Which is more economical with resources, an e-tick or a paper ticket?

According to this study by the IATA, it costs \$1 to process an e-ticket, versus \$10 to process paper ticket. I assume this is only from the airline's point of view. You could argue that this is not the "environmental cost" you care about, but I just wanted to point it out.

I'm a bit confused by your phone charging assumptions, but according to this post, it costs close to 1 cent to charge a phone (10,000 mAh, at least). Also, you won't need your phone at full charge to use e-ticket.

On top of that, paper tickets also have a lot of costs associated with it, from the manufacturing process, logistics of getting the paper to your local store, getting it from the store to your home. Then we need to talk about recycling costs, etc.

So without the cost of the phone taken into account, I still think it's more efficient to use e-ticket.


Difference between merchandise trade and trade in goods?

As far as I understand merchandise trade and trade in goods are the same.

The IMF Balance of Payments Manual breaks down international transactions according to standard categories. Trade is one of these categories. Trade is defined as international transactions involving products, i.e. exports and imports of goods (or merchandises) and services.

Merchandise or good trade are transactions involving the transfer of ownership of a tangible and moveable object from a seller to a buyer.
Services are transactions in which a consumer benefits from actions taken
by the service provider.

There is a subtle sub-category, which is

Goods for processing that are transactions in which raw goods (e.g. cotton thread) from country A are exported to country B, where
they undergo processing, and are then imported back into country A (e.g. t-shirts) without a transfer of ownership.


Does Average Total Cost (ATC) include implicit opportunity costs?

Generally, ATC does include implicit costs unless otherwise specified by regular accounting


Do discontinuous preferences imply no continuous utility function?

The easiest way to prove it is using the 'old' definition of continuity.
$\succ$ is continuous iff whenever $x\succ y$, there exists neighborhoods of $x$ and $y$, $B_x, B_y$, such that all $z\in B(x)$ and $z'\in B(y)$, $z\succ z'$.

Suppose $x\succ y$. Because $u$ represents $\succ$, $u(x)>u(y)$. Let $2\epsilon=u(x)-u(y)$. Because $u$ is continuous, there exists some $\delta>0$ such that for all $z\in B_{\delta}(x)$, $u(z)>u(x)-\epsilon$. Similarly, for all $z'\in B_{\delta}(y)$, $u(z')>u(y)+\epsilon$. But then for all $z\in B_{\delta}(x)$ and $z'\in B_{\delta}(y)$, $z\succ z'$ as required.


Nash equilibrium - mistake in proof of paper?

The objective is to show that, as long as $f'(n\hat\theta)\ne \alpha$, a firm can always engineer a package $(p',\hat\theta')=(p+\Delta p,\hat\theta+\Delta \hat{\theta})$ such that (i) a caring consumer strictly prefers this package, and (ii) the firm makes positive profits.

1) Your maths are correct: the inequalities are always false when f is increasing and strictly concave. In that case, the proof in the paper is unclear.
The package $(p',\hat\theta')=(p+f'(n\hat\theta)\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ does not satisfy (i).

2) You can show the proposition with the following reasoning.
Assume $f'(n\hat\theta)>\alpha$. Take any $h$ within the interval $(\alpha,f'(n\hat\theta))$.
For $\Delta \hat{\theta}$ small enough and positive, a firm that offers the package $(p',\hat\theta')=(p+h\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ satisfies (i) and (ii). With a first-order appoximation, we show (i): $$b-p'+f(\hat\theta'+(n-1)\hat\theta)>b-p+f(n \hat\theta),$$
and (ii):
$$\Delta p - \alpha \Delta \theta>0.$$

Assume $f'(n\hat\theta)<\alpha$, and take $h$ within the interval $(f'(n\hat\theta),\alpha)$.
For $\Delta \hat{\theta}$ small enough and negative, a firm that offers the package $(p',\hat\theta')=(p+h\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ satisfies (i) and (ii).


Minimization of costs combination of factors of production

Working with $\min$ (or $\max$) functions can be tricky, especially if one wants to write down the general solution in a fully rigorous way.

But in your case you have simple linear functions and a numerically given fixed production level ($200$) for which you want to minimize costs.

First, make sure you understand the production function $\min$  expression and what restrictions it poses: A frequently seen initial approach here is to think

"ah, this means either the one, or the other. So I will either use $x$ only, *or* a combination of $y$ and $z$ and perhaps only one from the two. There is no point in buying, say, $x$, if production is to be determined by the a combination of $y$ and $z$".

What is the fallacy in this line of reasoning? And what the resolution of the fallacy implies?

Next, think what it means to want to examine only a given fixed production level: on the right hand side you have the $\min$ function. On the left hand side, you have a specific number, not a general label/symbol anymore.
So you have an equation, not a function anymore. And then you have the cost function. And you want to minimize the cost function given the equation, and given what the "fallacy resolution" mentioned previously implies.

Take it from there.

And please, once you figure it out, post it as an answer here (you can answer your own question is perfectly fine, and even encouraged).


Do you consider a house a normal NECESSITY good or normal LUXURY good?

I would expect necessity goods to have an income elasticity between 0 and 1 in keeping with Engel's Law, which was originally observed for food (an obvious necessity good).

Housing indeed exhibits this feature because poorer households spend a larger fraction of their income on housing. That makes housing look like a necessity good. E.g.,

(source: http://www.lao.ca.gov/reports/2015/finance/housing-costs/housing-costs.aspx)


Does the value of a pure strategy Nash equilibrium(if exists) equal the value of the mix strategy Nash equilibrium in two-person zero-sum game?

Pure strategy Nash equilibria are a subset of mixed strategy Nash equilibria, so as long as your statement

  a mixed strategy Nash equilibrium always exists and all such equilibria have the same value

is true, all pure strategy Nash equilibria will have the same value as well.


Why does a buyer value a dollar in a small purchase (e.g. TV) differently to a dollar in a large purchase (e.g. car)?

One explanation from psychology and behavioral economics is "mental accounting": people put different transactions into different mental accounts.

When buying a house, people are imaging (probably unconsciously) a bank account dedicated to that purpose. The account's balance is pretty large (including mortgage loans). \$5000 in or out of this account does not really have any significant impact on the account.

When buying a cellphone, people are imaging a separate bank account, of which the size is much smaller than the home purchasing account. \$500 in or out of this mental account makes a big difference.

If we pool all accounts together, then \$500 saved from buying a cellphone should be the same as the \$500 saved from buying a house.


What guarantees that for every credit there is a debit in an electronic fund transfer?

Double-entry bookkeeping guarantees that for every credit there is a debit.

It's not really different for a bank than for you or I.

I can claim that I've got an account with myself for 10 squillion pounds, but everyone will ignore me. The only balances that matter are the physical currency that I hold, and the accounts I have with other people or institutions; e.g. my bank account.

It's the same for a bank: the balances that matter are the physical currency that it holds, and its accounts with other people and institutions. Predominantly, other banks, including Central Banks.

Clearing Houses are institutions, or groups of institutions, that act as intermediaries to ensure that when one bank has a bunch of transactions with another, the net balance is transferred correctly while minimising counterparty risk. And the Central Bank offers a settlement service: for example, banks with sterling deposits will (typically) have an account with the Bank of England, which carries out settlement between banks.


What would be the economic impact of disallowing the renting of housing?

If the required downpayment is substantial there will be fewer households if renting is illegal. This means the average number of people per household would be greater. Some people would be living in larger households as they save for a downpayment. If the required downpayment is zero then price volatility would imply that some home sellers have a debt from an earlier home ownership experience. If experience shows that debtors tend not to pay then the required downpayment will be substantial. The most likely outcome is substantial required downpayments and a greater average number of persons per household.


Does a strong US Dollar hurt American Manufacturing?

Most manufactured products are sold domestically, but even so a strong dollar will (usually) cause a decline in sales and profitability of US manufacturers. The reason for this is that a strong dollar makes foreign products relatively cheaper, not just manufactured products, but all products. Therefore it becomes cheaper for person to buy, say a foreign-made washing machine compared to an American-made washing machine.

What news articles like this don't mention is that a weak dollar hurts Americans because it makes everything they buy, like gasoline, more expensive, not just manufactured goods.

The exception to the rule is during wartime. For example, as a result of World War II the United States dollar became stronger AND manufacturing profitability increased at the same time. But this was because competitors in Europe had all their factories in ruins, which is not the case today obviously.


Is there any good textbook for theoretical economics mathematical modelling?

Your question seems a bit broad. What kind of “theoretical economics” are your reading?

Nevertheless, I infer from what you write that you need to start from the beginning with maths for economcis (and maths in general), (this is because of your comment not knowing when to use an exponent  and when to multiply.

Having said this, I would recommend:

“Essential Mathematics for Economic Analysis” by Hammond et al

This book starts at the basics and builds up mathematical tools to study economic ideas.

Having mastered this book, I would try

“The Structure of Economics” of Eugene Silberberg. This books is more advanced, but it is “economics with mathematics”, so having aquired the basic mathematical tools in the first book, this book focuses con economic models from a mathematical standpoint.


Every lagrangian with two goods has the same two steps basically. First, find the optimal relationship between the two goods (often through derivatives). Second, plug that relationship back into the budget constraint to solve for each good.

This problem is no different and you have essentially only done step 1. So you just need to do step 2.

The idea is that now you have 2 unknowns ($c$,$c'$), so you need 2 equations. One equation is the relationship you found between consumption today and tomorrow from the first order conditions. The second equation is the the budget constraint (or equation 3 as you called it). Your equation 3 has still not been used toward solving for consumption.

Plugging in the equation for $c'$ that you found into your equation 3 you get:

$c + \frac{c(1+r)}{1+r} =y+\frac{y'}{1+r}$

So $c= 0.5 *(y+\frac{y'}{1+r})$

Now plug in this equation back into the relationship you found through the lagrangian, which was:

$c' = (1+r) c$.

Then you'll be done.


Are there any analysis of what is the net benefit of NASA program to US economy?

I had a look, and there's not a alot, and not much that is current. The problem is, the question (What is the impact of NASA on the economy?) is too big and too difficult to measure. I think you'll have better luck by focusing on particular angles to this question.
For example, when referring to the impact of NASA, this could be anything from impacts on the aerospace industry, manufacturing industry, procurement, R&D, IP and patents, services and local economies at the different NASA sites, during different time periods, etc.
I'm sure you've already done some searches, but in case you haven't tried this, have a look at Google Scholar, and see if any of your search results bring up papers that a) sound topical and b) are widely cited.
For papers that fit both criteria, click on the "Cited by..." link and you will find a list of all papers that reference this result, from which you can get a better idea of different research angles people have taken, as well different key words that you can use to look for what you ultimately are trying to answer.
Below I list the titles of a couple of relevant papers to get you started. Try entering them yourself into Google Scholar and have a look at where they have been cited.  Good luck!

"Assessing the full effects of public investment in space"
"Evidence from Patents and Patent Citations on the Impact of NASA and Other Federal Labs on Commercial Innovation"
"Economic Effects and Spin-offs in a Small Space Economy: The Case of Canada"
"Measuring the Economic Returns from Successful NASA Life Sciences Technology Transfers"
"Estimating the Impact of Shuttle Launches on Regional Economic Activity"


It's well known that a convex preference implies quasiconcave utility functions. Since quasiconcavity need not imply concavity, it's easy to find examples of a non-concave utility function representing a convex preference.
For example: $u(x,y)=(x+y)^3$. The preference this function represents is convex (though not strictly so), as can be seen from its linear indifference curves. The function is quasiconcave, as evidenced by the convex upper contour sets. Lastly, the function is not concave, as betrayed by the exponent.


Normally when we think about value added, we think of marginal revenue product of labor: how much more revenue does adding another worker add to the firm?

If you assume that the market is competitive, then the MRPL should be exactly equal to what you said: wage, benefits, etc. Otherwise another worker would be willing to work for an epsilon less.

In other words, the "value added" by your definition (their "productivity" minus payments) should be zero. From my experience, researchers measure worker's productivity by wages.

If you want to find where this does not hold, then I guess you'll need more detailed data. Perhaps comparing workers with same job description but different compensation.


Well in principle you could. Let's suppose there are two countries that do not trade a lot. Then even if their economic growth is relatively similar, we should not expect their exchange rate to be 1:1.

The exchange rate is a price, so if country A likes more the goods and services from country B than vice-versa, you should expect currency B to have a larger purchasing power than currency A.

However, The demand for goods from abroad is correlated with income, so if both countries have relatively similar economic growths, and they trade a lot with each other, the relative purchasing power of their currencies will tend to be similar.


Ambiguity on a relation between inflation, interest rates, and central banks

The sentence:

  Therefore, the central bank controls interest rates in both ways to keep inflation close to its target rate.

does not make sense to me, and I am having a difficult time parsing the three suggested alternatives. (“Money over/under-valued” does not appear to make sense, since the value of money is always 1.)

In most cases, a central bank has a symmetric inflation target (e.g., the Bank of Canada has had a 2% inflation target). It will attempt to raise the inflation rate if it is below 2%, and reduce it if inflation is above 2%. Thus it is concerned with deviations “both ways.”

The inflation target need not be symmetric. E.g., one interpretation of the ECB mandate is to keep inflation below 2%. In which case, policy direction does not work “both ways”, since inflation does not need to be pushed up.

It will raise and lower interest rates (as well as other policy actions) in an attempt to steer inflation. That is, interest rates move “both ways” to hit its target. However, it is unclear why that would be specified.


What does poorest country by population mean?

"Poorest country by population" doesn't really mean anything: I think this is just sloppy writing by the website. The table on the website is actually ranking countries by

$$\text{GDP Per Capita}=\frac{\text{GDP}}{\text{population}}.$$

This is a measure of the value of goods or services that the average person in a country produces.


I presume, that you are familiar with the subject and need clarification only upon the expression in question.

I also preume that you know what volatility is. Volatility is generally unobserved in GARCH models, i.e. it can not be measured directly in which sence it is latent volatility. Usual GARCH model derives an estimate for this latent volatility from squared errors.

In this paper the way to incorporate a different estimatior of the latent volatility is sujested. As it is not entirely accurate to call it an estimator, that refer to it as measure. They consider measures from the family that is commonly referred to as realized measures of volatility, e.g. realized varianse and bipower variance. So we have realized measure to latent volatility. However we can not observe a random variable, which this realized measure actually. Instead we observe one realization of it - an observation. So the numbers that we got from using these procedure are called "observed realized measure to latent variation".

If my explanations were unclear or something else needs to be explained please specify it.


Does returns to scale always lead to economies of scale?

Excellent question (I am assuming your intended question is "does increasing returns to scale always leads to economies of scale"):
The two concepts are related but Returns to scale (RS) is much restrictive than the Economies of scale (ES).
The concept of RS is embedded in production function. If $Q=F(K,L)$ then increasing returns to scale simple means:
$$F(\alpha K, \alpha L) > \alpha F(K,L)$$
For example, in Cobb-Douglas production function: $Q=AK^aL^b$, we have that
$a+b>1 \implies \text{increasing }RS$
The concept of ES is much, much wider and goes beyond production function. All it says is that average cost (AC) decreases with $Q$:
$$\frac{dAC}{dQ}<0$$
Notice the use of $d/dQ$ instead of $\partial/\partial Q$. This is where everything changes. In partial derivative we are interested in the mathematical relationship that: ceteris paribus whether changes in quantity produced changes cost.
To illustrate the relationship and the difference consider derivation of $C=f(Q)$ using the Cobb-Douglas production function:
Given, wage rates $w$ and cost of capital $r$:
\begin{align}
C=wL+rK
\end{align}
To first solve for output maximization constraint by cost function:
\begin{align}
\max_{L,K} \;{AK^aL^b} \;\; s.t \;\; wL+rK=\bar{C}
\end{align}
Solving the langrangian would give us:
\begin{align}
K=\frac{bw}{ar}L \tag{2}
\end{align}
Substituting $(2)$ in production fuction gives us:
\begin{align}
Q=A\bigg(\frac{bw}{ar} \bigg)^bL^{a+b} \tag{3}
\end{align}
Re-arranging $(3)$ and getting $L$ in terms of $Q$ and then substituting it back to $(2)$ gives us:
\begin{align}
L=\bigg(\frac{ar}{bw}\bigg)^{b/(a+b)} \bigg(\frac{Q}{A}\bigg)^{1/(a+b)} \tag{4}
\end{align}
\begin{align}
K=\bigg(\frac{bw}{ar}\bigg)^{a/(a+b)} \bigg(\frac{Q}{A}\bigg)^{1/(a+b)} \tag{5}
\end{align}
Substituting $(4), (5)$ in $(1)$:
\begin{align}
C=\eta \cdot w^{\frac{a}{a+b}} \cdot r^{\frac{b}{a+b}} \cdot Q^{\frac{1}{a+b}}
\end{align}
where $\eta$ is a constant in terms of $a$ and $b$.
For average cost:
\begin{align}
\frac{C}{Q}=\eta \cdot w^{\frac{a}{a+b}} \cdot r^{\frac{b}{a+b}} \cdot Q^{\frac{1-(a+b)}{a+b}} \tag{6}
\end{align}
Now you see if there are increasing returns to scale, i.e., $a+b>1$,
$$\frac{\partial AC}{\partial Q}<0$$
So you see, if $w,r$ can be taken as constant then of course
$$\frac{\partial AC}{\partial Q} = \frac{dAC}{d Q}  <0$$
On the other hand, this is rarely true. In a general equilibrium model, $w$ and $r$ are also variables. For instance, say the Labor and Capital market is perfectly competitive:
$w=MP_L\equiv \frac{\partial Q}{\partial L} = a\frac{Q}{L}$ and similarly, $r=MP_K\equiv \frac{\partial Q}{\partial K} = b\frac{Q}{K}$
Substituting these in $(1)$ (or equivalently in $(6)$), we get:
$$\frac{C}{Q}=(a+b)$$
So interestingly, the average cost is constant despite having increasing returns to scale.
So, increasing RS can at best ensure:
$$\frac{\partial AC}{\partial Q}<0$$
But what ES requires is:
$$\frac{dAC}{dQ} = \frac{\partial AC}{\partial w}\frac{\partial w}{\partial Q}+\frac{\partial AC}{\partial r}\frac{\partial r}{\partial Q}+\frac{\partial AC}{\partial Q}<0$$
So it is perfectly possible that $\frac{\partial AC}{\partial Q}<0$ but $\frac{d AC}{d Q}>0$


How much the wealth inequality increased in the USA in the last decades?

This statistic shows the Gini coefficient, an index for measuring
  income distribution, for U.S. households from 1990 to 2016. A Gini
  coefficient of zero expresses perfect equality, where all would have
  the same income,a Gini coefficient of one expresses maximal inequality
  among values. In 2016, the Gini coefficient for household income was
  0,48. This is an increase since 1990 of 0.5 points, indicating an
  increase of income inequality in the United States.

Source: Statista

In case you are interested in how data for such indeces are gathered, a recent study explaining its data process is Wealth Inequality in the United States since 1913.


Examples of behavioural breakdowns in game theory?

Here is an article that reports about several laboratory experiments showing large inconsistencies between theoretical predictions and observed behavior for some payoff structures:

Goeree, J.K. and Holt, C.A., 2001. Ten little treasures of game theory and ten intuitive contradictions. American Economic Review, 91, 1402-1422.
https://doi.org/10.1257/aer.91.5.1402

There are also papers studying variants of the ultimatum game in order to quantify players’ preferences for fairness. One approach to do so is to compare (i) games in which two human players interact with each other and (ii) games in which a human player interacts with a computer (so there is no scope for social preferences):

Hoppe, E.I. and Schmitz, P.W., 2013. Contracting under incomplete information and social preferences: An experimental study. Review of Economic Studies, 80, 1516-1544.
https://doi.org/10.1093/restud/rdt010


If assets = liabilities + equity, what do we mean by overcapitalization?

A firm's Output $Y$ (in value-added terms, i.e. over and above the value of purchases of third-party services and materials) is distributed as reward to the factors of production. Denote $K$ the book value of the company's assets.  Use the standard notation $L$ for payments to the labor/human capital input to production. Denote $r$  the capital rate of return as measured with respect to the book value of firm's capital and denote $w$ the average remuneration of human capital services. Then we have

$$Y = rK + wL$$

Depending on the market, the industry, and a few other things, we can estimate  $r^*$, the rate of return that the firm "should" have on average, if it should be judged as an efficient user of capital allocated to it, for given output and human capital costs. $r^*$ is usually estimated by what are the returns on the market and the industry (we do not go outside the industry since we recognize the existence of "asset specificity"). Associated with $r^*$ goes the efficient level of capital, $K^*$. So we also have

$$Y = r^*K^* + wL$$

leading to

$$rK = r^*K^*  \implies r = r^*\frac {K^*}{K}$$

If we have that

$$r < r^* \implies r^*\frac {K^*}{K} < r^* \implies K^* < K$$

and we can say with some certainty that the inequality does not show a tendency to change in the near future, then we can argue that the company does not make efficient use of the current amount of capital it appears to have, either due to "functional inefficiency" (how things are organized and work inside the company and in the delivery of of its business, a qualitative aspect), or due to plain under-utilization of the existing capital (a quantitative aspect).

The "over-capitalization" story essentially assumes "functional efficiency" but it sees under-utilization of capital, and so it advocates the reduction of the capital/assets currently under the control of the company.

To measure it we have the relation ($K_E=$ Excess Capital)

$$ K_E = K - K^* = K- \frac {r}{r^*}K = \left (1-\frac {r}{r^*}\right)K$$

$r$ and $K$ is measured from the firms financial statement, while $r^*$ from Industry weighted averages.


Do large public companies ever "quit while they're ahead"?

Assume a company decides to "wind up". What would happen? Its competitors would want to get hold of its tangible and intangible assets (like customer base, patents etc). How is this different from "fighting to a buyout"? In this second scenario, the company has always available the option/threat of continuing competition (possibly harmful to the competitors), while in the "wind-up" scenario it does not -it has committed to leave the business. So "fighting to a buyout" gives the company better position in negotiations, and so chances to increase its selling price compared to the "wind up" scenario.

You could spice this with theories about management, and how "management will not let go" etc, but the above provides an interpretation that appears more widely applicable.

Compared to smaller companies, large companies have more market power and so the "threat" of continuing competition weighs more heavily with competitors.


Does the 2nd welfare theorem involve government intervention?

The second welfare theorem does not necessarily involve a government as such. It does not matter for the theorem who redistributes the resources. Nevertheless, in practice this will most likely often involve the government.


How to find change in the optimal choice with a utility function in general form?

It depends what you know about your utility function and the "space" of goods $x_1$ and $x_2$.

You ask in your question what the change of optimal choice is with respect to the parameter information.
This analysis is called comparative statics, and you can use a number of mathematical tools to achieve this. The most common are

The Implicit Function Theorem

If the space of goods in convex and everything nicely differentiable, quasi-concave etc. etc., then this is easiest. I will not discuss the theorem, you should work through it yourself (good references are in De La Fuente's book or in MWG).

First, what is the optimal choice for the actor? Since everything is differentiable, AND quasi-concavity implies that the first-order conditions characterize the optimum (this is important), you can derive the conditions for the optimal choice by the first order equations
$$ D_{x_1,x_2,I}U(x_1,x_2;I) = \mathbf{0}$$
where $D$ is simply the differential operator aka in your case the vector of $\frac{\partial U}{\partial x_i}$.

This characterizes the optimal choice. It is not yet "solved" as the form of $x_i=\ldots$, but this is often neither possible nor necessary. For example, maybe you don't really know what $U$ is except that it has certain properties, so you can never solve for $x$ explicitly. That is why we use the "implicit" function theorem.

Applying the Implicit Function Theorem allows you to calculate
$$\frac{\partial x_i}{\partial I}$$
simply by using the "ratios" of first-order conditions you derived above.

Again, you need to carefully check some properties of the functions, so I will give you the formulas but urge you to check the literature.

Monotone comparative statics

As it turns out, we can do comparative statics in an even more general world than a convex space of goods with differentiable utility.

Suppose we are only interested in "directions" of choice. That is, we care only about monotonicity. For example, the IFT does not work for discrete goods ("1 car", "2 cars", "3 cars") since the space is not convex. But the ideas should work similarly!

Doing optimization on these non-convex, and yet ordered sets it called lattice optimization.

Now, we merely need to assume that $X$, the space of goods, is a lattice - a set where the possible combinations of $x_1$ and $x_2$ have a certain order, and that the information values $I$ constitute a partially ordered set.

Now, if we assume that the utility function is at least supermodular (think similar to concavity in this more general space), then we can use a similar result to the implicit function theorem, called Topkis's theorem to establish how $x_i$ changes as response to $I$.

But in fact, this is not even the most general case. Remember the difference of concavity and quasi-concavity? Well! Here, too, we can invoke quasi-supermodularity. This brings us in the word of Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics”.

In particular, they claim that this form of comparative statics is the most general, and in a sense, necessary and sufficient from of "optimal choice" and "comparative statics".
I found that a very good reference is the book by Xavier Vives.

Well, you asked for generality, so there you go.

Edit: If you are an undergrad, then perhaps just file this response away and I will say that, yes, your understanding of the problem also works. You can differentiate the demand function to see how it changes with respect to the parameter ;-)


Can you please help me find this topic in Mechanism design/Rational choice theory?

I think you're referring to the decoy effect. A popular example of this is the Economist subscription puzzle, popularized in Dan Ariely's TED Talk (starting at 12:22).


If the Number of Workers in an Economy Doubled, Would Productivity Fall to Half its Former Value?

Since you say holding other variables as they were and double $L(t)$ I assume you have a production function that may include capital $(K(t))$, including this and assuming a Cobb-Douglas production function:

$$ Y(K(t), L(t)) = K(t)^{\alpha} L(t)^{1-\alpha} \; \; 0< \alpha < 1 $$

Doubling $L(t)$ gives, (and omitting $t$ from the writing for clarity):

$$ Y(K, 2L) = K^{\alpha} (2 L)^{1-\alpha} $$

$$ = 2^{1-\alpha} Y(K,L) $$

$$ \Rightarrow \frac{Y(K, 2L)}{2L} = \frac{2^{1-\alpha} Y(K,L)}{2L} $$

$$ \frac{Y(K,L)}{2^{\alpha} L} < Y/L $$

So yes in this sense as you described doubling labour decreased productivity.

You mention $Y$ having constant returns to scale, this implies if you double both $K$ and $L$ then $Y$ will double as well, then productivity will stay the same


Does it mean that the value of currency decreases?

This is a constant source of confusion in international macreconomics. There are two ways to define the exchange rate.

$$E = \frac {\text {home}}{\text{foreign}}$$

Here, we express "units of home currency per unit of foreign currency" (or basket of foreign currencies). In this definition
$$E \uparrow \Rightarrow \text {home currency depreciates}= \text {value} \downarrow$$
so it can be confusing at times, since usually the object of study is a specific "home" currency.

The other way is the reciprocal

$$S = 1/E = \frac {\text{foreign}}{\text {home}}$$

Here we measure "foreign currency units per unit of home currency", and so
$$S \uparrow \Rightarrow \text {home currency appreciates} = \text {value} \uparrow$$

Some argue that this second expression is more "natural" since it expresses the "price" of home currency in terms of foreign currency, i.e. "what is the price of 1 euro ? 1.22 USD", "what is the price of 1 USD?  0.82 euro".

But not even the notation is standard (for example, the wikipedia article uses the symbol $S$ to denote the exchange rate that we denoted by $E$).

One way to avoid confusion is to write the various relations with subscripts, and use just one symbol for the exchange rate. Using $S$ as this symbol, the subscript $f$ for "foreign" and the subscript $h$ for "home" we have

$$S_{h/f} = \frac {\text {home}}{\text{foreign}},\;\; S_{f/h} = \frac {\text{foreign}}{\text {home}}$$

For the specific question, since interest rates are in focus, it appears that the underlying law is the Uncovered Interest Rate Parity (UIRP), where

$$1+i_h = (1+i_f)\left(\frac {S_{h/f, t+1}}{S_{h/f,t}}\right)$$

$$\Rightarrow_{\text{approx}}\;\; i_h = i_f + \Delta s_{h/f,t+1}$$

or

$$i_h= i_f - \Delta s_{f/h,t+1}$$

where $s$ denotes the natural logarithm, and $\Delta$ is the first-difference operator. By looking at the UIRP, we see that if the home interest rate is higher than the foreign interest rate, $i_h > i_f$, it must be the case that $\Delta s_{h/f,t+1} >0$ to satisfy the relation. $\Delta s_{h/f,t+1} >0$ is expected percentage change in the "home per foreign unit" exchange rate. So we say that the home currency is expected to depreciate...

But hey, intuition appears to say otherwise:
Assume $i_h$ rises above $i_f$. Then foreign investors have a motive to deposit their funds in the home economy, since the interest is higher. But in order to do so, they increase the demand for the home currency, since they need to convert their funds in home currency units. Assuming constant home money supply, this leads to an increase in the value of home currency compared to other currencies. But it does that now so mathematically it decreases the current exchange rate, $S_{h/f,t}$:

$$i_h > i_f \Rightarrow S_{h/f,t} < S_{h/f,t+1} \Rightarrow \Delta s_{h/f,t+1} > 0$$

In other words a higher home interest rate increases the value of the home currency now, creating expectations for home currency depreciation in the future.


Syndicated Loans Terminology - Borrower & Sponsor

It sounds like you're talking about a leveraged buyout. In these transactions, a private equity firm (or sponsor as you call them) will acquire an operating company. The debt for the acquisition - raised via the capital markets - will reside on the target company's balance sheet. This debt will be used to pay the previous owners (another PE firm, founders, etc.. whoever owned the company previously). The PE firms will never take debt onto their own balance sheet, and during the course of ownership often raise additional debt on the company to pay themselves dividends.


How does low population growth increase the effect of accumulated wealth

Piketty's essential argument is that when the $r$ is the rate of return to wealth and $g$ is the economic growth rate, you will see wealth and economic power concentrate in the hands of the few when $r \gt g$. This presumes that the wealthy will reinvest their returns from wealth rather then spending it on consumption.

Low productivity improvements will lead to low growth in GDP per capita, and combining this with low population growth will then lead to low growth in total GDP, so these combined will lead to smaller values of $g$ than seen previously when both productivity and population were growing more quickly.  This is probably not particularly controversial.

Harder questions include

the impact of low productivity improvements and low population growth on $r$, and therefore on whether the gap between $r$ and $g$  would in fact widen
empirical evidence of whether $r \gt g$ has been true historically and globally
whether further concentrations of wealth may in future change $r$
whether Piketty's preferred solution of global wealth taxes and taxes on very high incomes (designed to reduce $r$) might also have an effect on $g$, and if so then in which direction and by how much


How would a 2D model of Hotelling's law outweigh the benefits of a 1D model?

There are several scientific papers analyzing this on topologies other
than the unit interval. A 2D example is Spatial competition of firms in a two-dimensional bounded market by Aoyagi and Okabe. Unfortunately there are usually no equilibria in these cases. The intuition behind this is that the consumers arriving from different directions are usually not "balanced", and one could gain by deviating to a direction where a higher number of consumers are coming from.

Also, Hotelling models are not used in practice (to my knowledge) when one is assessing where to open a new shop.


Your basic assumptions seem to be
$$
p(q_a) = AVC(q_a)
$$
and as the difference between price and average variable cost is decreasing before $q_a$ and increasing after it you also assume
$$
\frac{d \ \left(p(q) - AVC(q)\right)}{d \ q} < 0.
$$
(More on the necessity of this assumption later.)
 We can rewrite this second assumption to
$$
\frac{d \ p(q)}{d \ q} - \frac{d \ \frac{VC(q)}{q}}{d \ q} = \frac{d \ p(q)}{d \ q} - \frac{MC(q) \cdot q - VC(q)}{q^2} < 0.
$$
Multiplying by $q>0$ yields
$$
\frac{d \ p(q)}{d \ q} \cdot q - MC(q) - AVC(q) < 0.
$$
Adding $p(q) + AVC(q)$ we get
$$
\frac{d \ p(q)}{d \ q} \cdot q + p(q) - MC(q) < p(q) - AVC(q).
$$
The left hand side is now $MR(q) - MC(q)$. It follows from our two initial assumptions that
$$
\forall q > q_a: \ p(q) < AVC(q).
$$
Then for such values of $q$
$$
MR(q) - MC(q) < p(q) - AVC(q) < 0.
$$
Because of this there can be no $q_b > q_a$ such that $MR(q_b) = MC(q_b)$.

On the necessity of the second assumption:
Seems to me that without this assumption the goal function $p(q) \cdot q - C(y)$ would not be concave and hence $MR(q) = MC(q)$ would not be a sufficient condition for optimum.


How to find corner Pareto efficient allocations

Let us take your example:

First, we note that both utility functions are differentiable and quasi-concave.
Noting this, we also know that the necessary and sufficient condition for internal Pareto optimality is that $MRS_{x_1,y_1}$=$MRS_{x_2,y_2}$ (as you have already correctly stated).

This condition will clearly coincide with the portion of the solution identifying the locus on internal P.O. allocations.

Now, for the P.O. points along the right edge:

We can find the bound of internal solutions by identifying the range over which the MRS condition noted above fails.
Because the equality fails, we know a strict inequality must prevail.
The directionality of the prevailing strict inequality identifies the edge along which we find our P.O. allocations.

So, there are two ways to answer your question, I think.

1.) For this type of graph, where one agent has linear preferences and the other has curvilinear and convex preferences, it is easy to see that the locus of P.O. allocations shifts toward the right edge of the Edgeworth box. Thus, corner solutions run along that edge across the range where points of tangency are no longer interior.

2.) If you encounter a situation where, for example, both agents have linear preferences, you can use the directionality of the strict inequality to identify along which edges of the Edgeworth box you have P.O. allocations.

An example for 2.) :

$U_1(x_1,y_1)=X_1+2Y_1$ and $U_2 = 2X_1+Y_1$

Now, you should do the following:

ensure you understand why $MRS_1 \neq MRS_2$
determine the direction of the strict inequality between the two
use this to identify along which edge of the box you will have P.O. allocations.

Hints:

$MRS_A<MRS_B$
the P.O. allocations are along the left and upper edges of the Edgeworth box

Edit:

I also think this is a good reference:

http://nicolalimodio.com/wp-content/uploads/2015/11/ec202d.pdf

Hope something in there helps. This was an issue for me as well when I began learning such things and it took me a bit of reading and practice to finally become comfortable with it all.


The intuitive way

Draw the function $L_t$, the graph below is build with $\gamma = 0.8$ and $\pi_t^\otimes = 1$

Convex in this context means that the function $L_t$ looks like a bowl, just one well defined minimum. And if you cut it with a horizontal plane, the result is an ellipse.

  The formal way

Just for convenience, define the variable $p_t = \pi_t - \pi_t^\otimes$ so the function becomes

$$
L_t(p_t, Y_t) = \gamma p_t^2 + Y_t^2 \tag{B1}
$$

First note that if $\gamma > 0$  then $L_t(p_t, Y_t) \ge 0$. To prove convexity you just need to show that for any pair $(p_{t,1},Y_{t,1})$ and $(p_{t,2},Y_{t,2})$ the condition

$$
L_t(t p_{t,1} + (1 - t)p_{t,2}, t Y_1 + (1 - t)Y_{t,2}) < tL_t(p_{t,1}, Y_t) + (1 -t)L_t(p_{t,2}, Y_{t,2}) \tag{B2}
$$

holds for $ 0 < t < 1$. This is actually fairly easy to show

\begin{eqnarray}
L_t(t p_{t,1} + (1 - t)p_{t,2}, t Y_1 + (1 - t)Y_{t,2}) &=& \gamma[t p_{t,1} + (1 - t)p_{t,2}]^2 + (t Y_1 + (1 - t)Y_{t,2})^2 \\
&\vdots & \\
&=& t [\gamma p_{t,1}^2 + Y_{t,1}^2] + (1 - t) [\gamma p_{t,2}^2 + Y_{t,2}^2] \\
&&\quad + t(1 - t) [\gamma(p_{t,1} - p_{t,2})^2 + (Y_{t,1} - Y_{t,2})^2] \\
&=& tL_t(p_{t,1}, Y_t) + (1 -t)L_t(p_{t,2}, Y_{t,2}) \\
&&\quad \underbrace{t(1 - t)L_t(p_{t,1} - p _{t_2}, Y_{t,1} - Y_{t,2})}_{> 0} \\
&< & tL_t(p_{t,1}, Y_t) + (1 -t)L_t(p_{t,2}, Y_{t,2}) \tag{B3}
\end{eqnarray}

So $L_t$ is convex. To show that indifference curves are ellipses, just fix $L_t$ to a constant value, let's say $A > 0$ then we have

$$
A = \gamma p_t^2+ Y_t^2 ~~\Rightarrow~~ \frac{p_t^2}{A/\gamma} + \frac{Y_t}{A} = 1 \tag{B4}
$$

which are just ellipses centered at $p_t = 0 = \pi_t - \pi_t^\otimes$ and $Y_t = 0$ with axis lengths $\sqrt{A/\gamma}$ and $\sqrt{A}$


How buying bonds indirectly from the government prevents the central bank from financing government deficit?

Even though central banks are part of the government they are in most developed countries independent, this is similar to way how courts operate. For example US government has Judicial Branch, where the supreme court justices are part of the government and appointed by president (the same way as chairman of Fed is) but this does not mean government can order supreme court or Fed around. Government could wait for Fed's chairman term to expire and appoint new one but it is expected that these appointees are technocrats and thus its not guaranteed they will act according to party line (i.e. line of the party currently in power).
In principle government could ask ('order') Fed to buy its bonds, however Fed could (and should) say no if it conflict with its dual mandate of price stability and full employment. Or they could say yes if they believe that is consistent with their mandate. This being said as Krugman points out in his book Fed wont go this route in normal times in order to keep it's 'distance' from the government. This really has no economic rationale but rather political one. The more entangled Fed (or any other central bank) gets with the government the more likely it is that there will be pressure for politicization of the institution.
This idea is made explicit by Fed in its FAQ, where it states:

Conducting transactions in the open market, rather than directly with the Treasury, supports the independence of the central bank in the conduct of monetary policy.


“Capital” is not an asset (as that would be on the right hand side of the balance sheet) they are long maturity instruments on the right hand side of the balance sheet.

Common equity is one major component. It’s the residual you get after you subtract all other items on the right hand side of the balance sheet from total assets - which is what forces the balance sheet to balance. (There are presumably some accounting details I am skipping over.)

If we simplify things, it’s how much more its assets are worth than its liabilities, and so it cannot get put anywhere. All that can happen is that assets it owns (a loan to another bank) can lose value, and so the credit loss reduces equity (and hence capital).

(I think perpetual preferred shares are the other major component of Tier 1 Capital. These are issued by the bank in question, and so the failure of another bank has no effect on them.)


Why do banks need to write-off the debt against Shareholders capital invested

"Double-entry" bookkeeping is the resulting methodology of the fundamental approach of Accounting science to what a productive mechanism does (be it a company or whatever): Accounting perceives productive entities as mechanisms to transform capital from one form to another, irrespective of what their economic goal is for doing this (profits, charity, whatever).

The Liabilities+Equity side shows "how much capital and from whom" has  the firm sourced and put it under its control, and the Assets side shows to what forms the firm has transformed this same capital . But if it is the same capital, it necessarily has the same value : this is why accountants write the same number twice: they write it once and link it to a code that shows from whom did the firm get this amount of capital, and they write it a second time and link this to a code that represents to what has the firm transformed the capital.

So I took a capital of 1,000 from a source called, say, Shareholders, (or Deposits, or a combination) and transformed it in another form of capital called "Loans to customers". I write the number down two times, one linking it to the code that Shareholders have, and a second time to link it to the code that "Loans to customer has".

At the same time "Liabilities + Equity" shows how much capital must the firm return back -because in the theoretic Accounting model, the company itself "owns nothing", it is just a capital transformation mechanism. It may has capital under its control, but it has eventually to return it.

Now the customer appears and says, "no payback". It is the same thing as a fire destroying a building - an amount of capital has been lost. So the company can no longer return it. Given the general legal/social arrangements, in such cases, irrespective of the actual source of capital (say, Deposits), those who will now must expect to get lower capital back, are the shareholders. So the double-entry write-off of the bad loan reflects exactly that fact: "I, the transformation mechanism, inform the world that a) I just lost an amount of capital and consequently that b) given the rules, my shareholders have now a lower claim from me, as regards the return of capital to them".

Not doing this, won't "be better for everyone", because it would be misleading - by looking at the bank's book, the shareholders would still expect to receive back their 1,000 investment, but they would eventually find out that only 100 are there in reality.

I hope that helps.


Representing preference orderings over a finite set of outcomes by two payoffs

Say you have preference over $N$ deterministic outcomes, say $N$ bundles of goods, with $1\succ 2\succ\cdots\succ N$. The usual way to represent this preference is by assigning utility numbers $u_i\in\mathbb R$ to each bundle $i$, such that $u_1>u_2>\cdots>u_N$.

In contrast, you can also interpret the "outcomes" as lottery tickets that pay either a high payoff (say $\$1$) or a low payoff (e.g. $\$0$) with varying probabilities. So each lottery $L_i=(p_i,1-p_i)$ is a distribution over the two payoffs, where $p_i$ is the probability that you get $\$1$ and $1-p_i$ is the probability of getting $\$0$. Thus, by choosing $p_i$'s that satisfy $p_1>p_2>\cdots>p_N$, you have a preference representation over the $N$ lotteries in the expected utility framework.


To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on constant elasticity of substitution. In particular,

The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969).

Then, the CES utility function was popularized by Dixit and Stiglitz (1977) in their study of optimal product diversity in a context of monopolistic competition.
If you want to understand how the CES utility function behaves when $\sigma=\infty$ or $\sigma=1$ here is a nice discussion of the basics of the CES utility function, which is widely used in trade.
If you want a deeper understanding of the elasticity of substitution and a discussion of the special class for which the elasticity of substitution is a constant $\sigma$, here is an insightful discussion by Ted Bergstrom.


Negative Interest Rates and Pension Funds

This is two questions.

How do negative interest rates impact the future prospect and business model of (defined benefit) pension funds?

Defined benefit pension funds have a discount rate and assumed returns to determine the required contributions. The exact mechanism for determining these depends upon the jurisdiction, but they are normally related to market interest rates, possibly on an averaged fashion.  Lower discount rates - regardless of whether they are positive or negative - implies greater contributions needed for the same level of benefits. This makes defined benefits more “expensive.”

What could be winning investment strategies for pension funds in a negative interest rate environment?

This can only be answered as an opinion, which is not in line with guidelines.


Inflation, Future, And Value of Money - deciding to buy a house

If a country has a lower inflation rate than your country, it's currency will tend to appreciate against your currency over time, see interest rate parity.

It is true that if inflation is low, house prices will tend to increase less over the next ten years, however, the currency will tend to appreciate, making it more expensive to buy a house in ten years time.


When a central bank 'pays' interest rate X / year, what does it mean
  exactly?

It means that by lending to the central bank \$100/(1+X) at the beginning of the year, the central bank will pay \$100 at the end of that year.

  A popular way to fight high inflation is to increase interest rate,
  but doesn't that mean printing more money -> more inflation?

Not necessarily. Increasing the interest rate may attract cash flows from foreign lenders/investors. The effects of it on the exchange rate (appreciation of the local currency), currency reserves, and/or trade balance tend to alleviate the need to merely print more money in the local economy.


Well this is 'exponential discounting'. An infinite sum of geometric series:
$$\sum R+ R\delta + R\delta^2.... R\delta^t = \frac{R}{1-\delta}, \text{ if } |\delta|<1$$
Now lets call the denominator rho $1-\delta= \rho$.
The exponential discounting is there since its an infinite sum of geometric series.
Edit: In response to Giskard's +1 comment I tried to look deeper into the paper and while I think that the Dixit is actually implying the above and just not being clear on terminology it is also possible that he means simply net present value of infinite constant stream as:
$$\sum \frac{R}{(1+\rho)^t}= \frac{R}{\rho}$$
However, I think the former interpretation is the right one because its more closer to how discount rates are treated in some other papers I have seen and I think he was just sloppy with terminology.


How would the risk premium and a simplified taylor rule change the MP curve?

I get it now

Under traditional monetary policy
MP Curve is set by the central bank and follow the fisher equation
$$i_t = R_t + \pi_t$$

But traditionally the taylor asks that when the central bank considers monetary policy that they consider short run output and inflationary gap.
$$i_t = R_t + \pi_t+\bar m(\pi_t-\bar \pi)+\bar n \tilde Y_t$$
Where the monetary policy rule is
$$R_t-\bar r = m(\pi_t-\bar \pi)+n \tilde Y_t  $$
Now
$$R^{CB}_t-\bar r = \bar n \tilde Y \space  where \space \bar n>0  $$

and
$$R_t = R^{CB}_t+\bar p $$

I assume that in the original taylor rule
$$R_t-\bar r = m(\pi_t-\bar \pi)+n \tilde Y_t  $$

Hence in the original taylor rule
$$i_t = R_t + \pi_t+\bar m(\pi_t-\bar \pi)+\bar n \tilde Y_t$$
$$i_t = R_t + \pi_t+R_t-\bar r$$

But in this simplified taylor rule $R^{CB}_t-\bar r=n $
$$R^{CB}_t-\bar r = \bar n \tilde Y  $$

To which i assume also that
$$R_t-\bar r = \bar n \tilde Y = m(\pi_t-\bar \pi) $$

Hence
$$i_t = R_t + \pi_t+R_t-\bar r$$
$$i_t = R_t + \pi_t+n \tilde Y $$

If $R_t = R^{CB}_t+\bar p$

$$i_t = R_t + \pi_t+n \tilde Y $$ $$i_t =R^{CB}_t+\bar p + \pi_t+n \tilde Y $$

Hence the $ n \tilde Y $ component of the simplified taylor rule makes the MP curve upward sloping.


What are the justification of using 1990 or 2010 as benchmark in studies?

First, something that was not explicitly mentioned in your question is that in both cases you refer just to indexation, so answer will also be given in context of indexation.
Second, your premise is false. Datasets & research do not use just multiples of 10 but also multiples of 5 often. For example, for HICP Eurostat choose year 2015 as base year. Moreover, you will occasionally see other numbers as well.
There is no reason to choose round numbers or multiples of 10 or 5 other than round numbers and multiples of 10 or 5 are aesthetically pleasing.

Choice of base year in any index does not matter analytically, it only changes visuals. Analysis won't be affected by converting GDP into 2009 or 2010 or 2011 etc dollars. As long as whole analysis uses the same dollars for everything there will be no change.
Only time when choice of base year in index matters is when you are doing a visualization. For example, if you want to know how productivity in Russia increased since fall of USSR (in 1991), you could choose base year 1991 making productivity index in 1991= 100 which makes it easier to see if productivity declined or increased just by looking at the number itself (although even if you would choose year 1990 a correct comparison would take no more than few extra seconds of your time). Even here it is context depended as the graphs will look almost identical.

Most of the reasons you state are false except the first one:

These numbers look round and are convenient.

This is actually correct reason in most of these cases. For some reason people like numbers that are multiples of 10 or 5, why this happens is question for psychology not economics.

If you used the benchmark as 1991 or 2011, the readers will think you had some reasons to omit 1990 or 2010, you will be questioned about cherry-picking data.

This is false. If choice of index would make actually a difference for an analysis then choosing 2010 would be equally questionable and people could equally accuse you of cherry picking. In any case where choosing different year would actually affect data analysis, the choice of year needs to rigorously justified or your research will be sooner or later questioned.

Multiple-of-10 years do have an advantage in the USA and some other countries because these are the census years, so the macro-level data would be up-to-date and better in demographical.

Again false.

Most countries around the world do not even conduct censuses in round years. See here database for dates of all censuses around the world. For example, the Netherlands where I live conducts census every 10 years with years ending with 1 (i.e. 1991, 2001, 2011, 2021), yet the Netherlands does not indexes its statistics with years ending with 1. For example, as you can see here CBS (Statistical office of the Netherlands), indexes CPI with year 2015 (following Eurostat). You will not see virtually any indexes in year ending with 1 in any research produced by Dutch research centers such as Centraal Planbureau or De Nederlandsche Bank, even when research is only done on Dutch level without any international comparison.
You can still combine all census data with GDP, CPI or whatever you want being indexed or expressed in dollars from any year you choose. For example, if you want to regress real GDP on population, trying to see if larger population leads to higher GDP, and you have census from round years (e.g. 2010, 2020...) you can still choose to express everything in 1997 dollars or 1998 dollars. It literally does not matter. Only thing that does matter is that once you choose some base year you should express all dollar quantities in that base year and not use let's say consumption in 2005 dollars if income is expressed in 2010 dollars.

To really drive this point home, below I extracted from Fred dataset Consumer Price Index for all Urban Consumers. I made 2 graphs. In one of the graph the benchmark year is 2007 in the other one the benchmark year is 2010. I removed title from the graph telling you which index uses which year as a benchmark. Can you see any difference, can you tell which one is indexed to 2007 and which to 2010?


I am not sure whether the question asks about the function being convex, or simply the demand correspondence/function being a convex set. The latter is often asked and if that is the case here is an answer.

This is a general answer to your question which may be more useful to you and future readers than a specific one.

The following theorem holds in general:

Theorem: If preferences are convex, then the marshallian demand x(p,w) is a convex set for every p>>0 and $w \ge 0$.
Note: p>>0 here means that every element of the price vector is strictly positive.

Proof:

The Budget set $B(p,w) \equiv \{ x \in \mathbb{R}_+^n:px \le w \} $ is a convex set.

If two points $ x, x' \in x(p,w )$, then x ~ x' (the consumer must be indifferent between the two as they are both part of x(p,w), i.e. his best choice. If one of them would be better than the other, then only the best one would be part of x(p,w) as the marshallian demand is derived from maximization.)

For all $\alpha \in [0,1]$ we have that $\alpha x + (1-\alpha)x' \in B(p,w)$ by convexity of B(p,w). Furthermore, $\alpha x + (1-\alpha)x'\succeq x$ by the convexity of preferences. Therefore $\alpha x + (1-\alpha)x'\in x(p,w)$
Q.E.D.

Your preference relation is convex, so the theorem applies and your set x(p,w) is convex. Note that a preference relation is (strictly) convex if and only if the utility function  is (strictly) quasiconcave. Your utility function as far as I can see is concave, which implies it is quasiconcave, which implies convex preferences.

Further note that if preferences are strictly convex, then x(p,w) is single-valued for every p>>0 and $w \ge 0$. Hence in that case x(p,w) would be a convex set as a point is a convex set.


Example of a rational and continuous preference relation that does not admit a utility representation

A necessary condition for a preference relation to be represented by a utility function is that the preference relation is rational (where a "utility function" is a real-valued function that assigns a higher or equal numerical value to bundle A than to bundle B, when A is weakly preferred to B).

Preferences are rational when they are complete (I can express my preferences for all conceivable bundles), and transitive (If I weakly prefer A to B and B to C, then I weakly prefer A to C).

Given rationality, a sufficient condition is that the rational preference relation is continuous, i.e. if it is preserved under limits.

Then, the related utility function is also continuous.

Monotonicity is a property where we go from quantities to preferences, after assuming desirability ("goods" rather than "bads"). Local nonsatiation is actually the weaker assumption that is required for most of the theory.


Why do central banks currently favor cheap money policies?

Most central banks' primary goal is maintaining a moderate inflation (around 2 %). In some cases, they also have a secondary and lesser mandate to care for employment numbers.

Since inflation is not taking off but persists at a low level, the central banking system has flooded the markets with money. The thought is that the money will "trickle down" from banks and institutions with easy access to money. However, the effects on the real economy have been somewhat limited.

The central banks' idea is that when capital is cheap to come by, companies will invest and thus hire more people to the workforce, who will in turn consume more. This isn't really happening, which is likely due to demographic and technological factors. After employees were released during the 2008 financial crisis, companies haven't really started to employ people again in the same magnitude. To a higher degree, production processes are now outsourced or automated. Companies who are hiring, typically are looking for people with higher education or specialised skills. Most unemployed people available to the labor market do not meet these requirements. So we have deflationary pressures in the world economy that are keeping inflation down.

If central banks really wanted to get inflation going, it would probably be more effective to just hand out money directly to the consumers' bank accounts, than having money pass through intermediaries. The exception to this is that we have seen pocket bubbles in real estate and some other asset classes. Despite interest rates being record-low, people who already have some wealth aren't really consuming much more, neither are they saving funds in their bank accounts, but have instead tried to find asset investment opportunities to increase wealth. This is creating lending and speculation bubbles in real estate and an upswing in stock and bond prices.

Some economists are also starting to question the objective of pursuing inflation and if it really should be the paramount objective of central banks, when the problems created by massive asset bubbles could become so much worse.


Multidimensional product differentiation and density functions

Probably, the author probably is assuming $x_i$ and $y_i$ are independent in addition to the stated assumption on pg. 7 that types "are orthogonal" (which merely means uncorrelated.) (However, since they are also uniformly distributed, we have, $f_{X,Y}(x,y)=f_X(x)f_Y(y)\equiv 1, \forall x,y$, meaning they are independent.)
Assuming independence,
$$
\iint_{\{(x,y):0\le y\le \tilde y(x),0\le x\le 1\}} f_{X,Y}(x,y)\,dydx = \int_0^1 f_X(x)\left[\int_0^{\tilde y(x)} f_Y(y)\,dy\right]dx.
$$


What does "steep incentive contract" mean in the context of adverse selection?

"Steep incentives schemes" are synonymous with "high-powered incentives". A steep incentive scheme is thus one in which there is a large or high-powered performance incentive. This can take either the form of a small fixed payment and a (potentially) large bonus, or it can be a large fixed payment and a (potentially) large penalty for deviation.

Normally we think about using steep incentive schemes when we wish to have a separating equilibrium in which high-skilled (or risk-seeking) workers choose steep incentive schemes and low-skilled (or risk-averse) workers choose flatter incentive schemes. Here is one paper I'm aware of that discusses this. You will find more examples if you look at optimal contracts under risk-aversion. https://www.econstor.eu/bitstream/10419/30169/1/620692227.pdf

In your example, I believe the decreasing nature of $a(T)$ is capturing the movement from a low-cost (less risky) project to a high-cost (more risky) project. As the cost (risk) for the project increases, the principal will want a steep incentive scheme to ensure that only a high-skilled contractor will choose to perform the task. This is because we are shifting the weight away from $a(T)$ (the fixed payment) and towards $a'(T) \times (T-C)$ (the bonus/penalty) as $T$ increases. Moving the other direction, as the cost (risk) for the project decreases, $a(T)$ increases and the incentives have less power.


How much does a public sector pay rise actually cost the UK government?

Unison (2014) - Net cost

The Public sector trade union did an analysis of lifting the cap in 2014. The report is here. Unlike the IFS analysis presented above, Unison estimates the net cost of an increase in public pay by 3%, using a microsimulation model. They estimates are summarised in the table below:

The key here is that a gross increase in the cost by approx. £4 billion reduces to a net increase between £1.7 and £2 billion (or £2.2; see Table 6a). The difference between scenarios depends on the assumption of the fiscal multiplier, which can range between 0.34 (used by the Office of Budget Responsibility), and 1.02 (upper bound used by the IMF). Naturally, the larger the multiplier is the higher the benefit from the wage increase, and the lower the net cost.

Institute for Fiscal Studies (2017) - Gross cost

The Institute for Fiscal Studies provided an analysis of the 2017 general election Labour and Lib Dem manifiesto, which included a lift to the 1% public pay cap increase. As the report states:

  Our analysis of the Labour plan implies that, compared to the current government’s plans, by 2021–22 a Labour government would need to provide departments and local government with an additional £9.2 billion per year to pay for the higher costs of employing public sector workers. Of this, £2.9 billion would be for the NHS. Comparing the Liberal Democrats’ plans to the current Conservative government’s plans, it would necessitate an extra £5.3 billion per year, of which £1.6 billion would be for the NHS.

The table below summarises their conclusions:

The key difference between the two parties is that the Lib Dem suggested to increase pay in line with inflation, whereas the Labour party suggested to increase pay in line with the private sector (more precisely, based on the recommendations of Pay Review Bodies).

As the report states, any forecast is highly dependent on the state of the economy (uncertainty due to e.g. Brexit):

  These Figures are highly sensitive to the forecasts of the economy. For the Liberal Democrats, the cost will be higher if inflation is higher in 2018–19 and 2019–20 than is expected by the OBR currently, or lower if inflation is lower. Under Labour’s policy if private sector earnings growth performs less well over the coming years than is forecast by the OBR (it has consistently underperformed OBR forecasts in recent years), then it would cost less than is implied by Table 1. Conversely, if private sector earnings growth performs better than forecast, and public pay is increased faster too, then the cost to departments and local government of Labour’s policy would be higher too.

Notice that these figures are gross cost estimates. I.e. they do not include indirect costs or benefits associated with it. For example, further costs might arise due to how the spending is financed. Such increase in spending would certainly require (i) more taxation, and/or (ii) more borrowing (cuts also possible, but less likely, due to party manifestos). The former could change incentives to work or invest (affecting growth), whereas the latter means higher borrowing costs, which have to be paid too. Regarding the benefits, if you have some belief in the fiscal multiplier, you could expect that part of that increase might be offset by extra revenues (income, VAT, etc) due to higher GDP (e.g. via consumption) from public workers. If this is a permanent increase, then such change in consumption is likely to be higher (at least according to the permanent income hypothesis).


How would unlimited free energy affect the global market?

This would represent an extreme increase in long-run aggregate supply. In the AS/AD framework, this would result in a long-run increase in production growth and a decrease in the inflation rate.

How "badly" it will disrupt today's global market? You've built an assumption that it's bad into your question, but sure, it will be bad for anyone who has invested in current energy sources. It will be fantastic, however, for anyone involved in literally any other entrepreneurial activity.

The only disruption in such a sudden and unexpected technology improvement would be that some financial institutions would find themselves overinvested in the energy sector. Insofar as they cannot handle the financial capital losses, there may be a private debt crisis that pushes politicians to bail them out again (ideally using the increases in tax revenues caused by increased productivity), or to simply declare their debt null and void. This would cause some consternation, but given that people tolerated financial bailouts after the housing crisis (where financial assets were wasted without any obvious social benefits at all), it's unlikely that there would be overwhelming protest in such a scenario.

The negative impacts of such a productivity shock would only be temporary, however. Energy is one of the limiting factors of production, so free energy would cause production to grow as fast as entrepreneurs can find a use for it, which would be very quickly.


Is the locus of consumption in Ramsey growth model a vertical line?

In the standard/canonical Ramsey model for discrete time, the Euler equation is of the form (notation is standard here, I won't write the whole model),

$$\beta (1+r_{t+1})u'(c_{t+1}) = u'(c_{t})$$

To obtain a zero-change locus for consumption, we must satisfy

$$c_{t+1} = c_t \implies u'(c_{t+1}) = u'(c_{t}) \implies \beta (1+r_{t+1}) = 1$$

$$ \implies r = (1/\beta)-1$$

-irrespective of the form of the utility function assumed (as long as it does not change, as a function, from period to period), and irrespective of the level of consumption. The locus  therefore cannot have a slope in the consumption-capital space but it has to be vertical to the capital axis at a level of capital for which $r$ (which equals the marginal product of capital) equals the above constant.


Intuition

Well, when $r$ is zero, you are just getting value $A$ per time unit, so total present value will be $A \cdot t$.

The approximation

You are asking about the approximation
$$
\frac A {-r}(e^{-rt}-1) \approx A \cdot t.
$$
This is equivalent to
$$
\frac 1{rt}(1-e^{-rt}) \approx 1,
$$
which is true for small $rt$ values. You can see this by taking the limit of the left hand side and using L'Hospital's rule.
$$
\lim_{rt \to 0} \frac{\frac{\text{d} (1 - e^{-rt})}{\text{d} rt}}{\frac{\text{d} rt}{\text{d} rt}}
= \lim_{rt \to 0} \frac{e^{-rt}}{1} = 1.
$$

The approximation in another form

The approximation
$$
1-e^{-h} \approx h
$$
seems to be a variant of the more well known approximation
$$
\ln(1+h) \approx h
$$
for small values of $h$.

How good is this approximation

To get a sense of the error term for large values using your approximation, you can use Taylor-approximation for the function
$$
f(x) = \frac{1-e^{-x}}{x}
$$
starting from some very small $x$ value.


Can a complex interaction term mean more than what it's composed of?

What you describe as "complex interaction term" is typically referred to as composite variable.  Interaction terms are products of two or more variables.  You also use terms like "effect" and "meaning", which in more technical sense, seem to refer to issues of unbiasedness, accuracy, and interpretation.

If a composite variable is a linear combination of other variables (e.g., their sum), then including that variable over its components will not change the predictive power of a linear regression model, though it will change the interpretation of regression coefficient(s).  If it is a non-linear combination, such as the case with your WCI variable, then you will obtain different predictions in a linear regression model.  If you are interested in interpreting the effect of WCI on your dependent/outcome variable (or if you believe it lies in the causal pathway and affects the interpretation of another variable of interest), then you should include WCI in the model.  The interpretation will be standard for regression models, i.e., by how much the value of dependent variable is expected to change for a one-unit increase in the value of WCI, holding other variables constant.  If, however, you are interested in the effect of wind velocity or temperature on the outcome, then you will have a hard time interpreting the results with WCI used as explanatory variable.  Including interaction terms or composite variables can also lessen the precision of your coefficient estimates, just like it does when you have multicollinearity.

In purely predictive (machine learning) models, you are most interested in the predictive power of your model, not in the interpretation of variables.  Those models readily accept "engineered" variables (hence, "feature engineering" is super important for models like random forests, though not for deep learning).  Accuracy will depend on a lot of factors, but using composite variables can certainly help.


Yes. For example, the nationalisation of industries brought expanded reproduction into the state function. So do public private partnerships. The generalisation of private sector management standards and language have made obvious the service provisions of commodities to customers in areas like welfare, defence procurement, archival or parliamentary services.

I commend Volume 1 of Capital to you, in particular chapters 1 to 6, because your use of the term in commentary differs significantly from that present in the literature you reference.  The value-form, as in "the commodity," is a social relationship not an object.


How will non-rich citizens make a living if jobs keep getting replaced by robots and are outsourced?

This is an interesting question a lot of good labour economists have been thinking about for a while. There are a few conflicting theories as to what will happen. You could base a whole career on this question.

This IGM survey will give you some idea as to what leading economists think.

The prevailing opinion seems to be that increased automation is not going to come at a cost to employment. There are countless examples of advancement lowing the returns to labour occurring throughout history (the plough, the steam train, industrial revolution). None of them has shown a long-run reduction in employment. The Solow Swan model for example,  includes inputs to labour, capital and technology. They show technology and labour being complementary. I know of no empirical evidence suggesting that this has changed.

This HBR article suggests that we aren't really seeing a cost in jobs, more a benefit in productivity. It also mentions Robert Solow's famous remark (which was correct at the time):

  you can see the computer age everywhere but in the productivity statistics

This MIT article presents a bleaker perspective with the concluding sentence:

  In other words, in the race against the machine, some are likely to win while many others lose.

Another article suggests that "this time it's different".

The analogy used is that humans are horses and we are reaching peak human

  demand for the labour of horses today is vastly less than it was a century ago, even though horses are extremely strong, fast, capable and intelligent animals. “Peak horse” in the U.S. came in the 1910s

In my opinion, this analogy is intuitively satisfying but is not particularly useful. We will see fewer humans working the supermarket checkouts and more in entertainment in the same way horses are no longer our ploughs and our taxis but are more likely to be racing and performing. Past that I think the comparison is a pretty big logical leap.

The Economist has an opinion somewhere in between

  [Keyne's] worry about technological unemployment was mainly a worry about a “temporary phase of maladjustment” as society and the economy adjusted to ever greater levels of productivity. So it could well prove. However, society may find itself sorely tested if, as seems possible, growth and innovation deliver handsome gains to the skilled, while the rest cling to dwindling employment opportunities at stagnant wages.

More likely, we will see a (potentially painful) transition in the uses of labour. A factory that used to employ thousands will now employ hundreds, possibly eventually only a dozen. These people will seek employment elsewhere and probably find it, either in existing industries or ones that do not yet exist.


What is the effect of Quantitative Easing on the US budget deficit?

No, the deficit is not getting larger due to QE. Certainly not directly, because that's impossible, and also not indirectly, either.

Quantitative easing is a policy of purchasing government bonds with the intent of decreasing yields while injecting cash into the economy. It affects only the demand for government bonds, not the supply of bonds created by government borrowing.

This shouldn't be a surprise, as the monetary authority (the central bank) and the fiscal authority (Congress) are independent of one another. Congress cannot (without rewriting the Federal Reserve Act) directly control monetary policy, nor can the Fed force the government to borrow more.

Indirectly, Congress could choose to borrow more as a result of QE, due to lower yields on government debt, as the lower debt payments make it cheaper to do so. However, there's not much evidence that this is the case.

The deficit has fallen since 2009 (when it significantly increased due to passage of a large-scale stimulus program consisting primarily of tax cuts but also including direct federal spending, in response to the financial crisis and recession), both in nominal terms and as a percent of GDP. As a percentage of GDP (which is the appropriate long-term comparison, as taxable income is the best indication of a nation's ability to repay a debt), the deficit returned to a normal historical range by 2014.

So it's difficult to conclude even that QE encouraged spending, thereby increasing the deficit indirectly. Given two facts— that the Federal Reserve's QE programs started in earnest in 2009 and continued until late 2014, and that the federal deficit steadily fell during this time— it would be very strange indeed to conclude that QE had increased the deficit.


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


Meaning of Additively Separable, Linear in X

A utility function is additively separable if it can be written as:
$U(x,y) = u(x) + v(y)$

Examples:

*$U(x,y) =ax + by$ is additively separable by inspection.

*$U(x,y) = ax + bx2 + cy$ is also.

*$U(x,y) = x^a y^b$ is additively separable, because you can write it as  $U(x,y) = log(x^a)+log(y^b)=alog(x)+b log(y)= u(x) + v(y)$

*$U(x,y) = \frac{xy}{x+y}$ is not additively separable because there's no way to transform it into an independent sub-function of $x, y$. Even if you take logs, you're down to $U = log(x) + log(y) - log(x+y)$ - notice that the third term can't be 'split'. Generally, mixing addition, multiplication and exponentiation will destroy additive separability.

And so we can see, The actual definition of additive separability is:

A function $f(x_1,...,x_n)$ is AS if it can be rewritten as $f(x_1,...,x_n)=f_1(x_1)+...+f_n(x_n)$

The assumption is usually one of mathematical convenience. For example, when utility is additively separable in $x,y$, then the marginal utility of $x$ does not depend on the level of $y$, and vice-versa. And so anything requiring the use of partial derivatives is made easier. Is it a reasonable assumption? Sometimes. For example, if your utility depends on apples and horses, we can probably assume additive separability. If Instead your utility depends on two closely related things, it might be a bad assumption.


"Who moves first" can be conveniently detached from any causal inference, since there may be some third variable influencing both. Certainly one could build a logical and reasonable argument that links increases in investment to increases in GDP (or vice versa), but it is not necessary.

Then, examining cross-correlation coefficients on the levels or on an appropriate transformation (like first differences, or first log-differences) is a simple and valid way to examine the issue, as long as the chosen transformation will provide stationary series.

Specifically, one can examine (sample) ${\rm Corr} (X_t, Y_{t})$, ${\rm Corr} (X_t, Y_{t-k})$ and ${\rm Corr} (X_{t}, Y_{t+k})$, where, again, $X$ and $Y$ can represent the chosen transformation and not necessarily the levels of the variables. Also, what calendar time length will $t$ represent should also be decided according to the phenomenon under study. In turn, the distance represented by $k$ is a matter of experimentation, but it is advisable to start with $k=1$.

If there is a temporal order, then the contemporaneous cross-correlation ${\rm Corr} (X_t, Y_{t})$ should be negligible. Then, the magnitude and sign of the other two cross-correlations should provide statistical evidence as to "who moves first".


Why does a model that is used for policy analysis needs its shocks and parameters to be structural (= invariant) to the shock under consideration?

This is due to the famous Lucas critique.

To make long story short, in the past in the heyday of Keynesian macroeconomics it was quite normal for macroeconomists to just postulate some relationships based on relatively casual empirical observations like for example the Philips curve which says that there is positive relationship between inflation and employment.

However, this ignores some fundamental facts that such relationship only holds as long as people do not expect inflation. The (original) version of Philips curve did not really postulate any deep relationship between the two variables just assumed that it holds.

In short Lucas harshly criticized such approach as the (old) Keynesian approach gives you the correct predictions only as long as you assume that people are passive agents that do not adjust their behavior when government changes its policy. But that is not really reasonable assumption as we know from microeconomics people are not just passive actors, when government changes its policy people try to find ways how to take the maximum advantage of the new policy regime.

The Lucas critique is basically a generalisation of the Goodhart's law which states that "when a measure becomes a target, it ceases to be a good measure"

If you add structural microfoundations you create more rich model which actually has an ability to take account for peoples reaction to changes in policy.

For example, the New Keynesian version of Philips curve instead of assuming that there is positive relationship explicitly models peoples/firms expectations and shows that when  they do not expect inflation it is rational for them to increase employment but when they do it is rational for them to just adjust contracts in a way to account for inflation (for example linking pays to inflation). Hence in such model no matter what is the government policy (with respect to inflation) the answer will be always 'correct' in the sense that it does not just assumes there is a relationship but it explicitly models the reasons/mechanisms why there is sometimes positive and sometimes no relationship between inflation and employment.

  Isn't it reasonable to assume that, in addition to many economic variables, also many parameters that govern micro behaviour will be influenced by a policy shock?

Not really in the sense the microfoundations are set in the New Keynesian macro models. In such models when we talk about microfoundations we talk about proving the results of the model assuming that for example people are rational, have rational expectations and interact in the markets.

To affect such micro behavior the policy would have actually change fundamentally how people behave.

I suppose from sociological perspective you could argue that in a long term policy can change how people behave (I am not expert on that). But from economic perspective it is completely valid to say that model with solid microfoundations is policy invariant.

Moreover, note an extremely important caveat is what the authors from which you quote say here:

  Must be invariant with respect to the policy interventions considered.

Hence, it is not even necessary to make structural model that is invariant to any conceivable policy that you can imagine but rather the policies that are important for the model, like in Philips curve transitioning from market economy to centrally planned economy where property rights are abolished would certainly invalidate many micro foundations and probably changed conclusions, but different monetary policy doesn't and thats what is important.

Regarding your edit of a question, when it comes to microfoundations its not really about rational/adaptive expectations per se - it is about justifying people having such expectations based on some fundamental model of human behavior.

De Grauwe (and others) in his recent work takes a lot of inspiration from behavioral economics. In a sense his models are microfounded they just tweak the classical rational choice theory by adding some behavioral flavor to it. But I used the word in 'a sense' on purpose because to my best knowledge there is no general behavioral theory of human behavior. Rather he takes the rational choice theory and tweaks it by assuming that people are sometimes rational but in other situations not (or many of his models can be also interpreted as saying some portion of population is rational while other not).

Hence he uses some sort of microfoundations in a way that he has some model of fundamental human behavior based on which results are derived. But I would say there is some room for disagreement about that because since there is no general behavioral theory (at least not one accepted in macro for sure) you could argue that his assumptions on human behavior are ad hoc and not proper microfundations as he just assumes some behavioral regularities without really proving that they are not  policy dependent or giving some ‘deeper’ justification why some behavioral assumptions are made and not other other than showing that there is some support from them from behavioral economics or evidence (although I am just playing devils advocate - I actually really like the work of De Grauwe).


This question can be answered once a few facts are realized.

The Central Bank has a monopoly on the production of currency;
The Central Bank can purchase foreign currency by using currency it has either already produced but stored in its vault or by producing new units of the currency;
The currency that the Central Bank has produced but stored in its vault or not produced yet are not in circulation;
By purchasing foreign currency the Central Bank brings its currency in circulation;
This currency can be brought out of circulation by the Central Bank purchasing this currency with assets such as foreign currency and destroying it afterwards or by anyone else doing exactly the same.

If we therefore define the domestic money supply as the currency in circulation, then purchases by the Central Bank of foreign currency will bring currency in circulation from either its vault or its press.


What is the assumption behind "indifference curve does not cross"?

If only weak-ordering and continuity is assumed, ICs can definitely intersect.

This is not true. First, if you're speaking of indifference curves, you'd already be assuming either local non-satiation or monotonicity.

Let's speak of indifference sets instead. The analogue of two sets, $I_1$ and $I_2$, "crossing" each other can be formalized as $I_1\ne I_2$ and $I_1\cap I_2\ne \varnothing$.

Take two alternatives $x_1,x_2\in X$ and define two indifference sets as follows:
\begin{align}
I_1:=\{x\in X:x\sim x_1\}\quad\text{and}\quad I_2:=\{x\in X:x\sim x_2\}.
\end{align}
WLOG, assume that $x_1\succ x_2$, so that $I_1\ne I_2$. If we allow $I_1$ to "cross" $I_2$, then $I_1\cap I_2$ must be non-empty. Let $\bar x\in I_1\cap I_2$ be an element in the intersection. By the transitivity of the indifference relation $\sim$, we have $x_1\sim \bar x$ and $\bar x\sim x_2$, implying $x_1\sim x_2$. But this is contradictory to our assumption that $x_1\succ x_2$. The contradiction therefore suggests that transitivity, a property of the weak ordering, is violated.


equilibrium, optimum, and decentralizing the optimum

Q1 Optimal usually means one of two things:

Pareto optimal (there's no way to change the outcome such that everyone is at least as well-off, but someone is strictly better-off).
Optimal in the utilitarian sense (we maximise the sum of everyone's profit/utility).

If utility/profit is quasilinear then the two concepts coincide.

This paper appears to be using the utilitarian standard: the optimum is the allocation that minimises the sum of everyone's costs (which, in this case, are the time spent travelling and parking).

Q2 To understand what is meant by decentralisation, it is useful to first know what we mean by centralisation. We say that a decision is centralised if there is one agent (often called a "social planner") who makes all of the decisions and imposes them on society.

We know from experience that governments are not very good at centrally planning things (cf. the USSR). Thus, an interesting question is whether we can set up a market, system of prices, or other kind of mechanism such then we can leave each individual to his/her own devices and still end up with the optimal outcome (such as how we get the efficient outcome under perfect competition, without any kind of central planner).

When everyone is left to their own devices like this, the market is said to be decentralised. We say the optimum is decentralised if we find a way to decentralise the market such that the end result is optimal.


Correct Equation for Pooled OLS Regression (with Time Dummies and Interaction Terms)

I am not sure if I understand your research question correctly, but you may be trying to determine if the effect of critical reception on a film's revenue changes over time.  That means you only have one observation per film with its revenue and film-specific control variable values (including year of release and single critical reception value) saved in multiple variables.  In that case, this is the model equation you should have:

You should not have both i and t subscripts for any film-specific variables if you only have one set of values per film.  The i subscript is used on all film-related variables, including year of release.  I included t superscripts to make it clear that there are multiple Year variables.  And if you are not doing a fixed effects model (film fixed effects), there should not be any subscript on the intercept term.


Does a store suffering increased theft generally cause higher prices?

prices should have already been set to maximize the trade off between profit-per-sale and volume sold

But profit-per-sale depends on costs, which depends on the theft numbers, so if theft increases, the equation changes.


Important details are missing, but just think: whose expectations about inflation are included in the monetary policy rule? Model consistency should help you answer your question.

If you provide more details on your model, I will be happy to expand.


Have the Greek bailouts been included in state budgets Eurozone members?

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

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

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


How do reserves move between the 12 federal reserve banks?

This is surprisingly subtle.

When, for instance, when bank A in the Richmond Federal Reserve district sends $1000 in reserves to bank B in the Minneapolis Federal Reserve district, reserves are taken out of bank A's account at the Richmond Fed and placed into bank B's account at the Minneapolis Fed.

Now, bank A's reserves are a liability on the books of the Richmond Fed, while bank B's reserves are a liability on the books of the Minneapolis Fed. Without any offsetting change, therefore, the process would result in the Richmond Fed discharging a liability and the Minneapolis Fed gaining a liability - and if this continued, regional Fed assets and liabilities could become highly mismatched.

The principle, then, is that there should be an offsetting swap of assets. It would be too complicated to swap actual assets every time there is a flow of reserves between banks in different districts. (There's over $3 trillion in transactions every day on Fedwire, the Fed's RTGS system - and if even a fraction of those are between different districts, the amounts are really enormous.) Instead, in the short run the regional Feds swap accounting entries in an "Interdistrict Settlement Account" (ISA). In the example above, the Minneapolis Fed's ISA position would increase by \$1000, while the Richmond Fed's ISA position would decrease by \$1000, to offset the transfer of liabilities.

So far, this is all very similar to the controversial TARGET2 system in the Euro area, in which large balances between national banks have recently been accumulating. The American system is different, however, because ISA entries are eventually settled via transfers of assets. Every April, the average ISA balance for each regional Fed over the past year is calculated, and this portion of the balance is settled via a transfer of assets in the System Open Market Account (the main pile of Fed assets, run by the New York Fed). Hence, if in April the Minneapolis Fed has an ISA balance of +\$500, but over the past year it had an average balance of +\$2000, its balance is decreased (by \$2000) to -\$1500, and it has an offsetting gain of \$2000 in SOMA assets.

As this example shows, since it is average balances over the past year that are settled, not the current balances, ISA balances do not necessarily go to zero every April. Historically, they were fairly tiny anyway, but since QE brought dramatic increases in reserves, these balances have sometimes been large and irregular. In the long run, though, the system prevents any persistent imbalances from accumulating.

(Note: the process in April is a little bit more complicated than I describe, since some minor transfers of gold certificate holdings are also involved. Basically, gold certificates are transferred between regional Feds to maintain a constant ratio of gold certificates to federal reserve notes; the transfers of SOMA assets are adjusted to account for this. Wolman's recent piece for the Richmond Fed is one of the few sources that describes the system in detail.)