, in some respects, the fact that the CPI has doubled doesn't tell us very much.

There are two ways to deal with this:

the approach used when computing the CPI is to say that since people aren't buying cinema tickets we need to update the basket of goods we use to calculate inflation (remember the basket is supposed to be representative of peoples' consumption habits so if they buy fewer cinema tickets it should contain fewer cinema tickets). The problem here is that the basket is only updated infrequently so it takes time for the CPI to "catch up" with consumption habits.
the other approach (used in chained CPI) is to statistically adjust the standard CPI measure to account for the fact that we know people will substitute to different goods when the goods they were buying become more expensive.

To do this, the statistical authority gathers data on expenditure for each category in the basket of goods then then weights that category's contribution to the index by its share in expenditure. In the US, this is done on a monthly basis, so the chained CPI uses a more up to date measure of substitution between product categories.


I will give a hint. To increase/decrease utility by 1 unit, you can do two things. Increase $x_1$ and $x_2$ by $\frac{1}{2}$ units and this would cost you $\frac{p_1+p_2}{2}$. The second option is changing $x_3$ by one unit that would cost $p_3$. One other option is of course using a linear combination of the above changes. If you remember from well behaved utility functions, you would like to buy things for which utility per price is higher. Now you would go for a similar reasoning here and would try to buy extra utility at a cheapest price.


Optimal Production Input in Relation to Cost Minimization Problem

I`d propose you to follow these steps:

Set up the minimization cost problem (i.e. for a given output quantity $y$ minimize costs):
\begin{align}
\min_{H,L,K}&  \quad sH + wL + rK \tag{1} \label{1}\\
\text{such that} &\quad  \min\{H,L\} + \min\{H, K\}\geq y \tag{2} \label{2}
\end{align}
In principle you have 3 cases, depending on price of factors $(s,w,r)$.

1) When $r>s+w$, then buying a unit of $L$ and one of $H$ is going to cost you strictly less than buying a unit ok $K$. Consequently the monopolist opts for producing only using $H$ and $L$ ($K=0$). Thus $H = L = y$

2) The second case happens for $s>r+w$ which is, anyway, ruled out by the assupmption $w>s$ (if this was not the case, then producing only using $H$ and $K$ would have been optimal, i.e $H = K = y$ and $L = 0$).

3) Finally, when none of the above occurs, you have $L = H$ and $K = H$ because of the fixed proportions production function. Then the constraint \eqref{2} becomes $H + H \geq y$ and the inequality sign become equality. The equality sign comes from the fact that producing more than $y$ (the LHS of \ref{2} being strictly larger than the RHS) is just overshooting and inflates production costs for a given $y$. $L = H$ and $K = H$ comes from the Leontief production set-up of the problem: inside the $\min\{...,...\}$ operator is never convenient to exploit more one of the two factors (because by decreaing the most used factor one still gets the same amount of output but a lower cost). Therefore, each of the two $\min\{...,...\}$ can just be seen as one of its inner arguments, namely, for convenience, $H$. So you finally get $H+H = y$ and, from there: $H = y/2$ and $L = K = y/2$

Substitute the optimal values ${\big(H^*\small{(s,w,r)},L^*\small{(s,w,r)},K^*\small{(s,w,r)}\big)}$ in \eqref{1} so to get an expression only in $(s,w,r)$. This expression is the cost function $C(s,w,r,y)$
Now solve the monopolist's pricing problem for given factor prices:
\begin{equation}
\max_p \quad p\cdot D(p) - C\big(s,w,r,D(p)\big)
\end{equation}
and find the profit maximizing price, $p^*$ for output level $y^* = D(p^*)$

I hope this helps, comment below if you need any more hint.


Bayesian Nash Equilibria: Strong and Weak Types

Modeling the game as four separate matrices does not capture the fact that each general knows his army strength but not the strength of the other army.

Given the small action space, it may help if you could first visualize the game in an extensive form, with Nature deciding army strengths at the beginning, and each army general observing their type before making the attack or not decisions. (Of course, this step is not necessary if you have a good mental image of what the game looks like.)

You can then try to convert the extensive form game into its Bayesian normal form. Hint: each general has four pure strategies: attack or not when their army is strong and attack or not when their army is weak. The BNE can then be solved easily in the Bayesian normal form.


Low interest rates - who benefits, rich or poor?

https://www.bundesbank.de/Redaktion/EN/Downloads/Publications/Monthly_Report_Articles/2016/2016_09_effects.pdf?__blob=publicationFile

That should fit your question.

Imho we have never seen these kind of interest rates, and It will be very entertaining to see how this works out.

Could lead to another black swan event, which is nothing you can calculate/prepare for.


You can make $P$ the subject of the equation:
$$P=\frac{M-L} {ky} $$ then find the derivative $$\frac{\mathrm d P} {\mathrm d M} = \frac 1{ky}>0$$ to show there is a positive relationship (when $M$ increases, $P$ increases). The elasticity answers your question about proportionality: \begin{align*}
\frac{\mathrm d P} {\mathrm d M} \cdot \frac MP &= \frac 1{ky}\left( \frac {kPy} P + \frac {L \cdot ky} {M-L} \right) \\
&= 1 + \frac L{M-L} \end{align*} shows that $P$ increases more than proportionally (because the elasticity is greater than 1).


How long into the past can we trace the effect of institutions?

Institutions are likely having a permanent effects as they create path dependency which implies that even temporary shocks would have permanent effect. At least that is what is argued by many developmental economists including Acemoglu and Robinson (see Why Nations Fail). Furthermore, they claim that this hold not just for development itself but that the political institutions are themselves path dependent and reinforcing.

However, permanent effects and path dependency should not be confused for some sort of historic fatalism. Just because some countries had (or were imposed on them) extractive (i.e. in Acemoglu and Robinson's view 'bad') institutions  does not mean that they are forever doomed to be underdeveloped. It just means that the loss of welfare that occurred as a result to the 'shock' (i.e. adoption of extractive institution itself) wont be recovered.

For example, below I simulated an example of an one time adoption of extractive institution. I tried to simulate the sort of extractive institutions imposed by the king Leopold II on Congo which arguably both affected the production capabilities of Congo and its growth.

Below you can see that the effect is permanent even if country will eventually grow and attain the level of output it had before the extractive institution was implement.

Even if the old institution will be reversed and replaced by better one the effect of the old institution will be still permanent as you can see the simulated examples of growth where the old extractive institution was replaced by inclusive (left picture) vs where extractive institution was never in place (right picture) but inclusive was implemented show that the GDP per capita will be permanently affected.

Hence to sum up, if you buy into the theory that development is path dependent which Acemoglu and Robinson do then the answer is that the effect of those institutions is permanent. Thus you can trace the effects as back as recorded history goes. Even some ancient event such as the Bronze Age Collapse or the fall of Roman Empire (or as Brian mentions in his +1 comment even ancient road network patterns) and changes in institutions caused by those have still significant effect today as if we could avoid them today's output would arguably be significantly different.


Consider an exchange economy with two goods $x$ and $y$ and two agents $A$ and $B$. The utility functions are
$$
U_A(x_A,y_A) = x_A \cdot y_A, \hskip 20pt U_B(x_B,y_B) = \min(x_B,y_B).
$$
Let the initial endowments be
$$
\omega_A = (\omega_A^x,\omega_A^y) = (z,0), \hskip 20pt \omega_B = (\omega_B^x,\omega_B^y) =  (0,4),
$$
where $0 \leq z \leq 8$ is left as a parameter. In equilibrium
$$
x_A = \frac{z}{2}, \hskip 20pt y_A = \left(4 - \frac{z}{2}\right), \hskip 20pt x_B = \frac{z}{2}, \hskip 20pt  y_B = \frac{z}{2}.
$$
(A detailed calculation showing this is at the bottom of this answer.)

The utility of agent $A$ is
$$
U_A\left(\frac{z}{2},4 - \frac{z}{2}\right) = \frac{z}{2} \cdot \left(4 - \frac{z}{2}\right).
$$
The maximum of this is at $z = 4$. If the initial endowment of agent $A$ is larger than 4 he has incentive to destroy part of it, because assuming a competitive equilibrium outcome this will improve his utility.

As @Foobar pointed out in the comments further utility could be gained by not destroying but simply withholding part of the initial endowment. There are two arguments to be made for destruction:

Agent $B$ sees the supply and refuses the monopolistic outcome, insisting on the competitive price ratio.
One could divide agent $A$ into several 'smaller' agents. Agents who would have the same utility functions and equal shares of the initial endowment. In this case withholding part of the endowment is a cartel move. It is not individually rational. One could argue that destroying the 'surplus' is a commitment necessary for coordination.

In equilibrium total demand equals total supply in both markets.
\begin{eqnarray*}
x_A + x_B & = & \omega_A^x + \omega_B^x \\
\\
y_A + y_B & = & \omega_A^y + \omega_B^y.
\end{eqnarray*}
Our next step is determining demand. Here is the information we gather from the utility functions:

Let $y$ the numeraire good, i.e. let $p_y=1$. Let us denote the price of $x$ by simply $p$. The value of the initial endowment of agent $A$ is $z \cdot p$. Using the Cobb-Douglas property
$$
x_A = \frac{z\cdot p}{2\cdot p} = \frac{z}{2}, \hskip 20pt y_A = \frac{z\cdot p}{2}.
$$
Using this we have
\begin{eqnarray*}
x_A + x_B & = & \omega_A^x + \omega_B^x \\
\\
\frac{z}{2} + x_B & = & z + 0 \\
\\
x_B & = & \frac{z}{2}.
\end{eqnarray*}
It follows from agent $B$'s utility function that in equilibrium
$x_B = y_B$. Using this we have
\begin{eqnarray*}
y_A + y_B & = & \omega_A^y + \omega_B^y \\
\\
y_A + x_B & = & \omega_A^y + \omega_B^y \\
\\
y_A + \frac{z}{2} & = & 4 \\
\\
y_A & = & 4 - \frac{z}{2}.
\end{eqnarray*}


Geometric intuition behind the point where the indifference curve meets the budget line

No. This is not in general true.
Maximizing the rectangle formed in the described manner would maximize the product $x_1 \times x_2$. In the special case where your utility function takes the form $u(x_1,x_2)=x_1x_2$, utility maximizing implies you pick a point on your budget constraint where the rectangle you described is maximized.
Suppose however, that instead your utility function is given by $v(x_1,x_2)=x_1^{1/3}x_2^{2/3}$ And suppose you are currently maximizing $x_1 \times x_2$ on your budget constraint. Suppose that $p_1=p_2=1$, then maximizing $x_1 \times x_2$ implies $x_1^*=x_2^*$. However maximizing $v$ with these prices gives: $x_1^*=\frac{1}{2}x_2^*$. Thus not maximizing the rectangle subject to your constraint, but still defining a point of tangency between the indifference curve and budget constraint.