Calculating a precise strike time without additional details  is - in my opinion quite clearly - impossible. On the question of how to do this with additional info/assumptions:

A straightforward solution would be to make M time dependent - the larger the time elapsed, the smaller M is. Sounds like you do not want to do this.
Base problem
Let us now assume M is not time dependent, and B maximizes the present value of her payoff (assume a constant interest rate $r$) minus the cost of the effort of completing the task at time $t$ (denoted by $C_t$), so the goal of B is
$$
\max_t \frac{\text{M}}{(1+r)^t} - C_t.
$$
$C_t$ not dependent on time
If we also assume that $C_t$ is constant in time, then this is not only decreasing in $t$, but decreasing at a decreasing rate. (The second derivative w.r.t. $t$ is positive.) This means that B will either not want to lose any present value and solve the puzzle at the earliest possible time (when $r$ is quite large), or that B will not care about the loss very much and solve at the deadline (when $r$ is quite small).
B also puts money down
This is unchanged by B also escrowing (locking up) an amount of money ($\text{M}_B$), as the loss
$$
\text{M}_B - \frac{\text{M}_B}{(1+r)^t}
$$
resulting from this is also decreasing at a decreasing rate.
$C_t$ dependent on time
Let us now assume that $C_t$ is time dependent, e.g. B has to be at the movies for the next three hours and can only solve the puzzle afterwards, or the computational costs make it costly to lease additional computing power and cut down on time. If there are instances where $C_t$ increases at an increasing rate, that is $C_{t+2} - C_{t+1} > C_{t+1} - C_{t}$, then it is possible that the maximization problem will have an interior solution, i.e. the optimal time will be larger than zero but smaller than the deadline. However, one can only calculate this if one knows the interest rate $r$ as well as the cost function of B, so one needs to get quite a bit of information from B.
$r$ dependent on time
Oddly if we allow $r$ to vary, that is the interest rate is $r_t > 0$ at time $t$, not much changes. In the $C_t$ not dependent on time case, the problem will again be solved immediately or just at the deadline. If $C_t$ is dependent on time, an interior solution is possible, but to calculate the optimal time we need a bunch of additional information/assumptions.


GDP (Is this a part of GDP Calculation)?

GDP is a measure of the value of final goods produced in a country over a period of time. Final goods are those that are sold to consumers. Therefore, the intermediate step of a supplier selling to another company would not be counted in the GDP calculation. Since it wouldn't have been counted in US GDP when the US supplier was selling to the Canadian company, there is no change in US GDP when the Canadian company changes to a Canadian supplier.

Consider car production across many companies in the US (some make tires, some make engine parts, etc.) that then sell these parts to the final manufacturer. We wouldn't count the production of each part separately, we just count the final value when the car is sold to a consumer.

EDIT: Apparently, I was mistaken. What I said regarding intermediate/final goods is generally true. However, after doing a little more research on this, it turns out exported intermediate goods are an exception to this rule. The value of these exports IS included in the calculation of GDP. For more details on this, see https://research.stlouisfed.org/publications/page1-econ/2018/09/04/how-do-imports-affect-gdp. I apologize for the confusion.

Given this revelation, I am somewhat stumped by your original question. Are there any other details provided in the statement of the question?


I think the answer depends on the space $Y$ is coming from. Consider if $Y \in \{ 1, \frac{T+1}{T}+\epsilon \}$, where $\epsilon > 0$ and the probability $p$ of the better result is

$$p > \frac{T^2}{\epsilon T + 1}$$

(Which just comes from solving:)

$$1\cdot (1-p) + \left(\frac{T+1}{T} + \epsilon \right)\cdot p > T+1$$

Then no matter what utility function you have, risk averse or risk loving, you will always take the gamble (option B).

Did you mean to have some sort of restriction on $Y$ for this question? Otherwise we'd have to consider both $Y$ and $u$ to determine whether the player will take option B.


How to derive the standard euler equation from the Expanding Variety Model

The way you would go about solving this problem is as the ChinG said is by setting up the Hamiltonian. In this case this is:
$$\mathcal{H}:e^{-\rho t} \frac{C(t)^{1-\theta}-1}{1-\theta}+\mu(t)\left[Y(t)-C(t)-X(t)-Z(t)\right]$$
Taking the first order condition for this problem we get:
$$\frac{\partial \mathcal{H}}{\partial C(t)}:e^{-\rho t}C(t)^{-\theta}-\mu(t)=0 $$
or
$$e^{-\rho t}C(t)^{-\theta}=\mu(t)$$
Taking the natural logs of this equation and differentiating with respect to time we get:
$$\ln\left[\frac{\partial \mathcal{H}}{\partial C(t)}\right]:-\rho t -\theta \ln [C(t)]=\ln[\mu(t)]$$
$$\frac{\partial\ln\left[\mathcal{H}_{C(t)}\right]}{\partial t}:-\rho-\theta\frac{\dot{C}(t)}{C(t)}=-\frac{\dot{\mu}(t)}{\mu(t)}$$
Rearranging the above and noting that $\frac{\dot{\mu}(t)}{\mu(t)}=r(t)$
we get:
$$\frac{\dot{C}(t)}{C(t)}=\frac{1}{\theta}(r(t)-\rho)$$
Which is the Euler equation.
Hope this helps.


What happens if money supply grows more slowly than inflation?

There are three big reasons for this:

1) It takes time for businesses to price inflation in. As a business owner I am unlikely to raise my prices from $1.0 in year 1 to $1.04 in year 2 to account for 4% increase in the money supply. Instead I am likely to wait until I can justifiably raise the price to $1.25 or some other price that makes more sense for marketing. As a result there will be a lag between inflation and the money supply. In the long run I will choose to price money supply into my prices thus driving inflation but on a yearly basis in the short run businesses will not perfectly price in the money supply.

2) Money supply primarily drives inflation in the long term however short term price shocks can significantly affect inflation. For example if the price of oil suddenly drops due to discovery of new shale reserves in the United States that will be reflected in a drop in price levels across the entire country. For any business the cost of production, transportation, etc. has dropped which will be reflected in a drop in the price level (which dictates inflation) despite no change in the money supply.

3) Economic growth drives deflation even if the monetary base stays the same size. If there is economic growth through an increase in efficiency (i.e new technology) then the value of money will go up. If there is $100 in an economy and the only product within the country is 100 widgets which are produced every year with the past year's 100 widgets being consumed then the price of a widget would be $1. If suddenly a new type of tech allows us to build widgets for half the cost we could now produce 200 widgets in this economy using the same resources and capital stock. With the same $100 you can now buy 200 widgets or $.5 per widget. Effectively this means that the price level has gone down and that the value of a dollar has gone up. This happens in the real world through increases in technology and is the driving reason for a long run increase in the monetary base for almost all economies.


Take any $a\in(0,3)$.
Since $a>0$, X is a best response (BR) to x. Since $a<3$, Y is BR against y, and Z is BR against z.
Similarly, x is BR against X, z against Y, and y against Z.
Hence, your only pure strategy NE is (X,x). The latter is also true if $a=3$.
If $a>3$, X and x are strictly dominant strategies such that (X,x) is the unique NE.
Now back to case $a\in(0,3)$.
Suppose 1 randomizes with probabilities $p_X,p_Y,p_Z$, and 2 randomizes with $q_x,q_y,q_z$. If all those probabilities are positive, both must be indifferent over all three options. Let $\pi^i_a$ be the expected payoff of player $i$ playing action $a$.
$\pi^2_x=a, \pi^2_y=3p_Z, \pi^2_z=3p_Y$ and the $\pi^1$ look similar. You need $\pi^2_x=a = \pi^2_y=3p_Z = \pi^2_z=3p_Y$ if 2 shall randomize over all three. Hence, we need $p_Y=p_Z=\frac{a}{3}$, which is a probability as $a<3$. But since $p_X+p_Y+p_Z=1$ we also need $a\leq \frac{3}{2}$. From the same exercise for player 1, we get $q_y=q_z=\frac{a}{3}$ and also condition $a\leq \frac{3}{2}$.
Hence, if $a<\frac{3}{2}$, we have a mixed strategy Nash equilibrium in which both players randomize over all three of their actions with probabilities $q_y=q_z=pY=p_Z=\frac{a}{3}$ and $q_z=p_Z=1-2\frac{a}{3}=\frac{3-2a}{3}$. Such that both of them expect equilibrium payoff $a$. Any unilateral deviation would also lead to payoff $a$.
We also have an equilibrium in which both players only randomize over $Y,Z$ and $y,z$, respectively, all with probability $\frac{1}{2}$. In this case, equilibrium utility of $\frac{3}{2}$ cannot be improved upon by a deviation to $X$ or $x$, respectively, as $a<\frac{3}{2}$, and the other two pure strategies yield the equilibrium utility by construction.
Both these equilibria would not exist if $a>\frac{3}{2}$, as a deviation to  $X$ or $x$, respectively, would be better or the indifference condition could not be fulfilled.
For any $a>0$, there cannot be an NE in which 1 mixes between X and Y and 2 between x and y, because 2 can only be indifferent between x and y if 1 also plays Z, $p_Z=0 \Rightarrow \pi^2_x=a>\pi^2_y=0$.
Similarly, the other combinations (such as X and Z and x and z) would violate the indifference condition.


Why is it that the "right" inflation is positive?

In addition to the reasons provided by EnergyNumbers, there are possibly a few more, such as:

Deflation (even if small) is broadly considered worse that moderate inflation. This is because deflation is very difficult to fight. If consumers expect prices to fall (because of deflation) they will wait before consuming and push consumption into the future. This fall in current demand reduces prices again, which increases deflation. The cycle then keeps going on, which is a phenomenon known as the deflationary spiral. One reason deflation is bad is for the same reasons inflation is bad (e.g. menu costs). On top of that, deflation is an issue because it increases real wages, which increases costs of firms, which can have negative effects on employment. Japan has tried to fight deflation unsuccessfully for some time for example, leading to what is called "Japan's lost decade". So once you get a little deflation it may persist for a long time. Policymakers are potentially scared, that if they are always at 0 inflation that a small shock could push them into deflation, which is very bad. So having small, positive inflation is a buffer against deflation.
Many countries have estimated or continue to estimate inflation using a Laspeyres index. It is well known, that such an index has an upward bias, i.e. overestimates inflation. Let's say we want to have 0 inflation. If the Laspeyres index says we have 0 inflation, we probably have negative inflation, which is also not optimal. So measurement error provides a case for positive inflation numbers.
Inflation provides government revenue through what is called "inflation tax". Further, most governments are in debt and holders of debt profit from inflation.


Differences between Hicksian and Slutskian approaches

The income effect (IE) is about assessing purchasing-power impacts of a price change, while the substitution effect (SE) is about the impact of that price change on the relative attractiveness of the different goods. In reality these effects are not observable - when a price changes, your consumption choices will change for both reasons. But we can conceptually decompose the overall effect into IE/SE. In the SE we will be looking to isolate the impact of relative prices changing, without the purchasing-power effect.

Now, there are two ways of thinking about this:

-Slutsky: what if price changes but my purchasing power were (literally) to remain constant (i.e. I could still buy the exact same bundle as before)? How would the relative price change by itself affect my decision?

-Hicks: what if the price changes but my purchasing power were adjusted so I am still able to achieve my initial level of utility (i.e. I would be as well-off as before)? How would the relative price change by itself affect my decision?

Either way, the SE is always negative, that is, a higher price for one good will tend to make consumption of that good lower.

After you understand the SE, the IE is just the income adjustment needed in either case to get us to the actual final decision. It can reinforce the SE or contradict it, depending on whether the good is normal or inferior.

Hope this helps.


The effect of future increase in production on current market equilibrium

A key distinction is whether the good is storable.

Example 1: Milk (non-storable)

Q: Imagine that there is new technology being developed that will increase the supply of milk ten years from now. What will happen to the supply of milk today?

A: Nothing. Each supply and demand curve is specific to the time period.

Example 2: Oil (storable)

Let's assume oil producers currently have access to oil reserves that they are voluntarily leaving in the ground. That is, firms are effectively storing oil (by leaving it in the ground).

In such a case, increasing the supply of oil in the future is similar to increasing the supply of oil today. You can draw from future supply by storing less today.

Some intuition:

Imagine you have 10 cookies and were planning on eating 5 today and storing 5 to eat tomorrow.

Q: What happens to your cookie supply today if I tell you that I'll give you another 4 cookies tomorrow? Will eat 5 cookies today and 9 tomorrow?

A: No. You'll store fewer cookies for tomorrow.

If you're storing a good for the future, an increase in future supply will increase supply today.


Yes. That's tricky for newcomers to figure out. The concepts to keep in mind are:

a) to buy a bond is essentially to lend to the bond issuer

and

b) bonds are tradable.

So let's say that, in 2010, you bought a 30-year bond at \$10,000 face value that was yielding a 5% rate (known as the coupon rate) and today, in 2020, a 20-year bond is being issued also at \$10,000 par value but only offers a 4% return ($400 annual coupon).

Because bonds are tradable, that means your 30-year bond with 20 years remaining and an annual coupon of \$500 should now have buyers willing to pay a bit more than the $10,000 you paid for it.

Specifically, they can pay up to around $1.135 and still get the equivalent yield to maturity as the newly issued 20-year bond, a 13.5% profit for you.

The reverse also applies. If your 2010 30-year bond had a 4% coupon and the 2020 20-year bond is yielding 5%, you would only be able to sell the 2010 today at a price lower than $10,000.


The Credit-to-GDP, say  ratio measures the relative size of the outstanding debt of non-financial private sector, say $D_{p,t}$ with respect to (yearly) Gross Domestic Product, say $Y_t$

$$\text{Credit-to-GDP ratio} =\frac { D_{p,t}}{Y_t}$$

Note that GDP should be measured in nominal terms here, since so is debt. It is a metric related to how much burden WE have placed on our future from our current decisions and activities.

The "Credit-to-GDP" gap is, as the OP notes, "the difference of the Credit-to-GDP ratio from its trend". How do we extract that?

Look for example at the, informative with data and literature, page of the Bank for International Settlements. If you hit the "data" link, you will see that what they do is
a) calculate the Credit-to-GDP ratio (per quarter but always using past-12-months GDP in the denominator), and then decompose the obtained time series by using the HP filter.

The Credit-to-GDP gap is what it remains if from the actual "Credit-to-GDP ratio" series we subtract the (non-linear) trend as calculated by the HP filter. One can understand why then this "gap" can be negative: we are currently below the trend. Finally note that the Credit-to-GDP gap is also measured in GDP percentage units. But sometimes it may be reported as percentage deviation from the trend (i.e. "we are 30% above trend" does not mean that we are "30 percentage points of GDP above trend"). So read carefully any source.

The question is: why use the "gap" and not the ratio itself?

Because we are accepting that the trend likely reflects a sustainable evolution of the economy over time (sustainable by the very ongoing existence of said economy), while the "gap" reflects short-term tendencies that may not be sustainable and may lead to crises if left unchecked and un-managed.

A higher positive gap means that the private sector borrows at a level that is perhaps "not justified" by the current output-producing abilities of the economy. Banks may tend to experience abnormally high rates of loan defaults etc, which can lead to a banking crisis.

A negative gap supposedly implies that there is a "safe" amount of additional borrowing that could be done currently (for consumption or investment purposes), and we are leaving it "unexploited".

But never read too much on a single metric, economic systems are too complex.


How a country buy stuff from other countries? and what is its means of exchange?

In general one country cannot just print currency of other country. At least not legally, I suppose some countries could engage in counterfeiting.

A country has to go to foreign exchange market where they purchase the currency of the other country first and then make the transaction in that currency. In practice the foreign exchange market is ‘accessed’ just by making bank transfer where banks converts the money at the current exchange rate, same as you would when you order stuff from other country on internet or when on vacation.

Gold was used in international trade before modern age where gold and silver was kind of an universal currency. The only difference was in the gold and silver content of the coins so once that was accounted for, for all practical purposes, countries were basically in a monetary union - in fact Euro is often compared to gold standard see Eichengreen and Temin’s work on this  here. But this system is no longer used as it’s impractical to ship gold between countries when floating exchange rates can do the same if not even better job with just paper fiat money.


Robust Standard Errors in Fixed Effects Model (using Stata)

Use -areg- in Stata, and the standard errors will come out as in the textbook. Specifically, the command

areg lpassen lfare ldist ldistsq y98 y99 y00, absorb(id) vce(robust)

will produce the desired result.

-xtreg- with fixed effects and the -vce(robust)- option will automatically give standard errors clustered at the id level, whereas -areg- with -vce(robust)- gives the non-clustered robust standard errors. The latter seems to be what Wooldridge estimated.

Moreover, -xtreg- assumes that the number of -xtset- groups (id in your example) grows when more data is added to the sample. -areg-, however, assumes that the number of groups is fixed. I.e., the two estimators have different asymptotic properties. The point estimates will be identical, but standard errors will be different, sometimes substantially so.

Old versions of Stata (e.g. Stata 9) did not make the appropriate degrees of freedom adjustment when -xtreg, vce(robust)- was called, which is why you get a bigger standard error when specifying -version 9-. In fact, those standard errors are identical to -areg, absorb(id) vce(cluster id)- in newer versions of Stata.

As a side note, it is puzzling that Wooldridge got non-clustered robust standard errors when calling -xtreg, vce(robust)- in version 9, but perhaps I have a flawed understanding of what the call -version 9- does.

For more information on -xtreg- vs -areg-, see the blogpost and comments here.

(I recognize that this is a year-old thread, and that the question might have been answered on the Statalist. Consider my answer as "for future reference".)


Max Roser (2014) – ‘International Trade’. Our World in Data provides data on overall international trade growth during the requested time period, citing  as their sources:

International Historical Statistics (by Brian Mitchell)
Correlates of War Bilateral Trade (as mentioned in the comments)
The Maddison Project (which I love)
The Rise and Fall of World Trade, 1870-1939 (2003)
The contribution of intercontinental trade to Europe’s expansion (O'Brien 1982) (Which I could only find a discussion of)

and many other sources...

I also enjoyed Trade and Empire, 1700-1870 and Sailing Away from Malthus: Intercontinental Trade and European Economic Growth, 1500-1800

Hope those help!


Let $TR(Q), TC(Q): [0, \infty) \to [0, \infty)$ be continuous and twice differentiable functions w/ resp derivatives $MR(Q), MC(Q)$.

It is not always the case that

$$TR(Q) > TC(Q) \ \forall Q$$

If it were, we would have

$$\{Q | TR(Q) = TC(Q)\} = \emptyset$$

Even if we had

$$TR(Q) > (or \ge) \ TC(Q) \ \forall Q$$

meaning our profit is positive for any quantity Q:

$$\pi(Q) := TR(Q) - TC(Q) > 0$$

we still want to find $Q^{*}$ s.t. $\pi(Q^{*}) = TR(Q^{*}) - TC(Q^{*})$ is maximised.

Analogy: $e^x > x \ \forall x \in \mathbb R$, but $f(x) := |e^x - x|$ is not constant. Some values of $x$ give greater distances between $e^x$ and $x$ than others

This is not done by solving

$$TR(Q) = TC(Q)$$

which merely gives the quantities that give us zero profit ie

$$\pi(Q) := TR(Q) - TC(Q) = 0$$

We want to maximise $\pi(Q)$ so we get the first derivative and set it to zero:

$$0 = \frac{d}{dQ} \pi(Q) = MR(Q) - MC(Q)$$

Btw, solving

$$0 = MR(Q) - MC(Q)$$

gives us some value $Q_0$, but this is not necessarily give $Q^{*}$. We must check the value of $\pi''(Q_0)$


Math based proof for currency changes (microeconomics)

So a US consumer picks their non-car bundle of goods, $x$, with prices $p$ denominated in dollars. The price of an American car is $a$ dollars, the price of a Japanese car is $y$ yen, and the price of a dollar in yen is the exchange rate, $r$ (how many yen does a dollar buy?). If the consumer buys an American car, $z = 1$, yielding utility $u_a$ at a price of $a$, and if the consumer buys a Japanese car, $z=0$, yielding utility $u_j$ at a price of $j$, $z \in [0,1]$.

Then the consumer is solving
$$
\max_{x\in \mathbb{R}_N,z\in [0,1]} u(x) + u_a z + u_j (1-z)
$$
subject to $p'x + z a + (1-z) r j \le I$ where $I$ is the consumer's wealth.

The Lagrangian is
\begin{eqnarray*}
\mathcal{L} &=& u(x) + u_a z + u_j (1-z) - \lambda (p'x + za + (1-z)rj - I) \\
&=& [u(x)-\lambda (p'x-I)] + z(u_a-\lambda a - u_j + \lambda rj) -\lambda rj + +u_j
\end{eqnarray*}
The part in $[...]$ is the standard consumer problem: pick a basket of goods $x$ with income $I$ and prices $p$. The $z$ must be in $[0,1]$, with 1 meaning "buy American" and 0 meaning "buy Japanese". So if $u_a - \lambda a > u_j - \lambda rj$, the consumer buys American, and otherwise Japanese.

Now, if we increase $r$, what happens to the consumer's maximized utility?  The envelope theorem tells us that if we want to see how welfare varies with a parameter, we partially differentiate with respect to that, then evaluate the Lagrangian at the optimum. Let's do that:
$$
\dfrac{\partial \mathcal{L}}{\partial r} = (1-z) \lambda j
$$
So if the consumer is buying American, $z^*=1$, and this equals zero: the dollar becoming strong doesn't affect his welfare, because he is buying the American car anyway.  But if $z^*=0$, he is buying Japanese, and a stronger dollar makes it easier to pay the $j$ price denominated in yen.

If you think about the Japanese consumer, everything is the same, except $r$ is going down and not up: he is worse off if he is buying an American car, but the same if buying a Japanese car.

Now, how is $r$ determined in equilibrium?  The above problem gives you a demand curve for yen by American consumers: the ones who want to buy Japanese cars, i.e., they have a high $u_j$ relative to $u_a$.  A similar problem gives demand for American cars by Japanese consumers, i.e. those Japanese households for whom $u_a$ is large relative to $u_j$. As the price $r$ varies, people will opt in or out of the market for the cars made by the other country. Equilibrium in the currency market occurs where the flow of dollars from Americans to Japanese and the flow of yen from Japanese to Americans is equal, just like your regular Supply-And-Demand diagram. This means that if a dollar buys fewer yen in equilibrium, it must also mean that yen buy more dollars in equilibrium, and vice versa.


What does "the event of economic consequences" mean?

This text is basically about the use of assets. They can be sold once generating some cash or they can be put to use generating cash. The latter can be done multiple times (which is what the repeatability phrase is about), and in the long run should generate more cash than just selling it.
When writing that the latter is the event of economic consequence, the authors mean that this is the process that generates the most money.
Incidentally you seem to be thinking of natural assets, but the authors are thinking of man-made capital assets such as machines or buildings. Those can be put to use.


As hinted in a comment, the problem with using the difference instead of the ratio, is that the difference is not free of the units of measurement while the ratio is. And in general we want to not depend on units of measurement (whatever it is that we measure).

Although I am not familiar with the literature, I understand that here utility is a fully quantitative, "cardinal" concept. Moreover, there exists the network $G$ whose characteristics generate utility, and then there are the "idiosyncratic" losses, which is the component we want to get out of our sight.  Then why not define a $\text {PoA}$ net of idiosyncratic losses, especially since they appear not to depend on the decision making framework, and so they are the same both in the decentralized set up and in the centralized one?
I mean,

$$\text {nPoA} \equiv \frac {\sum_iU_i -\sum_i\log(l_i) }{U_s - \sum_i\log(l_i)}=\frac {\sum_if_i(G)}{f_s(G)}$$

This seems to me as a meaningful concept to reflect the network contribution to utility and how it changes under the two decision-making cases.


What kind of economic freeware exists for economics students?

I never understood why discussion about specialized software should be off-topic in the specialty's website. And of course I don't agree. So:

Before diving into R, which indeed appears to be the dominant (and rich) freeware for statistical computing, one can try Gretl. It is an Econometrics freeware, with a lot of functionality, a very good Random Number Generator, and has both menu-driven implementation but also code-writing by the user. It is easier for beginners - but it is serious stuff. The wikipedia page for Gretl lists some reviews about Gretl.

The other kind of "economics" (not mathematics) software apart from econometrics packages would be specialized simulation software, for say Dynamic Stochastic General Equilibrium Models, or micro-applications.  For DSGE, one such is Dynare.  A lot of DSGE-code for various other software can be found at International Network for DSGE Modeling, Monetary and Fiscal Policy.


Why is the risk premium always positive for risk averse individuals?

Suppose that the vector $W=\left(w_1,w_2,\dots,w_n\right)$ represents wealth in $n$ possible states. In addition, assume the probability of each state occurring is represented by the vector $\pi=\left(\pi_1,\pi_2,\dots,\pi_n\right)$. We can express this as the simple gamble:

$$g = \left(\pi_1\circ w_1,\pi_2\circ w_2, \dots, \pi_n\circ w_n\right)$$

The expected value of the simple gamble $g$ is:

$$\mathbb{E}[g]=\sum_{i=1}^{n}\pi_iw_i$$

Suppose a consumer has a sub-utility function $u(w)$. Further, suppose that the consumers preferences over gambles are such that they can be represented by a von Neumann Morgenstern utility function (has the expected utility property). The utility of the gamble can be expressed as the expected utility across all states:

$$u(g)=\sum_{i=1}^{n}\pi_iu(w_i)$$

Now, if we assume that the sub-utility function $u(w)$ is strictly concave (risk averse consumer), by the concavity of functions it must be the case that:

$$u\left(\mathbb{E}[g]\right) > u(g)$$

This implies that the consumer would prefer to receive the expected value of the gamble with certainty rather than the gamble itself. A question we could ask; how much would I need to give to the consumer such that they are indifferent between choosing the gamble and receiving some amount of wealth with certainty (Certainty Equivalent)? The certainty equivalent $CE$ must satisfy the following:

$$u\left(CE\right) = u(g) \implies u\left(\mathbb{E}[g]\right) > u\left(CE\right) \implies \mathbb{E}[g] > CE$$

Since $u(w)$ is increasing in wealth, there exists a $P>0$ such that the following will hold:

$$u\left(\mathbb{E}[g]-P\right)= u\left(CE\right) \implies \mathbb{E}[g] -P= CE$$

$P$ represents the risk premium. It is the amount you would be willing to pay to get rid of uncertainty (receive the certainty equivalent). The fact that the risk premium is positive does arise from the fact that we assume the sub-utility function is strictly concave.


Well-informed citizen can contribute better to country's economy?

how can a person from the general public contribute to country's
  economy by being well-informed?

Primarily, by promoting policies and pursuing models which the informed person identifies as premised on realistic arguments versus those that are detrimental or make no sense.

A well-informed individual on his or her own is unlikely to outweigh activism or lobbying that pursues adverse interests. Nonetheless, that individual has more elements [than uneducated people] to scrutinize, analyze, and --if need be-- dismantle others' arguments. At the bare minimum, the well-informed individual can reasonably discern what would be most convenient (whether domestically or for the economy as a whole).

  I understand one of the reason would be tax. If a person understands
  the importance of paying tax, then the person would be understand his
  responsibility and pay his tax promptly.

Not necessarily. Being well-informed is not deterministic of the rationale and conclusions at which the person will arrive. The well-informed person could reach a conclusion which is the opposite of the conjecture that he or she will become enthusiastic about taxes.

For instance, the individual could identify (as I personally do) that heavy taxation curtails a person's freedom, significantly complicates his or her financial planning, destroys incentives for productivity/innovation/self-improvement/etc., and even is ineffective toward the purposes of "well-fare" state with which politicians attempt to justify brutal levels of taxation.

Additionally, notions of utility as well as time value of money might still prompt a pro-tax advocate to delay his payment of taxes instead of paying them sooner.


Ricardian vs Non-Ricardian policy identification

I think you're misreading what Sala wrote. The first sentence in the image you linked is just saying that in the worlds where either Case 1 or Case 2 occurs, the response of $b_t$ to a shock in $s_t$ allows us to differentiate between Ricardian and non-Ricardian regimes since there are no identification issues. In other words, if either the Case 1 inequalities or the Case 2 inequalities hold, then we can be confident that we have identified a non-Ricardian regime from a Ricardian regime.

According to Sala, it seems that we cannot as easily identify Ricardian regimes from non-Ricardian regimes in Case 3 due to identification issues related to co-movements in real debt.


Relation between Nash equilibrium , subgame perfect equilibrium, backward induction, perfect Bayesian equilibrium

1- Backward induction solution is Nash equilibrium solution. TRUE

2- Not all Nash equilibria are sequentially rational. TRUE; NE allows for sequentially irrational behavior off equilibrium path.

3- All Backward induction solutions are sequentially rational. TRUE

4- SPNE solutions are Nash equilibria. TRUE

5- SPNE solutions are sequentially rational if game has at least one proper sub game. FALSE (Counterexample below: the game has one proper subgame; $(XC,L)$ is an SPNE but involves a sequentially irrational strategy $L$ by player 2). If only has improper sub game then it may not be sequentially rational. TRUE (Treat the subgame after $X$ as a game of its own.)

6- PBE solutions are Nash equilibrium and SPNE. UNCERTAIN / TRUE (in most cases); it actually depends on how exactly you define PBE (there are many definitions). See Watson (2017) for more details.

7- PBE solutions are sequentially rational no matter what. TRUE

8- WPBE solutions are Nash equilibria. And sequentially rational no matter what. TRUE


Multiproduct firms and horizontal product differentiation

Here is a link to Schmalansee's classic paper.  Use SSRI to look forward in time from this paper.  Useful search terms in Econlit or Google Scholar would be "crowding the product space" or "product space congestion."


Which are commonly used economics dictionaries?

For most part different 'schools of thought' do not have different definitions of terms. New Keynesian definition of GDP is the same as Keynesian definition of GDP or Neoclassical definition of GDP or any other 'school of thought' that you can think of. Marginal utility also has the same definition across economics.  This is because terminology is not about interpretation but about definitions and most of the time it makes sense for people to agree on one definition so everyone can understand each other.
This is because in a modern context distinguishing among schools is much less useful than in past (save for classification) since nowadays any idea that can get empirical support will eventually be accepted. For example, it is no longer even valid to say that modern mainstream is just neoclassical synthesis (neoclassical economics + New Keynesian economics) since now it comprises of behavioral economics, institutional economics etc. In fact even historically mainstream economics always absorbed ideas from everywhere (you will find Austrian influences in theories on prices, Marx is why we use $K$ for capital etc.) although less so than in the present.
When terms have different definitions it is usually not across 'schools of thought' (although there might be some exceptions) but across sub-fields. For example, the definition of technology is more narrow in the growth theory than in microeconomics.
Good economic dictionaries are:

The New Palgrave Dictionary of Economics (this is probably the most comprehensive one)
The Economist: Economics A-Z
Oxford A Dictionary of Economics

The above provide reasonably well overview of definitions used in economics. Most textbooks, especially undergraduate ones, will have 'glossary' at the end where you can find terminology as well.


Does utility in economics also refer to producer's surplus ? How to balance the consumer surplus and producer surplus?

Utility is not the same as consumer surplus. Consumer surplus is the difference between price an individuals pay and their individual reservation price.

Utility is a measure of gratification that can be completely different from consumer surplus and depending on what kind of utility we are talking about it might not even be possible to assign integer value to it. For example, with ordinal utility you can’t say some good gives you 20 utils from its consumption but with cardinal utility you can.

However, utility should be monotonically increasing function of consumer surplus so it is valid to say that at a point where total surplus is maximized also utility should be maximized.

Also producers also get utility because the higher producer surplus implies larger difference between actual price and their reservation price. Again utility should monotonically increasing function of this.


Sign of substitution and income effect of a price change

The income effect is negative for normal goods and positive for inferior goods. That is, you buy more normal goods when you are richer and less inferior goods.

In contrast, the substitution effect is negative when price increases and vice-versa. It always moves opposite to the price sign.


Decreasing Costs, Increasing Returns to Scale, & C''(q)

Second derivative of cost function is actually the first derivative of marginal cost function. i.e.
$$
\frac{\partial^2C(q)}{\partial q^2} =\frac{\partial}{\partial q}\frac{\partial C(q)}{\partial q}=\frac{\partial}{\partial q}MC(q)
$$
Now if $\frac{\partial^2C(q)}{\partial q^2}<0$, this means that marginal cost is decreasing in output. If marginal cost is decreasing then that implies that firm's average cost is decreasing and hence it exhibits increasing returns to scale.


How does inflation affect the value of savings?

It's not so grim: interest rates on savings tend to be higher than inflation, because lenders factor in expected inflation and won't lend unless they're compensated for the risks they face and the fact that they can't use the money in the meantime. So while Joe isn't likely to be lending directly, by saving money in a bank, he's lending money to the bank that the bank will then lend to others, and all Joe needs to do is make sure that his money is in an account that will pay him a decent rate of return (i.e., one that is higher than inflation).

If Joe put \$100,000 in a mattress or in a bank account that pays a 0% rate, then yes, in 20 years if inflation has risen by 1.5% annually, then the money will only be worth about \$74,000 in today's dollars. But if he saves the money in an account that pays a higher interest rate than inflation (or buys a financial instrument like TIPS, a type of government bond that pays an inflation-adjusted return), his kids will have an amount of money equal to more than \$100,000 in today's dollars.

The yield to maturity on a TIPS bond maturing in 2040 is currently about 0.9%, which isn't very high, but that's a nearly risk-free investment that is protected against inflation. If Joe bought one of these, his kids would have about \$100,900 in today's dollars. If he instead took on a certain amount of risk, his kids could have potentially much more.


Fraction of labour force providing income redistribution services

"What % of the labour force for an OECD country like the UK (mostly civil service I'm guessing) deals primarily with income redistribution, rather than income generation?"

It seems like you are talking about the   Volunteer sector .

in the UK particularly  NVCO reports in its 2012 almanac:

  What is the relative size of the voluntary sector workforce?

  The voluntary sector is a major employer. An estimated 765,000 people work in voluntary sector organisations. This is 2.7% of UK workforce, and compares with 25% who work for public sector employers, and 72% who work for private sector employers. The voluntary sector paid workforce is roughly the same size as the number employed in restaurant and catering in the UK (around 770,000).

For your second question:

  "Would a simple flat income tax at 40% and a universal basic income have the extra benefit of letting more people enter the value adding economy?"

I see no connection between an income tax and creation of jobs. Labour supply models would indicate that we  will see changes in the amount of labour supplied by a given worker in the presence of income taxes and will end up with some deadweight loss.

This is not fully explained but a clearer understanding of the effect of income taxes  on welfare will come from reading up on labour economics models.

Hope this helps.


Empirical estimates of CRRA and CARA utility

There are many estimates in the literature. For example, Havranek (2013) does a meta-analysis of avalible results and argues for a value of intertemporal elasticity (inverse of sigma in your notation) around 0.3-0.4. But it might also depend on what your goal is - the single parameter in CRRA utility controls both risk aversion and intertemporal smoothing motive, so a calibration for asset-pricing model might need to differ from let's say a deterministic growth model.


Difficult quantity theory of money question

The velocity should just be the amount of times the average unit of currency exchanges hands per year. That is $10$ if we take the highest possible number.

Money supply is $100$ units of island currency ($IC$), so that the left side of

$$MV=PY $$

turns into $100*10$ Nominal GDP,
so $YP$, is \$$100$ per Islander, or
$$\$100/XR*n $$ in $IC$,
where $XR$ is the exchange rate of $USD/IC$ and $n$ is the amount of people.

Then we get
$XR=1/10*n$.

However, I'm a bit lost for what $n$ would be in this case. I'm guessing you got this far yourself?


Link between elasticity and marginal utility

We can break this into three parts: (1) price elasticity, (2) substitutes, and (3) marginal utility.

Price elasticity measures price sensitivity (how much a change in price affects quantity consumed). In the example in the book, the product is elastic, which means that a decrease in price increases consumption proportionally more than the decrease in price.

Substitutes are any products that can be substituted for another. This is important because a change in price of a product will not only affect it's consumption, but also the consumption of substitute goods. For example, people who want soda have a choice between Coke and Pepsi. Assume that when priced the same, $10$ people by Coke and $10$ people buy Pepsi. The total consumption is $20$. Now, assume that Coke lowers its price while Pepsi keeps the price the same. Some people who once consumed Pepsi will now by Coke, say $5$ people switch to Coke. The total consumption remains the same, $20$, but its distribution is different. It is the case that a drop in price will encourage consumers to purchase Coke who wouldn't normally do so at the initial price. because a price decrease makes the product available to new consumers the total consumption of soda may increase to $25$.

Marginal Utility is the amount by which utility changes with each unit of consumption. The text doesn't seem to say this, but marginal utility is subject to the law of diminishing returns. This means that at a certain point, consuming more of a good makes a person less happy. For example, if I eat 2 cookies, I'm happy. With each cookie I consume, I grow happier in total, however this is happiness increases at a decreasing rate.

Utility is maximized when marginal utility is zero. After this point one loses...experiences disutility.

Essentially, the text is saying that if the price of an elastic good decreases its consumption will increase. The consumption increase comes from (1) "stealing" consumers from other similar products and (2) new consumers that didn't buy the product at the initial price. Since the total consumption increases as did total utility, the increase was small and thus marginal utility decreased. This isn't saying that utility was taken away, its saying the the rate at which utility increased was slower.


In my view, this is not a well defined question since it isn't very clear what you mean by 'feasible':

If you mean 'are developed countries in principle able to pay for a UBI?', then the answer is trivially yes.
Perhaps you mean 'politically feasible'? Obviously, that depends on the political constraints on redistribution, attitudes towards 'handouts', etc. - though I'm not sure this is the appropriate forum to discuss such issues.
Perhaps you mean 'A Good Thing'? Well, that's not really what feasible means...


Is there any specific distribution that is recommended for modelling individual income?

I find it awfully odd to use logistic regression since income is clearly not binary. Sometimes people use logistic to explore if people are in the "top X%" or other sorts of ad-hoc quantile regression. Using logistic seems sub-optimal to me, but perhaps passable. I would prefer quantile regression.

If everyone is employed and you have no unemployed doctors, then you have a wide range of incomes, it makes sense that the distribution would be approximately log normal.  Income distributions have heavy(ish) tails. This is commonly held belief, tied to "Gibrats Law".

Here's an interesting paper on it- carefully note they chose to argue more log normal than. It implicitly confesses of the widely held belief that both consumption and income is log normally distributed.  There are probably more basic papers, but JPE is pretty respected and this has 100+ cites and a good place to dig.

Battistin, Erich, Richard Blundell, and Arthur Lewbel. "Why is consumption more log normal than income? Gibrat’s law revisited." Journal of Political Economy 117.6 (2009): 1140-1154.


This does not directly answer your question, but I will try to explain why I don't think that such well known papers exist.

If submitted I don't think that such a paper would be accepted by the top five journals. This is because the journals also compete to stay relevant, to give surprising information. The phenomenon is known as publication bias.

This was also a big problem for the medical community. The solution they arrived at is that they only accept trials if the trial and its hypothesis is reported to the authorities at the beginning of the trial. Because of this the result will be stored no matter what it is. Even though unsurprising results are unlikely to get much publicity, a researcher looking for them can find them. The same solution cannot work for economic regressions because running a regression on the same data always gives the same result. Hence false positives are not a result of lucky samples, but of tweeking the model to particular data.


Why could interest rates go up if Brexit happens?

What would happen in the event of a "Leave" vote in the referendum? Well, the pound would quickly fall in value against its major trading partners - and some falls have already happened as the "Leave" vote appears to increase in probability. That makes imports more expensive, which is directly inflationary. Which pushes the Bank of England (the UK's Central Bank) to raise interest rates in response.

On top of that there are second-order effects which would pull in the other direction, but over a longer time period: there's the directly deflationary effect of the significant increase in unemployment. And the increased risk of further disintegration of the EU, with all the consequential risks to peace and prosperity that that would bring.

And as Alexis L. points out, the UK leaving the EU adds significant risk to the UK's currency; and that corresponds to a need for a higher return on the currency: that return is the interest rate. That's because if interest rates are too low, then people would keep selling the pound sterling, and it would keep dropping in value, pushing up inflation (requiring higher interest rates), and harming UK companies that rely on exports (creating political pressure to do something about the exchange rate - and that "something" is higher interest rates).


Measuring productivity with the Törnqvist index

The formula you have above to solve for multifactor productivity (or total factor productivity) is correct; however, after reading your post, you are interpreting it differently.

The terms with the natural logs in the formula is asking you to obtain the rate of change (growth rates) [same as the difference in logs seen in formula] of a particular period for capital, labor , and output and the average cost shares for labor and capital.     The term on the left indicates the log difference of MFP which is the same as the rate of change (growth rate) of MFP.

To answer your first question:
 If you are calculating firm level productivity you should be using firm level labor input (number of hours worked of all persons), capital input , output data.

For your second question:
The number of hours should not be used as labor input cost share. That term would be labor compensation or total wages.

The term ~s is actually the two period average cost share of labor input [cost share of labor input is the labor cost (or labor compensation) divided by the cost of both labor and capital (which should be the same as total revenue/output/sales)].

-**So your statement is correct; you should multiply the hours worked with an average base wage to get the labor cost in monetary units. total employment costs should also be the same and work. **-

For average capital cost share, the formula provided is actually showing you a shortcut in coming up with average capital cost share.   Just take the average labor cost share and subtract it by 1.

For your third question:
Capital should reflect the real value all physical assets related to the firm’s final output which should include the use of:

Equipment

Structures

Inventory- items that are in the process of being prepared for sale

Intellectual property products (software, R&D, artistic originals)

Land

These fixed production assets costs should include depreciation.

To calculate MFP growth in 1987-1988 (or 1988%)

Real output % = 4.3

Labor input% = 3.2

Capital input%= 3.9

Average labor cost share =  ( .663 + .671) /2  = .667

Average capital cost share= (1- .667) = .333 [shortcut]

4.3 – (.667*3.2) – (.333*3.9) = .87

Actual value of MFP% is  =  .8

*note that doing it this way in few periods will cause rounding issues that may show a .1 (tenth decimal) difference.  But in most periods you will be right on target.


Theoretically, the fair value of a company is based on its earnings prospects (or its expected dividends, which is related to earnings). Since not everyone agrees on those prospects, and technical issues hit (investors need to sell for some reason), stock prices fluctuate. However, the fundamental value of a company is relatively stable from day-to-day, regardless of the volume of trading. This valuation stability is quite different from your proposed lottery system, where the value is entirely based on previous purchase volume.


Optimizing Lagrangian Function Subject to 4 Input/Output Constraints:

I think what it’s referring to optimal $y$ (i.e. what would be normally in calculus denoted as $y^*$- but you already use that for the initial stock).

It makes sense that to maximize the utility MRS should be equal to marginal productivity since MRS should be equal to ratio of prices and in case like this where the person produces goods for themselves the prices will equal to the marginal costs of production which in turn will also equal to the marginal product (essentially producing for yourself is like being in a perfectly competitive market). The intuition is that person will produce good only to the extent the marginal utility of consumption becomes equal to marginal disutility of production, as if you are producing less you can increase utility by producing more and if you are producing more than that you can increase utility by producing less. At least that’s my understanding of the intuition behind this kind of model.


Why don't researchers use regularized probit model?

I have spent three days thinking about how to answer your question.  I still haven't really decided how it should be answered, so this answer is more in the way of a set of observations.  I can readily imagine a formal treatment of general linear models or limited dependent variables models could do better than this answer.

The first thing I would observe is there is probably no answer to your question.  Instead, you should consider constructing such a solution.  It is probably impossible to talk about why something did not happen.

The second is that every academic tradition tends to have traditions about what and how to use a statistical model.  There is a lot of discretion in a pretty wide range of problems.

Some of that tradition is influenced by computability and the size of data sets.  Many fields have been heavily influenced by the role punch-card computing had on a choice of solutions.  For older fields, where tables or slide rules were used, you see its influence.  At least some of the question is likely due to a field feeling the need to explore some methods more than others.

Other parts are influenced by the frequency of some type of problem.  If a field commonly runs into censoring, for example, it still influences the thought even when censoring isn't present.  Economists generally cannot perform randomized experiments and are being asked by policymakers to make decisions almost for them.  The economics profession loves to assume normality, even when it cannot be present, because it makes for tractable solutions.

Another observation is that all regularization methods map to some Bayesian point estimator under highly stylized assumptions.  That is not trivial because it biases the estimator.  The small loss in accuracy can lead to a substantial improvement in precision.

That would matter more for sampling distributions that have heavier tails than others.  The mean of the normal distribution has better sampling properties than the mean of the logistic distribution.  To see this, I created ten thousand samples of size one thousand and found the mean of each sample and plotted its density plot.  Each sample has a mean of zero and a variance of one.

The density plot of the means of the normal distribution are in red and the logistic are in blue.

The purpose of regularization is to make the leptokurtic sampling distribution of the mean for the logistic distribution become closer to the mesokurtic distribution of the mean of the normal distribution.

That,  of course, can be the side effect of a well constructed Bayesian prior distribution.  Indeed, it is one of the reasons Bayesian models are less likely to overfit.  Of course, they have their own unique problems that Frequency-based models lack.

Each of these densities is the "test" distribution given a known distribution.  Because the slope of the regression, or the mean here, is easily pulled away from the true parameter value when compared to the normal mean you can reduce the width of the test distribution by not weighting all values on the real number line as equiprobable.

So one of the likely answers would be that the advantage of regularization would produce smaller gains for probit.  Nonetheless, Stein's paradox would still hold, though in a strange form, and there would be gains.

Indeed, the entire point of Bayesian estimation is to impute real outside knowledge into the estimator.  If something is unexpected, then the weighting implicit in a Bayesian estimate pulls any point estimator away from the sample on the logic it is an outlying sample.  Of course, the potential conceit is that the researcher would be able to identify a sample that is unusual.

Regularization automates that conceit without asking the researcher if they do have actual knowledge from prior research.

There is no reason not to find a regularization for higher dimensional probit, but there is a need to do it with a high dimension logit.

It is certainly a feasible thing to do because Bayesian priors for logistic regression are discussed quite a bit in the literature.  Finding one that works well so that an analytic solution accepted by your peers can be found is quite another issue.


Maximization problem FOC and Euler equation

Taking a variable as given comes from the assumptions of the model and cannot result from any optimisation problem (think simply that when a variable is taken as given it is treated as a constant in your problem so one cannot lead to the other). Here, taking the public good as given means the household is not able to choose the level of public good to maximize their discounted lifetime utility, and instead this is taken as given (i.e. for a given level of public good I can simply choose my lifetime consumption and bond holdings path to maximize my utility). As for why this is assumed it really depends on the model you are looking at (the level of public good might be fixed (although note that in your problem there is a time subscript) or any individual might be to "small" to exert any influence on the level, etc.) but unless you are looking to endogenously determine the optimal amount of the public good the taken as given assumption is pretty standard.

As for the maximization problem, what is exactly your doubt? Have you tried setting up the Lagrangian and obtaining the FOCs to get the consumption Euler equation?


There seems to me no relationship between the general direction of
  price movements and market volatility.

Clearly the market disagrees with you. As the saying goes, "markets go up in an escalator and down in an elevator"---just take the recent mid-March correction and subsequent rebound
of, say, the S&P500 index.

Price and its properties are outcomes of market participants' trades.
Direction of price movement is determined by direction of trade and volatility by size of trade.
It is a consistently observed fact that when traders perceive that their positions are in danger of being wiped out (the perception possibly spurred by some exogenous event), they head for the exits en masse and quickly. This can also be seen during a "short squeeze", where shorts scramble to close their positions, or a bank run (e.g. the margin calls faced by Lehman Brothers, and subsequent liquidation of its positions, during the precursor to the financial crisis).
In comparison, a crowded trade usually becomes crowded gradually.

The economic explanation for this market behavior is risk-aversion.

Much of finance is built on the equilibrium relationship between the first and second moment of prices, i.e. expected price movement and volatility.
What matters is not just expected return $\mu$, but risk-adjusted return---this is precisely the Sharpe ratio $\frac{\mu - r}{\sigma}$ that arises in CAPM, where $\sigma$ is volatility.
This is because risk aversion leads to market participants putting more weight on the downside than upside. Therefore high risk (volatility) means market demands high risk premium.

The flip-side of this coin is that, in a bull market, it's the upside that is being priced and risk-premium demanded by the market is low. Therefore volatility is low and price escalates up gradually.
On the other hand, when perception of market risk changes, traders reduce their holdings as prevailing expected excess return falls short of required risk-premium.
As more and more market participants exit their long positions, this cascades and becomes a self-fulfilling phenomenon.

Price becomes more volatile on the down-leg because the market is expecting more volatility, and vice versa.

(To be slightly more technical, the Sharpe ratio $\frac{\mu - r}{\sigma}$ is proportional to the degree of risk-aversion. Assuming the degree of risk-aversion remains constant, then expected excess return $\mu - r$ is proportional to $\sigma$. Therefore, low volatility, low return---escalator. High volatility, high return---elevator.
If people become more risk-tolerant during bull market and more risk-averse during bear market, this would exaggerate the escalator/elevator contrast even more.)

The change in market perception of risk---implied volatility---can be large compared to (already dramatic, during a crash) change in price.
See, for example, the comovement of the S&P500 and VIX indices during the mid-March episode.
The drop in S&P500 is tame compared to the increase in VIX.

Further Comments

In economic theory, the fact that agents "put more weight on the downside than upside" is reflected by the stochastic discount factor (SDF). This is a standard object in asset pricing. The SDF is basically agent's state-dependent marginal utility, up to a multiplicative factor. Because of risk-aversion, in bad states of the world where consumption is low, marginal utility is high. Therefore agents put more weight on bad states than good states. A closely related object in mathematical finance is the risk-neural measure. The risk-neutral density and SDF differs by a discount factor.
VIX is the volatility of S&P500 implied by the Black-Scholes formula (one example of risk-neutral pricing). High VIX means market expects high volatility, as reflected by options prices.
The typical "volatility smile" is skewed towards the put side---one such example here. Put options  are more expensive relative to calls, i.e. it is more expensive to buy insurance against price falling than betting on price rising. This again points to the fact traders put more weight on the downside.
In a world where market participants are risk-neutral, what you say would be true. But, as discussed above, the market tells us otherwise.


You are right. The problem here is a corner solution.

Let us define the optimal combination $X^* = \beta_1e_1^* + \beta_2e_2^*$.

The first derivative gives you

$$X^* = \dfrac{l_1}{2\beta_1} $$

whereas the second gives you

$$X^* = \dfrac{l_2}{2\beta_2}$$

They can only be true "by chance". This is, the existence of an interior solution cannot be assured for every possible parameterisation allowed by the parameter space.

Actually, if you take a look at the two functions, this is clear. The cost function generate linear isoquants. You can check this here. This is because the cost function is equivalent to $c = X^2$ and $X$ is a linear combination of both $e_1$ and $e_2$.

Similarly, the value function also gives linear isoquants. This is because the function has the form $e_1 = a + be_2$.

Then, the most likely case is that the isoquants will join in a corner. The interior solution happens just by chance. Notice that when this is the case, any interior solution is optimal! The equilibrium set is then infinite.


Relationship between Interest Rates and Supply of a currency?

Maybe it will be useful for you to start:  [http://www.wisegeek.com/what-is-the-relationship-between-money-supply-and-interest-rate.htm][1]"
 As asked I summarize the core ideas, written in the link.  So, higher interest rate decreases the supply of the money:

The central bank can set the reserve percentage, which tells how much percentage of the total money of each commercial bank must be hold back. The higher this rate, the less money the banks can give to the consumers as loans.
Central bank also can set the target percentage, which is the interest rate between commercial banks or between commercial banks and the central bank. The higher this rate, the more the banks must pay when making loans among them, so the less money remains to loan to the consumers.
Central bank can set the consumer interest rate, too. The higher rate the consumers have to pay after their loans, the less money remains at themselves. So the supply of the money decreased. (And as I think, but not in the link: if the interest rate after deposits are high, more people will think that it is better for them to save their money in the bank and not spend it for something else. If it is combined with point 1., it will also decrease the supply of money.)


Should Prices (or Price Indices) be modelled with deterministic trend?

Often yes. Prices could have time trend for various reasons. For example, the most obvious one is inflation. While inflation is the increase in aggregate prices not necessary in every individual one, aggregate price level would not increase if significant number of prices would not. Thus you would expect prices to have upward time trend in many prices in any country experiencing inflation. For this reason when people analyze CPI they typically test unit root hypothesis against trend stationary (often even with non-linear trends like in Bierens 1997)
The CPI logic might not be applicable to every single individual  price, some of them might happen to be stable, but it can be extended to many individual prices.
The rationale for trend in CPI is simply that central banks target increase in CPI, so you would expect it to have positive trend because based on economic theory central banks are able to control (with more or less precision) inflation (see various rigorous models of this in Romer Advanced Macroeconomics). If on the other hand central banks would target $0\%$ inflation you would not want to include the trend.

Edit: In response to question about inclusion of drift term where there is no trend it does not lead to bias in general. To explain this consider the Dickey-Fuller test and its mechanics:
The DF test with drift is given as:
$$\Delta Y_t = \delta + (\theta-1) Y_{t-1} +\epsilon$$
Now this is at its hart estimated just by OLS. Consequently, if the model is supposed to have drift but you exclude it and estimate $$\Delta Y_t =  (\theta-1) Y_{t-1} +\epsilon$$ you will bias your estimation of $(\theta-1)$, but the converse will not hold. If you estimate  $\Delta Y_t = \delta + (\theta-1) Y_{t-1} +\epsilon$ but series has no drift then in expectations $\delta$ will just be zero and estimation of $(\theta-1)$ will not be biased. This is crucial because even though we do not care about $(\theta-1)$ per se but about value of the DF test, if $(\theta-1)$ is estimated in a biased manner any subsequent DF statistics is useless (also note running auxiliary DF test and examining if $\delta$ is significant and close to zero or not is cheap way of quickly assessing if it is important).
Now when it comes to $t$ values from DF test you have to realize that you are not using critical values based of DF distribution. Under the null the DF test distribution is non-standard so appropriate critical values have to be tabulated using Monte-Carlo simulations for the non-standard distribution that holds under the null (see Verbeek, A Guide to Modern Econometrics. pp 292).
The correct critical values for DF test with trend and with drift are given in the table below (from ibid).

The table does not include appropriate value for specification with no drift but I found that here.
You will note if you apply these appropriate critical values to the distribution of $t$-statistics of DF test from your simulations you will find that areas in the tail are virtually identical between the DF test with and without drift. The area left of green line  $t^* = -2.87$ below red dist. is very close to the area left of orange line $t^*=−1.942$ below blue dist*.
The areas wont be exactly same of course, and using DF test with drift will in marginal cases lead to incorrect inferences, so you are right into being interested in theoretical reasons for whether there should be drift or no if you want to be pedantic about it (which you should - don't get me wrong). However, this being said, whenever in doubt you should use DF with drift, because excluding it might lead to bias in $\theta-1$ itself and at that point any inferences are completely useless and it is not possible to even get right $t$-stat since $DF_t= \frac{\hat{\theta}-1}{se(\hat{\theta})}$ - if you get numerator wrong then that is the end.

*: note I modified the code to have only 500 observations so I can apply appropriate critical values. In addition, I change the series to be just random walk because there is no qualm that non-drift version is wrong if there is a drift.


Interpretation of regression coefficients with and without logs

A possible answer if it's not too late.

Q1: Yes, if you mean "per years of schooling" at the end of your question. So, the unit of $\beta x_i$ is $\$/hr$.

Q2: The unit of $\beta$ is $\log(\$/hr)/years$ literally. Then the unit of $\Delta (\beta x_i)$ is $\Delta \log(\$/hr)$, nothing strange, which is approximately the % change times 0.01. This is "dimensionless" if you like to say so, or would "dimension-irrelevant" be better? We interpret $\Delta (\beta x)$ as % change, not $\beta$.


Practical advice for calculating optimal prices

A price-response function differs from a demand curve. A price-response function appears in a model where prices are selected by the firm, instead of output. A demand function appears when firms are price takers and cannot influence the price of the product on the market. They will have to sell it at whatever the going rate is.
I will try to answer the questions but they will be very general answers, hopefully they are useful.

The price-response function can be estimated from historical data. I don't know what you mean by transaction data. But if your company is a manufacturer of stuff then transactions could mean the cost of your raw materials and intermediate goods purchases. Then it is possible to estimate a price-response function from this. But it is important to remember that the price-response you estimate is based on your own companies pricing strategy in the past. If your company has made mistakes in it's pricing before, how will estimating this price-response function help that? I think it is better to use a method that takes data from the entire industry instead and benchmark your company against that.

You don't need to do this in very much detail. Just setup your problem generally. Such as, Production Function = Y = F(x1, x2, ...) and cost function = C = G(x1, x2, x3, p1, p2, p3). Solve that problem, what do you end up with? Focus on estimating what is needed in that function in reduced form. I would think what you need is a good measure of your marginal costs.

Estimating the price-response function will certainly have error. But that depends on a lot of things: data availability, model specification, measurement error, etc. You can benchmark against other studies in your field or outside your field instead of trying to removing error completely. You can eventually compare the model's predictions against out-of-sample values.

Unless there is some interdependency between the production of these products within the factories, this should not be a problem I think. For instance, are these products made in the same factories? Does the production of one affect the other? If not, you can get probably get away with solving the problem for each product's price response separately.


Does Singapore's computation of the Gini coefficient exclude low-paid foreign workers?

More generally, how is Singapore's Gini coefficient computed?

According to the Singapore's ministry of finance (MOF):

Singapore’s Gini coefficient is typically reported based on household income from work per household member. The calculation by the Organisation for Economic Co-operation and Development (OECD) is based on the Square Root Scale. ...
... Singapore’s Gini is based on household income from work whereas data on OECD economies is based on income from all sources (which includes non-work income from investments and property).

Their full report on "Income Growth, Distribution and Mobility Trends in Singapore" has more details and you can find even more details in the sources reported therein.

To what extent is it comparable (or not comparable) to those reported by other countries?

There is no simple way of measuring 'comparability'. Generally speaking, any income distribution related cross-country comparisons are problematic and can't be taken at a face value even if the final statistics comes from the same source where the same methodology is followed (like OECD/World Bank) because even if the same methodology is followed when constructing the inequality statistics the raw data that are produced on national level might have some systemic biases as the data come from either household surveys or tax records and different countries might systematically differ in their response rate, rates of tax evasion and other sampling problems/errors (see discussion of this in Atkinson's Inequality book).
This being said despite above difficulties scientists use these statistics in cross-country comparisons (see one such example in Frazer; 2006).  So as long as you just don't take these statistics at face value they are still useful for research purposes and also just plain international comparisons. As in the Economist article you linked you can make reasonable guesses about how the differences in methodology or any underlaying data issues affect the statistics and take that into account.


How to calculate the average wage increase?

This is a very standard % change over time formula.

It is simply:
Change Between Years / Original Value

$ΔW_t$ means the change in wage from year $t-1$ to year $t$ (where $t$ represents an arbitrary amount of time). You can easily calculate that by: $ΔW_t=W_t−W_{t−1}$.

Thus it is simply saying % change over time is $\frac{New - Old}{Old}$.

Middle school math with more symbols.


How do you achieve quantative tightening?

When a U.S. Treasury bond matures the principle is repaid in money. If the owner of the bond is the Federal Reserve, it can reinvest the money in an asset like a U.S. Treasury bond. If the Federal Reserve wants to reduce the money supply the receipt of the money is coincident with a reduction in the account of the federal government. Since it is computerized they do not have to destroy any currency and coins; they change a number in a computer.


As this page states:

  International Development Association (IDA) countries are those with low per capita incomes that lack the financial ability to borrow from the International Bank for Reconstruction and Development (IBRD). Blend countries are eligible for IDA loans but are also eligible for IBRD loans because they are financially creditworthy.

Therefore the classification of countries can be representd with the following Venn diagram:

The final category, IBT, compromises IDA + IDBR, equivalent to IDX + IDB + IBD.

Note: this is my interpretation, and not based on an official definition (which I couldn't find).


After substituting for $c_{1t}$ and $c_{2t}$, the problem can be rewritten as :
\begin{eqnarray*} \max_{q_t^e} \ & \frac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} + \frac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}

which is same as
\begin{eqnarray*} \max_{q_t^e} \ & \alpha \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma}  \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}
where $\alpha  = \dfrac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma}  + \dfrac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} $

Observe that this is equivalent to solving :
\begin{eqnarray*} \max_{q_t^e} \ &  (1-q_t^e)  + z_tq_t^e  \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}

This is an objective that is linear in choice variable $q_t^e$, which is increasing in $q_t^e$ when $z_t > 1$, decreasing in $q_t^e$ when $z_t < 1$, and is a constant for $z_t = 1$. Consequently, the solution is
\begin{eqnarray*} q_t^e \in \begin{cases} \{1\} & \text{if } z_t > 1 \\ \{0\} & \text{if } z_t < 1 \\ [0, 1] & \text{if } z_t = 1 \end{cases} \end{eqnarray*}


The consequence would be confusion, if someone is further east of you it's natural to think they're ahead of time (relative to you).  Especially if you're used to the ubiquitous maps of the globe (with Europe/Africa in the middle, the Americas to the left and Japan/Oceania on the right).  Confusion then might arise in some economical contexts that are highly standardised.

For example, spot transactions on the forex markets are delivered (cleared) in 2 days.  If someone behind the date line trades on your behalf you might have to wait 3 days, or just one day for the funds to clear, depending on what side you're on.  Also in your example, think of recurring events (like quarterly reports, dividends, etc.), you might still be on the day before they're due but time-wise (just the hours on the clock) you're ahead.  Or if you're west of the date-line and waiting for events from someone east thereof you have to wait another day.

If the explanations above are confusing then that's exactly the point I'm making.  So, my opinion, if you wanted to pull off something like that without confusing people you'd also have to establish a new prevalent mapping of the globe, with Europe/Africa far to the "west" and the Americas on the far "east" side.  Of course, established terms, like far-east, western society, etc. have to be recoined as well.


What is the true source of dynamic(Pareto) inefficiency in OLG models?

Shell 1971 argues (in a ten page paper, so read it!) that the dynamic inefficiency stems from the double infinity of traders and goods, and not the dynamics. This allows us to do the Hilbert hotel switch.

  Therefore, even when all souls are able to transact business in the same
  Walrasian market, the absence of Pareto-optimality persists in the competitive-equilibrium model. This absence of Pareto-optimality is implicit in Gamow's hotel problem. An innkeeper has committed each of the denumerably infinite number of beds on a certain rainy night. A guest asks for a bed when all are occupied, but a bed can be found if the innkeeper requires each guest to move down one bed. In our little chocolate game, the imposed allocation can produce one extra chocolate. In the hotel problem, on the other hand, the innkeeper by imposing an allocation will be able to produce a denumerable infinity of extra beds.

Consider the following static setup.

A countable number of agents $i$, indexed $-1, 0,1, 2, 3, \ldots$.
A countable number of goods $j$, indexed $0, 1, 2, 3, \ldots$.
Agent $-1$ has preferences $u^i(c_0, c_1, \ldots) = c_0$.
Agent $i\ge 0$ has preferences $u^i(c_0, c_1, \ldots) = c_i + c_{i+1}$.
Agent $i\ge 0$ is endowed with one unit of good $i$. Agent $-1$ has nothing.

Then autarky (everyone consumes what they were endowed with) is a Walrasian equilibrium (even though everyone can meet at the same time). This Walrasian equilibrium is not Pareto efficient (for the same reason as always). The fact that people can meet and trade does not alleviate the inefficiency.

Note that in the static setup, introducing money cannot help (because who would be so dumb as to hold money after the static trade?).

  The fragility of intergenerational social contracts shows up in our example. If the $k$th generation repudiates the money of its elders, it can avoid passing chocolate backward. Furthermore, the $k$th generation can gain seigniorage by printing new money. Under this monetary reform the reforming generation may have chocolate passed to it, but itself passes none backward. If all souls had met in an atemporal Walrasian market, the monetary reform would have been revealed to be a nonequilibrium solution. In the actual dynamic world, "monetary reform" merely leads to frustration of the $(k - 1)$th generation's expectations. The most notable instance of such frustration is when the peasants' sons leave home for the city and fail to provide adequately for their parents' old-age.


Would the market stabilize if companies paid taxes on revenue instead of gross income?

If taxes were paid on revenue, wouldn't the money flow stabilize somewhere else? (prices would go up, and so would salaries in order for people to afford the new prices).

This sentence is unclear. If you mean to ask whether market would eventually reach some stable equilibrium the answer would be yes.
However, this is a bit of a moot point. There actually already are taxes on revenue as pointed in comments. VAT and sales taxes are de facto taxes on company's revenue even if they might not be de jure taxes on company revenue. This is because company's revenue is defined as price times quantity sold $R=pq$ so any tax that is levied on price company charges is de facto a revenue tax.
I make a purposeful distinction between de facto and de jure tax because in most tax codes you will see written that firms collect VAT/sales tax from consumers on behalf of the government. So de jure these taxes are applied to consumers, however from economic perspective it does not matter if VAT or sales tax is de jure levied on consumers or producers as tax burden is always allocated by the market (see Mankiw Principles of Economics).
The intuition why they are equivalent can be seen in the pictures below I taken from the Mankiw's book. The intuition is that whether tax is levied on consumers or suppliers (firms) it will have the same effect on new market equilibrium price and quantity supplied.

So de facto VAT and sales taxes are also revenue taxes. Also since tax burden is split by the market it does not really matter from distributional/welfare perspective if government puts this tax de jure on businesses, so there is no opportunity here to do any redistribution just by changing the de jure designated payer of a tax. This does not mean the burden of the tax is distributed equally or that even there might not be cases where only one side of the market bears the burden. The point is that the burden is determined by the market not government.
In practice the designation of who pays might have political significance. That is an economically equivalent tax on producers might be more palatable to voters to the same tax on consumers but that does not matter for economic analysis.
Furthermore, an important caveat is that sometimes it might be more practical to tax one party because they might have for example lower administrative burden. This is why VAT and sales tax are both collected by companies so in fact they pay the tax even though de jure these are taxes on consumers and de facto the tax burden is always split between consumers and producers depending on parameters of supply and demand regardless of designated payer stipulated in law.


(Mostly) ignore money for this growth issue; it's by and large a red herring that's distracting you.

Instead just think of technological progress for instance. Assume everyone is washing their clothes by hand. That takes a fair bit of time. Now someone invents a washing machine. Everyone (who can get a washing machine) will then have more time on their hand to do other stuff, like (say) invent and produce faster means of transportation, saving people time in yet another way and so on. All this is leads to improvements in efficiency (productivity).

If you apply this process of improvements to (say) agriculture, you get to produce more food with the same amount of resources, like with the same manpower and even the land being the same. Now there is a literal pie (of food) that gets bigger, which can feed more people. (The inventions would have to be appropriate, of course, like irrigation, mechanization [tractors etc.], or genetic selection/modification of crops.)

The additional people you can feed in turn can produce more stuff [until you run into limits on natural resources]. This is true even if you don't have technological progress. That's why demographic growth also matters for economic growth.

Of course, money becomes useful as a means of exchanging all this different stuff that people invent and produce, but it's not money that grows the economy, by and large, although financial services (including the basic presence of money) are also a way to make some processes more efficient. As an example of the latter, a loan allows someone to get machinery or even housing up-front, before they've produced enough stuff to pay for the item. That in turn buys them time (in their life) to produce stuff faster (using the machinery sooner rather than later) or even make babies, which would have housing where to live. I.e. financial services can speed up things which otherwise would happen slower or maybe not at all. (Do note that loans are money in the broad sense, in the modern economy; I'm not getting into details on that here; the first link explains it pretty well.)


Autocorrelated is the opposite of independent, which is a term easier to understand (and explain). If you throw one die, the probability of getting the number any number is 1/6. If you throw it again, the probability of guessing the result number is 1/6. If you do it again and again and again the probability of getting it right is always 1/6 (provided that the die is «fair»). This means that the next outcome is independent to the previous one. Whether you get a 3 or a 5, the outcome of the next «experiment» is not affected.

The opposite would be, for instance, if you have a set of 20 cards, and then you draw one, the probability that the card drawn is the one you predicted is 1/20th. But this time, things have changed: you are left with only 19 cards on your hand. So the next time, you have a probability of 1/19th of guessing the card. If you do it again and again the probability of guessing the next card is greater. Since the outcome of the previous event helps to predict the next one, we say that this process is autocorrelated.

One doesn't need to be an expert on statistics to see that these two examples are different. This is why it is important.

Now, when it comes to prices, it simply means that the price of a good today helps predicting its price for tomorrow. Take currencies: If 1€ = USD \$1.1193 today, it would be very unlikely (but never impossible) that 1€ = USD \$10 tomorrow. It is more likely that the price will be around USD \$1.1193.


Why are different currencies able to purchase different amounts?

Let EUR 1 = USD 1.17. Also, let INR 1 = USD 0.015 (these are the actual figures reported by Google on 10/2/2017)

Before we can make any meaningful comparison, we need to express the values of the dozen of eggs in a common currency. For simplicity we choose as that common currency the USD.

A dozen of eggs in Frankfurt cost USD 4.142
A dozen of eggs in Delhi cost USD 0.99

How did we get the USD equivalent prices?

It's straightforward: we multiplied the price of the commodity (dozen of eggs) denominated in some currency (EUR or INR) with the price of that currency denominated in USD.

Therefore, the price of a dozen of eggs is lower in Delhi than it is in Frankfurt, when the common currency in which the prices are quoted is the USD.

Let's turn now to the compensation figures:

wages of INR 10,000 are equal to USD 152.58
wages of EUR 1,000 are equal to USD 1,170

A employee in Germany can afford to buy 282 dozens of eggs while an employee in Delhi can afford to buy 154 dozens of eggs (divide the wage figure in USD by the price of the eggs again denominated in USD).

Therefore, if we are quoting wealth in terms of dozens of eggs that can be bought, then the employee in Germany is 1.8 times wealthier than the corresponding employee living in Delhi.

What are the contributing factors to this unequal distribution of wealth (measured in dozens of eggs-potentially-bought)?

There are three factors in play, namely

the price of the commodities quoted in domestic currency (ie the
price level in eg Germay or India)
the price of the currencies quoted in USD (the common currency we
used as a common measure)

and lastly
the level of the compensation quoted in domestic currency that each employee can command, in different countries


Acessible sources for Hierarchical Bayes?

Here are some good guides on this topic:

Bayesian Inference 2019 from Ville Hyvönen & Topias Tolonen. - this is good guide to Bayesian inference in general. They have separate chapter on hierarchical Bayes and all their explanations include worked out step by step examples in R.
Bayesian Data Analysis by Gelman, Carlin, Stern, Dunson, Vehtari, and Rubin - is a textbook that is very accessible (at least relatively) and well regarded.
Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan by John Kruschke - this is really the most introductory text on Bayesian analysis I know. It has lot of worked out examples in R, I would even say it’s written on bachelor level, but it’s very good text

———————————————————

Edit: I actually just discovered that the 2nd source Bayesian Data Analysis was made available for free by the authors in PDF form that you can access here.


is there a way in NIPA accounts to see a measure of size/change in private debt stock?

What you want is the Flow of Funds (which measures transactions and levels of assets and liabilities), not the National Income and Product Accounts, which primarily measure output and (mostly, though not entirely, physical) capital stock.


How is AVC(Average Variable Cost) ‘U’ shaped because of the principle of variable Proportions?

I obtained the definition of the principle of variable proportion from this site
It states:

... means that upto the use of a certain amount of variable factor, marginal product of the factor may increase and after a certain stage it starts diminishing.

The following is a technical derivation. See the picture at the end for an illustration of the various results.
Denote by $X$ the variable input and by  $Q = {\cal Q}(X)$ the output produced by the amount $X$. The marginal product can be expressed using the first derivative $MP(X) = {\cal Q}'(X)$. The increase or decrease of the marginal product then says something about the second order derivative $MP'(X) = {\cal Q}''(X)$.
The principle of variable proportions then states that $MP'(X)$ is first positive and then negative.

period
change
sign of $MP(X) = {\cal Q}'(X)$
sign of $MP'(X) = {\cal Q}''(X)$

1
marginal product increases
$>0$
$> 0$

2
marginal product decreases
$> 0$
$< 0$

Given that ${\cal Q}'(X) > 0$ we can invert ${\cal Q}(X)$ to obtain  $X = {\cal X}(Q)$ which gives the amount of variable input that produces $Q$ Then as the following identity holds for all $X$:
$$
{\cal X}(Q(X)) = X,
$$
We can take the derivative on both sides and obtain:
$$
{\cal X}'({\cal Q}(X))\,\, {\cal Q}'(X) = 1 \iff {\cal X}'(Q) = \frac{1}{{\cal Q}'(X)}.
$$
and taking derivatives once more gives:
$$
{\cal X}''({\cal Q}(X)) \,\,({\cal Q}'(X))^2 + {\cal X}'({\cal Q}(X))\,\,{\cal Q}''(X) = 0 \iff {\cal X}''(Q) = -\frac{{\cal X}'({\cal Q}(X))\,\, {\cal Q}''(X)}{({\cal Q}'(X))^2}.
$$
This allows us to complete the table:

period
change in $Q$
sign of ${\cal Q}'(X)$
sign of ${\cal Q}''(X)$
sign of ${\cal X}'(Q)$
sign of ${\cal X}''(Q)$

1
marginal product increases
$>0$
$> 0$
$> 0$
$<0$

2
marginal product decreases
$> 0$
$< 0$
$> 0$
$>0$

Now, let's have a look at the variable cost. If $w$ is the unit price of an input $X$,  then total variable costs is:
$$
VC(Q) = w {\cal X}(Q).
$$
Marginal costs is the derivative of the variable costs:
$$
MC(Q) = VC'(Q) = w {\cal X}'(Q)
$$
Th Marginal costs are first decreasing and then increasing.

proof: follows from the previous table as ${\cal X}''(Q)$ is first negative and then positive.

Average variable cost is variable costs divided by output:
$$
AVC(Q) = \frac{w {\cal X}(Q)}{Q}.
$$
Now let's have a look at how the  $AVC(Q)$ varies with $Q$ (which can be expressed using the derivative $AVC'(Q)$). We have:
$$
AVC'(Q) = \frac{(w {\cal X}'(Q))Q  - (w{\cal X}(Q))}{Q
^2} = \frac{MC(X) - AVC(Q)}{Q},\\
$$
Th $AVC(Q)$ is decreasing if  $MC(Q) < AVC(Q)$ and increasing if $MC(Q) > AVC(Q)$. If $AVC(Q)$ is at a minimum (or maximum) then $MC(Q) = AVC(Q)$.

proof: follows from the equation above as the sign of $AVC'$ depends on the sign of $MC - AVC$.

The following shows that for $Q = 0$, average and marginal costs are identical.
Th If $MC(0)$ exists, then $MC(0) = AVC(0)$.

proof: We have that $AVC(Q) = w {\cal X}(Q)/Q$. Using l'Hospital, when taking the limit of $Q \to 0$, gives  $AVC(0) = w {\cal X}'(0) = MC(0)$.

The following shows that initiall, for low values of $Q$, the $AVC$ will be decreasing.
Th for $Q$ small enough, $MC < AVC$, and therefore $AVC$ is decreasing.

proof, Assume not, then $MC > AVC$ in some neighbourhood $]0, Q[$ which means that $AVC$ is increasing. Also, we know that $MC$ is initially decreasing. As $MC(0) = AVC(0)$, this means that $AVC > MC$ in $]0,Q[$, a contradiction.

The following shows that the $MC$-curves cuts the $AVC$-curve from below.
Th let $Q^\ast$ be the first point where $MC(Q^\ast) = AVC(Q^\ast)$, then $MC'(Q^\ast) > 0$ , so the $MC$-curve cuts the $AVC$-curve from below.

proof: From the previous result, we know that slightly before $Q^\ast$, $MC < AVC$  and at $Q^\ast$, $MC = AVQ$ so $(MC - AVQ)' > 0$ at $Q^\ast$.  Working out this derivative, shows that $MC' - \frac{MC - AVC}{Q} > 0$. As $MC = AVC$ at $Q^\ast$, we have $MC'(Q^\ast) > 0$.

The following shows that the $MC$-curve cuts the $AVC$-curve at most once.
Th let $Q^\ast$ be the first point where $MC(Q^\ast) = AVC(Q^\ast)$ and let $Q > Q^\ast$  then $MC(Q) > AVC(Q)$ and therefore $AVC$ is increasing. As such, there is at most one point where the $MC$ is equal to the $AVC$.

proof: As at $Q^\ast$ , $MC$ is in its increasing part, we have (from the table above) that $MC(Q) > 0$ for all $Q \ge Q^\ast$,  Also, as  $AVC(Q^\ast) = MC(Q^\ast)$ and $MC'(Q^\ast) > 0$ and $AVC'(Q^\ast) = 0$, we have for $Q > Q^\ast$ close enough to $Q^\ast$ that $MC(Q) > AVC(Q)$. So $AVC(Q)$ is increasing for all $Q> Q^\ast$ close enought to $Q$.
Next, for such $Q$,  $(MC(Q) - AVC(Q))' = MC'(Q) + (MC(Q) - AVC(Q))/Q > 0$, so the distance between $MC(Q)$ and $AVC(Q)$ is increasing. this means that the distance between $MC(Q)$ and $AVC(Q)$ will grow bigger and bigger. As such, the two curves will never again be equal so $MC(Q) > AVC(Q)$ for all $Q > Q^\ast$.

The final situation can be seen in the picture below. The $MC$ curve is first decreasing and then increasing. At $Q = 0$ the two are the same. First the $MC$ curve obtains its minimum. Then in its increasing part, it cuts the $AVC$ curve at the point where the $AVC$ curve is at its minimum. After this point the distance between the $MC$ and $AVC$ curve grows bigger and bigger, so the two curves never intersect again, and both courves keep increasing.


I think you misunderstand what "electronic money" is - moving electronic money around isn't simply a matter of sending the right "codes" - it is ultimately about asking the central bank of that currency to move money around.

Sure you can open up excel and write in it "I have \$100" but that isn't USD, much as writing \$100 on a piece of paper doesn't make it a \$100 bill.

In order to lend you dollars I actually have to have some dollars to lend you.  That means I need a reserve account with one of the 12 Federal Reserve banks.  There are no electronic USD that are not ultimately in a Federal Reserve Account.

The "ultimately" in that sentence is there because banks can (and do) form hierarchies.  Only the largest ("Tier 1" banks, sometimes called "clearing banks") actually have a reserve account with the central bank in a given currency.  Other, Tier 2, banks will simple hold accounts with Tier 1 banks.  Smaller local banks may even be Tier 3.  When it comes to foreign currency dealing, you could be even further down the chain (i.e. a small bank in Australia may be Tier 4 for USD)

Because of this hierarchy a certain amount of electronic banking can be done without moving reserves.  A transfer only needs to progress up the hierarchy until it gets to a common bank between the two customers.

So for example:

A payment between two accounts at the same bank, can be done on that bank's systems.
A payment from an account at a Tier 2 bank, to a different Tier 2 bank might be done using a shared Tier 1 bank.  The Tier 2 bank's are "customers" of the Tier 1 bank, and so the Tier 1 bank can do the transaction on their system.
A payment between customers that ultimately fall under two different Tier 1 banks must be done by at a Federal Reserve bank.
The US adds an extra complication, because there are 12 Fed Reserve banks - if two Tier 1 banks don't bank with the same Fed Branch, then the Fed Reserve System also has to go "one step higher" and the New York Fed acts as the very top bank: "How do reserves move between the 12 federal reserve banks?" explores exactly how that is done.

So the problem in your scheme is you couldn't get your central bank "into" this pyramid of banks.

Your central bank's computer can show $1tr on screen but the only way to transfer to another bank, would be to instruct YOUR Fed bank to take money out of YOUR fed reserve account and put it into the other bank's account.  Your central bank can't make reserve dollars.

Aside:  An alternative way of looking at this, for those familiar with how BitCoins work, is to note that the important feature of electronic money is to prevent double spending.  BitCoin does this with the BlockChain; the USD equivalent of the block-chain is held on the Fed Reserve computers.  You can't just declare yourself to have billions of bitcoins - to spend them you have to get your transactions onto the BlockChain.  With electronic USD you have to get your transactions onto the Fed ledger.  Only the Fed has a copy of this and their say is final.  There would be no way for your central bank to change the Fed ledgers.