Converting a GDP series to constant international dollars

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

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

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

See this document or this website for further references.


What are the main differences in a macroeconomic model, i.e. New-Keynesian DSGE, under a currency substitution (full dollarization) scheme?

You are not likely to find New Keynesian DSGE models where money or currency plays an explicit role. These models mostly abstract from these considerations altogether and monetary policy is simply represented by the interest rate. I understand your situation as I myself have tried to write papers and searched for literature where dollarization plays a role.

If you can, try to circumvent the explicit use of currency in your model. Think about what the currency's role is in your model, e.g. if it is to have different inerest rates in different countries or in different currencies, then simply use different interest rates instead of different currencies.

Another way would be to go outside of the standard DSGE framework and use other international finance, open economy models, as you would find in the Obstfeld/Rogoff book.

If you really want to stick with DSGE, I would recommend looking into open economy DSGE models as well as DSGE models with "Money in the Utility Function".


I cannot understand Sobel & Crawford 1982 (cheap talk)

$U^S(\bar y,\bar m,b)=\max_{y\in Y}U^S(y,\bar m,b)$ is standard notation that says $\bar y$ is the action (taken by receiver) that would maximize sender's utility given message $\bar m$ and bias parameter $b$. In other words, sender would prefer receiver to choose $\bar y$ when he sends message $\bar m$. Sender's maximized utility is $U^S(\bar y,\bar m,b)$.
For a simpler introduction to Crawford and Sobel, you can read Chapter 18 of Tadelis's Game Theory textbook on cheap talk games.


Reference Request - Dynamic Optimization with More Than One State Variable

As pointed out by mark leeds, the chapter appears to be part of an incomplete text by Peter Thompson - see here.


Finding demand functions for an unusual utility function

I assume you know how does $\min\{x,y\}$ look like? In order to draw utility function of interest, you need to consider cases: $u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x > y\end{cases}$

With $x$ on horizontal and $y$ on vertical axis:

Not sure about the "usual" perfect complements. It is more like a combination of substitutes (below $y=x$) and complements (above $y=x$).

Also, take a look here: Identifying utility function and Algebraic approach towards convexity where you can see more graphs.

EDIT.

As for the demand function, Finding demand function given a utility min(x,y) function.


What does it mean that households provide the four factors of production?

While it’s true that firms can pay other firms for land or capital (and in fact firms can pay another firms for entrepreneurship - you can hire consulting firms, and there are also firms that ‘borrow’ workers to other firms). All firms are in the end owned by some household.
Even if legally you can have both physical and legal persons (i.e. limited company for example) in the end any company has to be owned by household. A building or a machine cannot own property. In the end after some chain of legal ownership there must be some physical person (i.e. household) that owns those companies.
From economic perspective it does not really matter how long the paper chain between company and some household is, only thing that matters is that it exists. Hence in the end it always must be some household providing that capital or land (by providing equity to the firms).
This is really just obscured in modern world and hidden under all the legal arrangements but in the end there can be no company that is just owned by itself there must be some household behind it. If you own a plot of land and decide to build factory on it you are supplying that land to your firm where you are the household and as a result your compensation (profit you will get from operating the company) will include the rent from land as well as return on capital you supply that firm. If you put your own entrepreneurship or labor profit will include that as well.


If all indifference curves are parallel lines, then preference has linear representation

The indifference curves are constructed by viewing the utility function as an equation (for a fixed utility index value per curve).  So from

$$U = U(x_1,x_2)$$

where the left side is just a symbol, we move to

$$\bar U = U(x_1,x_2)$$
where now the left side is a specific number.

Take the total differential on both sides to obtain

$$0 = U_1dx_1 + U_2dx_2 \implies \frac {dx_2}{dx_1} = -\frac {U_1}{U_2}$$

Any straight line in the two-dimensional plane has a constant slope so also

$$\frac {dx_2}{dx_1} =c$$

Same for $y$-bundle. Etc.


How do I obtain a correlation between two variables?

For the above equation is differentiable, and hence the partial derivatives will exist:
$\frac{\partial Y}{\partial X_2}= 2\beta_2X_2$.
Adding a dummy variable should not affect the above derivative for all practical purposes - the function will be differentiable a.e.
(note that you can't differentiate with respect to a dummy variable though)
Edit:
After the question was modified, the new expression of the partial derivative would be:
$\frac{\partial Y}{\partial X_1}=\beta_1+ 2\beta_2X_1$.


Very interesting question, I'll try to summarize some aspects and I probably won't be exhaustive !

Venezuela has long been a country relying on oil (95% of their exports). Certain experts have estimated that Venezuela holds just shy of 25% of worldwide oil reserves, but the oil extracted in Venezuela is very heavy and hard to refine and then sell to other countries, which makes it more expensive than oil coming for example from Saudi Arabia. When prices boomed, the former President Hugo Chávez, who died in 2013, used oil exports to quadruple the foreign debt. He also led a some nationalisations. The main factor for Venezuela's current crisis is therefore the counter-shock on oil prices which lowered exports. This quote comes from an FT article

  If oil prices remain at January’s average levels, exports in 2016 will be less than USD 18bn, while servicing the debt will cost over USD 10bn. This leaves less than USD 8bn of current income to pay for imports, a fraction of the USD 37bn imported in 2015

One of the direct consequences of that is that the country is relying on its dollar reserves to pay its imported goods. The country's foreign reserves are already withering at 12-year lows. Venezuela has even paid Switzerland with gold! But the problem is that on the land the demand was still the same and the supply could not follow, which led to a massive inflation. Inflation will surge to 720 percent in 2016 from 275 percent last year, according to a note published by the IMF.

The huge outstanding debt is also worrying investors.Creditors are now demanding a high premium for their trust in that promise. The yield on Venezeula’s dollar bond that matures in 2020 is 37% (an investment mainly made by hedge fund). This is the more financial aspect of the question: it's hard or costly for Venezuela to get financial help from investors right now. For this aspects, I encourage you to take a look at this article by The Economist:

  About half of Venezuela’s foreign debt is explicitly owed by the sovereign; the rest is owed by PDVSA. There are important differences. Most of the sovereign-debt contracts have collective-action clauses (CACs), under which a restructuring, if accepted by holders of an agreed proportion of debt, can be imposed on all of them. PDVSA, Venezuela’s main source of foreign exchange, would have a harder time restructuring its debt. Its bond contracts do not have CACs


You are asking about the expected value of the leading term of a sequence. There can be no general answer given the assumptions you make, since for example

$$x_{t+1} = x_{t} + v_{t+1}, \;\;\;E(v_{t+1}) = 0,\;\;\; x_0 \;\; \text{given}$$

is a unit root process, and its expected value is equal to $E(x_{t+1})=x_0,\;\;\; \forall t$, while

$$x_{t+1} = a +x_{t} + v_{t+1}, \;\;\;E(v_{t+1}) = 0,\;\;\; x_0 \;\; \text{given}$$

is also a unit-root process, but $E(x_{t+1}) = x_0 + a\cdot(t+1)$

which is very different compared to the previous case.


Say two countries have the same amount of resources. A has an absolute advantage relative to B if A has technology such that A produces a larger quantity of output using that given level of resources relative to B. So, lets say potatoes are inputs and mashed potatoes are outputs. A has an absolute advantage in mashed potato production if it can produce more pounds of mashed potatoes than B. Comparative advantage, on the other hand, deals with the opportunity cost. For example, the french fries foregone by each country when they decide to produce mashed potatoes.

It is worth noting that comparative advantage is the basis of trade. Not absolute advantage.


Is $(\mathbb{R}^m)^n$ the real coordinate space of dimension $m\cdot n$?

No. And yes. For any set $X$ we have (by definition) $$X^k=\underbrace{X\times\cdots\times X}_{k\text{-times}}=\{(x_1,x_2,\ldots,x_k)\mid x_i\in X\text{ for }i=1,\ldots,k\}.$$
Now let, for example, $m=2$ and $n=3$. Then
$$(\mathbb{R}^m\big)^n=(\mathbb{R}^m\big)^n$$ $$=\big(\mathbb{R}^2\big)^3=\Big\{\big((x_1,x_2),(x_3,x_4),(x_5,x_6)\big)\mid (x_1,x_2)\in\mathbb{R}^2,(x_3,x_4)\in\mathbb{R}^2, (x_5,x_6)\in\mathbb{R}^2\Big\}$$ $$=\Big\{\big((x_1,x_2),(x_3,x_4),(x_5,x_6)\big)\mid x_i\in\mathbb{R}\text{ for }i=1,\ldots,6\Big\}.$$
This is different from $$\mathbb{R}^{m\cdot n}=\mathbb{R}^{2\cdot 3}=\mathbb{R}^6$$
$$=\Big\{(x_1,x_2,x_3,x_4,x_5,x_6\big)\mid x_i\in\mathbb{R}\text{ for }i=1,\ldots,6\Big\}.$$
So they are not the same, formally $(\mathbb{R}^m\big)^n\neq \mathbb{R}^{m\cdot n}$ However, there is clearly a natural way to associate points in one space with points in the other space and this way to connect them preserves a lot of structure. They are equivalent as vector spaces, topological spaces and whatnot. For this reason, one might often identify these two spaces and unless one does very strange things, little can go wrong.


Moral hazard with linear effort and two possible outcomes

principal: $\max_{w_h,w_l,e}p(e)v(q_h-w_h)+(1-p(e))v(q_l-w_l) $ subject to

(IR)       $~~~~~p(e)u(w_h-c(e))+(1-p(e))u(w_l-c(e))\geq \bar u$

(IC)$~~~~~~e\in arg\max_{\tilde e}p(e)u(w_h-c(e))+(1-p(e))u(w_l-c(e))$

Replace the IC with the FOC approach:

(IC'))$~~~~~~p'(e)u(w_h-c(e))+p(e)u'(w_h-c(e))(-c'(e))-p'(e)u(w_l-c(e))+(1-p(e))u'(w_l-c(e))(-c'(e))=0$

Now consider the principal's foc wrt $w_h$:

$p(e)v'(q_h-w_h)(-1)+\lambda(p(e)u'(w_h-c(e))+\mu [p'(e)u'(w_h-c(e))+p(e)u''(w_h-c(e))(-c'(e))]=0$

So the difference between this and first best is that third term in the FOC. Using the standard assumptions, we know that that third term is positive and so:

$v'(q_h-w_h)_{first best}<v'(q_h-w_h)_{second best}$

Now v concave and (assuming inverse exists):

$q^f_h-w^f_h>q^s_h-w^s_h$

So the spread decreases,


Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable?

I would suspect that if  $X_1$ is independent of $X_3$, then it must imply $E[X_1*X_3]$ is equal to $E[X1] * E[X_3]$, given this you can conclude that the Covariance is 0. $Cov(X_1,X_3) = E[X_1*X_3] - E[X_1]* E[X_3]$. If the Covariance is 0 between these two variables then so is the correlation. Thus I would believe that removing $X_3$ from the regression and having it's effect enter the error term would not result in having $γ_1$ be biased as the $X_3$ in the residual term does not have correlation with $γ_1$.

Are you looking for a rigorous mathematical proof?


Symmetric mixed-strategy equilibrium: Entering markets [Solved]

I solved it, haha. Turns out I was being silly. I had completely neglected the fact that this was a 3-player game, and $\text{not entering}$ still had probability values attached to it. Let $A$, $B$, and $C$ be the three firms. So the game looks like this:
$$
\text{P(A enter;B,C enter)+P(A enter;only B enter)+P(A enter;only C enter)+}\\
\text{P(A enter;neither enters) = P(A dne;B,C enter)+P(A dne;only B enter)+}\\
\text{P(A dne;only C enter)+P(A dne;neither enters)}
$$
It's a giant mess of $p$'s and $(1-p)$'s, but at the end of the day, the final version $8=9(1-p^2)$ is derived.


What value does one set, in general, for the null hypothesis of $\beta_0$ ("Y-intercept")?

If you simply want to estimate the values of Y corresponding to particular values of X, you can calculate the regression to obtain estimates of $\beta_0$ and $\beta_1$, and need not be concerned with hypothesis testing.

If you want to test one or more hypotheses, then the choice of hypotheses should be driven by the purpose of your research. A common aim is to assess the relevance of an independent variable by testing the hypothesis that its parameter equals zero, but that is just one choice.  In this case you might want to test the hypothesis that $\beta_1=0$, and you can do so without testing any hypothesis about $\beta_0$.

Where matters possibly become confusing is that regression software often, by default, generates t-values for all the independent variables and the constant. If you want to test the hypotheses that $\beta_0=0$ or $\beta_1=0$, these t-values are useful.  But you can also ignore them if you are not interested in testing those hypotheses.  The estimated parameters don't depend on these hypotheses or t-values, so you can still use them in estimating values of Y.


Fed must hold collateral equal in value to notes in circulation

Section 16 of the Federal Reserve Act, 1913.
No, the Fed’s accounting generally follows the standard convention, where assets=liabilities+equity. All of its assets count as collateral; no particular assets are “earmarked” as collateral. By construction, the Fed holds assets against all its liabilities (mostly deposits), not just Federal Reserve Notes.
Yes, you can see the assets and liabilities of the Fed in its H.4.1 release.


In Economics, to define/describe "demand for $X$", whatever $X$ represents, we need

a definition for $X$
a measurement unit for the quantity of $X$
a measurement unit for the price of $X$

Given these, the "demand schedule" for $X$ (terminology dating back to A. Marshall), is the geometric locus of pairs (quantity, price) that reflect the quantity demanded for the corresponding price in the pair.

And that's that: why is the quantity demanded what it is for a given price, is (at least) one level deeper in the economic structure and analysis.

As regards "money", I just record here the usual approach taken:

We have in mind something that is perfectly liquid (and in practice we may include in the definition "essentially liquid" assets), where "liquid" means "immediately acceptable by anyone in any exchange"
The quantity of money is measured in some currency unit (like USD, Euros etc),
We use the interest rate as "the price of money"

To go this one step deeper, the reason why "demand for money" is not fixed at infinity, has to do with the question that we pose so as to avoid a useless answer. And the question is essentially a question of allocating a given level of assets between different forms/different degrees of liquidity, money included: Given that my wealth is $W$ (measured in a currency unit), and given that the interest rate I will enjoy if I save it all in an interest-bearing form is $i$ (unitless), how much of this wealth do I want to keep in "money form"?

Enter arguments as to why we want to keep money, functional forms that formalize these arguments, etc.

In dynamic models, it is understood that this question is posed and answered by the demand schedule for money for each time-period of the model. In static models, there is a single time period.

Another usual aspect is that we examine "real  money balances", i.e. the deflated value of the nominal money balances.

"Current holdings of money" is not the demand schedule but equilibrium quantity demanded, given whatever constraints we postulate that they exist.

This gives rise to the concept of a possibly "liquidity constrained" economic agent: given his wealth, the interest rate and his preferences, the agent would want to allocate $W_m^*$ of his wealth to money holdings. But we may observe that it allocates something less, for example because the existing markets where the agent can transform non-liquid assets to liquid ones are "incomplete" (as they say), and so the agent finds himself with non-liquid wealth that he would want to transform into money but he cannot (perhaps due to asymmetric information, risks that cannot be priced, etc). While this is a constrained equilibrium also, it is important for economic analysis and economic outcomes to realize that here we may have an additional "binding constraint" on the agent's behavior, and as this constraint changes, the behavior of the agent will also change.

Detecting the existence of such exogenous liquidity constraints is difficult, because it is difficult to separate their effects on money holdings from the effects of other constraints (like the level of existing wealth of the agent).

So "current money holdings" is tautologically equal to "current equilibrium money demanded", but the challenge of economic analysis is to determine which constraints are binding and so characterize the observed equilibrium.


What would be a an accurate metric of the real economy without the financial services sector?

This question is difficult to answer. You can start by simply finding the value of the manufacturing sector of any country's economy. But this quickly becomes very hard to define. Let's take Tesla for instance, Tesla manufactures cars, but they also are deeply involved in software engineering for their company. Is Tesla a software firm or a manufacturing firm?
The question gets even more complicated when you take a company which manufactures some of its own supply chain and also has investments in other companies, which will include basically every manufacturer in America, because almost any firm in the world will diversify its holdings when it has extra cash on hand.
The best answer you have will be to look at the latest GDP report and then subtract whichever sectors you don't consider "real"... which is available here for the United States: https://www.bea.gov/data/gdp/gross-domestic-product
But given how many large companies have financial investments as part of their corporation you have to have a good definition for what a "real economy" is in reality, which is not an easy question in your terms. Ultimately it is all real, just as real as the value of intellectual property such as patents or copyrights.
If you don't want to measure it in dollars you can always use xe.com to calculate the value in gold. Nobody measures economic data in gold, silver, or oxen nowadays.


Why was activity analysis abanadoned as a field of research?

In  Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society: Series A (General), 120(3), 253-281.
which is widely seen as the starting point of systematic Efficiency and Productivity Analysis, the author writes, referring to what he is about to present:

Similarly, although the treatment of the efficient production function is largely inspired  by  activity  analysis (see for example
Koopmans 1951), no  reference is made  to  this in the exposition. The
professional economist can easily draw the  necessary parallels for
himself, as indeed,  he can  note  the similarity  of the measure  of
"technical efficiency" and Debreu's "coefficient of resource
utilization"(Debreu,1951).

I would suggest that "activity analysis" has been subsumed into "Efficiency and Productivity Analysis", eventually living someplace inside Data Envelopment Analysis.
As for the reason why it did not stay in the "general microeconomics core" I believe it is because it was not a framework for economic models (that include behavioral hypotheses, incentives etc), but mostly a framework for quantitative measurement and analysis. As such it was more of a tool for applied decision making than for providing theoretical insights.


Price discrimination- how much is optimal?

Varian has a paper on Price Discrimination and Social Welfare in which he gives some necessary and sufficient conditions for (third degree) price discrimination to increase welfare.

A necessary condition is that the total level of output (i.e. the total number of consumers served) increases as a result of the discrimination.

A sufficient condition is that the profitability of the new output (i.e. after discrimination) exceeds the profitability of the old output (before discrimination) evaluated at the new prices.


Why can't the Land-Value Tax (in some cases) be passed on to tenants?

Because the supply of land is essentially fixed, land rents depend on what tenants are prepared to pay,

  That doesn't make any sense. If more people want to live in a particular place, rents will increase.

And if rents increase, then obviously tenants are prepared to pay more.  If they aren't willing to pay increased rents, then they don't really want to live there more than another place.  I.e. your statement is entirely consistent with the statement that you are criticizing.

    rather than on landlord expenses, preventing landlords from passing LVT to tenants.

  Only if there's somewhere else that tenants want to be, that also has lower rents. The real estate boom in San Francisco shows that when people with lots of money really want to live in a particular place, rents rise.

I think you missed the point here.  Presumably landlords are already charging what the market will bear.  How does a Land Value Tax (LVT) change what the market will bear?  As it's a tax on landlords, obviously it has no direct effect on tenants.  The tenants are not prepared to pay more rent as a result of the LVT.

If rents will go up by \$100 a month without an LVT and \$100 a month with an LVT, then landlords can't pass the LVT to tenants.  That's true even if the \$100 is enough to pay the LVT.  Without the LVT, the landlord would have raised the rent and kept all of the money (minus income taxes, etc.).  With the LVT, the landlord pays the LVT and may still have to pay more income taxes (depending on the deductibility of the LVT).

The only way that an LVT can influence rents from the landlord side is if it causes the number of units available for rent to either increase or decrease.  In the long term, that happens.  Landowners can choose to build more or fewer units on their land.  But in the immediate term, the number of units is fixed.  A landowner can't take a single family home and instantaneously convert it into a hundred-unit apartment building, even if there is sufficient land to support that.  Thus the LVT does not change unit availability or the rent that tenants are willing to pay.

Now, it is possible that the LVT enables spending that tenants find valuable.  And that spending causes demand and therefore rents to increase.  But that spending would be just as valuable to the tenants if it were financed by a different tax with less impact on landowners.  Of course, tenants might be negatively impacted by that tax, e.g. if they pay it.  But those effects are aside from the impact of an LVT on landlords.

The LVT itself does not increase rents in the short term.  It may increase rents in the long term if it drives away landowners by making some places economically infeasible to rent.  But that assumes that landlord profits are less than the LVT.  Or landowners may choose not to expand the number of units in the long term.  Or the LVT may finance a benefit that increases the value of the units to tenants.

So the landlord can only pass on the LVT if tenants benefit from the spending financed by the LVT, shifting the demand curve such that tenants will pay more.  So the more that the LVT finances benefits for tenants, the better able the landlord will be to pass on the LVT.

If those things don't happen, the landlord can't pass on the LVT.  The landlord can only get the regular market rent.  That may be an increasing market rent, but the increase is not due to the LVT.  It's only determined by movement in the tenants' demand curve, which is not directly impacted by the LVT.


Find a subgame perfect equilibrium and a Nash equilibrium

As afreelunch only answered in the comment section, I'll answer it here.

The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call $S$ the set of all SPNE and $N$ the set of all NE in a game. Then:

$$S \subseteq N.$$

If the sequential game has only one subgame (see the subgame definition here), the game itself, then we will have.

$$S = N.$$

And my reasoning was right.


Natural real interest rate and output gap

Regarding you first question, the answer is 'yes', at least in your model. For if the real interest rate, $r_t=i_t-E_t\pi_{t+1}$, is equal to its natural level $r^n_t$, then in your model $x_t=E_tx_{t+1}$ and $x_t=0$ for all $t$ is a solution which is consistent with the dynamic equation. (Note that other solutions are consistent with the dynamic equation in the case $r_t=r^n_t$ for all $t$, such as $x_t=1$ for all $t$.)

According to my experience, the natural level of real interest rate is usually defined as the level of real interest consistent with output being at its 'natural', 'long-run', 'equilibrium' or 'potential' level, given stable inflation, absence of shocks to demand, etc. I.e., natural real interest rate, $r^n_t$, is such that whenever $r^t=r^n_t$ and certain stability/regularity conditions are fulfilled, then $x_t=0$. Read a FRBSF Economic Letter for more information.

We may also view the natural level of real interest rate as the real interest rate at which the output gap $x_t$ is following its trend $E_tx_{t+1}$, i.e. $x_t=E_tx_{t+1}$. This definition is similar to the one given by Knut Wicksell.

Regarding your two last questions, I may say the following. In your model, a ceteris paribus increase in $r^n_t$ increases $x_t$ by $\frac{1}{\sigma}$. (Just take the partial derivative with respect to
 $r^n_t$ on both sides in the dynamic equation, holding everything else constant.) This means that either output, $y_t$, is increasing, or the natural level of output, $y^n_t$, is increasing. However, if such ceteris paribus changes cannot be made, then taking the partial derivative holding all else constant is not meaningful. Indeed, in new keynesian models, the natural real interest rate is often a function of $y^n_t$, and in such cases we cannot increase the natural real interest rate without affecting the natural level of output. One such case is when $r^n_t=\rho+\sigma(E_ty^n_{t+1}-y^n_t).$

More concrete examples may be provided, I will see if I can find some.


Proof on weak axiom of revealed preferences

As far as I can see this comes just from definitions:
As given in MWG definition 1.C.1:

The choice structure $(\mathscr{B},C(\cdot))$ satisfies the weak axiom of revealed preference if the following property holds:

If for some $B \in \mathscr{B}$ with $x,y \in B$ we have $x\in C(B)$, then for any $B'\in \mathscr{B}$ with $x,y\in B'$ and $y\in C(B')$, we must also have $x\in C(B')$.

This basically states that under the weak axiom of revealed preference (WARP) if there is any choice set which contains both $x$ and $y$ and $x$ is preferred then there can be no set containing both $x$ and $y$ where $y$ would be chosen over $x$. Or we can say that "if x is revealed at least as good as y, then y cannot be revealed preferred to x."
Next according to MWG definition 3.B.2:

The preference relation $\succeq$ on $X$ is monotone if $x\in X$ and $y >> x$ implies $y \succ x$. It is strongly monotone if $y \geq x$ and $y \neq x$ imply that $y\succ x$.

In addition, according to MWG definition 3.B.4:

The preference relation $\succeq $ on $X$ is convex if for every $x\in X$, the upper countour set $\{y \in X: y \succeq x\}$ is convex; that is, if $y \succeq x$ and $z \succeq x$, then $\alpha y + (1-\alpha) z \succeq x$ for any $\alpha \in [0,1]$.

Lastly, utility maximization is often used to imply rationality (Simon 2001) and preference relation can be rational only according to MWG definition 1.B.1:

The preference relation $\succeq$ is rational if it possesses the following two properties:
(i) Completeness: for all $x,y \in X$ we have that $x\succeq y$ or $y \succeq x$ or both.
(ii) Transitivity: For all $x,y,z \in X$ if $x \succeq y$ and $y \succeq z$ then $x \succeq z$.

Given the definitions above if they are satisfied then if $y$ is revealed to be at least as good as $x$, $x$ cannot later be revealed as better.  So it follows from the definitions of those properties. For example, WARP would be violated if the following choices are made $C(\{x,y\}) = y$ and in $C(\{x,y,z\}) = {x}$, by a person holding their preferences constant - such situation would clearly violate several of the above definitions.


LIBOR rates: Are these monthly or yearly interest rates?

To answer the first part, it's an "annualised" interest rate convention - like all other quoted interest rates. For example, if a one-month money market rates are unchanged at 4%, you would receive approximately 4% in interest after a year, or roughly 1/3% a month. (Note that those numbers are ignoring compounding, further details below.)

As for the calculations, things are complicated. The high level answer is that a monthly interest rate is roughly $r/12$, with $r$ being the quoted rate. This is a simple interest rate.

The true calculations are complicated by the nature of LIBOR. Technically, LIBOR (London Interbank Offer Rate) is a polled rate of "large" banks in a number of currencies. There are similar "fixes", such as Euribor (Euro area), TIBOR (Japan), CDOR (Canada). The polled rate is based on the money market convention in each currency for a certain class of interbank borrowing.

Interest rate calculations depend upon the daycount convention, and the calendar used. If we borrow on June 1st, one month later is normally July 1st. What happens if July 1st is a weekend or holiday? There are standard comventions in each currency that determine which days are holidays, and how to move the due date relative to such dates (before/after).

There are a wide variety of day count conventions that are used across markets. I believe that act/365 (GBP) or act/360 (other currencies) are standard. (Based on this ICE LIBOR document: link.) Anyone who needs to know the exact calculations should refer their queries to their banking counterparties, as there can be a large number of small modifications to the generic description I give below.

In act/360, the amount of interest is given by the quoted rate times the fraction (actual number of days in the lending period)/360. For act/365, the interest is given by the quoted rate times (the number of days in the period/365).

Therefore, if the one-month maturity period is 30 days in the future, and the currency uses an act/360 convention, the fraction used for the one-month rate is 30/360 = 1/12. However, it is clear that the true annual compounding rate will vary from compounding $(1+r/12)^{12}$ since months are not exactly 30 days, and the number of days in the year is either 365 or 366.


Throw-away paradox with independent goods

It seems (from the first page, at least; I don't have JSTOR access right now) that the essence of Aumann and Peleg's example is that

  this rise in price of the commodity [trader A] holds is more than enough to compensate for the drop in the amount.

One similar situation from history occurred during the New Deal, when farmers were paid to plough over cotton fields and send hogs to be slaughtered for no reason, in order to take excess quantities of these goods off the market.

Obviously, the difference here is that this was a centralized intervention in a decentralized (and huge) market. The set of incentives are more complicated, and the many "trader A's" in this scenario were compensated for throwing away their goods. But the sentiment is very similar.


Solution to maximization not Pareto efficient

An example with two agents and two goods: let
$$
U_1(x) = 0, \hskip 20pt U_2(x) = x_1+x_2, \hskip 20pt w = (1,1).
$$

In this case allocating all the goods, so (1,1) to the first consumer solves the above problem. Even though any other feasible allocation fulfills the conditions, none of them gives a higher utility to the first consumer.

Yet this allocation is clearly not Pareto-optimal, allocating (1,1) to the second consumer would be a Pareto-improvement.


is endogeneity a problem in the first-stage regression in a two-stage least square regression?

Yes it is a problem. The first stage itself has to satisfy the same assumptions that standard OLS would and $cov(Z,\epsilon_1)\neq 0$ would violate them (see A Guide to Modern Econometrics by Verbeek).
Furthermore, actually the two conditions you mention are not enough. The instrument should also not be 'weak', that is the first stage should have $F$-statistics above $10$ (as a rule of thumb). Also the instrument should have an effect on $Y$ only through $D$ - the so called exclusion restriction (see Mostly Harmless Econometrics by Angrist and Pischke).


What is the difference between knowledge-based and R&D?

A knowledge-based company sells products or services that are primarily knowledge-based. This doesn't have to be new knowledge - it can be existing knowledge.

An R&D company earns its way by developing new knowledge.


My work focuses on the Economics of Education, although from a more micro perspective. I can honestly say it's a complicated question and there are many people working on optimal education spending.

First, a bit of simple math. Let $EdExp$ be education expenditure, $GDP$ be Gross Domestic Product, $E_{students}$ be enrolled students, $S_{pop}$ be school-age population, $N_{pop}$ be non-school age population. One can decompose Education Expenditure by GDP ratio in this way:

$$ \frac{EdExp}{GDP} \equiv \frac{E_{students}}{S_{pop}} \times \frac{S_{pop}}{(S_{pop}+N_{pop})} \times (\frac{EdExp}{E_{students}} \times \frac{(S_{pop}+N_{pop})}{GDP})$$

Note this isn't a "made-up formula", it's an accounting identity. This is true in the US, China and in a tribe of herdsmen. I'll translate its terms to english. $\frac{EdExp}{GDP}$ is Education Spending to GDP ratio, which Tabarez thinks should be 6%. $\frac{E_{students}}{S_{pop}}$ is the enrollment rate, the fraction of people who are school-aged and attend school. $\frac{S_{pop}}{(S_{pop}+N_{pop})}$ is the dependency ratio, the fraction of the population that is school aged. Lastly, $\frac{EdExp}{E_{students}} \times \frac{(S_{pop}+N_{pop})}{GDP}$ is education expenditure per student divided by GDP per capita. This is the proportion of average income that is spent on the average student.

Countries like France have high GDP per capita and low dependency ratios, which means they can sustain high expenditure per student without having to compromise a large share of GDP on education. Uruguay and Brazil, on the other hand, have high dependency ratios and somewhat low GDP per capita. Enrollment rate isn't very high too. So, in order for these countries to sustain high expenditure per student (which is what ultimately counts), they need to reserve a larger share of GDP for education.

That's one part, the one having to do with demography and aggregate spending. I'd like to talk about the one some of my research is concerned with, spending efficiency. Uruguay is not an extreme case as Brazil, but it shares many problems with its northern neighbor.

Out of the ~6.5% of GDP that Brazil spends in education, only about 1% goes to public fundamental/elementary schools. About 1% goes to public high schools and the rest goes to public universities. This is just terrible. You can't spend 2% of GDP on dozens of millions of students and then 4% on 1 million of them and expect the first system to have reasonable quality. Then, we have the institutional problems. In most states, schools aren't allowed to pay bonuses to better teachers and many poor municipalities can't even hire trained teachers because of low pay, depending on self-taught locals to teach. Access to federal resources such as financing for new materials and data to better understand your school's demands is somewhat easy but mostly untrained, unqualified school managers don't know what to do with these resources.

In the cities and school that do have somewhat qualified personel and more money, public tenure rules in Brazil make it impossible to fire low productivity teachers and managers and even promote the good ones. Leading to overall lack of incentives.

On one hand, spending matters. It's hard to imagine schools giving good education if there aren't blackbords, chairs and basic equipment in it. On the other hand, poor institutional design and lack of incentives can make tons of money go to waste.

As a rule of thumb, anything in the 4-5% education spending to GDP ratio is reasonable. More than that means mostly resources going to waste if there aren't serious attempts to create better incentives and improve school management.


What is the “entry price” in relation to tariffs?

Entry price system (EPS) is a EU-specific term relating to their food import tariffs, more specifically to fruits and vegetables. It's a somewhat complicated system designed to strongly discourage imports below a certain price. This is a summary from 2009 paper:

  The EU protects growers of 15 kinds of selected fruits
  and vegetables against international competition not
  only by the means of ad valorem tariffs of up to 20%,
  but also by the EPS. The EPS came into effect on 1
  July 1995, replacing the former RPS. Analogous to a
  minimum import price, the EPS is designed to restrict
  imports below the product-specific, politically
  designated EP plus ad valorem tariff (Table 1). If the
  EP is undercut, an additional specific tariff is levied,
  which proportionally varies depending on the gap
  between the product’s actual import price and the EP.
  When the EP is undercut by 8% or more, the
  maximum specific tariff, referred to as the maximum
  tariff equivalent (MTE), of up to 80% of the EP is
  charged. For example, the EPS is applied to oranges
  during the EU orange harvest season in the time period
  December 1 to May 31. The MFN tariff for oranges
  seasonally varies between 3.2% and 16.0% whereas
  the MFN EP remains constant at a level of 354 €/t. If
  oranges are exported to the EU at a price of 336.3 €/t,
  the EP is undercut by 5%. This implies that the
  exporter has to pay an additional specific tariff of 17.7
  €/t which is equal to the gap between the import price
  and the EP. If the entry price for oranges is undercut
  by 8% or more, an additional specific tariff at the level
  of the MTE of 71 €/t is charged. [...]

  Monitoring compliance with the EPS faces the
  difficulty that a large share of fruit and vegetable
  imports in the EU is on commission, implying that the
  import price is not determined until the product is sold
  in the EU market. Therefore, the EC calculates a
  synthetic import price, the standard import value
  (SIV). Fruit and vegetable prices, surveyed for each
  product and export country individually, are collected
  on representative fruit and vegetable wholesale
  markets in all EU Member States. The daily SIVs are
  calculated as a weighted average of collected
  wholesale market prices, less a marketing and
  transportation margin and applied tariffs.
  The EPS can be circumvented (both legally and
  illegally), so that some product is finally sold at prices
  below the EP (García-Álvarez-Coque, 2002).
  According to information from importers, illegal
  circumvention (e.g. based on false invoicing) is more
  prevalent in small-scale trading, particularly between
  related trading partners. Storage can offer a means of
  legal circumvention, as storable products can be
  imported at any time while customs clearance is
  delayed until some later date when the SIV is above
  the EP. Once cleared at a favourable SIV, the product
  can be sold later on EU markets at any price (Cioffi
  and del' Aquila, 2004).

There were some 2014 changes (not covered in that quote).


Labor market model with subsidies homework

Does this make sense?:
$$\frac{\partial L_d}{\partial W} = \frac{-\partial g/\partial W}{\partial g /\partial L}=\frac{1}{PF_{LL}(L_d)}$$
Now from solution of (a) we know that as $S \uparrow$, $L_d \uparrow \implies F_{LL} \downarrow \implies \partial L_d/\partial W \uparrow$
Therefore, $\partial^2 L_d/\partial W\partial S >0$
EDIT: Note that I have assumed above that $F_{LLL} <0$. Clearly this is not given in your question statement. So as per the answer above, what happens to the slope should depend on sign of $F_{LLL}$. If, for example, it is zero, then the curve simply shifts with no change in slope.


Why can’t we know the specific outcome of an exchange economy with initial resource endowment?

The idea of the contract curve is just restricting the Pareto set so that no one player is worse off than the initial allocation. There's no concept of "price" involved here. When your endowment is $(x, y)$ and you're asked whether you'd like to get another bundle, $(x', y')$ instead, you only check if your utility is higher. If it is, then you'd take it.

The concept of price comes in with the Walrasian equilibrium (p. 654 of MWG) which is yet another level of refinement on the contract curve (i.e. the "core").


Why is Keynes attacking the (neo)classical theory of interest rates on the grounds that it is indeterminate?

You can use some some initial $Y_0$ to find an interest rate, but this is precisely the indeterminacy problem.  We would like to be able to derive the equilibrium interest rate without assuming some initial values.
Furthermore, note the indeterminacy problem is actually present even in canonical Keynesian models (See Hansen, 1951) and Neo-Keynesian models (see Beyer & Farmer 2004).
However, the Keynes critique is now a bit outdated. Keynes was writing in a past before macroeconomic was even given proper formal treatment (by modern standards), and at a time when macro had no micro-foundationds. Nowadays you can solve the indeterminancy problem in macroeconomic model of either variety by assuming central bank follows some interest rate setting rule, such as Taylor rule or making some specific assumptions about model's micro-fundamentals (see Romer's Advanced Macroeconomcis for discussion of this). Although as the Beyer & Farmer point out in their paper even central bank interest setting rules won't solve this issue always so the problem is somewhat open.


Growth regressions over 3-year periods; regressors in average or year values?

I have never seen any "serious" (i.e. peer-reviewed) study defining independent variables as the end of the period under consideration (if you could provide some link to the ones you are referring to, it would be helpful). Normally, they are used as the average over the period or at the initial year. For instance, in the big field of finance and growth, that is the common approach. Thorsten Beck - among the most renown researchers in this field - says it himself (bottom half of page 2). There are many examples of this in other fields too. For instance, when estimating inequality on growth (top of page 282), or the famous paper by Xala-i-Martin, where he run 4 million growth regressions. Look in page 7 and 8, in sections "Data" and "Choosing the Fixed Variables".

Now, why not to use final period variables? Because of reverse causality concerns. An economy which grows fast over a given period might end up with a higher savings rate and human capital level at the end of the period. As such, the order of causality is in fact the opposite at the one your equation is assuming (even though the sign coming from estimations might be the same!), leading to endogeneity and therefore to inconsistent estimates.

Even the use of average period variables is subject to some controversy. As Xala-i-Martin puts it in the aforementioned paper (page 8):

  I reluctantly use some variables of this sort (the average savings or investment rate and the DeLong and Summers measures of equipment and non-equipment investment are examples in this category). The reason for being reluctant to the inclusion of such variables is that these may be “more endogenous” than the variables measured at the beginning of the period.

To conclude, using end of period variables is a very risky practice, and in my view very hard to justify theoretically and empirically. Even more, it seems unnecessary. Why not use beginning of period? Or average?


What is the equilibrium price in this case?

I have quickly skimmed the chapter and there does not seem to be much about bargaining models so as the question says you have to solve this using supply and demand.
We know that an equilibrium price on the market will occur when supply intersect demand. Given that the supply is fixed at $24$ and $25$ apartments respectively it will be just straight vertical line in each case. Now when it comes to demand, if we assume each person wants only one single apartment the demand will be given in step-wise fashion, with two steps given by reservation prices ${\\\$}500$ and ${\\\$}200$ respectively. So this situation will look like shown on the tikz picture I made below (supply is blue, demand is red):

At point where supply is equal to $24$ the equilibrium price given by intersection of supply and demand ($S=D$) is clearly ${\\\$}500$. However, when supply is equal to $25$, supply and demand will intersect at a whole range of different prices $[200,500]$.
Consequently, in this case using this simple supply and demand model we cannot say anything more that price will be something between ${\\\$}200$ and ${\\\$}500$. There are more advanced ways how we could model this situation which might yield unique price instead of range (maybe some models with explicitly modeled bargaining process or some matching models), but here the author of the book gives you hints by asking how demand would look like etc., that you are supposed to use this solution concept (also the first chapter uses only simple supply and demand models so that is another hint).


Do capitalists in the Marxism world view create value?

I'm not a Marx expert, but I've read The Capital some years ago and as far as I can remember Marx said that only the workers produce value. All others' wages (and the workers' wages too!) and other costs are covered by the produced values of the workers, so the others decrease the "profit", which could remain at the workers otherwise. Therefore Marx suggested the workers to make collectives without any capitalists.


Literature Recommendation: Advanced Undergraduate Economics

I would suggest Mankiw and EconTalk (despite my Keynesian leanings).

Specifically Mankiw's text "Principles of Economics" is a great math-light introduction that covers both Macro and Micro. I recommend this edition particularly because you can get it new off Amazon for under $20, though of course the latest edition is best, but the field hasn't changed that much that I'd spend 15x as much to get it.

Additionally Stiglitz, Piketty, Buchanan, and Fama (particularly Fama, given your interest in finance), are all great authors, and in my opinion fairly accessible (in book, not in paper, they're each just as jargonated in paper as any one else).

Hope that helps.


World as a society of interlinked multiplayer individuals

Chess is EXPTIME-Complete, which makes it significantly harder than NP-Complete problems.

Perhaps you are interested in the study of economic networks. Strategic network formation sounds like a good starting point. A lot of the work examines when certain classes of graphs arise under pure strategy Nash equilibria. There are exponentially many pure strategies. Enumerating the vertices of the polytope is likely not feasible.

Edit: Some of the big names in the area include Matthew O. Jackson, Rachel Kranton, Sanjeev Goyal, and Hans Haller. I would start with their papers. In particular, Matthew O. Jackson has a book on the subject.

Here are links to their homepages/CVs so you can scout out their publications. Economic networks are a pretty hot area at the moment, so you can likely look at journals like Econometrica to see what is being published.

http://web.stanford.edu/~jacksonm/papersarticles.html
http://econ.duke.edu/people/kranton/networks
http://www.econ.vt.edu/cvsandresearch/hallercv.pdf


What does it mean if the controls in my IV model are correlated with my instrument?

This would not make instrument necessarily invalid. For some 2SLS instrument model of form:
$$y_i  = \beta_0 + \beta_1 \hat{x_i} + \beta_2 k_i +\epsilon_i$$
$$x_i = \pi_0 + \pi_1 z_i + \pi_2 k_i +e_i $$
where $y$ is dependent variable, $x_i$ is the endogenous regressor, $k$ some controls and $z$ instrument the main conditions for instrument validity are:

$z$ has to be correlated with the $x$ variable $cov(x,z)\neq 0$ and have causal effect on $x$

$z$ can affect the $y$ only through $x$ and $z$ itself has to be exogenous $cov(z,\epsilon)=0$

Furthermore, instrument should not be weak which would invalidate inference from the regression and so 1st stage should have $F$-statistics higher then $10$ as a rule of thumb (although some new results suggest that actually correct rule of thumb would be having $F>105$ - see Lee, McCrary, Moreira, & Porter 2020).
You can find more rigorous proof of these conditions in econometric hanbooks such as Verbeek (2008) guide to modern econometrics or Angrist & Pischke (2009) Mostly Harmless Econometrics. The point is that none of the conditions directly say instrument cannot be correlated with other controls as long as it does not violates any other of the conditions. In fact omitting them could lead to omitted variable bias violating the condition that instrument should be endogenous if they are important. However, having highly correlated variables can inflate the standard errors of your model due to multicolinearity, this being said multicolinearity does not lead to bias it only reduces the precision of the estimates and reduces statistical power.


Is there a good way of assessing an estimated absolute value of a currency?

Actually there is an effort by IMF that indirectly offers a common unit of measurement to express the value of the major world currencies homogeneously.

It's called SDR (Special Drawing Rights) sometimes abbreviated as XDR. It's a sort of meta-currency whose value is calculated as follows:

  The value of the SDR in terms of the U.S. dollar is determined daily
  and posted on the IMF’s website. It is calculated as the sum of
  specific amounts of each basket currency valued in U.S. dollars, on
  the basis of exchange rates quoted at noon each day in the London
  market.

  The basket composition is reviewed every five years by the Executive
  Board, or earlier if the IMF finds changed circumstances warrant an
  earlier review, to ensure that it reflects the relative importance of
  currencies in the world’s trading and financial systems IMF website


Unusual perfect complements utility function min{ax+y, x+2y}

You can solve this question by breaking the Utility function into 2 parts.

Use U(x,y) =

i) 6x+y if 6x+y < x+2y

ii) x+2y if x+2y < 6x+y

This would simplify into the Utility Function

U(x,y) =

i) 6x+y if 5x < y

ii) x+2y if 5x > y

When you graph the function you'll get 5x=y as the line of kinks. When 5x>y, that is, to the right of the line of kinks, the corresponding IC will be have a slope -(1/2) since it lies on the line x+2y. Points to the left of line of kinks correspond to the line 6x+y and have the slope -6.

Given point (8,9) clearly lies to the left of the line of kinks in the area where 5x>y, therefore this point lies on the Utility line x+2y having slope -(1/2)


Elasticity when the demand function is given

The definition of elasticity of demand with respect to price is: $\varepsilon_{q,p} = \frac{dq}{dp} \cdot \frac{p}{q}$. So in your demand function we have:

$$q = kp^{-\epsilon}$$
$$\frac{dq}{dp} = -\epsilon kp^{-\epsilon - 1}$$
$$\varepsilon_{q,p} = \frac{dq}{dp}\cdot \frac{p}{q}  = -\epsilon kp^{-\epsilon - 1} \cdot \frac{p}{kp^{-\epsilon}} = -\epsilon$$

And we have the answer you're looking for.


Why do firm output decreases when fixed cost decreases in Long Run?

The quantity produced does not depend on the fixed costs. Fixed costs only influence the decision of whether or not to produce, not how much to produce. The individual firm's level of production is determined by MR=MC, independent of fixed costs. Firms will produce less when fixed cost decrease, because, as you point out, more firms will produce (entry in the market). On the aggregate market supply shifts out, so that price declines. For each individual pricetaking firm, the exogenously determined price has declined. Put differently, MR has shift down, resulting in an optimum (MR=MC) at a lower level of production.


Does investing in the stock market contribute to economic growth?

tl;dr
Yes, investing into stock market has positive impact on growth. First, before going any further, investing into stocks is from an economic perspective just a form of saving. Consequently, following the literature this answer will be repeatedly referring to saving in general.
In economics, there are two main theories of economic growth. An exogenous Solow-Swan growth theory and endogenous (eg Romer model) - see Romer Advanced Macroeconomics Ch 1-3. In both of these strands of literaterature saving impacts economic growth positively but under Solow-Swan increase in saving rate causes level increase in economic output which will show up as higher economic growth but in order to raise economic growth permanently a one would have to continuously increase. On the other hand under endogenous growth models increase in savings rate can have permanent positive effect on economic growth.
Effects of Savings Under Exogenous Growth
To illustrate the mechanisms at play consider augmented Mankiw-Weil-Romer (1992) Solow-Swan model given by:
$$ Y(t) = K(t)^{\alpha}H(t)^{\beta}(A(t)L(t))^{1-\alpha - \beta} \tag{1}$$
Where $Y$ is economic output (change of which is economic growth), and economic output varies positively $K$ is stock of capital, $H$ is stock of human capital (i.e. education) $A$ is the stock of technology (assumed to be growing at an exogenously given rate $(A (t) = A (0)e^{gt})$ and $L$ the stock of labor (growing at exogenously given rate $(L(t) = L (0)e^{nt})$. The Greek letters are exogenously given parameters that depend on the nature of society's production function (with $\alpha +\beta <1$). Hence, when stock of capital, education, technology level or labor force expands economy will experience economic growth.
Now in order to see how savings plays a role in all this we can first define everything in terms of effective labor (e.g. $Y/AL= y$, $K/AL=k$ and $h/AL=h$), and using the fact that capital (and human capital as well) is determined by investment which is macroeconomically equivalent to saving we get that:
$$ \dot{k}(t) = s_k y(t) - (n + g + \delta_k)k(t), \tag{2}$$
$$ \dot{h}(t) = s_h y(t) - (n + g + \delta_h)h(t) \tag{3}$$
where $s_k$ is share of saving invested in physical capital and $s_h$ share of saving invested in education, whereas $\delta_k$ is depreciation of physical capital and $\delta_h$ erosion of human capital (e.g. loss skills over time). For simplicity it will be further assumed that $\delta_h = \delta_k=\delta$. Substituting the evolution of capital and human capital into the per capita version of (1) we get:
$$\ln y =  \ln A(0) + gt -  \frac{\alpha + \beta}{1-\alpha -\beta} \ln(n + g + \delta) +  \frac{\alpha }{1-\alpha -\beta} \ln s_k + \frac{ \beta}{1-\alpha -\beta} \ln s_h$$
Now stock market is stereotypically associated with investments into physical capital so I will focus mainly on relationship between output and physical capital. Hence, let us set growth rate of technology, population and to simplify calculation also depreciation to zero and let us assume technology stock and saving rate for human capital are fixed so their time derivates are zero. We end up with expression:
$$\ln y = \ln \bar{A} +  \frac{\alpha }{1-\alpha -\beta} \ln s_k + \frac{ \beta}{1-\alpha -\beta} \ln \bar{s_h} $$
Taking time derivative gives us:
$$ \frac{\dot{y}}{y} = \frac{\alpha }{1-\alpha -\beta} \frac{\dot{s_k}}{s_k}  $$
$\frac{\dot{y}}{y} $ is just continuous version of growth rate and consequently we can see that output growth depends on growth in savings rate. Furthermore, since $\alpha + \beta<1$ the relationship will be strictly positive}.  However, a caveat here is that saving rate cannot growth forever as it has to be between $0\leq s \leq1$ since it is a proportion of income not consumed, so any increase in saving rate will increase the growth rate but you can only increase your saving rate to the point you devote your whole income to it. Visually a simulation of an increase in savings rate would produce the following result (see Romer Advanced Macroeconomics pp 20 - please note the simulation is based on simplified version of an above model so the precise shape of those curves might be different but the point carries over):

However, this being said  higher level of saving will still increase level of output which will lead to temporarily higher growth just the effect won't be permanent and the only thing that can permanently increase the per capita growth rate in exogenous growth rate model is an increase in rate of technological progress.
Effects of Savings Under Endogenous Growth
While the theory above is more widely accepted, recently this theory was expanded on by endogenous growth theory. I won't go into such deep detail into endogenous growth theory as models there get far more complicated, but I will try to explain in in relation to the exogenous growth model above.
In short, the gist of endogenous growth theory is that it rejects the exogenous growth theory assumption that technology develops at some exogenously given rate and society cannot do anything to speed up technological progress. Endogenous theory rejects that view and argues that investments in capital, education, or having larger labor force (and thus be able to devote more labor to R&D projects) can increase the rate at which technology accumulates and so $g$ will be also function of investments into capital, education and share of labor devoted to R&D (see discussion in Romer Advanced Macroeconomics Ch 3).
Now since you are interested primarily in saving via stock market the first relationship will be of greatest interest to you.  Now endogenous growth literature argues that the link between growth and saving here is mediated by either learning by doing it (e.g. investing money into building a cargo ship allows firms to learn how to better produce cargo ships - see Arrow, 1962). Next there will be link because firms can use portion of that investment to finance R&D that then improves productivity of either capital or labor depending on what sort of production function industry has (e.g. see Romer 1990).
In economics there is still debate between the exogenous and endogenous growth camp, because given data we have it is very hard to statistically distinguish between endogenous and exogenous growth empirically (we have solid evidence for Solow-Swan model but it is difficult to distinguish transitory effect of saving from a permanent one with limited data), but recent Nobel Prize in Economics awarded to Romer in 2018 for his work on endogenous growth theory shows that this theory is becoming more dominant.
Summary
To sum it up, both major strands of economic literature shows saving whether through investment in stocks or in various other ways (putting your money in bank etc) has positive effect on economic growth. However, under exogenous growth theory this effect is just transitory whereas under endogenous growth theory there might be long lasting effect.
However, given that the economic literature on endogenous growth suggest that the link between saving and economic growth is mediated via investments in R&D and learning by doing, if you want to minimize economic growth of society you can do your research and invest exclusively into stocks of companies that devote as little of their budget to new research as possible. This won't avoid effect due to learning by doing but it would almost certainly result in lower long term economic growth than investing into stock of companies that invest heavily into research.


How does interest rate impact exchange rate?

Your first paragraph, except the last part of the sentence, is the statement of covered interest rate parity (CIP), which holds perfectly if not absolutely, except after the 2008 financial crisis (See this paper). However, in the last part of the sentence, if you use forward exchange rate to represent expected future exchange rate, this is where the uncovered interest rate parity comes in, which does not hold according to many papers.

About your second paragraph, the confusion arises from the benchmark that you want to compare. According to CIP, if the domestic interest rate is higher than foreign interest rate, then the spot exchange rate is less than the forward exchange rate (in domestic currency per foreign currency). In other words, if you use forward rate to represent expected future exchange rate, you expect that the domestic currency will depreciate and the foreign currency will appreciate. In this case, you are comparing spot rate and forward rate at the same point in time.

If, suddenly and unexpectedly, the domestic central bank raises interest rate, immediately you will observe a sharp drop in the spot rate (in domestic currency per foreign currency). So if you compare the spot rates before and after raising the interest rate, you find that the domestic currency appreciate. However, whether you stand at the time before or after raising the interest rate, you always observe that the spot rate is less than the forward rate (in domestic currency per foreign currency) as long as the domestic interest rate is greater than the foreign interest rate.


How to justify the treatment and control groups for Difference-In-Difference with staggered implementation of laws?

[W]hy do they need to write down "adopted a leniency law at some later point of time"? Because in Korea case, the word "our sample period" means "1995-2002" already.

Assuming Korea is the early-adopter country, then all countries theretofore untreated before 1997 may serve as a counterfactual. This includes the countries never adopting a leniency law and those with impending treatment adoption periods.

Does it mean that only $()_{}$ of the treatment group after the laws being passed received a value 1? Other than that, the control group and the treatment group before laws being passed receive a value of 0?

Correct.
The binary treatment variable in this more general setting is not the same variable as in the 'classical' difference-in-differences case. Suppose a leniency law is espoused by all firms within treated countries in the year 2000. In this setting you could write this equation more simply as the interaction between a treatment-control dummy and a post-treatment indicator equal to 1 after the law goes into effect in both groups, 0 otherwise. However, once we move away from this setting and the roll out of treatment is staggered or even switching 'on' and 'off' over time, then the "post-treatment" variable is no longer well-defined. To proceed, we must use the 'generalized' difference-in-differences estimator which defines the product term in a different way.
In the equation you reference, the interaction term is implicit in the coding of $LL_{kt}$ (i.e., leniency law). It is equal to 1 in all firms that are headquartered in countries that have passed a leniency law by year $t$, 0 otherwise. Imagine you have a column of zeros. Input a value of 1 once the firms in country $k$ adopt by year $t$, 0 otherwise! Thus, any country never adopting a leniency law is left as 0 for the entire observation period.

Imagining two cases: (1) Both Germany and Brazil implemented this law in the year 2000 and (2) Only Germany implemented this law in 2000. So whether the numbers of observations of the control groups for these two cases are identical?

From the perspective of this estimator, the treatment variable $LL_{kt}$ makes no distinction between treatment/control groups. We only have countries $k$ switching on their policies by year $t$. Any country $k$ never adopting a leniency law is equal to 0 in all $t$.
Suppose a panel runs from 1995–2002. In example (A) both Germany and Brazil will 'turn on' (i.e., switch from 0 to 1) in the year 2000 and stay on. Thus, Brazil and Germany have 5 pre- and 3 post-treatment time periods. They have "identical" treatment histories. The countries never espousing the new law would serve as a counterfactual.
In scenario (B) only the within-group observations in Germany would switch on in the year 2000, while the within-group observations in Brazil would all equal 0. The countries never espousing the new law, which now includes Brazil, would serve as a counterfactual.

Australia and Belgium passed the law in 2003 and 2004, accordingly. So, firms in Malaysia only exist once in the control sample? I mean, whether one observation can be the control for many treatments in this DiD setting?

Yes.
Suppose Malaysia never adopts a leniency law—ever. Malaysian firms may serve as a counterfactual for Australian and Belgian firms up until the time periods in which they are treated. In other words, the Malaysian sector is a counterfactual for the early- and late-adopters before each of their respective exposure periods.
The data frame below shows how we code $LL_{kt}$ in practice. Note this is a little different since we only see country-year observations. I assume based upon my cursory review of the paper that leniency laws affect all firms within each country. If so, and the authors observe all firms before and after the laws goes into effect, then we can estimate the equation at the firm level or at the country level.
For simplicity, the fictitious data frame below observes three countries from 2000–2006. Germany espouses the law early; $LL_{kt}$ switches from 0 to 1 in 2003 and stays on. Australia adopts late; $LL_{kt}$ switches from 0 to 1 in 2005 and stays on. The Malaysian market never adopts a leniency law. Note how $LL_{kt}$ is 0 for the entire observation period. If we also observed firms $i$ over time within Malaysian industries, then all firms should be coded 0 in every firm-year period.
Again, the variable $LL_{kt}$ is not delineating a specific subset of treated or untreated firms/countries; rather, it simply 'turns on' (i.e., switches from 0 to 1) if a jurisdiction was treated and only during the periods $t$ when the law was actually in effect, 0 otherwise.
The following data frame should help with your intuition:
$$
\begin{array}{ccc}
country & year & LL_{kt} \\
\hline
 \text{Malaysia} & 2000 & 0 \\
 \text{Malaysia} & 2001 & 0 \\
 \text{Malaysia} & 2002 & 0 \\
 \text{Malaysia} & 2003 & 0 \\
 \text{Malaysia} & 2004 & 0 \\
 \text{Malaysia} & 2005 & 0 \\
 \text{Malaysia} & 2006 & 0 \\
\hline
 \text{Germany} & 2000 & 0 \\
 \text{Germany} & 2001 & 0 \\
 \text{Germany} & 2002 & 0 \\
 \text{Germany} & 2003 & 1 \\
 \text{Germany} & 2004 & 1 \\
 \text{Germany} & 2005 & 1 \\
 \text{Germany} & 2006 & 1 \\
\hline
 \text{Australia} & 2000 & 0 \\
 \text{Australia} & 2001 & 0 \\
 \text{Australia} & 2002 & 0 \\
 \text{Australia} & 2003 & 0 \\
 \text{Australia} & 2004 & 0 \\
 \text{Australia} & 2005 & 1 \\
 \text{Australia} & 2006 & 1 \\
\end{array}
$$
Note, before Germany adopts a leniency statute in 2003, their counterfactual history is the Malaysian and Australian sectors. It is also worth highlighting that previously treated countries may also serve as counterfactuals for the late-adopter countries. In other words, when Australia adopts a leniency law later in 2005, the Malaysian firms (i.e., non-adopters) and the German firms (i.e., early-adopters) may approximate their counterfactual history. This estimator is actually averaging all possible 2x2 difference-in-differences estimates.
The downsides of this estimator may serve as a whole separate discussion. But let's address one concern briefly. For instance, the model assumes the absence of time-varying effects. For example, the effect of a leniency law on Germany's outcome trajectory is assumed to be instantaneous and constant. This assumption is tenuous and difficult to defend statistically. Note how we still assume a "common trend" across all groups before their exposure periods. Imagine a scenario where Germany's outcome trend was changing over time after 2003. Once Australia moves into the treatment condition in 2005, then Germany's outcome trajectory may be offset by their earlier exposure. When 'already-treated' countries are allowed to act as controls for 'soon-to-be-treated' countries, then changes in their treatment effects over time get subtracted from the difference-in-differences estimate. So while the two-way fixed effects estimator is yielding a weighted average of treatment effects across all groups and times—some of the weights may be negative. The negative weighting only arises in the presence of heterogeneous treatment effects. Peruse this working paper for more on this topic.

Above is how he grouped the control and treatment based on his identification, but on page 2600, Dasgupta, 2019, section 3.1 said that "control firms are all firms in the same industry in countries that had not passed a leniency law in the 7 years surrounding the event date". This identification confused me because now we have one more dimension about the industry. I am wondering why they did not put it right in the baseline identification but instead in results section? What is the reason behind that?

It is my understanding that the timing of leniency laws did not vary across industries, but I can't be sure until I give the paper a thorough read.
Usually authors may specify a base model and then expand upon it in their results section. For example, they may report lead/lag coefficients in tabular form but do not explicitly show their equation. The authors may also try a series of alternative specifications such as including industry-by-year effects even though this term was excluded from their main empirical equation.


How to estimate loss of customers due to customer support inefficiency?

The best way would be to use a regression analysis, What i would do as an economic major would be use two independent variables such as in your case level of customer support, customer feedback to see if the dependent variable "customer retention rate". also you need to use parameters for customer support evaluation, customer feed back and customer retention rate for the years. Then run a regression analysis to find if the independent variable are co-related to your dependent variable, thus you can arrive at a statistical justification for your problem.


Why utility should be bounded (or unbounded)?

tl;dr: Short answer is that cardinal utility of rational person is in large part of a literature is derived from the von Neumann and Morgenstern expected cardinal utility framework. In such framework utility must be bounded by our definition of what rationality is. So in such framework the short answer is simply that utility has to be bounded for person to be considered rational.
There are also expected utility frameworks that allow expected utility to be unbounded (see review by Fishburn, 1976). However, these are not very widely used as they can often lead to paradoxes and do not seem to offer any special insight. I suppose this is due to the strong influence of instrumentalism on economic thinking. A researcher would prefer a utility framework that is able to provide some testable predictions rather one that just results in a paradox and hence offers no useful testable prediction. So even if someone could perhaps consider concept of unbounded utility more elegant its instrumental value in doing research would be close to none (if in particular case it leads to unresolvable paradox).
Full Answer:
The utility needs to be bounded in order to avoid paradoxes such as the St. Petersburg paradox (for more nuanced overview see this entry in Stanford encyclopedia of philosophy). As a matter of fact a rational person's utility should be bounded as suggested by Arrow (1970) precisely in reference to the paradox above.
More general actually cardinal utility will be bounded by the axioms that were used to derive the expected cardinal utility in the first place. Following Neumann and Morgenstern (1947) Theory of Games and Economic Behavior,  an expected utility from gamble can be described by the so called von Neuman-Morgenstern equation:
$$E[u(g_i)] = \sum_j u(X_{ij})p_{ij}$$
where $u$ is utility $g_i$ is gamble $X$ is an outcome and $p$ is a probability. Furthermore, for the above to be utility we must have some continuum of gambles for which:
$$g_i,g_j \in \mathbf{G}: g_j \succeq g_i \implies E[u(g_j)]\geq E[u(g_i)] $$
Now given this we can ask (as the author of the paper did) what would be properties of such utility function?
Now it turns out that basic rationality constraint where preferences satisfy transitivity, completeness, continuity and independence imply that the utility has to be bounded.
The completeness axiom states:
$$ \forall x,y \in \mathbf{X}: x \succeq y \vee y\succeq x \vee y \thicksim x $$
that is we can order all our options in terms of preference in some way.
The transitivity axiom states:
$$ \forall x,y,z \in \mathbf{X}, \text{ if } x \succeq y \wedge y \succeq z \implies x \succeq z$$
So if someone likes $x$ more than $y$ and $y$ more than $z$ then $x$ must be preferable to $z$
The continuity  $z \succeq y \succeq x$, then there must be some probability $p$ such:
$${px,(1-p)z} \thicksim y $$
This implies that no outcome $x$ is so terrible that you would not take up some gamble involving $x$.
by independence axiom if $y \succeq x$ then for $z$ and some probability $p$
$${px,(1-p)z} \preceq {py,(1-p)z} $$
This axiom states that if two outcomes have the same probability, we should evaluate the two alternatives independently of what we think that the outcome is.
The above axioms are how we define rationality in expected cardinal utility. There are of course different possible specifications of utility functions but most modern research relies on the von Neuman-Morgenstern type (or related utility functions).
Now these rationality requirements - which are axioms so they are by definition what rationality is in this context, simply demand that no outcome can produce infinite utility. In order to see this we can try to do proof by contradiction - suppose that there is a gamble wehere: $x = 1€$, $y=100€$ and $z= \infty €$ and that $u(X)=x$. In this case clearly $z \succ y \succ x$ but there is no $p$ for which the continuity axiom holds. Since the continuity axiom would be violated our basic axioms of what rationality is would not hold and a person with unbounded utility would cease to be rational (within the context of von Neumann and Morgenstern framework).


Interpretation of Difference-in-Differences Regression Results when Only the DID Coefficient is Significant

I would heed the advice in the comments.
In settings where policy adoption is standardized and you have multiple pre- and post-treatment time periods, you could also interact your treatment indicator with separate time dummies specific to treated and untreated units. In the comments you indicate you observe multiple units $i$ across months $t$ from 1990 to 2005. Your model would look something like this:
$$
y_{it} = \gamma T_{i} + \lambda_{1} (T_{i}*\mathbf{I}_{t = \text{Jan-2000}}) + \lambda_{2} (T_{i}*\mathbf{I}_{t = \text{Feb-2000}}) + \lambda_{3} (T_{i}*\mathbf{I}_{t = \text{Mar-2000}}) + \lambda_{j} (T_{i}*\mathbf{I}_{t = j}) + \epsilon_{it},
$$
where the $\lambda_{j}'s$ represent post-exposure effects after the policy goes into effect. Note, to estimate all post-period effects up to period $j$ you must create a 'month-year' variable to distinguish, for example, between January in 2000 and January in 2001. To do this, concatenate categorical indicators for month and year together and append it to your data frame. Interacting $T_i$ with individual post-treatment time dummies results in separate estimates for each post-exposure period (e.g., Jan-2000, Feb-2000, Mar-2000, ... , Dec-2005, etc.). This approach is helpful for many reasons. First, treatment may grow or fade over time; the policy might be perceived immediately upon implementation, or with delay. Due to its flexibility, this approach allows you to assess the relevant time dependencies. Second, if there is a treatment withdrawal period, you can estimate effects in those periods beyond policy/program termination; effects might persist even in the absence of treatment.
I admit this results in 72 interactions which might be more than your data can handle. Notwithstanding this concern, it is still a useful way to model effects in the post-treatment epoch. You can still estimate unit and time fixed effects in this setting. As already noted, the unit fixed effects will absorb $T_i$ and the time fixed effects will absorb your 'month-year' main effects. But I wouldn't worry about this. Your interactions (i.e., $T_i \times \mathbf{I}_{t=j}$) should be your focus.


Why is there a positive risk-free interest rate?

The interest rate is (1) the price needed to take on risk and (2) the price needed to delay consumption.

The reason there is a positive risk free rate, even though there is no risk, is because of the time preference typical of any economic agent. It is preferable to consume today, than to consume tomorrow. To put off consumption today and invest in the risk free asset instead, the agents want to earn interest, otherwise they will consume instead of saving.

Normally, if the risk free rate was zero, nobody would buy the risk free asset, but spend everything on consumption. E.g. nobody would buy government bonds. The government has to offer a positive interest rate to find buyers.

Current negative interest rates are actually a reflection of how our classical assumption, that agents prefer to consume today rather than tomorrow, has been turned on its head. Nobody wants to consume today despite zero or negative interest rates contrary to the classic theory above (and people buy government bonds with negative interest rates). This can happen if agents are apprehensive enough of the future to want to increase their precautionary savings instead of consuming. If they are worried enough, they will do this even if they lose money on their savings (due to the negative interest rate).


Where to learn financial modelling concepts and skills?

When I was studying for my bachelors we learned finance from textbook Corporate Finance by
Jonathan Berk. I think thats good entry into learning some basic financial analysis.


Verify Indifference curves of two utility function to be the same

Here is a way to approach the problem:

An indifference curve is a set of consumption bundles all of which yield the same utility (the consumer being, of course, indifferent to which she consumes). Nonetheless, the numerical level of utility is inconsequential, it is just a placeholder for the more fundamental (and ordinal) preference--any monotone transformation of a utility function represents the preferences.

So let $I_U(x_1,x_2)$ denote the indifference curve (of $U$) that contains $(x_1,x_2)$. That is $$I_U(x_1,x_2) = \{(x'_1,x'_2) \in X \mid U(x'_1,x'_2) = U(x_1,x_2)\}$$
Define $I_V$ in an analogous way.

So showing the indifference curves are the same amounts to showing $I_U(x_1,x_2)= I_V(x_1,x_2)$ for all $(x_1,x_2) \in X$.


Why are lower cost retailers going bankrupt but premium chains are not?

I think your anecdotal evidence is not enough to support your generalisation. In many markets there are just a few big firms (well-known to many), and many small ones (lesser-known). You could imagine a distribution of firm size being left-skewed (like a Chi-Squared or Beta distribution). Actually, this is a table showing the distribution of size by employment for 2014 for the US:

(Source here)

Thus, even if some big companies are closing down, what about the so many other smaller ones? Maybe some of the are also closing down, but this is less salient because they are less known (and less big thus less "important").

Now, if your claim were to be true, several factor could contribute to it. Most of shoppers in mass-market stores want quality, but also cost-efficiency. Conversely, luxury chains rely particularly on the value of brands. Their shoppers want quality, but also status. Consumers of the former are mainly worried about cost, and therefore loyalty to brand/stores is more or less inexistent. If they can buy cheaper online many would do so. Thus, internet has certainly increased competition for these stores. Conversely, consumers of exclusive shops care about store experience, relations, visibility, things which Amazon cannot give. There might even be an issue of power. You would not spend \$5,000 on a dress buying it online. You would go to the store, where others can see you, and buy it, so other can see your purchasing power.

Greater inequality might be a factor too. Inequality in the US went up since the 1980's. This has been "driven" particularly by a stagnation of income for most of households and a fast increase in top 10 and top 1% income shares. You can imagine that this means great pressure on prices for mass retailers, without an equivalent effect for exclusive retailers.

Finally, big (ergo popular) stores and small (ergo exclusive) stores are not direct competitors. Since brands are precisely a mean of differentiation, many luxury goods face lower competition than mass-market oriented goods (I am sure you can name more brand of cars directly competing with Ford than with Aston Martin). This, together with the status element mentioned earlier, mean that luxury producers (and retailers) are likely to face a higher price-elasticity of demand. Under this setting, they can change higher markups than standard goods producers. This increases the profitability of their business, giving their brand/company higher life expectancy. That is why they are less likely to go bankrupt.


Budget Constraint in Utility Maximisation Problem with Lagrange Multipliers

Objective function of Lagrangian can be set up either with $+\lambda$ or $-\lambda$, depending on how you solve the budget constraint.

Actually, for the solution it does not matter if $\lambda$ has negative or positive sign in the equation. You can clearly see it from the formula if you expand the second term:
$$
\mathcal{L}(x,y, \lambda) = U(x,y) + \lambda(m -p_{x}x - p_{y}{y})= U(x,y) +\lambda m -  \lambda p_{x}x - \lambda p_{y}{y}$$

Using the alternative approach you get exactly the same result as:

$$
\mathcal{L}(x,y, \lambda) = U(x,y) - \lambda(p_{x}x + p_{y}{y}-m)= U(x,y) -  \lambda p_{x}x - \lambda p_{y}{y} +\lambda m$$

So the resulting equations are 100% equivalent.

However, it’s easy to confuse when to use which sign so many people just memorize one of the two ways of solving Lagrangian and never use the other, because it’s easier to just memorize one of the ways than every single time thinking about the sign of lambda. There is really nothing deeper about it.


Show that the elasticity of substitution is σ

You cannot find $\partial \ln\frac{q(\omega)}{q(\omega')}$, it has no well defined meaning here. However, by the first relation you mention, it follows that $$\ln(q(\omega)/q(\omega'))=-\sigma\ln(p(\omega)/p(\omega')).$$ Taking the partial derivative on both sides with respect to $\ln(p(\omega)/p(\omega'))$ gives $$\frac{\partial \ln\frac{q(\omega)}{q(\omega ')}}{\partial \ln\frac{p(\omega)}{p(\omega ')}}=-\sigma.$$


EMH: Difference between market price and fundamental value?

The fundamental value of an asset over an infinite horizon is the sum of dividends discounted over time.

You are stumbling into the literature on rational and irrational bubbles. In general, the difference between the fundamental value of an asset and its market price constitutes, to some degree, bubble behavior.

Edit — I can probably expand this for clarity.  Each of those definitions tries to get at the same thing — there is no room for systematic bias in the prices of assets traded on the market. Markets, more or less, are powerful because they aggregate massive amounts of information. They incorporate ever trader’s wants, fears, expectations, etc. and distill it all into a single thing — a price. And markets do this very quickly. We can see this, for example, when markets adjust to incorporate information shocks that occur around FOMC statements. Because of this, it should be true that prices do not deviate systematically from fundamentals. The argument boils down to one of arbitrage opportunities, I suppose.

However, many economists (many people, really) openly discuss bubbles. A bubble, rather loosely stated, exists if prices deviate systematically from fundamentals. The common theme of each definition above is that we should see systematic deviations on prices away from fundamentals. They just vary a bit in how emphatically they say this. Nevertheless, we all talk about bubbles.

Also — how do we know if prices have deviated from fundamentals? Well, there is a nice literature built around this idea, but I am not convinced this is possible to do with empirical data. However, economists have used the laboratory to show that this happens in markets. In fact, generating bubbles in an asset market is probably among the most reproducible laboratory phenomena I can think of.


Certainty equivalent and risk aversion parameter given utility function

For the certainty equivalent and the "risk compensation" (which I am interpreting as probability premium because it's the only thing that intuitively makes sense in this context to me; feel free to correct me), think more intuitively about the concepts. The certainty equivalent is the amount of cold hard cash you'd be indifferent to taking in lieu of the uncertain outcome.

In the good outcome with $\frac{1}{3}$ probability, you'd end up with a wealth of $10 + 12$, and in the the bad state with probability $\frac{2}{3}$ where the lottery does not give you anything, you still end up with wealth of $10$. Thus, we are looking for wealth that I will denote $w^c$ where

$$v(w_c) = \mathbb{E}(v(w)) \implies \frac{w_c - 1}{w_c} = \left(\frac{22-1}{22}\right)\cdot \frac{1}{3} + \left(\frac{10-1}{10}\right) \cdot \frac{2}{3}$$

Solve for $w_c$ and that will be the certainty equivalent. You'll notice that trying to solve this out, we get:

$$\frac{w_c - 1}{w_c} = \frac{101}{110}$$
$$\implies \boxed{w_c = \frac{110}{9} \approx 12.2}$$

As for probability premium, it is the shift in probability towards the better lottery outcome that would be needed to make the expected utility of the new lottery equal to the utility of the expected value of the old lottery.

$$\mathbb{E}_{\text{new}}(v(w)) = v(\mathbb{E}_{\text{old}}(w))$$
$$\implies \left(\frac{1}{3} + \pi\right) \cdot \left(\frac{22-1}{22}\right) + \left(\frac{2}{3} - \pi\right) \cdot \left(\frac{10-1}{10}\right) = \left(\frac{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3} -1}{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3}}\right)$$
$$\implies \frac{101}{110} + \frac{6}{110} \pi = \frac{13}{14}$$
$$\implies \pi = \frac{8}{770} \approx 0.01$$

(assuming I calculated my fractions correctly)

Edit: I see that the question was asking for a risk premium, not probability premium. Notice that the expected wealth due to the lottery is 14, but the certainty equivalent is 12.222...

The difference between the two is the risk premium.


Why the stock market went up when the currency being weakened?

Quoting from the question

When the yen being weakened [...] "The automobile won more order as a result..."

and more orders made the automobile sector more profitable, hence the higher stock price.


What is meant by an inflation rate spoken in the typical news?

In media the inflation number that you see is generally the Year-on-Year inflation rate, i.e.,
$$\frac{I_t-I_{t-12}}{I_{t-12}}$$
where $I_t$ is the price index (CPI or WPI). For most countries, the index is a fixed base index. So the absolute value of the index gives you the idea of where the prices are relative to the base year.
Inflation rate on the other hand gives you idea of how much the index has increased as compared to same month previous year. The main reason for using YoY over MoM is to prevent seasonality misleading you.
For example, if media reported MoM, every December you'd see high inflation because of holiday season. So inflation of December is not strictly comparable with other months because of inherent seasonality. To prevent this YoY is used.
On your second point: No inflation does not keep decreasing with time. This would be true if price increases with a decreasing slope. There is no economic reason to believe that.
On third point: if we only looked at inflation rate from a historic point then that number would rather be an increasing number. Then again you would not be able to compare it with previous months' or years' numbers as then the difference of inflation would be of interest.


Why are demand and supply curves shown as concave up?

This is so because usually the marginal costs of firms are increasing in production and many economists think also at increasing rate.

Intuitively you are only willing to provide good Q if the marginal costs are less or equal to price $MC \leq P$. Providing goods to the market at price below marginal cost is irrational and hence inconsistent with basic assumptions behind supply and demand.

The reason why marginal costs are thought of as usually increasing is that the more you use up some factor of production let’s say capital the scarcer it becomes and hence the return to capital increases and hence the MC increases. Moreover, many economists believe that they also do this at increasing rate (That is if you increase the consumption of labor twice the labor costs will increase more than twice) and hence you get convexity. But this being said you can find concave cost functions and also linear cost functions as well which are neither convex or concave in many textbooks as well.


What is the difference between solving for utility maximization separately or aggregately?

Under what constraints is $u_1+u_2$ maximized? The exercise may be about the First Welfare Theorem: a competitive equilibrium is Pareto efficient. In this case, maximizing the sum may be designed to highlight that the equilibrium you found also maximizes the sum of utilities.

In general, the maximum of a sum of functions need not equal the maximum of any of these functions.


Persistent changes in relative prices due to productivity differences

When comparing industries it is almost intuitive that a relative productivity increase causes a relative price decrease in the absence of anything else happening. The reason it is intuitive is that all the competitors within the same industry probably benefit from the same productivity increase so together you can all raise prices slowly or not at all or decrease prices.

You might be looking for the Baumol effect. Wikipedia says...

  The rise of wages in jobs without productivity gains is from the
  requirement to compete for employees with jobs that have experienced
  gains and so can naturally pay higher salaries, just as classical
  economics predicts. For instance, if the retail sector pays its
  managers 19th-century-style salaries, the managers may decide to quit
  to get a job at an automobile factory, where salaries are higher
  because of high labor productivity. Thus, managers' salaries are
  increased not by labor productivity increases in the retail sector but
  by productivity and corresponding wage increases in other industries.

According to this, a necessary condition for the Baumol effect is competition between industries for resources.

This is the relevant link.


gibbard-satterthwaite theorem and median voting

Gibbard and Satterthwaite insist that the social choice function must be defined over all rational preferences over outcomes. That is, if voters' preferences could be anything (subject to the constraint of completeness and transitivity), then we have the Gibbard–Satterthwaite theorem.

On the other hand, if preferences were single-peaked, then the assumptions they make do not hold, and the theorem does not apply.


Features of the set of Bayes Correlated Equilibrium in a single-agent decision problem with uncertainty

Yes,  $Q$ is convex and closed precisely because $P$ is and Q is its image under a continuous function. Therefore, $Q$ is compact.

The function you are maximizing is continuous as well, so the maximum must exist.


What does it mean by saying someone is "effectively risk averse/loving"?

"Effectively" has two definitions:

1: in such a manner as to achieve a desired result.

2: actually but not officially or explicitly.

O+C are using the second definition here. This is because the risk averse/loving behaviour that is being exhibited does not come from the concavity of the agent's utility function, but instead indirectly, from the concavity of the c.d.f of the random cost $\eta$ that the monitor will have to pay if they misreport information. So the agents are not "officially" risk averse/loving but as a result of the concavity/convexity of the monitor's cost cdf they will behave as a risk averse/loving agent would when the cdf is linear.

Hope this helps!


What does this phrase about economics mean [Lionel Robbins]?

To me, the quote seems to be about what economics is, and, more subtly, what it isn't. Robbins claims economics is about scarce resources that people can use for multiple things. The implication, and you will find this in undergraduate economics textbooks, is that economics is concerned with efficiency. Economists are simply looking for ways to make the most out of scarce resources. An outflow of this focus on efficiency is the paradigm of utility or profit optimization under constraints.

Perhaps more interesting is the subtle claim of what economics is not about: the distribution of resources. This does not seem particularly radical now, but it was much more so at the time. Keynes and Marx still dominated the political discourse in the early 20th century, and both were very much concerned with the distribution of resources. By focusing on efficiency rather than distribution, Robbins puts himself firmly in the neoliberal paradigm.


What additional axiom to GARP do we need to generate a differentiable or smooth utility function

None if the axiom is to be on preferences, as any smooth utility representation can be monotone transformed into a non-smooth utility function.


Swanson and Williams (2014): restricting based period response to one

I'm no empiricist (surprise), but skimming through the paper, I'm pretty sure that just means that the $\delta^{\tau_i}$'s are chosen so that

$$ \frac{1}{11}\sum_{i=1990}^{2000} \delta^{\tau_i} = 1 $$

as I stated in a comment above. This assumption is necessary in order to separately identify (estimate) $\delta^{\tau_i}$ and $\mathbf{\beta}$.

Without this assumption, all you are able to estimate is $\delta^{\tau_i}\mathbf{\beta}$, so you need additional assumptions to solve for the two factors separately.


What is an equilibrium generated by replication?

For each agent in the economy, create n agents with the same endowment and preferences. The result is the replication economy. Sometimes n is taken to infinity in proofs.
Chapter 17.I in Mas-Colell explains in detail. Related material is the regularizing effects of aggregation, appendix of Chapter 4.


increased demand but no increase in price?

Such a situation can be explained in the following ways:

If supply is perfectly elastic (horizontal supply curve), then an increase in demand does not increase the price. It's highly unlikely that supply of petrol here is perfectly elastic. This could be the case under perfect competition with constant marginal costs of petrol. However, typically petrol has increasing marginal costs. Alternatively, it could be the case with constant marginal costs and Bertrand competition (price competition, with few firms) meaning they also set prices equal to marginal costs as in perfect competition.
There may be costs to increasing the price. So-called menu costs make prices sticky and therefore changes are not as often as we expect. E.g. if you're a restaurant and want to increase prices, you have to reprint all menus, which may not be worth it for a temporary demand increase. However, this is probably not tbe case in your example.
When setting the price before the shortage was announced, the increase in demand was expected and the price already set accordingly high. Was there an increase in the price (shortly) before the announcement? If not then the next explanation is the only possibility.
Government regulation of the price.


Cryptocurrencies: Is it possible to estimate price from market cap?

The standard definition of market capitalisation is:
Market capitalisation = (number of units in existence)*(price).
Update: VARulle gave a link to the definition they use for “circulating supply”: “The best approximation of the number of coins that are circulating in the market and in the general public’s hands.” They need to eliminate “burned” (destroyed?) coins as well as those that might be lost or sent to unrecoverable addresses.
This is an estimate, and so different sources might get different values. Adjusting market cap in this fashion is not normally done in other asset classes, as people generally want to avoid the use of estimates in discussions.