Does price cap in emergency situation (COVID-19) reduce the availability of a good? Does unregulated market increase it instead?

Capping prices at a low level permanently would presumably lower supply in the long term, but that is not germane to this question.

I will attack a slightly narrower version of this question: does capping prices in an emergency lower investment?

I would argue that the definition of “emergency” means that it is short term, and activity is disrupted. It is extremely difficult to launch a fixed investment project during an emergency. Fixed investments take time to finish, and the emergency is likely past before completion.

What we are seeing in 2020 is existing capacity being re-purposed to new uses. E.g., distillers are switching from producing whiskey to producing hand sanitiser. I doubt a survey has been completed, but anecdotally, many manufacturers are selling at a slim or no profit. (However, they benefit from switching from non-essential to essential production, allowing them to keep workforces employed. This is a benefit, even if profit margins are lower than their usual output.)

Most emergencies are geographically limited; e.g., hurricane. It typically makes no sense to switch production like is happening in 2020, capacity in other regions can be accessed more quickly.

The question then shifts: does a lack of controls create an incentive to have extra capacity/inventory during normal times to take advantage of an emergency?

This is perhaps plausible, but a good portion of manufacturers attempt to reach a global market. The existence or non-existence of price caps in a few countries would not radically shift their investment decisions.

Keep in mind that there is an aversion to price gouging during emergencies. Very few companies want to be publicly associated with the practice. As such, most news stories about price gouging involve largely invisible intermediaries that do no have a public “brand”. (There is an academic literature on price gouging, but I am not familiar with it. However, the reality that there is a public aversion to price gouging is non-controversial.)


expected payoff of an American put option

An explanation behind this could be that, as you mention

  [...] the payoff of an American put option at maturity is given by
  $$\max\left\{K-S(T),0\right\}$$

and

  [...] sources [...] say that $$(S,t)=\sup_\tau E[e^{-r(\tau-t)}\max\left\{K-S(\tau),0\right\}]$$

Thus the difference between the two appears to be the discounting factor considering how far (in time) we are from the exercising/stopping date (which is the same as the maturity date if $\tau=T$), which implies to compute (continuously) the present value of this futur payoff. If  $(\tau-t \rightarrow0) \land (\tau=T \lor t \rightarrow T)$, both equations become equal.

If one decides (for some reasons) that the option will be exercised at maturity, i.e. $\tau=T$ and that "now", $t$, is "turning into this futur date $T$", i.e. $t \rightarrow T$, one gets

$T-t \rightarrow 0 \implies \sup_T e^{-r(T-t)} \rightarrow 1$

$\implies \sup_T E[e^{-r(T-t)}\max\left\{K-S(T),0\right\}]$

$\rightarrow  E[\max\left\{K-S(T),0\right\}]\rightarrow\max\left\{K-S(T),0\right\}$

Where the last above row shows that the expected payoff becomes certain.


Why is railway electrification in North America far less common than in Europe?

Essentially, the question of electrification is a trade-off: electrified railways are more energy efficient, the engines often supply more power and they are quieter and without local emissions while at the same time requiring a large up-front cost and additional maintenance. The benefits of electric railways are best played out when running many trains on the same line, especially if they have many stops and accelerate often.
On the other hand, while diesel is less energy efficient, has slightly higher running costs and a different complexity of moving parts, there is no additional up-front cost other than laying the rails, acquiring and distributing the fuel is easy and there are no problems with different standards if the network has multiple owners.
Electrification of (sub)urban and underground railways as well as electric tramways date back to the 19th century. At the time, the principal technology was steam; the main reasons for experimenting with electric systems was probably noise complaints and air quality (the latter is of particular concern in tunnels such as the London Underground). The advantages of electric traction over coal-powered steam engines were, however, quickly noticed and attempts to extend the technology of urban railways to main lines were made early. The first electrified full railway in Germany (and possibly the world) was opened in 1905 (it had run on steam for the previous five years due to lack of operating permission and the initially developed system being insufficient) between Murnau and Oberammergau. This line did not touch any major population centres but the landscape is rather hilly so the power advantages of electric traction could be exploited.
By the way, this is a pattern in Europe: most early electrifications were in hilly or mountainous areas, where large amounts of power were needed to climb, a situation that steam engines struggled with. On the other hand, these areas often provide ample opportunites for hydroelectricity which became the primary power source of many railways – especially in Switzerland which wanted to be more independent of coal (and later: oil) imports. In areas where electrification provided less of a benefit because less power was needed, it often progressed far slower or stalled altogether as coal was cheap overall.
After the Second World War, diesel traction started becoming a thing that would eventually displace steam traction on all non-electrified lines typically by the 1970’s. Diesel provided immense benefits over steam (no reliance on water, greater power output, only one person required in the cab all the while at a similarly cheap price as coal) while being able to use the same infrastructure – most notably, it also does not require any power lines to be built or catenary to supply the power to the engine. Now, the trade-off outlined above began to really matter: while coal was more obviously inferior in everything except price of the raw material and ease of construction, diesel could actually challenge electric traction on equal grounds, each having a use case where they are superior.

After all this backdrop, the actual answer follows here.
The mostly freight-only railways in the United States use most of the advantages diesel provides to the fullest. They typically run very long trains with lots of cargo (maximising profit per train while minimising driver cost). They tend to use more of the airspace above (e.g. double-stacking of containers, larger cars) as they have essentially unlimited clearance in most places. They tend to run fewer trains per line and the trains tend to run very long distances. There are many different lines that criss-cross the US (more than one might expect when looking at the Midwest’s population density), typically each belonging to a different company that will run trains along its own line. As the lines are mostly freight-only, the frequency of service doesn’t matter as long as there is sufficient capacity.
If these were electrified, that would mean they could either not use as much vertical space (or would need far taller pantographs). Each company would have to electrify its own lines at a high cost but the return in fuel savings would be far lower as there are not as many trains that can benefit. The power needs to be generated somewhere and distributed to the line; this is a significant problem in the sparsely populated Midwest. Finally, they would likely have to agree on some kind of standard as interoperability across companies exists in the current diesel-based system.
Contrast this with Europe: Most railways are or were nationalised allowing for a single national standard. A nationalised company has an easier time investing large sums for a slow return as electrification would provide. Most lines are primary passenger-oriented; the benefits of electrification for passengers are much higher than for freight (acceleration, noise and pollution are all more important). Thus, freight tends to be handled a little bit as an afterthought and has to follow passenger rail requirements rather than vice-versa. In addition, most freight runs along a smaller number of key trunk routes that are used by all freight companies; meaning that the single investment generates a return at a much faster rate. Finally, due to the more densely populated nature of most of Europe (Germany has almost 20 times the average population density Kansas has; even Lower Saxony still has 10 times the density of Kansas) pollution is more of a concern and electric infrastructure is more widespread.
A similar line of reasoning applies to Russia: the Transsiberian is the one single trunk railway that connects European Russia to Siberia and the Far East. The electrification of this one single line benefits so many trains including probably 50 % of long-distance and freight trains. The benefits are enormous and train density is large. Furthermore, the Transsiberian not only connects the major cities of Asian Russia but also actively developed the areas it was built through, leading to a string of higher population density all through the country which not only means more accessible infrastructure but also a greater ease of electrification.


First it would be about the index used to take account inflation.

There are different indexes for calculating inflation, for example there could be by national prices for consumer or national prices for producer or other references as index calculations. Every index has its own methodology.

To take this example and crunch it with the numbers provided we'll assume the inflation was calculated with the National prices for consumer with a basic market basket of prices for swiss on the different periods and then the inflation was calculated by the change of values in the index for every period.

Taken this on account note that inflation is a percentual change of a wide selected prices or quantities change depending on the methodology.

So the inflation shown in your data comes from a change on index, the base index from wich the changes will be calculated generally has to be a period of realtive economic estability.

In this case we'll assume the base period as year 2000 because is the starting period of your data. The Index for base period will be 100.

  Year        2000
  Inflation   .98%
  Index       100

From there we will calculate the value of the other indexes.

To do this we aply the aritmethic formula

  (%Inflation+1)*(Prior year index)

    For example

  2001 Index will be = (1+.0266)*(100) = 102.66

  2002 Index will be = (1+.0206)*(102.66) = 104.77

  2003 Index will be = (1+.0127)*(104.77) = 106.11

The running totals are :

Having the index values then we can take the price of a commoditie or a service we want to take inflation on by account. The price with inflation is called "Nominal Price" and the one without inflation is "Real Price To do this we employ the arithmetic formula:

  (Nominal Price for Selected Year)*((Index of year selected)/(Index of base year))

    For 2000 Real Price = (81.21)*(100/100) = 81.21

    For 2001 Real Price = (89.9)*(102.66/100) = 92.29

    For 2002 Real Price = (96.54)*(104.77/100) = 101.15

The Running Totals would be

I hope the explanation was not that technical or that thedious, we can enter in more detail or less detail but i guess this would be right.

You can apply the same methodology for montly or quarterly basis

Now you can see than given the general level of prices on swiss economy 2012 was the Peak year on real prices for the selected period.

But 'ill add that given that we live on a world economy and the price of gold is fixed by international exchanges betweeen coutries a theoritical calculation of the real price involves currency exchanges and more theory.


Gale-Shapley Follow-up Literature and General Questions

How does social welfare change when the two entities do not have the same number of individuals in them, so that not everyone can be matched?

The algorithm works fine in this case. The important assumption is still that the underlying graph is bipartite (and that preferences are transitive). If we allow for members of a given type to propose to each other, then we lose convergence. Suppose we have three players: $A, B, C$. We have $A$'s preferences as $(B, C, A)$; $B$'s preferences as $(C, A, B)$; and $C$'s preferences as $(A, B, C)$. So $A$ proposes to $B$, which accepts. Then $B$ proposes to $C$, which accepts. This unmatches $A$. Next, $C$ proposes to $A$, which accepts. This unmatches $B$. We then repeat.

    Are there faster ways to match individuals with each other (lower runtime?)

Gale-Shapley has a runtime complexity of $\Theta(n^{2})$. In graph theory, most bipartite matching algorithms take $\Omega(|E| \sqrt{|V|})$ time. In the worst case, $|E| = |V|^{2}/4$. So I wouldn't bank on much better.

Additionally, this isn't a problem that lends itself well to a divide and conquer approach. It's really a greedy problem. You might be able to get the expected runtime down by selecting a random ordering for proposing. The analysis would be involved though.

    It seems like a very weird game theoretic model just because when I read the proof for the validity of the algorithm, it doesn't seem like the problem is "discrete"; that is, optimizing who gets matched with who and finding a Pareto efficient result can't be solved through maximizing a continuous function.

Actually, it's a discrete model. In more traditional fields of economics, we have continuous variables. So linear programming makes sense. We could actually formulate an integer-linear program for the stable-marriage problem if we look at it from a graph theoretic perspective. Matching problems can be formulated in the language of flows. And we can write flow problems as LPs. The flow constraints form a convex polytope. If the graph is bipartite, the vertices of the polytope are the matchings. The trick here is that we have to add additional constraints to ensure we end up with the correct matching (since the matching from the Gale-Shapley algorithm is unique, regardless of the order in which the men propose).

Even if we go to the trouble of formulating the linear program, solving an LP for this problem by hand or by a computer is less efficient than using the algorithm.

    (As a side question, is the Gale-Shapley allocation Pareto efficient?)

The matching generated by the Gale-Shapely algorithm is in the core. An imputation belonging to the core satisfies the property that no coalition can collude their initial endowments and improve its members' outcomes. An imputation is Pareto optimal if no set of agents can improve their outcomes without another player's outcome being less favorable. So core allocations are Pareto optimal.

For more on Gale-Shapley, check out my blog: https://michaellevet.wordpress.com/2015/05/22/algorithmic-game-theory-stable-marriage-problem/


Note that strictness of quasi-concavity is not required, unless you want to obtain strict concavity as a result.

Take any $x,y\in\mathbb R^n$. Observe that homogeneity of degree 1 (HD1) implies that
\begin{equation}
f(x/f(x))=f(x)/f(x)=1=f(y/f(y)).
\end{equation}
For any $\alpha\in(0,1)$, let
\begin{equation}
\theta=\frac{\alpha f(x)}{\alpha f(x)+(1-\alpha)f(y)}.\tag{1}
\end{equation}
Note that $\theta$ also lives in the (open) unit interval.
Thus, by quasi-concavity, we have for every $\theta\in(0,1)$,
\begin{equation}
f\left(\theta\frac{x}{f(x)}+(1-\theta)\frac{y}{f(y)}\right)\ge\min\left\{\frac{x}{f(x)},\frac{y}{f(y)}\right\}=1.
\end{equation}
Expanding the LHS using $(1)$, we get
\begin{equation}
f\left(\frac{\alpha x+(1-\alpha)y}{\alpha f(x)+(1-\alpha)f(y)}\right)\ge1.
\end{equation}
Invoking HD1 again, we have
\begin{equation}
\frac{f(\alpha x+(1-\alpha)y)}{\alpha f(x)+(1-\alpha)f(y)}\ge1
\quad\Leftrightarrow\quad
f(\alpha x+(1-\alpha)y)\ge \alpha f(x)+(1-\alpha)f(y),
\end{equation}
which means $f$ is concave.


Modelling involving sum of random variables: Simple CDF?

So we have

$S_n \thicksim^{iid} \ ?$

$\bar{s} = \frac{1}{n}\sum{s_n}$

Exponential

Suppose $S_n$ follows the exponential distribution.

$$f(s|\beta) = \frac{1}{\beta} e^{-\frac{s}{\beta}} \quad , \quad 0 \leq s < \infty \quad , \quad \beta > 0$$

Take the simple bivariate case. Say $Z = \frac{S_1 + S_2}{2}$ and $W = S_1$.

So $S_2 = 2Z - W$ and $S_1 = W$. The determinant of the Jacobian: $\frac{\partial s_1}{\partial z} \frac{\partial y}{\partial w} - \frac{\partial x}{\partial w} \frac{\partial y}{\partial z} = -2$. The absolute value is $2$.

Thus, given the join marginal of $S_1 , S_2$

$$f(S_1, S_2) = \frac{1}{\beta ^2} e^{-\frac{S_1 + S_2}{\beta}}$$

we then find the joint marginal of $W, Z$, and integrate out $W$ to find the density function for $\frac{S_1 + S_2}{2}$

$$
\begin{align}
h(W,Z) = & \ f(W, 2Z-W) \cdot 2 \\
= & \ \frac{2}{\beta ^2} e^{-\frac{w+2z-w}{\beta}} \\
= & \ \frac{2}{\beta ^2} e^{-\frac{2z}{\beta}} \\
\end{align}
$$

Integrate with respect to $w$ from $z$ to $0$ gets you the density:

$$f(Z) = \frac{2}{\beta ^2} z e^{-\frac{2z}{\beta}}$$

Then you do a lot of garbage integration by parts (let the buyer beware if I have made an error) to get the cumulative distribution function evaluated from $z$ to $0$:

$$\boxed{F(Z) = -\frac{2z + \beta}{2 \beta} e^{-\frac{2z}{\beta}} + \frac{1}{2}}$$

Substitute $q = -\frac{2z}{2\beta}$ and you can make this look more tolerable:

$$\boxed{F(Z) = \left(q - \frac{1}{2}\right) e^{-2q} + \frac{1}{2}}$$

You may now umm...generalize the case for $n$ agents LOL. It may be more viable to compare this to the CDF of a single exponential distribution, then maybe try the case for $n=3$ and see if you can see a pattern.

Cauchy :P

One thing I looked at for fun online, take a simplified Cauchy distribution:

$$f(x) = \frac{1}{\pi(1+x^2)}$$

By the convolution formula and then a lot of work, the density of $Z = X + Y$ is:

$$\frac{2}{\pi(4+z^2)}$$

You can also verify that if you want the density of $Z = \frac{X + Y}{2}$, it is:

$$f(z) = \frac{1}{\pi(1+z^2)}$$

It's the Cauchy again! Now, the Cauchy has that pesky property where umm...it has no moments. But I'm pretty sure for this Cauchy distribution, the density is symmetric around 0. And technically you did not specify that your signals would have mean and variance, just that they were iid. :)

The cdf of this ends up being:

$$F(z) = \frac{\arctan(z)}{\pi} + \frac{1}{2}$$

So if the average for three variables or four variables also gives the density of the Cauchy, you may feel so bold as to use this.

Final Comments

Uhh, if you are willing discretize the noisy signal (you mentioned Bernoulli), you can try using the Poisson distribution or something. If you are looking into another continuous option you could try Beta. If I have more time I may edit this answer with some investigation into one or the other.


How central banks raise interest rates

This is a common misunderstanding of central banking.  The popular media portrays things as if the act of increasing interest rates prevents hyperinflation.  In truth, "preventing inflation" and higher interest rates are both side effects of restraining the growth of the money supply.

What central banks set is not generally interest rates but a target for the interest rate.  In the United States, this is called the federal funds rate target.  Let's break this down.  Funds are simply money loaned from one bank to another, generally on an overnight basis.  The funds rate is the (weighted) average interest rate at which they loan to each other.  It's the federal funds rate because it is at the national level, not limited to one state.  That number is calculated based on market actions.  It is not government controlled.

The Federal Reserve sets a target for that statistic.  If the statistic is below the target, the Fed sells bonds.  Too high, the Fed buys treasury bonds.  That puts more money in the system.  With more money, more banks have excess reserves (to loan to other banks) and fewer banks are short on reserves (needing to borrow from other banks).  As a result, the average rate falls.  Inflation increases.  Too low, and the opposite happens:  sell bonds; less money; less excess reserves; more borrowers; higher interest rates; lower inflation.

Alternative tools of monetary policy

Are there other tools that a central bank can use?  Sure.  A central bank can change the reserve fraction, pay interest on excess reserves, or change the rate at which it loans to banks.

In the US, the rate at which the Fed loans to banks is called the discount rate.  It is generally set to around the same value as the federal funds rate target.  Banks hardly ever borrow from the Fed, as the rules around it make it undesirable.  They would generally prefer to pay a premium and borrow from other banks.  But there is a limit to the premium that they will pay.

Changing the reserve fraction means changing the amount of deposits that banks must hold.  The higher it is, the less that banks can lend.  This is hard to change suddenly.  If the reserve fraction was 10% (as it currently is in the US) and is raised to 11%, that means that most every bank will have to borrow and few will have money to lend.  This would force many to borrow money from the Fed or be forced to suddenly sell assets.  They would prefer to have some notice.  Then they can save up to have 11% by the announced date.

Paying interest on excess reserves has a similar effect to changing the reserve fraction, as it increases the amount of money held as reserves.  It is more gradual than changing the reserve fraction.  However, it can also compete with banks that are trying to borrow from other banks.  The other banks may prefer the deposit interest to the loan interest.  Paying interest also costs money.

Monetary policy and hyperinflation

Let's get back to hyperinflation countries.  The governments of those countries don't want to pay interest on excess reserves.  Even if they didn't mind the other side effects, those countries don't have the money to spend.  And they generally aren't constrained by the reserve fraction.  In a hyperinflation, times are uncertain.  Banks don't want to get stuck loaning money at an interest rate that will be lower than tomorrow's market rate.  But rates are always increasing.  So banks often refuse to loan even if they are awash in reserves.

For the same reason, central banks don't increase the reserve fraction.  It won't help.  And in a hyperinflation, the central bank would want to increase the discount rate and loan less.  But as already said, central bank to bank loans are rare.  Even eliminating them will not have much of an effect.

We're back to the one tool remaining, open market operations to try to meet a higher interest rate target.  But for those to work, the central bank has to sell government bonds in competition with the government.  The central bank has a limited amount of assets that it can sell.  Once it is out, it's helpless.  It can set any interest rate target it wants, but it lacks any way to hit it.

Meanwhile, the government retains plenty of tools to cause inflation.  For example, they can print money.  And they are likely to do so, as everything is becoming more expensive.  So they need more money just to offer the same level of services as they did previously.  So they print it.  But then everything is more expensive and they need more money.

Effective policy

The solution of course is to lower the level of government spending instead of trying to maintain it against inflation.  But in the short term, that is extremely painful for all those that suddenly don't get the level of government support to which they are accustomed.

They still need to do that eventually.  Because in a hyperinflation, everything only gets worse.  Because they are trying to trade a permanent increase in inflation for a temporary improvement in the economy.  Obviously trading a permanent harm for a temporary benefit is a bad deal.  But in the short term, it's hard to make that decision and go for the permanent benefit in exchange for the temporary harm.  And the longer this goes, the larger the short term harm is.  And desperate people think more about the immediate term than the long term.  That's part of what it means to be desperate.

TL;DR:  in the short term, benefits of more money seem to outweigh the harm of inflation.  And in a hyperinflation, people tend to think in the short term, trying to survive today while letting tomorrow fend for itself.


Is elasticity purely an aggregate concept or can it be individualized

In theory, the elasticity quantifies in proportional terms the "reaction" of the dependent variable to a change of the independent variable, where the two are related through a functional relationship. So the concept is universal and mathematical -it applies to any univariate function, and not only in the field of Economics (when we have a multivariate function, then partial elasticities can be defined and used, but they face some constraints). So there is nothing "intrinsically aggregate" about the concept. In its theoretical conceptualization.

But in order to estimate/calculate the elasticity of any function, we must have the function in the first place. So the question boils down to "Can we obtain an individual demand function?"

It would appear that the matter is only one of data availability. If we somehow could have data on many transactions of the same individual related to the same good, then we could use these data points (price, quantity) =$\{(p_1,q_1),...,(p_n,q_n)\}$,  in order to approximate its demand function... and we would probably be mistaken.
And this is because these points do not necessarily represent movements along the individual demand schedule. They are equilibrium points, points reflecting not the preferences of the individual alone, but their "crossing" with the supplier's behavior. Only if we could be reasonably certain that for a given set of transaction data, the demand schedule have not shifted (due to say, income effects), we could then use these data to estimate the demand function (by the way at market level, this is the classic endogeneity problem: using market outcomes to estimate a relation we estimate neither market demand, nor market supply, but rather the locus of market equilibrium points).

So while realized transactions possess the strong element of "truth" (the individual did actually bought such quantity at such price), they do not necessarily reveal to us his demand function.

In your case, if you can have other data on the buyer, especially in order to assess any possible "income effects" (for the case you are examining, that would be turnover and/or profit data for a company), you could combine them with its purchases from you to arrive at an elasticity estimation (there are also possible issues with substitution effects, but for them you will have to have data on what your customer purchases from your competitors). Is it a listed company? If it is, I guess you can find quarterly Financial Statements over the web (meaning that you will have to transform the data on purchases to quarterly too).

This could take the form of a multiple regression model

$$E(\ln q_i) = \beta_0 + \beta_1\ln s_i + \beta_2\ln p_i, i=1,...,n$$

where $q_i=$ quantity purchased in period $i$ (month or quarter), $s_i$ would be the turnover of the customer (assumption = if the customer sells more to its customers, he purchases more from me), and $p_i=$ average price of $q_i$. One could also include Customer's profit data as an additional regressor, in order to control for their possible effect on purchases from us. And of course, any other regressor that you think can have an effect on the purchasing decisions of the customer.

The use of natural logarithms, means that

$$\hat \beta_2 = \frac {\partial E(\ln Q_i)}{\partial \ln P_i} \approx \eta_{q,p}$$

i.e. it is the estimate of the price elasticity of demand. The above model assumes a demand function that is characterized by a constant price elasticity. You can make it more complex or even non-linear, if you suspect that the elasticity of demand in your case may vary.


Set Theory Properties of the Budget Constraint

Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier.

Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed here and here.

Here is also a short, concrete example of something for which we need convexity of the budget set:

Suppose you want to prove that Marshallian Demand $x(p,m)$ is single-valued for every (larger than zero) price ($p$) and (non-negative) income ($m$) combination. That is to say, a Marshallian Demand Function exists, since without single-valuedness we would have a Marshallian Demand Correspondence (i.e. we'd have multiple good combinations as part of the optimal demand choice for a single given price and income level), which can be harder to work with. This is a pretty nice result that we (almost) always work with in undergraduate texts. For that you need strictly convex preferences and a convex budget set $B(p,m)$. The proof is as follows:

If two points $x_1$ and $x_2$ are both elements of the demand $x(p,m)$ at given prices $p$ and income $m$, i.e. if $x_1, x_2 \in x(p,m)$, then by definition we must have indifference between the two points, that is $x_1 \sim x_2$. For a single-valued demand function two different points $x_1$ and $x_2$ cannot exist  for single given level of prices and income, i.e. they would have to be the same point. In other words  $x_1 = x_2$. We prove it by contradiction. Suppose $x_1 \neq x_2$.

For all $\lambda \in (0,1)$, we have $\lambda *x_1 + (1-\lambda) *x_2 \in B(p,m)$ by convexity of $B$. By strict convexity of preferences we have $\lambda *x_1 + (1-\lambda) *x_2 \succ x_1$. However, this contradicts the fact that $x_1 \in x(p,m)$ in the first place. Hence, $x_1 = x_2$.

So, in words, some combination of $x_1$ and $x_2$ is better than just $x_1$ at the given price and income level. But such a combination is also attainable at that given price and income level because of the convexity of the budget set. This is where the budget set convexity assumption is important. Furthermore, if for a given price and income level there is a point (in this case a combination of $x_1$ and $x_2$) that is both strictly better and attainable, then this contradicts the original starting point that $x_1$ is a point in the demand function for the given price and income level.

As a side note, I think it is a bit misleading to talk about producers in your post when talking about budget constraints. If we are looking at a budget constraint, we should be discussing consumers.


Methods or models to predict activity of clients of a bank?

Some keywords you might search for will be profiling, RFM, CRM and customer analysis.  clv (or customer lifetime value) might be your best keyword though.

An example CLV calculator: http://www.clv-calculator.com/customer-lifetime-value-formulas/clv-formula/

Here is an interesting clustering model: http://www.iieom.org/ieom2011/pdfs/IEOM068.pdf

An article on how surveys are used to classify customers: http://www.finreghq.com/articles/bank-customer-profiles-and-why-they-are-important-sample-profile-questions/

Again, a lot of good stuff if you search for banking and clv:

https://www.google.com/search?client=ubuntu&channel=fs&q=banking+clv&ie=utf-8&oe=utf-8


Is Universal Basic Income compatible with free international trade?

There is nothing to prevent other countries "taking advantage" of an export market's UBI.  However, I don't think that that implies a country introducing a UBI would need to increase tariffs, and I have never heard this being advocated.

Since UBI hasn't been implemented on a large scale, there is little compelling evidence on what its effects will be, but a lot of speculation on what they might be.

First, it is important to observe that countries' cost bases already differ a lot in terms of firms' tax burdens and unskilled wage rates.  Countries with high taxes and high wages don't typically sustain them through protectionist tariffs.

Second, it is suggested that UBI may lower a lot of government costs, so that the impact on firms' tax burden would not be as large as you might expect.  For example, UBI would replace many benefits (e.g. job seeker and housing benefits) reducing administration costs, especially for means-tested benefits.  It is argued that it would reduce physical and mental healthcare costs related to poverty, which could be particularly important in countries like the UK that have free universal healthcare alongside substantial poverty.

Third, it is unclear what effect a UBI would have on wages.  Since workers' basic cost of living is (at least partially) covered, it is possible that wages could fall.  There would certainly be a political argument for reducing the minimum wage.  Depending on how the cost of UBI is distributed, this could more than offset the tax burden on firms.  It is also well established that a lot of traditional benefits result in very high effective marginal tax rates on part-time work (as earned income is offset by reduced benefits); a UBI is likely to improve the financial incentive for unemployed people to do small amounts of work in a lot of countries, even if wages were somewhat lower.

Finally, it is largely unknown how UBI might affect a country's comparative advantage and productivity.  For instance, some (not all) suggest that UBI would drive innovation by giving people the time and space to pursue an invention or start a business under less financial pressure.  This would lead to more value-added jobs, which is how countries typically manage to compete internationally while sustaining high wages and tax rates.


Log-linearization of the market clearing condition

We have:

$$y_t = c_t − w^u(1 − N_t) + \kappa v_t$$

We take the log of both sides:

$$\ln y_t = \ln \left[ c_t − w^u(1 − N_t) + \kappa v_t \right]$$

And then linearize around the steady states:

$$\begin{align}
\ln \bar{y} + \frac{1}{\bar{y}}(y_t - \bar{y}) & = \ln \left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right] + \frac{1}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}(c_t - \bar{c}) \\
& + \frac{w_u}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}(N_t - \bar{N}) \\
& + \frac{\kappa}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}(V_t - \bar{V})
\end{align}
$$

Cancel $\ln \bar{y}$ and $\ln \left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]$

$$\begin{align}
\frac{1}{\bar{y}}(y_t - \bar{y}) & = \frac{1}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}(c_t - \bar{c}) \\
& + \frac{w_u}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}(N_t - \bar{N}) \\
& + \frac{\kappa}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}(V_t - \bar{V})
\end{align}
$$

Multiply the first term on the right hand side by $\frac{\bar{c}}{\bar{c}}$, and similarly for the other terms:

$$\begin{align}
\frac{1}{\bar{y}}(y_t - \bar{y}) & = \frac{\bar{c}}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}\frac{(c_t - \bar{c})}{\bar{c}} \\
& + \frac{w_u \bar{N}}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}\frac{(N_t - \bar{N})}{\bar{N}} \\
& + \frac{\kappa \bar{V}}{\left[ \bar{c} - w^u(1-\bar{N}) + \kappa \bar{V} \right]}\frac{(V_t - \bar{V})}{\bar{V}}
\end{align}
$$

And we simplify, taking advantage of the fact that $\hat{v}_t = \hat{\theta}_t + \hat{u}_t$, to get what you have:

$$\hat{y}_t = \frac{\bar{C}}{\bar{Y}}\hat{c}_t + w^u \frac{\bar{N}}{\bar{Y}} \hat{n}_t + \left( \frac{K\bar{V}}{\bar{Y}} \right) (\hat{\theta}_t + \hat{u}_t) $$

I don't think this derivation is wrong. But you can see that $y_t$ and $N_t$ are directly proportional to each other. It's definitely plausible that $\frac{\bar{N}}{\bar{Y}} = 1$, but I've pored over the equations in the setup for 3 hours to no avail at showing that rigorously. It's probably something silly that I don't understand. I'll look back later and if nothing else, I'll turn this answer into a community wiki so anyone who has the right idea can edit it.


I would strongly recommend you to read Erik Brynjolfsson, Andrew Mcafee et al for some serious and real insights about this topic. The world economic forum this year had the same theme.
In my view (which are heavily derived from the works of the above mentioned dignitaries) the problems arising from complete automation will be the lowering or stagnation of income levels. As Erik mentions in his Ted talk, productivity levels are at all time high but income levels are not. This was not the case in previous technological revolutions. People and systems adapted to new technologies and income rose along with productivity. Now they don't share the same relationship. One of the reasons is that we are not changing our systems well to absorb the technology. For example, if people are saving a lot of time because of automation, they should be able to spend it in something constructive. May be entrepreneurship, maybe knowledge work. To make all those automation work, people are required to program, design, develop and create. There needs to be a shift from typical work to knowledge work. If such a shift is not there and that too at a speed that matches with the speed of technological changes then income levels will continue to fall, unemployment will rise.
In Andrew Mcafee's words, "we need to RACE WITH THE MACHINE".


There is more than one indifference curve. There is one belonging to every utility level. So for any utility level $c$, the points $(x,y)$ that satisfy
$$2\cdot \sqrt{x} + y = c$$ are an indifference curve.

For example let $c = 2$. Can you find a point $(x,y)$ that satisfies
$$2\cdot \sqrt{x} + y = 2$$
and is on the $y$-axis or the $x$-axis?


Can Apple use its off-shore cash to do Apple stock buy-backs tax free?

To the best of my knowledge - and I'm not an accountant, so I may very well be wrong: buybacks are not profits, but are instead considered as an 'investment' in Apple in its own company. Since they are not profits, they are not taxed.

Apple using its offshore cash to buy back its own stock is changing the accounting balance rather than actually generating any profits.

Consider the following balance sheet:

        Assets
Cash      100M
   Liabilities
Loans      70M
        Equity
C. Stock   22M
R. Earnings 8M

In this balance sheet, the retained earnings (what is taxed) is 8M.

With a buyback of 12M in common stock:

        Assets
Cash       88M
   Liabilities
Loans      70M
        Equity
C. Stock   10M
R. Earnings 8M

Retained earnings have not changed, the company's taxable income has not changed, and thus the company should not have to pay any taxes.
This should be true regardless of where the money 'physically' is - in the U.S. or elsewhere.


What does DOD stand for in this indicator?

It stands for "disbursed and outstanding debt", which is explained here, for example.


How to operationalize agglomeration economies?

Agglomeration economies is a large concept that includes any effect that increases firms' and workers' income when the size of the local economy grows.
The general idea is to estimate the overall impact of density on a local characteristic, such as the impact of local employment density on local productivity. The density estimate captures the agglomeration effect.
As a naive example to operationalize the concept of agglomeration economies, consider the following specification:
$$Y_c = \alpha Density_c + \beta Skills_c + \epsilon_c$$
where $Y_c$ is a measure of the local outcome in a city $c$, say the logarithm of local wage, $Density_c$ is the logarithm of density, $Skills_c$ captures local skills and $\epsilon_c$ is the error term. If there is no correlation between the random component and explanatory variables, then the ordinary least squares estimate of the elasticity of productivity with respect to density ($\beta$) is a consistent measure of total net agglomeration economies. For instance, a value for the elasticity of the local outcome with respect to density of 0.03 means that a city twice as large (knowing that a factor of 10 is often obtained for the interquartile
of local density in many countries) has $2^{0.03} - 1 = 2.1\%$ greater productivity,
because of either pure local externalities or market agglomeration effects that dominate dispersion effects of any kind.
You will find much more, including a discussion of the mechanisms, in Chapter 5 of the Handbook of Urban and Regional Economics, The Empirics of Agglomeration.


What is the exact relationship between the efficient-market and random-walk hypotheses?

They don't have anything to do with each other.

Use the superscript $e$ to denote "anticipated value" in whatever way it is formed. Then the Efficient Market Hypothesis states

$$X^e_{t+1,market} = E(X_{t+1}\mid I_t)$$

In words, the value anticipated "by the market" equals the true expected value conditional on all information available at time $t$. So first and foremost, the EMH is about what is the predictor that reflects the anticipation of "the market", and it asserts that it is the conditional expected value based on all information available (to state the obvious, there are many other predictors of future values. We can use the median, the mode, etc etc etc).

By the properties of the conditional expectation we have that

$$X_{t+1} = E(X_{t+1}\mid I_t) + u_{t+1},\;\;\; E(u_{t+1}\mid I_t)=0$$

Therefore the EMH claims that the value anticipated by the market, is off-target only by something ($u_{t+1}$) that cannot be predicted as anything other than zero, given the information available. But that is what it means to "use the information efficiently": to extract from it everything that it can possibly tell you.

The "cannot beat the market" assertion is reflected in the part of the EMH that "the market" forms an expectation as though it was a single mind having available, again, all information. This implies that while "inside", "private" information may matter for what individual agents do and accomplish, if we somehow "average/combine" over those individuals each having a different information set, we will obtain the conditional expected value we would have obtained if we have collected all information in one Information set.

Nowhere in the above do we mention, or need to mention, what is the Data Generating Mechanism for $X$. It can be anything.


Did the dollar depreciate when Trump won the election?

You might find the answer in this article on the Forex website. I hope it helps!


The foreign exchange market is a huge over-the-counter market, primarily intermediated by large dealer banks with subsidiaries around the world (sometimes referred to as “money center” banks).

A buyer or seller of a currency is almost certainly at or trading with one of these banks. The dealer banks make money via a “spread,” where every buyer or seller of a currency effectively pays the dealer they’re trading with. For example, if 1 cur1 = 1 cur2, a buyer of cur2 might have to pay 1.01 cur1 for it, and a seller of cur2 might receive 0.99 cur1 for it, with the dealer taking .01 on both sides.

The forex market is large and highly competitive, and balance between the prices of any two currencies is just determined by supply and demand— as people trade cur1 for cur2, the supply of cur1 at dealer banks will rise, and the supply of cur2 will fall, so the price of cur2 will rise relative to cur1.  From a practical standpoint, as more people trade cur1 for cur2, a dealer bank will have to replenish its reserves of cur2, which it will often do by borrowing it.

In my opinion, the simplest way to think about all this is to remember that the primary source of demand for any currency is people who want the stuff denominated in that currency (be it consumer goods, financial or real assets, etc.). When more people want that stuff, relative to the stuff in other currency areas, the price of that stuff (i.e., the value of the currency) rises. For example, Canada is a huge oil exporter, and if you want Canadian oil, you first have to get Canadian dollars. As a result, when demand for oil rises, the value of the Canadian dollar rises.


Repeated games without a beginning and an end

What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z_+$ have the same cardinality. So it makes no substantive difference which index set you use.

Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) Markov strategy in a continuation game with infinitely long history.

If a history of infinite length is what you really care about, then may be looking into the literature of continuous-time repeated game will provide some insight.


This article might be of help: https://ageconsearch.umn.edu/bitstream/122709/2/sjart_st0160.pdf


Macroeconomic model or knowledge base representing connected industries in the world

is it even possible to have such a model, even on the macro level?

Yes, it is entirely possible. There are large scale general equilibrium models that would be able to answer those sort of questions. For example, the questions you are asking could be answered with dynamic stochastic general equilibrium (DSGE) models.
Namely, you want some large-scale multisector DSGE model (there are smaller versions where you will lump multiple sectors together), such as the multisector DSGE model used by  Antosiewicz & Kowal 2016 for investigating effect of CO2 emissions reductions mandated by policy on the economy. In fact based on your description you likely want to go even bigger and include more equations - that is still doable, once you know how to make 2 sector DSGE model adding more sectors is just question of having data and enough time (usually large and detailed DSGE models are built by large teams not individuals).
You would likely want to first start by studying some simple cases though. Macroeconomic Theory a Dynamic General Equilibrium Approach by Wickens is good introduction.
Alternatively, you could also model it via old school input-output tables in some sort of Leontief model, but that would likely be less accurate than the above. However, if you are planning to do this just for fun that would be easier.

Is there a database/API/graph or any source of data showing this kind of connection between industries and the nature of that connections?

This sort of question cannot be answered with raw data. For example, if data would show that there is 500 mil ton of fish imported into Belgium from outside, you can't just assume that Belgium will have this amount of fish less because under autarky Belgium could increase its production of fish at the expense of other industry so you always need some model for tracking all the effects through the economy.
As for data, you have to be more specific. There is no single 'uber'-economics-dataset that would have all economic data in the world. For most EU countries you could get necessary data from their statistical offices. For example, for the Netherlands you would likely be able to find most data you need if not all at CBS (it is hard to say unless you precisely specify the model).


you cannot compute real exchange rates based on the data you have. What you can do, and what is often done (not sure what it is good for though) is to use relative PPP to calculate changes in RER using some base year.
The OECD publishes some statistics on P*/P, you could also use that, although I am not sure what methodology they use.
In practice, you would need the actual prices of a basket of goods to calculate the RER, once you somehow normalize prices within a country you can no longer calculate the RER.
hence the appeal of stuff like the Big Mac Index.


Why isn't National Net Worth of a country used to calculate its economic power, as opposed to GDP?

Your first country producing cars (possibly after receiving investment or loans from others) has an economy, and if it is selling cars then it can buy other things in exchange for cars. It could even buy some gold. So it has a degree of economic power, but so too do its investors, suppliers and customers.

Your second country sitting on a pot of gold and doing nothing (it may as well have left the gold in the ground) has no economy and no economic power until it changes that position.  For example it cannot buy more gold as it has nothing to use to pay for it except gold. Only after it has sold gold for something useful or productive might things change.

By the way, GDP (on the production measure) is actually the market value of all goods and services produced, less the value of intermediate goods and services consumed.  In a closed economy with no international trade, this is equal to all final goods and services produced (adjusted for the change in stocks of intermediate goods and services) but in an open economy in international supply chains it can be very different.


Does merger control really affect market structure?

Clifford Winston´s excellent book Government Failure versus Market Failure:
Microeconomics Policy Research and Government Performance surveys the literature on the effects of government regulation generally and antitrust merger control specifically. He doesn't exactly answer your question on economy-wide structural effects, but largely concludes that merger control is rather ineffective. A small quote:

  Evidence from the finance literature indicates that mergers that were
  challenged by the DOJ and the FTC were in general not anticompetitive
  and would have been efficient had they been allowed to go through
  (Eckbo 1992). Mergers that were challenged or opposed by antitrust
  regulators but were consummated anyway have often resulted in gains
  for consumers (Schuman, Reitzes, and Rogers 1997; Morrison 1996).
  Finally, Crandall and Winston (2003) analyzed the effects of merger
  policy on price-cost margins and found that the antitrust authorities
  have primarily attacked mergers that would enhance efficiency, either
  by blocking them in court or allowing them to proceed only if the
  merger partners agreed to conditions that turned out to raise their
  costs.


Principles of Economic Reasoning: How Are Expected Costs and Benefits Distinct from Ethical and Moral Judgments?

The following is simply not accurate statement about economics:

Terse principles of economic reasoning might be stated as follows:

Maximum benefits are good.
One should act to maximize benefits and minimize costs in the social context.

In order to answer this question, and see why the statement is inaccurate, it is important to recognize that inquiry in economics (an other sciences as well), can be split into two categories:
Positive Economics: Positive economics deals with questions of what is. For example, what is the elasticity of demand for market for potatoes? What is the effect of rising minimum wage on employment? etc. (see Mankiw Principles of Economics pp 27.). Most economics as practiced today (in academic research) is positive economics.
Normative Economics: Normative economics deals with question what ought to be? If we estimated a differences-in-differences model that shows small increases in minimum wage do not significantly affect employment should we increase it or not? (ibid. Principles of Economics). Normative economics is mostly dealt with in public economics or in practical public policy.
Now it is plain to see that positive economics and research about positive economics simply does not concern itself with maximum benefit or whether one acts to maximize benefits and minimize costs (from moral perspective that is).   When we model rational behavior of profit seeking firm  in positive inquiry we are not implying that firm should act that way - we are attempting to just be descriptive and describe the behavior of firm.  When we say people are trying to maximize their utility (or when behavioral economists tell us sometimes they don't) we are not claiming this is moral thing to do or they should behave in such way. This is done without any moral judgement or prejudice similar way as physicist would describe motion of a gas or planets (of course, the precise techniques will differ but in principle on fundamental level there is not much difference).
Hence already from onset we can safely say that whole corpus of positive economics is  distinct from making any moral or ethical judgements so let us turn our attention to normative economics.
Normative economics deals with ethical and moral issues but not by any means in such overtly reductionist way as suggested in the 'terse principles' you stated.
Normative economics will nowadays start with recognition that although economics originated in moral philosophy* it is no longer part of it and we economists are not anymore specialist in moral philosophy (of course, a person can have multiple specializations one can be both economist and moral philosopher the same way as one can be biologist and moral philosopher, or as someone who can be both physicist and chemist at the same time). Consequently, when we make normative inquiry we won't start by trying to deduce what is moral or not from first principles.
Rather in normative economics we start by looking at past work of moral philosophers and moral frameworks that are already developed (yes sometimes those moral philosophers happened to also be economists - e.g. Adam Smith - theory of moral sentiments, Amartya Sen - Capability and Well-Being etc.). The three widely popular moral frameworks used across public econ literature are moral philosophies based on Rawls' theory of justice, classical liberalism and utilitarianism but there are of course many more different frameworks used (see Ross (2014) Philosophy of Economics for short overview).
In both Rawls theory of justice or classical liberalism the objective is neither to maximize benefits or minimize costs in either private or social context (see Rawls (1971)
A Theory of Justice; Nozik  (1974)  Anarchy, State, and Utopia). Rather a Rawlsian economist would recommend to conduct economic policy in accordance with maxmin principle i.e. maximize welfare of the poorest member of society and nobody else (although once you raise one person someone else becomes the poorest so Rawls is not implying to take care of only one person). A classical liberal/libertarian economist would favor any policy that minimizes the use of force and coercion by state (even if it would not necessary maximize utility - classical liberals are not utilitarians). Utiliterians would indeed want to maximize benefits and minimize cost in terms of utility.
However, note in athe above cases it is not really the economist who postulates what is good or bad. Rather the economist will defer to moral philosophers. Furthermore, when doing normative economics an economist will not necessary even pick a side. Vast majority of the corpus of public economics literature usually makes comparative analysis that takes into account several different moral  philosophies. For example, in optimum income taxation literature people routinely estimate optimal taxes and transfers using Rawlsian, utilitarian, classical liberal (also known as conservative - due to US politics where conservatives tend to be economically libertarian) or other welfare function (see Saez 2001 as an example of a study that does this).
Of course, ultimately in a democracy it will be people to decide which social welfare function society adopts, and people should decide based on their own values and moral principles. I think most economists nowadays humbly acknowledge this.
This being said economists are also people and most people love discussing politics and moral philosophy so many economists will frequently make statements that go way beyond even normative economics. Some will even dabble in moral philosophy more professionally. This is not unique to economics. You will find many physicists such as Einstein for example who at some point started to dabble in moral philosophy. While in case of physics it can be easy for people to make distinction in economics this can be more difficult due to the historical roots in moral philosophy that economics has but that does not make 'principles of economic reasoning' any more close  to it.

* I am specifically putting emphasis on 'moral' as it can be argued no science can be divorced from metaphysics and other branches of philosophy that deal with epistemological questions (see Ross cited above).  How do we even know we can know anything? Is there even such thing as a 'fact' or 'objective truth'?


A collar strategy is only in part to protect your gains made on the underlying stock. The collar strategy consists of buying protective puts and at the same time writing covered calls.

You are right in saying "If you want to protect your gains, why would you not simply buy an OTM put?". This is known as the Protective Put and if you simply want to protect your gains, this is the way to do it.

When you also write a covered call, you once again got it right that it is usually used to offset the premium costs of the put.

However when you combine the two strategies (creating an equity collar), your view of the market would be that it is moving sideways. I.E. you are expecting the asset prices to not rise above your written call's strike price, and you are protecting your asset prices in case it falls below your put's strike price. And in the meantime, you are still holding on to the asset earning dividends while the covered call is financing the protective put.

On a side note, this question seems to be more in tune with the Quantitative Finance Stack Exchange community.


You have the right idea, make a 4 by 3 payoff matrix with fruit at the top and at the side weather/water yes/no, giving 12 different cells, then put the respective dollar payoffs.
You could also make another table and find the expected payoff by multiplying each payoff by its probability, although since all the probabilities are the same, the table wouldn’t show anything different/useful to the normal payoff matrix with nominal dollar values.
Then add up the values in each fruit column and the highest value will be the optimal strategy.


A closed set does not need to be bounded. For instance, the set $[0,\infty)$ is closed but unbounded.

Formally, a set is closed if it contains all its limit points. You can easily verify that it is the case for your $C$. Take a sequence of elements $x^m=(x_1^m,\cdots,x_n^m) \in C$ that converges to a vector $x^{*}=(x_1^{*},\cdots,x_n^{*})$ under any appropriate topology. The $j$-th coordinate of $x^{*}$, $x_j^{*}$, is the limit of the sequence $x_j^{m}$. Since all $x_j^{m}$ are nonnegative real numbers, so is the case for the limit $x_j^{*}$. Since this is true for all $j$, $x^{*}$ also belongs to $C$.


When would prices increase according to Malthusianism?

This question rests on a few assumptions.
So let's dig into them.
What Malthus Said
First, the Malthusian collapse you are referring to is similar to the Malthusian Trap, from whence it derives, so let's look at that.
From Wikipedia:

In accordance with the theory, cross-country evidence indicates that technological superiority and higher land productivity had significant positive effects on population density but insignificant effects on the standard of living, during the time period 1–1500 AD. In addition, scholars have reported on the lack of a significant trend of wages in various places over the world for very long stretches of time. In Babylonia during the period 1800 to 1600 BC, for example, the daily wage for a common laborer was enough to buy about 15 pounds of wheat. In Classical Athens in about 328 BC, the corresponding wage could buy about 24 pounds of wheat. In England in 1800 AD the wage was about 13 pounds of wheat. In spite of the technological developments across these societies, the daily wage hardly varied. In Britain between 1200 and 1800, only relatively minor fluctuations from the mean (less than a factor of two) in real wages occurred in Britain. They peaked at around 1450 and in 1800 they were actually significantly worse.

So to go back to the source, Malthus didn't predict a population explosion at all (or food price rises for that matter). In fact, his theory was specifically that population growth fluctuated with economic development such that the prices of wheat (in terms of wages) remained relatively stable over two millennia.
This was known as the Malthusian Trap, where technological advancement would always be matched by (not exceeded by) population growth, resulting in the stagnation of living conditions.
Since the time of Thomas Malthus, new evidence has emerged that, while the trap may exist, many societies break out of it, and achieve explosive growth in living conditions. This is generally termed, the "breakout". In A Trap At The Escape From The Trap?$^1$, Andrey Korotayev suggests that this breakout leads to social upheaval, and may have played a role in events like the revolutions in Latin America, or possibly even the formation of the US. Unfortunately if we delve too far down the social macroevolution road we wind up having to have the Guns, Germs, and Steel discussion, and that's straying dangerously off topic, though I also want to throw in a reference to Huntington, as another great and accessible read on the really big picture.
But More People...!
But moving back to your assumptions. Let's assume that population and technological growth became uncoupled by the "breakout". Now all of a sudden standards of living are at all time historical highs. Under Malthus we would expect the population growth rate to also be at historical highs.
But it's not.
In fact much of the developed world is at "peak population", facing major demographic shifts as baby-boomers age out of the workforce faster than they are being replaced.
Japan faces a "demographic crisis"$^2$$^3$, the US shows substantial aging and even China's growth rate has slowed dramatically and is now lower than the US's.
In fact, according to the UN, the projected world population over the next several decades looks like this:

Which you may notice is not an explosive growth. For more information on this, I'll defer to the experts at the Population Division of the Department of Economic and Social Affairs, and their technical paper
Cause and also Cause
Lastly, you're assertions rely on the assumption that increased population means food prices go up.
But that's not how numbers work.
Prices are denominated in currency.
More people (without a correlating rise in currency), actually means there's fewer dollars per person. In that scenario, food prices would go down (although probably slower than wages), leading to shortages. Price is a way of rationing scarce resources, but it's not the only way. The other way to ration is queueing, or shortage. This is what we see happen in developing nations. Rice is not more expensive, there just isn't enough rice there.
If everyone only has \$5 a rice merchant can't charge \$6/lb, or he won't sell any.
An Inflationary Theory
In the event that the population increased, and central banks also increased money supply, than standard economic models would predict that the prices of all commodities (not just food) would rise, since the relationship to amount of quantity over the amount money would have changed, or roughly
$$\rho (\Delta P_n, \Delta \frac{M_0}{Q_n})>0$$
where $\rho$ is correlation, and $P_n$ and $Q_n$ are price and quantity of $n$ respectively. Of note, there is some disagreement o whether $M_0$ or another measure of money should be used, but prices in the US (as measured by CPI) have a substantial correlation with $M_0$ ($\rho = 0.46$ for $M_0$ as compared to $0.086$ for $M_1$ from 1985-2013)$^3$
This would be conformant with standard economics, and not reliant on a Malthusian population collapse.
The only way we would expect prices to rise dramatically without central bank intervention would actually be after the collapse, as people accumulated currency, and would trade more currency for food.
TL;DR
In the event of a zombie or post-nuclear apocalypse, cash will decline in value relative to commodities, so stock up on rations and bullets because dollars won't help.$^4$
References
$^1$: Korotayev, A.; et al. (2011). "A Trap At The Escape From The Trap? Demographic-Structural Factors of Political Instability in Modern Africa and West Asia". Cliodynamics 2/2: 1–28.
$^2$: Aging of Japan. (2014, December 9). Retrieved December 18, 2014, from http://en.wikipedia.org/wiki/Aging_of_Japan
$^3$: Current research. You'll have to take my word or pull the data from St Louis Fed and BLS yourself.

$^4$: Fallout: New Vegas. (2010). United States of America: Bethesda Softworks.


What are the advantages and disadvantages to a country's people when the national currency is the world's reserve currency?

I cannot believe no one answered this. Someone could have easily given a one phrase answer, its called Triffins Paradox.

The core of Triffin's Paradox is that the issuer of a reserve currency must serve two entirely different sets of users: the domestic economy, and the international economy, but you will not be able to make both sets of users happy.

So our Federal Reserve needs to print USD to meet global demand for US dollars, but this money printing can lead to the devaluation of the currency which hurts the people of the issuing country by losing purchasing power, but does help foreign governments to be able to make their debt payments in USDs and certainly the people of these other nations like to stockpile USD in case their local currency goes kaput, but to meet their needs, you then have Americans who have to pay $4 a gallon for gas and are seeing prices in other commodities go up.


If a CDS is x%, why does this mean $(1 - x)\%$ probability of default?

Note that the 20% annual premium would imply a 20% annual default probability. Tim Bennett is then talking about the probability of default within five years.

He might have thought that
$$
1 - Prob(\text{no default for five years running}) = 1 - (1 - 0.2)^5 > 0.8
$$
but this is not true as
$$
1 - (1 - 0.2)^5 \approx 0.67
$$


It's important to consider what exactly the population is about which an inference is being drawn.  It's easy to overlook the time aspect in this context.

Suppose for example that the aim is to forecast the next two years' GDP for each country in the world. Then the population of interest is a set of pairs of the form "country, year".  It isn't simply "all countries", and even if a forecast model has been estimated by regression on current and past years' data for each country, that doesn't mean that the whole population of interest has been included.

If one really does start from a complete dataset for the whole population of interest, then all one can do is calculate summary statistics. That could include standard deviations, but it would be inappropriate to call these standard errors, since that term relates to a sampling distribution whereas the only "sample" in this case is the whole population.


Are these two derivatives the same with regards to TR?

Both of these approaches are correct and valid approaches, in the first approach you calculate the derivative of total revenue with respect to quantity in second with respect to price.

If you are interested in seeing how total revenue varies with quantity you can use the first one if you want to see how the total revenue varies with price then the second one.

Also, you can have multiple different elasticities. Elasticity for single variable function is just by definition $EL=\frac{f’(x)x}{f(x)}$. Depending on what types of elasticity you want to relate the total revenue and what  exactly is your ultimate goal of your study either of the formulas could be used.


Currency Values, Developing Economies and Balance of Trade

Currency values change relatively slowly over time, partly due to differences in the rate of expansion of the money supply over time between different countries, and partly due to changes in the terms of trade.

If we compare the Indian Rupee with the US dollar. M2 for the US dollar roughly doubles every 10 years, the Rupee is going up roughly 5 times in the same period. The Indian M3 measure is equivalent to the US M2 measure (M measures vary considerably between countries in what is being measured.)

US Dollar M2
Indian Rupee M3

This is why the Rupee is getting progressively weaker, and will continue to do so, especially if balance of payments are also negative. The ratio of rupees:dollars is increasing, and so the value of rupees in dollars is falling, as the market tries to maintain equilibrium.

When the central banks intervene in the currency markets, they essentially use their position to alter the relative value by buying/selling as appropriate. However, in this case they can only do this if they have foreign currency to sell, and with a negative balance of payments, and a rapidly expanding money supply, they are essentially in the position of standing on the sea shore and telling the tide not to come in.

While all of the above is the monetary side of things, as an exporter you're standing on the trade side of the equation, and what you say is quite correct. All things being equal, over time local producers will step in to provide the imports that are now too expensive, exports will flourish, and the situation should reverse itself as you explain. However, that implicitly assumes that money supply expansion rates are broadly similar, and that clearly isn't the case. Unless and until the Indian central bank gets its banking system under control, and slows down the expansion in the money supply, you can essentially rely on continuing inflation, and a weak currency.

It continues to be an open question in Macro-Economics as to what the 'right' degree of monetary expansion is. However, there is a very strong correlation between countries with low expansion rates, Germany for example, and economic strength. There are a lot of negative effects within the economy when inflation rates are too high due to monetary expansion, fundamentally because it leads to widespread distortion of the price signal.


Why GDP series always has an unit root problem?

Whether GDP has a unit root depends on whom you ask, and what they believe about the nature of business cycle.

The seminal paper that first applied a unit root test (the ADF test) to macroeconomic time series and extracted economic implication is Nelson-Plosser (1982).
They were unable to reject the unit root null hypothesis for GDP and other standard macro series. They offered the following conclusion:

"...macroeconomic models that focus on monetary disturbances as a source of purely transitory fluctuations may
never be successful in explaining a large fraction of output variation and that stochastic
variation due to real factors is an essential element of any model of macroeconomic fluctuations."

These are fighting words in the debate of Real vs. New Keynesian Business Cycles. They interpret their finding as evidence against the latter. Or at least that variation in output is more attributable to permanent real shocks rather than transient shocks to aggregate demand ("monetary disturbances").

Later, however, in a subsequent milestone result Perron (1989) found that a trend-stationary series with a break in its deterministic trend may be mistaken as unit root. In particular, he overturned Nelson-Plosser's non-rejection result for GDP (and other series), after taking into account the 1929 crash and 1973 oil price shock.

Another possible example of such trend-stationary series with structural break is the CBO GDP data series covering the Great Recession shown in answer by @1muflon1.

Today, the suggestion that, say, US GDP of the last 20 years has a unit root would probably not be too controversial.
In general, the answer you get depends on the data, the unit root test being applied (there are many), and whether the economist has a strong RBC or New Keynesian bias.

(For example, Nelson and Plosser were already RBC advocates before publishing their finding.)


The Economical "justification" of mass killings

I'm not really sure why this is tagged [social-welfare] or [welfare economics].  It seems to me that more appropriate tags would be [history] and/or [international economics] (since the question involves international relations), and/or [game theory] (since the question involves putatively rational behavior in a multi-agent context).

Having said that, here is my (uninformed) response.  In a case where a much stronger group murders, enslaves and expropriates a much weaker group, there can be a net economic advantage (for the stronger group).  The European conquest and colonization of the Americas and their abduction and enslavement of millions of Africans are probably the most obvious examples.  Of course, genocide, slavery and mass expropriation do not guarantee long-term economic growth, as can be seen by comparing the economic histories of North America and South America.  Furthermore, the influx of easy resources following an easy conquest can create something analogous to a "resource curse" or "Dutch disease".  For example, the massive influx of silver from Spain's American conquests initially made Spain "rich", but eventually it led to inflation.  It also caused the Spanish economy to stagnate relative to the dynamic economies of northern Europe (Corden, 1984).

However, in a military contest between actors who are roughly evenly matched, I believe the historical record shows that it is always a negative-sum game.  Even if there is a clear winner, the wealth which the victor gains from plundering and/or enslaving the vanquished state almost never justifies the amount of blood and treasure that was needed to obtain the victory.  So the upshot is that these military conquests cannot be explained in terms of the economic self-interest of the warring states.

But it might be a mistake to treat states as monolithic rational actors.  Every state has a governing class, and it may be the case that the war was economically beneficial for the governing class, even if it is economically detrimental for the country as a whole.  For example, it may be that the Iraq war was a huge net economic loss for the United States, as argued in the book The Three Trillion Dollar War, by Joseph Stiglitz and Linda Bilmes.   But at the same time, it may be that certain individuals and firms associated with people in the Bush Administration (e.g. Haliburton) reaped a large economic benefit.  So from the perspective of those people, the war was "economically rational".  (Note that I say "it may be....", because I don't have enough information to back up this speculation.)

It is also likely that some wars and mass-murders just have non-economic motives.  For example, Hitler was motivated by race-hatred at least as much as by economics.  Stalin's purges, massacres, and genocides were motivated by a desire to consolidate his political power, at least as much as by economics.  And the Crusades and the Thirty Years War were motivated (at least initially) by religion, not economics.  (Certainly, they were a net economic loss for almost everyone involved.)

Afterthought.  (added later)  Here is another data point.  At the beginning of the twentieth century, the economies of Western Europe  were closely integrated in a big trading block.  Many people thought that a major war between Western European powers was impossible, because it was so obviously against their economic interests.  We saw how that turned out....


There are issues with this question as currently written.

I will attempt to offer a partial answer by responding with a different question. If money did not exist, how would governments in modern developed countries impose taxes?

Historically, rulers would impose taxes in kind: forcing peasants to serve in the army, or taking a portion of the harvest. This is not particularly easy to see how a modern government could achieve this in a politically acceptable fashion.

So if you can find a way to allow taxation without money, you have found a way for money to not be necessary in a modern developed country.


Help understanding Lagrangian multipliers?

A constrained optimization function maximizes or minimizes an objective subject to one or more constraints. As I understand it, the Lagrangian multiplier approach transforms a constrained optimization problem (I) into an unconstrained optimization problem (II) where the optimal control values to problem II are also the optimal control values to problem I. Additionally, the the objective functions in problems I and II take the same optimal values. The trick is a clever way of putting the constraints into the objective function directly rather than using them separately.

I agree with your presentation of the consumer's maximization problem:
$\Lambda(x,y,\lambda) = x^{\alpha} y^{1-\alpha} + \lambda ((xp_x+yp_y)-w)$.

Now we take the partial derivatives with respect to x an y, set them equal to zero, and then solve for x* and y*.

$0=\partial\Lambda / \partial x  = \alpha x^{\alpha -1 } y^{1-\alpha} + \lambda p_x = (\alpha / x ) x^{\alpha  } y^{1-\alpha} + \lambda p_x$

$\Rightarrow -\lambda = (\alpha / (x p_x)) x^{\alpha  } y^{1-\alpha}$

$0 =\partial\Lambda / \partial y  = (1 - \alpha) x^{\alpha} y^{-\alpha} + \lambda p_y = ((1 - \alpha) / y ) x^{\alpha  } y^{1-\alpha} + \lambda p_y$

$\Rightarrow -\lambda = ((1- \alpha) / (y p_y)) x^{\alpha  } y^{1-\alpha}$

$\Rightarrow (\alpha / (x p_x)) x^{\alpha  } y^{1-\alpha} = -\lambda = ((1- \alpha) / (y p_y)) x^{\alpha  } y^{1-\alpha}$

$\Rightarrow (\alpha / (x p_x)) = ((1- \alpha) / (y p_y))$

$\Rightarrow ( y p_y ) / (1- \alpha) = (x p_x) / \alpha$ (eqn 1)

Recover the budget constraint equation by taking the partial derivative $\partial\Lambda / \partial \lambda = 0$.

$0 = \partial\Lambda / \partial \lambda = xp_x + yp_y - w \Rightarrow xp_x /  w + yp_y /w = 1$ (eqn 2)

We now have two equations and two unknowns (x,y) and can solve for x* and y*.

$\Rightarrow yp_y / w  = xp_x/w \cdot (1 / \alpha - 1) = xp_x/w / \alpha - xp_x/w$

$\Rightarrow 1 = yp_y / w + xp_x/w = xp_x/w / \alpha$

$\rightarrow \alpha = xp_x/w$ (result 1)

$\Rightarrow \alpha = xp_x/w = 1 - yp_y /w$

$\rightarrow 1-\alpha =yp_y/w$ (result 2)

Results 1 and 2 form the famous constant expenditure shares result for the Cobb-Douglas utility and production functions. Which can also be explicitly solved for x* and y*: $x^* = \alpha w /p_x$ and $y^* = (1-\alpha) w /p_y$ which are the optimal values for both the Lagrangian and the original problems.


Why cannot I compare AIC values obtained from nonlinear least squares and the ordinary least squares?

In short, because AIC takes the log of the max value of the likelihood function.  Except under strong conditions there is no a priori reason to assume that these values will be the same after log-transforming your model.  In fact, a priori one should assume they will not be the same, since one of the reasons people log-transform their models is to change the sample distribution to be a little closer to normal.

So, you can't simply look at the two AIC values and conclude that they convey the same information, because in general, they do not.


What impact does the paid vacation leave have on labor productivity? Do employees get re-energized and work better...

There are not many studies that examine this question directly, but it is argued in the literature that based on the indirect evidence that shows that vacation does have health and happiness benefits on the employees that it increases productivity (Strauss-Blasche et al. 2005; Bellet et al. 2020).  Studies that examined directly various paid leaves from work generally find positive but not always significant effect of provision of paid leave/vacation on productivity (e.g. Bassanini & Venn; 2008). Moreover, Zenger and Folkman (2015) provide some tentative evidence for increase in productivity from paid vacation that is also based on international comparison. However, the effect will also likely be heterogeneous across individuals, industries, time and place. One should not just assume just because the average treatment effect is positive the policy has the same effect everywhere every time.

Are American employers stupid or, on the contrary, prudent when they show reluctance to grant such leaves to their employees in the absence of employee-oriented labor laws in the US?

This question makes no sense. It is simply not possible to judge employers 'prudency' or 'stupidity' from their decisions to provide or not provide paid vacations and stupidity and prudence are not even opposites.

Prudence is defined as individuals proclivity to save more when faced with increased risk (Sandmo, 1970). Consequently, to the degree that there is any uncertainty about productivity gains from paid vacation a prudent employer would spend less on paid vacations not more.

Stupidity is not rigorously defined. If by stupid behavior you mean irrational behavior then game theory clearly shows that rational (and thus not stupid) behavior can often lead to failure of getting maximum possible payoff or profit in a strategic interaction like in a classic prisoner dilemma where fully rational actors typically end up play strategies offering much lower payoff than possible (See Tadelis Game Theory: An Introduction pp 48).

Increase labor productivity does not automatically means there is increase profitability. Labor productivity is defined as value of output over labor inputs. This does not take into account the costs that employer has to undertake to achieve this increased productivity. However, even if paid vacation would be directly and unambiguously linked to increased profitability, it is not possible to make any conclusions on whether employer is 'stupid' or 'prudent' for above mentioned reasons.

Amount of paid vacation days is part of total compensation. As such it is negotiated over by both employee and employers. A poll among American workers showed that significant amount of them do not fully use all their paid vacation days. Consequently, this provides at least some evidence that in the US workers are not interested in paid vacations. Hence, a rational employer would minimize provision of benefit that their employees do not want, and maximize provision of work benefits they do want, because benefits (provided they are wanted) are substitutes for higher wages. It would be irrational to provide benefit a worker does not want since then employer has to still pay for high wage in addition to unwanted benefit (see Borjas Labor Economics 223-224).


Balanced Growth Path (Qualifier Question)

(That one can use the log-difference approximation for the growth rates, can be glimpsed by the fact that while the model apparently is set in discrete time, the log-evolution of technology is expressed using the exponential, which is how we express a constant growth rate in continuous time).

From
$$\frac{y_{t+1}}{y_t}=e^\mu\bigg(\frac{k_{t+1}}{k_t}\bigg)^\alpha\qquad(3)$$

Taking logs you get

$$g_y = \mu + \alpha g_k$$

Where the "$g$"s are growth rates. One the balanced growth path, they have to be equal to some constant, say $g^*$, so you have

$$g^* = \mu + \alpha g^* \implies g^* = \frac {\mu}{1-\alpha}$$

which is the growth rate of the economy on the balanced growth path.

You should also be able (by simmply manipulating the equation with which you start showing your work), to arrive easily at

$$\left (\frac {c}{y}\right)^* = 1- (\delta +g^*)\left (\frac {k}{y}\right)^*$$
So you need only to determine one of the two to obtain the other.

Realize how you can write the left-hand side of $(1)$ while on the balanced growth path, then turn the equation around and solve for $k/y$. Don't expect to get something necessarily "simple-looking".


Homework Question - Price Floor / Elasticity of Demand

Price elasticity of demand of $-0.5$ means that if price increases by $1\%$ demand decreases by $0.5\%$ (and vice versa in case of decrease).
Consequently, if the equilibrium quantity with floor is $100$ and the price of lettuce is $25\%$ that means that eliminating the price flow - which will offset the $25\%$ will increase the quantity demanded by:
$$0.5\cdot25\% = 12.5\%$$
$12.5\%$ increase from $100$ is $112.5$ so that will be new quantity.
Regarding the price you don't mention what the price at the price floor originally was but it should decrease by $25\%$ from its original value.


Rational expectations econometrics: a theory or an excuse?

If you prefer, think of the rational expectations framework as a benchmark -- much in the same way that perfect competition is a benchmark.  It's not about whether the benchmark is wrong, so much as what the benchmark allows us to conclude when it is wrong.

The canonical example for rational expectations is the Barro-Gordon model of central banking (or more generally, the Kydland-Prescott model of policy-making), which essentially couches monetary policy in the framework of an infinitely repeated game.

The idea there was to consider the question of whether it is better to have a central bank that sets monetary policy "by discretion" or by adhering to a stated rule.  In this case, the idealized assumption being made is that agents are rational and possess full information about the past behavior of the central bank (as well as the true value of pertinent measures like inflation & unemployment).

In the model, the bank's objective function implies a tradeoff between unemployment and inflation via the Phillips Curve.  Both inflation and unemployment are costly, but setting both to the equilibrium value is not feasible.

In a one-shot game, the bank can improve the value of its objective function by "inflation surprises" -- basically, by injecting money into the economy without announcing it, so that in the short-run there is an increase to employment but no impact on inflation.  But since realized inflation is a function of inflation expectations (because if people expect their money to be worth less tomorrow, they will spend more of it today, thus leading to increased consumption demand which drives up prices and makes the expectation self-fulfilling), in the iterated game setting rational economic agents will come to expect the central bank to generate "inflation surprises".  What is interesting about the Barro-Gordon model is that this leads to an "inflation bias" -- realized inflation is always higher than the optimal value when the bank's actual policy targets don't match with the expectations they set out through statements, etc.

So rational expectations predicts that central banking done according to a rule is more efficient, because it keeps policy inflation in line with expectations and avoids this bias.  In the literature one hears about "credible banks" and this is essentially what this is about.

Getting back to your question, is this used to "explain" forecast error?  In a sense, yes:  if the model predicts a certain value of inflation under rational expectations, and that value of inflation isn't realized, then one either questions the rational expectation model itself or (more likely) that one or more of the assumptions behind it is violated.  Usually, this is the assumption of complete information, although one should also consider the possibility of additional shocks not accounted for explicitly in the model.


What should the value of a private sector company tell us?

A company's value, in financial terms is given by its equity, or its difference between its assets and liabilities. The well known expression $E=A-L$ reflects this. When you buy a share, you buy a part of the equity. The point is that since the world we live in is dynamic, decisions today affect outcomes tomorrow. As a result, when you buy part of this equity, you are buying part of the claims on the firms' expected future profits. The market rewards those goods that consumers demand more with a higher price and one can definitely make the argument that a firm's future profits are conditional on it producing goods/services that are valued by the public. Whether or not the goods themselves improve the lives of consumers is a separate question altogether. Do we always buy goods that improve our lives? What does that even mean?


The technique described in the question is almost correct. Consider a panel data set consisting of three cross-sections ($a$, $b$, and $c$) and three time-periods ($1$, $2$, and $3$). Let y denote the column vector with the observations of the dependent variable, x the column vector with observations of the first explanatory variable, and z the column vector with observations of the second explanatory variable. They take these forms respectively:

$\textbf{y} =
\begin{bmatrix} y_{a1} \\ y_{a2} \\ y_{a3} \\  y_{b1} \\ y_{b2} \\ y_{b3} \\ y_{c1} \\ y_{c2} \\ y_{c3} \end{bmatrix}$
; $\textbf{x} = \begin{bmatrix} x_{a1} \\ x_{a2} \\ x_{a3} \\  x_{b1} \\ x_{b2} \\ x_{b3} \\ x_{c1} \\ x_{c2} \\ x_{c3} \end{bmatrix}$;  $\textbf{z} = \begin{bmatrix} z_{a1} \\ z_{a2} \\ z_{a3} \\  z_{b1} \\ z_{b2} \\ z_{b3} \\ z_{c1} \\ z_{c2} \\ z_{c3} \end{bmatrix}$.

Let $\bar{y}_{i} = \frac{1}{3}(y_{ai} + y_{bi}+y_{ci})$, with $i = 1, 2, 3$, and equivalently for $x$ and $z$: $\bar{x}_{i} = \frac{1}{3}(x_{ai} + x_{bi}+x_{ci})$ and  $\bar{z}_{i} = \frac{1}{3}(z_{ai} + z_{bi}+z_{ci})$, both for  $i = 1, 2, 3$.

This is the correct way to get the CCEP estimator allowing for one common factor, as in the model described in the question:
\begin{equation*}
\begin{split}
\begin{bmatrix} y_{a1} \\ y_{a2} \\ y_{a3} \\  y_{b1} \\ y_{b2} \\ y_{b3} \\ y_{c1} \\ y_{c2} \\ y_{c3} \end{bmatrix} = \beta_{1} \begin{bmatrix} x_{a1} \\ x_{a2} \\ x_{a3} \\  x_{b1} \\ x_{b2} \\ x_{b3} \\ x_{c1} \\ x_{c2} \\ x_{c3} \end{bmatrix}+\beta_{2}\begin{bmatrix} z_{a1} \\ z_{a2} \\ z_{a3} \\  z_{b1} \\ z_{b2} \\ z_{b3} \\ z_{c1} \\ z_{c2} \\ z_{c3} \end{bmatrix}+
\gamma_{a} \begin{bmatrix} \bar{y}_{1} \\ \bar{y}_{2} \\ \bar{y}_{3} \\  0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
+\gamma_{b} \begin{bmatrix}  0 \\ 0 \\ 0 \\ \bar{y}_{1} \\ \bar{y}_{2} \\ \bar{y}_{3} \\  0 \\ 0 \\ 0 \end{bmatrix} +
\gamma_{c}  \begin{bmatrix}  0 \\ 0 \\ 0 \\  0 \\ 0 \\ 0 \\ \bar{y}_{1} \\ \bar{y}_{2} \\ \bar{y}_{3}  \end{bmatrix} \\
- \beta_{1}\gamma_{a} \begin{bmatrix} \bar{x}_{1} \\ \bar{x}_{2} \\ \bar{x}_{3} \\  0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
- \beta_{1}\gamma_{b}  \begin{bmatrix}  0 \\ 0 \\ 0 \\ \bar{x}_{1} \\ \bar{x}_{2} \\ \bar{x}_{3} \\  0 \\ 0 \\ 0 \end{bmatrix}
-\beta_{1} \gamma_{c} \begin{bmatrix}  0 \\ 0 \\ 0 \\  0 \\ 0 \\ 0 \\ \bar{x}_{1} \\ \bar{x}_{2} \\ \bar{x}_{3}  \end{bmatrix}
-\beta_{2} \gamma_{a}  \begin{bmatrix} \bar{z}_{1} \\ \bar{z}_{2} \\ \bar{z}_{3} \\  0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
-\beta_{2} \gamma_{b} \begin{bmatrix}  0 \\ 0 \\ 0 \\ \bar{z}_{1} \\ \bar{z}_{2} \\ \bar{z}_{3} \\  0 \\ 0 \\ 0 \end{bmatrix}
-\beta_{2} \gamma_{c} \begin{bmatrix}  0 \\ 0 \\ 0 \\  0 \\ 0 \\ 0 \\ \bar{z}_{1} \\ \bar{z}_{2} \\ \bar{z}_{3}  \end{bmatrix} \\
+ \phi_{a} + \phi_{b} + \phi_{c} + \mu_{it}
\end{split}
\end{equation*}
Here, $\phi_{a},\ \phi_{b},$ and  $\phi_{c}$ are cross-sectional fixed effects and $\mu_{it}$ a well-behaved error-term that does not require any restrictions.

An interesting remark is that if the six $\beta$s that appear together with the $\gamma$s are not restricted to be equal to each other and equal to the first two $\beta$s,   more than one common factor is allowed.


What is the name given to an exporting-only economy?

Is "autarky" the word you're looking for? That's been used to refer to a self-sufficient economy that does not need imports. As others have written, it's an ideological position that chooses to make a country worse off, and doesn't happen in reality, except for short periods in exceptional circumstances.


Best terminology for "inverse" preferences

@Oliv's comment lead me to the following wikipedia article https://en.wikipedia.org/wiki/Inverse_relation
which indicates that the term "inverse" is used to define the kind of relation between $R$ and $R'$ that I described in the question.

The wikipedia article recommends to use $R^{-1}$ to denote the inverse relation of $R$.


Solve the "consumer's problem":

$\max U(x,y)$ subject to $I=p_xx+p_yy$

Then you will have $x=x(p_x,p_y,I)$ so x is some function of the parameters. The same goes for y.

Then determine the Hicksian demand functions, either by using some duality result or solving the dual problem:

$\min p_xx+p_yy$ subject to $U(x,y)=\bar u$ and you'll have both x and y in terms of prices and $\bar u$. We denote these demands as $h_x$ and $h_y$.

Now Slutsky's equation for x with respect to $p_x$:
$\displaystyle\frac{\partial x}{\partial p_x}=\displaystyle\frac{\partial h_x}{\partial p_x}-\displaystyle\frac{\partial x}{\partial I}x(p_x,p_y,I)$

The substitution effect is the first term on the RHS, and the income effect is the third term.

The general formula for all cases is here https://en.wikipedia.org/wiki/Slutsky_equation


The best introduction is, in my opinion, the book by Matthew Jackson. It has a pretty nice introduction (even if you know nothing about social networks) and chapters about many of the applications of graph theory in economics.

Jackson, M. O. (2008). Social and economic networks (Vol. 3).
Princeton: Princeton University Press.

It sounds like you try to model a game on a network (i.e., using game theory with players on a network without changing the network). If this is true, then the following paper might be even more relevant:

Galeotti, A., Goyal, S., Jackson, M. O., Vega-Redondo, F., & Yariv,
L. (2010). Network games. The review of economic studies, 77(1),
218-244.


Real exchange rate of India and United State?

Presumably one bundle of goods and services which costs about USD 113 in the United States would cost about USD 29 in India

so the bundle which costs about USD 113 in the United States would cost about INR 18546 in India at a market exchange rate of 64:1

suggesting a purchasing power parity of about INR 16.4 : USD 1

The OECD calculations of PPPs suggest about 17:1 recently for India so this is the correct order of magnitude


Approximating the socially-optimal price in a monopolist market

The following paper apparently deals with a very similar question:

  Regulating a Monopolist with Unknown Costs   In Ecnometrica (1982),
  pp. 911-930, doi:10.2307/1912769 by P. Baron, Roger B. Myerson

This and other similar papers can be found here.


What do interest rates tell us about the state of an economy?

Very little. That is because interest rates are a mechanism aimed at influencing one of many variables of an economy, rather than to diagnose it/them. Interest rates are a one-dimensional dimensional parameter, whereas the state of an economy cannot be captured with just one number.

For instance, an economy might be analyzed from the standpoint of its GDP, its Gini coefficient, income per capita, inflation, an agent's (or sector's) purchase power parity, and other aspects which are not comparable to each other. A rate simply cannot describe all these aspects.

Your question itself reflects how interest rates are used for purposes which are not necessarily --or not always-- related: to aid governmental budget, and to influence consumption. In some situations, a cash-strapped government might wish for a reduced consumption (for instance, to be able to cope with public demand), whereas in other situations it might wish for greater consumption (to increase its tax revenues). The fact that two variables in an economy don't always go -or are not intended to go- in the same direction illustrates the ambiguity of how to interpret interest rates.

It should be added that interest rates are also a mechanism aimed at procuring currency stability. This goal can be at odds with patching a budget so that the government can fulfill its agenda. Hence the importance, at least among those of us with a neoliberal mindset, of keeping government and central bank separate; that is, to ensure that the latter is an autonomous entity.


In my university the choice of
program is considered generally irrelevant. We focus on results, and it is up to each student to determine which program is best suited for the task and user preference.

You will find that using one language translates very well to another. With resources like stackoverflow, I would not be too concerned about which.

I would very carefully consider the statement "two equal universities". My experience has suggested to me such a thing does not exist, granting personal ambitions, career goals, and topic preference. I think programming choices are perhaps only a proxy for technical rigor and budget, little more.  Given you are only accepted to a finite number of schools, take time to evaluate this criteria closely.


Are corporations essential components of a free market economy?

A classic reference that is somewhat related is Ronald Coase's "The Nature of the Firm". Coase set out to address the following puzzle: since markets are efficient, why do we need to organize into firms? Couldn't we just have each individual acting alone and rely on the market to put the pieces together into complete products? Coases answer to why this wouldn't work well was transaction costs. If there is a (small) cost to engaging in market transactions then the aggregate cost of the many transactions needed to assemble complicated products will be prohibitive. The solution, argues Coase, is to form an organization (a firm) within which all of these transactions can take place without recourse to the market.

Coase's reasoning has been expanded in the 80 years since its publication. Notably, Oliver Williamson did important work to dig into the details of the conditions that make organizing work within a firm especially attractive. He argued that firms should be most likely to form when work is relationship-specific and when there are significant difficulties in writing and enforcing contracts. A great summary of Williamson's work can be found here.

Both Coase and Williamson received the Nobel prize for this work.