Theoretical economy calculation of closed country

You are basically asking for a growth model and yes there are plenty of them the two most popular are:

Solow-Swan growth model.
Endogenous growth model.

I will go over simplified version of both in order to answer how 'G' depends on time and also how 'N' plays a role but I will change the name of all variables to follow standard economic terminology and notation as your non-standard terminology is bit confusing and all over the place. Furthermore, I will present only the most simplest versions of such models, skipping some derivations for the sake of brevity, for more advanced versions and full treatment I would direct you toward Romer Advanced Macroeconomics or Barro & Sala-i-Martin Economic Growth.
Solow-Swan Model
The Solow-Swan model is currently the more dominant model in mainstream economics as the latter endogenous growth model, while theoretically appealing, is very hard to empirically test.  A long story short in this model evolution of GDP (real output - denoted by $Y$) would depend on a) exogenously given technology (denoted by $A$) growth ($g$) and also population ($L$) growth ($n$). However, living standards depend not just on total of $Y$ but on $Y$ per head (country with 100 GDP but 10 people is arguably richer than country with 100 GDP but 100 people).
The model goes as follows:
We need to start with specifying some production function as we need to have some way of determining the output $Y$. A classical example would be:
$$Y = F(K,L) = K^{\alpha}AL^{1-\alpha}$$,
where $K$ is the stock of capital and $L$ the labor which is for simplicity assumed to correspond to population. Furthermore, the production function will be expressed per unit of effective worker ($AL$):
$$\frac{Y}{AL}= F(\frac{K}{AL},1) = (\frac{K}{AL})^{\alpha} = f(k) = k^{\alpha}$$
with $0<\alpha<1$ and $f(0)=0, f'(k)>0$ &  $f''(k)<0$. The evolution of population and technology in the model would be given by $\dot{L}(t) = nL(t)$ and $\dot{A}(t) = gA(t)$. Furthermore, the evolution of $K$ is given by $\dot{K}(t)= sY(t) -\delta K(t)$ where $sY(t)$ is the proportion of output saved under savings rate $s$ and $\delta$ is the depreciation rate -i.e. the rate at which capital breaks down.
I will skip most of the derivations but under the above conditions it can be showed that the capital per person has following dynamics: $\dot{k}=sf(k(t))-(n+g+\delta)k(t)$. Given this conditions in a steady state (a state where $\dot{k}=0$) it can be shown that the evolution of capital ($K=ALk$) will be given by:
$$\frac{\dot{K}}{K} = n+g \implies \frac{\dot{Y}}{Y}=n+g  $$
The reason why above implication holds is because in this case production function implies constant returns to scale. When expressed in per capita terms the growth of $Y/L$ will simply be $n$.
Hence, your closed martian economy will grow at the rate of $n+g$ in absolute terms and in per capita terms at the rate of $n$. Hence, In this model the output/GDP of your martian economy would grow as long as combined growth of population and technology is bigger than zero ($n+g>0)$ and would shrink if the inequality would go other way. In per capita terms only the growth of technology would matter.  The stock of $Y(0)$ and population $L(0)$ as well as capital $K(0)$ (which you dont mention) that people will bring to the mars will determine the initial conditions but not the growth rate of the colony.
As mentioned by Giskard in his +1 comments the exact solution depends on assumptions about the production function. For example, if we would not have constant returns to scale there would also the $Y$ would not grow exactly at $n+g$ but generally speaking the  growth rate of total output would still be a function of $n$ and $g$ and per capita output of $g$.
Endogenous Growth Model
I wont cover endogenous growth model in such explicit detail as Solow-Swan model as endogenous growth model is far more complex and even surface level treatment would take several pages. However, I will cover some departures from the above Solow-Swan model.
In endogenous growth model the rate at which technology grows ($g$) is not just exogenously given but determined by the model. For example, according to Romer's textbook mentioned above a simple endogenous model inspired by the works of Romer, Grossman and Helpman and Aghion and Howitt would use the following production function:
$$Y(t) = ((1-a_k)K(t))^{\alpha} (A(t)(1-a_L)L(t))^{1-\alpha}$$
with the change in technology given by:
$$\dot{A}(t) =B(a_k K(t))^{\beta} (a_L L(t))^{\gamma}A(t)^{\theta}$$
Translated to plain English the first equation says that the output will be given by the share of capital and people devoted to production of goods and services   ($(1-a_k)K(t)$ and $(1-a_L)L(t)$ respectively as well as technology $A$ - this is really just modified version of production function in the first part.
The next equation in plain English says that the change of technology in time depends on how much capital and labor is devoted to production of this new technology as well as on the current stock of technology ($a_k K(t)$,  $a_L L(t)$ and $A(t)$) respectively.
I wont provide rigorous solution to this problem but the solution would show that depending on exactly what are the parameters of the model - especially the $\theta$ parameter which tells us what the effect of current knowledge and 'success' of future R&D will be. If past knowledge makes generating future knowledge easier $\theta>1$ if harder $\theta<1$ and if its effect is constant $\theta=1$. I will grossly oversimplify here but generally if $\theta<1$ the model will predict virtually the same result as Solow model. If $\theta>1$ the model will exhibit ever increasing rate of economic growth and in such case having more people will have dramatic impact on speeding up economic growth. If $\theta=1$ the population growth will still matter but the effect wont be as dramatic as in the previous case and even if population growth is zero the saving rate will play a crucial role in determining long run growth.
To sum it up, in this case depending on what are the parameters of the model we can either say that the result will be similar to the one from Solow model or that here population growth and choose savings rate will also matter for the growth of economic output in your Mars colony. As in the previous case growth would cease or output would decline if either technology or factors of production would decline.
Furthermore, again endogenous growth models have many variants but general takeaway from these models is that population growth or peoples decisions to invest in R&D can also affect the rate of per capita growth of output. Hence in this case the rate of growth of your martian colony would also depend on the number of people on mars and how many of them are engaged in 'production' of R&D or more generally how much they decide to invest resources in such endeavor.

PS: You will note that none of the models above really care about how people are distributed between different sectors of economy (your A, B, C and D) - aside from the fraction of factors of production devoted to R&D in endogenous growth model or how much money economy has. Money stock should be not confused with output or GDP and since in the long run money is considered by most economists neutral most economic growth models will not even explicitly include stock of money as it is not considered important.
Moreover, the treatment here is surface level just to showcase some different ways how you could think about the problem. For more full treatment you should refer to sources I cited and sources cited therein.


Why is money in circulation a liability of the central bank?

Balance sheets always balance, so assets equal liabilities.

Imagine a commercial bank goes to the central bank and wants cash. The central bank provides the cash, but asks for some of the commercial bank's loans (or government bonds) in return.

The central bank now has the loans (or government bonds) as assets and the cash as liabilities.

The cash is a liability, because if the commercial bank goes back to the central bank and gives back the cash, the central bank will have to give back the loans (or government bonds).

So while it's true that cash is not backed by gold, it is still backed by something. You can take your cash to the central bank, exchange it for government bonds, earn cash interest on your government bonds and then use this cash to pay your taxes. More generally cash is backed by the goods that you can purchase for it.


If you check Stata's help file on regress you should understand how to do it. Particularly pp. 16-7 have specific examples of how to apply weights.

I will edit in order to be more detailed.

gen lnyl1y=ln(y)-l1.ln(y)
xi: reg lnyl1y i.country [w=y]

Notice that if the weighted regression is done by dividing all values for observation $i$ by $\sqrt{w_i}$, then
$$\tilde{x} =
\begin{pmatrix}
\frac{x_{1,1}}{\sqrt{w_1}} & \cdots & \frac{x_{1,k}}{\sqrt{w_1}}\\
\vdots &\ddots &\vdots\\
\frac{x_{n,1}}{\sqrt{w_n}} & \cdots & \frac{x_{n,k}}{\sqrt{w_n}}
\end{pmatrix}
$$
Call the diagonal matrix composed of $\sqrt{w_i}$ in the $(i,i)$ element $\sqrt{w}$. Then, the WLS estimator is given by
$$
\hat{\beta}_{WLS}=\left(\tilde{x}'\tilde{x}\right)^{-1}\tilde{x}'\tilde{y}=
\left[(\sqrt{w}x)'\sqrt{w}x\right]^{-1}(\sqrt{w}x)'(\sqrt{w}y)=(x'Wx)^{-1}(xWy)
$$
as $\sqrt{w}'\sqrt{w}=W$ and $\tilde{x}=\sqrt{w}x$ which is exactly the same as multiplying each row of $X$ and $y$ by the sqrt of your weight (analytical weight, here $\sqrt{w}$).

The thing here is to understand how your matrix relates to applying the weights and understanding the connection between your formula and the one which is in the help file. For other ways to calculate weighted regressions in relation with Stata, see e.g. this document


Here is a proof by induction on $n$, for any $(n,m) \in \mathbb{N}^2$.

We will need the following preliminary result, which strengthens the assumption on $\succeq$:

Lemma
\begin{equation*}
\forall x,y,z \in \mathbb{R}, x \succ y \Rightarrow x+z \succ y + z
\end{equation*}

Proof
Suppose that $x+z \preceq y+z$. By your assumption on $\succeq$, we can add $-z$ to both sides of this inequality (since $-z \in X=\mathbb{R}$) and obtain
\begin{equation*}
x+z+(-z) \preceq y+z+(-z)
\end{equation*}
i.e. $x \preceq y$. This proves the lemma (by contrapositive).

Now, let us prove the main result.

Basis ($n=0$)

We need to prove that $mx \succeq 0$ for any $m \in \mathbb{N}$. This part can itself be proved by induction on $m$. It is trivial for $m=0$ and $m=1$. If the property holds for $m$, then
\begin{equation*}
(m+1)x = mx +x \succeq mx + 0 \text{ since } x \succeq 0
\end{equation*}
Therefore $(m+1)x \succeq mx$, and since $mx \succeq 0$ by the induction hypothesis, by transitivity we obtain $(m+1)x \succeq 0$: the result holds for $m+1$. This proves the result for $m \in \mathbb{N}, n=0$.

Inductive step

Let us fix $m,n \in \mathbb{N}^2$ and suppose that $\dfrac{m}{2^n}x \succeq 0$. Let us show that $\dfrac{m}{2^{n+1}}x \succeq 0$. To this end, we proceed by contradiction and we assume, otherwise, that $\dfrac{m}{2^{n+1}}x \prec 0$. Our lemma yields
\begin{equation*}
\dfrac{m}{2^{n+1}}x + \dfrac{m}{2^{n+1}}x \prec 0 + \dfrac{m}{2^{n+1}}x
\end{equation*}
and therefore (computing each side of the inequality)
\begin{equation*}
\dfrac{m}{2^{n}}x \prec \dfrac{m}{2^{n+1}}x
\end{equation*}
By assumption $\dfrac{m}{2^{n+1}}x \prec 0$, and we just showed that $\dfrac{m}{2^{n}}x \prec \dfrac{m}{2^{n+1}}x$. By transitivity we obtain $\dfrac{m}{2^{n}}x \prec 0$. This is a contradiction, since the result was supposed to hold for $n$.


Transparent Fractional-reserve banking as an alternative to the current system

As mentioned by @BrianRomanchuk which system is "better" is matter of opinion as that includes value judgement. For a radical ascetic a system which forces people to live on a subsistence level is the best system possible.  However, what you propose is most likely not optimal from economic perspective.

What you propose here is basically transitioning to some sort 100%  fractional reserve system. There have been proposals for such system in the past - it is called 'the Chicago plan'. The idea first gained some traction in 30s at the University of Chicago (hence called the Chicago plan) and few years back the plan generated a bit of a buzz when Jaromir Benes and Michael Kumhof (IMF economists) published their paper Chicago Plan revisited where they tried to revive the idea. However, the idea was not really embraced by the profession. There are several reasons why.

You are right that if there would be 100% reserves and banks could lend only money that are in investment accounts (what you call setting aside), it would prevent the chance of bank runs (at least for the deposits itself). But evidence shows that deposit insurance by government does this equally well. You might not know that but in most developed countries including US all banks are by law required to have insurance for their deposits. Hence, even though your deposit money are technically being lend out you will not loose them if bank goes bust (at least up to certain limit, in the US the limit on deposit insurance is \$250.000, see here). Hence unless you are very high up in the income distribution you never have to make bank run as your money is safe, only if you are quite rich you might worry about it, but if you are rich enough to have such balance on your deposits there are other ways that you can protect yourself from this, and generally you would probably have some wealth manager who actually invests money for you. The only advantage of Chicago plan would be that there are always some 'shadow banks' that try to get around the deposit insurance rules by disguising themselves as other types of financial institutions, but it is generally believed this problem can be controlled by appropriate regulation.
Fractional reserve system has many advantages relative to the full reserve system. In a fractional reserve system money supply can expand and contract more flexibly based on what is the demand for loans in the economy and consequently government through the central bank can easily manage money supply just by setting interest rate instead of actually changing monetary base. However, in your system that would no longer be possible and central banks would have to manage money supply through changes in monetary base which is much harder to do with the same precision as when you just play with interest rates. Managing quantity of money in economics is extremely important for mitigating the boom-bust cycles. Hence, even if we would agree for the sake of the argument the Chicago plan could help mitigate some financial crises it could make non-financial crises more difficult to cope it.
Under the 100% reserve plan there would be, in  the most likelihood, an overabundance of sterile deposits. Money that are just deposited but never lent out to help finance investment which is extremely beneficial for firm creation, job creation and economic activity in general. In such system firms would have tougher time to invest into new factories R&D and so forth as most national savings would not be put into productive use.

Hence, generally mainstream economists are not very cordial to the 100% reserve system and it is believed such system would be worse for the economy as it has some serious drawbacks and only some tentative possible advantages.

The only country brave enough to try to implement such plan was Iceland, but to the best of my knowledge even though they announced in 2015 they want to implement it they did not really do it yet. According to most recent source I managed to find on their reform from 2016 it is still being reviewed by the committee set up to examine such system.

However, let me reiterate that again what system is a best is a matter of opinion. A system that delivers the highest possible level of material welfare might not be the best system for everyone. You might oppose fractional banking on moral grounds etc. but most of the profession supports the fractional reserve system, or at least does not really endorse any alternative as better. In fact as an undergraduate economist the only other monetary system you will hear about is the gold/silver standard and even that probably in history class - by this I am just trying to show that among mainstream economists fractional reserve is really taken so for granted that alternatives are often, for better or worse, considered not worth to even mention to undergraduates.


Why do businesses care about rate(s) of change?

In economics, a firm wants to maximize profit. Your ice cream shop has some costs that vary depending on how many half-gallon ice cream packages are made, and some benefits, the revenue from selling ice cream. The additional cost of making one more package of ice cream is called the marginal cost, and it can change based on how much ice cream you are already making. The marginal benefit of course the price that you sell each package for.

In order to maximize profit, you have to choose some quantity of ice cream packages to produce. You do this by setting marginal cost equal to marginal benefit, and then solving for production. So if you have a formula for what the price of ice cream will be over time, then you can predict your marginal benefit  of production over time, and can anticipate what your production will look like in the future.


Change in price and income using the Slutsky equation

The question is not asking what you solved. You literally gave how much income would be needed by buy $90$ bottles at the new price. What the question is asking is basically "calculate the income effect, and add it to the original income, in order to cancel it out".

That is, you solved for what he would have bought if there was no income or substitution effect, but the answer wants to know what he would have bought if there was no income but still a substitution effect.

How the slutsky equation comes into play? That formula you gave is the income effect. The difference of two bottles is the substitution effect.

From here on is an attempt at providing intuition that may be wrong in some aspect. I would not read it unless the above paragraphs did not make sense.

At original prices and income you want to buy $90$ bottles. At the new price and income you want to buy
$$
.02 \cdot 7500 - 2\cdot 40 = 70\text{ bottles}
$$
you the quantity demanded has changed by $20$. That is,
$$\frac{d x}{dp} = 20
$$
(counting a ten dollar price change as small, which is probably not the best assumption).

The slutsky equation says that we can decompose this change in quantity demanded into an income and substitution effect. That is,
\begin{align}
\Delta x \text{ price change } = \Delta x \text{ substitution effect} + \Delta x \text{ income effect}
\end{align}
Our income has decreased by the number of bottles we were buying times the change in price (This is important to understand, IMO). Think about it, if you were buying $90$ bottles, and the price went up by $\$10$, now you are paying $90\cdot 10 = 900$ dollars more. Therefore, after the price change it is like we are facing prices of $\$30$ and income of $7500-900=\$6600$, if we are discussing the income effect. That is, if we are discussing the idea/thought/alternate reality where, instead of prices changing, we had prices be the same but instead our income decreased by $900$ dollars, we would demand
$$
6600\cdot .02 - 2\cdot 30 = 72
$$
Which means that we have a substitution effect of $2$ and an income effect of $18$. So to cancel the income effect, we would need enough original income to buy $70 +18$ bottles, which is $8400$ dollars.

Now how/why these decompositions work I am still trying to better understand myself, so my reasoning may be off or wrong somewhere. There are also different ways to consider/define the income/substitution effects, which is something to take into consideration.


Introductory or Primer text on Economics?

If you were to read only introductory book in econonomics:
Economics: The User's Guide by Ha-Joon Chang.
It explains all basics concepts in economics, and the differences between the main school of thoughts. It is quite easy to read, it is not a textbook.

For an overview of financial markets:
Capital Ideas, and Capital Ideas Evolving by Peter L. Bernstein

For your interest in electricity:
The Grid: A Journey Through the Heart of Our Electrified World by Phillip F. Schewe, very interesting on how the grid came to be.

Smart Grid: Integrating Renewable, Distributed & Efficient Energy
by Fereidoon P. Sioshansi (Editor), Sioshansi is a reference in Smart Grid and the paradigm change from consumer to prosumer.

Renewable Economics: How to compare the costs of different technologies:
Energy Technologies and Economics


How can a company run out of cash when its stock prices is dropping?

It appears to me that it is the other way round: The RBS was running out of cash which is why the stock price was dropping.

Stocks usually don't affect the immediate operation of a company, since they are traded on secondary markets (stock exchanges) among stock owners, not bought from / sold to the actual company which issued the stock.


It's been a while since I've look at perfect competition and cost curves.

First off, it's helpful if you draw the standard diagram for a perfectly competitive firm, one with supernormal profits and the other with normal profits (and compare it to the 3 types of situations in Johns diagram)

From what I do remember, Long run equilibrium under perfect competition consists of:

1) P = MC = MR (profit maximisation)
2) P = Min ATC

Remembering that the market is the price maker, and the individual firm (johns farm) is the price taker.

In the long run, market price (indicated by demand and supply intersection) must be level with the lowest point on the LRATC curve (also lowest point on SRATC-2). This doesn't appear to be the case in these diagrams, thus it must be a short run equilibrium.

So in order to adjust to this long run equilibrium two things need to happen

1) Either demand for apples falls, or supply of apples rises in the market for apples. To bring about a price consistent with the long run equilibrium

2) John needs his ATC to move from SRATC-1 to SRATC-2. (Because any perfectly competitive firm not at its minimum LRATC will, in the LR, change its input combination to take advantage of lower average costs).

According to Johns apple farm, the firm is earning abnormal/supernormal profits because of the position of price on SRATC-1 i.e it is not the case that P=min ATC=AR. There is a distance between ATC (higher than minimum ATC on SRATC-1) and AR. This distance is abnormal profits.

If firms are able to make positive profits by moving outward on the LRATC curve, those profits will attract entrants into the industry in the usual fashion (shifting the supply cure outward). Profits will eventually dissipate to zero.

In the long run, a perfectly competitive firm is allocatively and productively efficient: P=MC & ATC=MC i.e lowest point on SRATC-2 and LRATC.

In regards to the high medium or low level of capital, because there are 3 options and 3 SRATC curves, I believe it is medium (greater than the quantity associated with SRATC-1, and lower that then quantity associated with SRATC-3).


Why is there Ricardian equivalence in Ramsey's model, but not in OLG?

Ricardian equivalence essentially means the timing of taxation doesn’t matter. Ramsey models are typically infinitely lived agent (or can be formulated as such) models with no heterogeneity. OLG models have age heterogeneity, so if people live for k periods an agent born in period 0 receiving a transfer in period t < k to be financed by a tax in period t+j, t+j > k, they don’t live to face the tax and the transfer is a permanent increase in income. Similarly, a person born in period k does not receive the transfer and views the tax as a permanent decrease in income. In an infinitely lived agent model (here a Ramsey model), the agent will always live to see both the transfer and the tax.


How do you formulate a distance constraint and a budget constraint?

Because you are talking about constraints, it appears you do not consider the case of inserting such "access costs" (because this is what they are) in the utility function. It implies that they do not create disutility directly, only direct or indirect monetary costs.
Let $d$ be the distance in some units. Let $C_d = c_dd$ be a linear (for simplicity) distance cost function,  where $c_d$ is the average cost per unit of distance. This cost may include fuel consumed, or opportunity costs of spending the time to go to the shop. It is realistic to also postulate the existence of a strictly non-zero minimum distance, $d_{min}$, because the shops are not inside your house.
Your budget constraint now looks like (the undefined symbols have the usual meaning)
$$\sum p_ix_i + c_dd \leq I,$$
while you also have the additional constraint $ d \geq d_{min}$. So the Lagrangian is here
$$U(\mathbf x) + \lambda \left(I-\sum p_ix_i - c_dd\right) + \mu(d- d_{min}),$$
with $\lambda, \mu$ being the non-negative Karush-Kuhn-Tucker multipliers. Your optimization problem involves maximizing the Lagrangian with respect to the $x_i$s and with respect to $d$.
One could specify reasonably a convex non-linear distance cost function. Finally, one could include $d$ in the utility function as generating directly disutility.
I guess you can take it from here.
Just note that with the formulation as above, you should obtain that at the solution, $\mu^* >0$ meaning that the constraint will be binding and so you will always choose to go to the nearest shop... which aligns, at least in this case, with intuition (don't start thinking about shop/product differentiation, that would be a whole different situation).


Understanding the effects of government deficit on international trade

No, your logic doesn't make a ton of sense. That doesn't mean that the Trump tax cuts are a good thing.

Foreign debt isn't the same as a country removing money from the economy. The way the U.S primarily finances debt is through the sale of treasury bonds (T-bills). Essentially an agreement with a private individual (sometimes foreign) that over the course of X time period the debt will be paid back with interest.

This is the same as making a loan with your local bank. They give you money, in this case $5 trillion and after a period of time you pay off the principal loan plus interest payments. The assumption that Trump (and all American presidents since Bill Clinton when we last had a current account surplus) is making is that in the period between making the loan and paying it off the economic stimulus from the tax cuts will outweigh the loan and its payment. The U.S gets extremely favorable terms on its loans with virtually 0 interest rates meaning that the 5 trillion stimulus in decreased taxes may very will increase tax revenue through increased economic activity by more than 5 trillion dollars which is the argument that Trump is making.

Whether or not this occurs depends on where we are on the Laffer Curve which essentially states that at a certain tax rate decreasing taxes will actually increase government revenue by stimulating businesses. Personally I, and many contemporary economists believe that we are not at that point on the Laffer curve and a tax decrease would in the medium run lower output, however clearly Trump disagrees and there is no true consensus among economists as to where we currently fall on the Laffer Curve.


What's the difference between a "consequential" and a "direct" externality?

According to the examples given in the paper, direct externality results from, among other things, unsolicited promotions, while the sources of indirect / consequential externality are price discrimination, etc.

In this sense, the externality on the individuals (whose information is collected and used) is deemed direct if the information about them is used directly back at them (in the form of targeted advertising, for example). In contrast, indirect externality only occurs if the individuals choose to engage in transactions with the firm which has (collected or purchased) their information, and that these transactions make them vulnerable to price discrimination, etc. In other words, externality of this second kind occurs consequently.


The first equation can be written as:

$$ r_E(Levered) = \frac{E+D}{E}r_E(Unlevered) - \frac{D}{E}r_D $$

Then, isolating the unlevered return gives:

$$ r_E(Unlevered) = \frac{E}{E+D}r_E(Levered) + \frac{D}{E+D}r_D$$

And this is the WACC.


Are goods that are wasted counted into the GDP of a nation?

It may depend on when they are wasted:

yes if a consumer purchases them, as it does not matter whether they are eaten or thrown away rotten from the consumer's cupboard (except on the consumer's waistline and perhaps therefore on other purchases);
no if a producer fails to sell them and writes them off as they rot in the producer's stores, as they have a value of zero
probably not if a retailer buys them from the producer but writes them off as they rot in the shop, as they count towards GDP from the producer's accounts but negatively towards from the retailer's accounts (either as negative profit in the GDP income calculation, or as consumption of intermediate goods in the GDP production calculation, or simply not counted in the GDP expenditure calculation)


Given a universe of choices $X$, subsets of alternatives $A \subseteq X$, and choice probabilities $\Pi_{A}(x)$ for item $x \in A \subseteq X$, the Block-Marschak polynomials can be defined recursively as (see McFadden, Revealed stochastic preference:
a synthesis, 2005) as

$K_{x, \emptyset} = \Pi_{X}(x) \,, \quad K_{x, A} = \Pi_{X\setminus A}(x) - \sum_{C \subset A} K_{x, C}$

When the choice probabilities are the result of utility maximization, Barbara and Pattanaik (Falmagne and the rationalizability of stochastic choices in terms of random orderings, 1986) provide a useful and intuitive interpretation of $K_{x,A}$ as the probability of the event that $x$ is ranked behind the elements of $A$ and ahead of all the remaining elements in $X \setminus A$.


This question overlaps with the question: link to question on quarterly/annual GDP.

Since you are referring to news commentary, I just want to add that the standard in the United States is to quote annualised quarterly GDP figures (e.g. 2.0% annualised roughly equals 0.5% on the quarter), while European countries (e.g., Germany) often quote the non-annualised quarterly growth rate. (These are the conventions used by the national statistical agencies when quoting growth rates, and the business press usually follows the local convention when reporting on the results.) This difference sometimes can cause confusion in international comparisons.


Why doesn't the trade balance equation take into account the value of products sold by the U.S.?

If foreign countries suddenly bought $\$1$ trillion worth of stuff, presumably the US would supply this out of existing inventories (i.e. stocks of goods).

Before anything else happens, this represents a reduction of investment in the US and your equation would continue to balance since the increase in $NX$ would be offset by a reduction in $I$ (which has a negative sign in your national accounts identity).

What would happen next is probably that the manufacturers would make more stuff to bring inventories back towards previous levels, increasing investment. They and their workers may pay taxes on their income.  They may import some more stuff.  What is left may add to savings.  But throughout this process the national accounts identity will remain true.


They are related, but not the same thing.

The yield associated with an asset is the ratio of the (usually annual) return produced by that asset to its price: $$\text{Yield (%)}=\frac{\text{Annual Return (\$)}}{\text{Price of Asset (\$)}}.$$

Simple case: when yield = interest rate

An interest rate is an example of a yield. For example, suppose a bank account pays an interest rate of x% p.a. That means that \$100 deposited in the account today would become \$(100+x) in a year's time. The annual return is therefore \$x. We can plug these numbers into our formula to calculate the yield: $$\text{Yield}=\frac{\$x}{\$100}=x\%.$$ In this example the yield is just the interest rate, so the two coincide.

More complicated case: yields and asset prices

Things get a little more complicated when the interest rate is fixed but the price of the asset can change over time.

For example, the government issues bonds when it needs to borrow money. Each bond has an initial price and a set annual return (known as the coupon payment) that does not change over the entire life of the bond. Suppose that we have a bond that initially costs \$100 and has a coupon payment of \$5 (meaning that you get \$5 for each year that you hold the bond). We can check the bond's coupon yield: $$\text{Coupon Yield}=\frac{\$5}{\$100}=5\%.$$
This is, again, just like an interest rate.

Now, though, suppose that banks start offering an interest rate of 10% on an ordinary savings account. Suddenly, your bond's 5% doesn't look so great! But the \$5 coupon payment can't change—it is permanently fixed when the bond is created. People are therefore going to want to try to sell their bonds to get their money back so that they can put it into a bank instead. As per normal supply & demand logic, lots of people selling bonds is going to make the price of bonds fall. So now you can buy a bond for, say, \$80. That means the yield has changed!
$$\text{Current Yield}=\frac{\$5}{\$80}=6.25\%.$$
Alas, people are still going to be reluctant to buy the bond because it still has a lower yield than the bank account—so the price is going to have to fall even further. Eventually, the price of the bond falls to \$50, at which point $$\text{Current Yield}=\frac{\$5}{\$50}=10\%.$$ Now the bond and bank account have the same yield (meaning that someone with money to invest would get the same return from either). People are now indifferent between holding the bond or a bank deposit and equilibrium is restored.

In practice, things are a little more complicated because people compare the returns on bonds to all kinds of assets—stocks, real estate, etc. But the principle is the same: the price of the bond adjusts to ensure that the yield on the bond is in keeping with the risk-adjusted return available elsewhere.

A final note: yields where there are no interest payments

The last important reason why we distinguish yields from interest rates is that some assets don't have any interest rate to speak of, but it nevertheless makes sense to talk of a yield. One example is stocks—there is no interest payment on such assets, but the board may decide to return some of the firm's profits to investors in the form of a dividend. A dividend is an amount that is paid to investors for each share that they hold. We can use the dividend to calculate a yield just as though it were an interest rate:$$\text{Dividend Yield}=\frac{\text{Dividend (\$)}}{\text{Stock Price (\$)}}.$$


Why do lump sum transfers affect prices?

Prices (and hence price ratios) carry information about what people want, and what is supplied.

You're only changing initial endowments, hence you're not affecting what is supplied. However, as you change the initial distribution, you change what each individual wants, and possibly (not necessarily) also what the economy as an aggregate wants. This change in wants and haves is reflected in prices.


Paul Samuelson wrote a series of papers formalizing Marxian economics, e.g.
https://www.sciencedirect.com/science/article/pii/B9780123567505500177


What's the difference (if any) between "fixed assets" and "fixed capital" in the UN SNA?

Put on your institutional hat. And make a distinction between fixed assets, like land, and the ownership of fixed assets: capital. See the European System of National Accounts 2010 p. 73.


As far as I recall, you're dead on: they don't have to. The contracts don't allow for "some countries kicking others out of the Euro Zone", they never envisioned this scenario when writing them.

On the other side, Greece can, and should, leave on its own. Sharing the currency is hurting them economically, the only reason they're not is that the continuous inflow of loans is preferred over leaving the zone. But if Greece really defaults, I don't think there will be any more lending in the short run. And then really there is no reason for Greece to stay.


A standard ("vanilla") interest rate swap is a contract in which one party pays a fixed rate of interest, and the other pays a floating rate (typically LiBOR, but it depends upon the market convention in the currency). (There are a lot of interest rates that deviate from this formula, but they have the same "system risk" characteristics as standard swaps.)

As a simplification, assume that the floating rate of interest is paid annually.

If you receive fixed, each year, you would net receive:

Net receipt = (contracted rate of interest - the "swap rate") - (latest 1-year floating rate).

There is no notion of "default" in the contractual payments, unlike a credit default swap.

From a risk perspective, this can be viewed as owning a fixed rate bond, and borrowing 100% against it at the floating rate. There is obviously leverage, which poses risks (see below).

The only way a default can affect you is if your counterparty fails. All that you lose is:

whatever value of future payments you would have received (the Net Present Value), that is not covered by collateral.
You lost the interest rate risk exposure that the swap provided. If that swap was a required hedge, you need to find a new hedge.

The normal procedure is that the net NPV exposure is covered by collateral; an entity would have have to be in a weak bargaining positiom (or badly informed) to enter into derivatives trading without a collateral agreement.

During the Financial Crisis of 2007-2008, I believe that it is safe to say that there was very little NPV losses that were incurred by inadequate collateral. That is, losses due to "defaults" was a very small number (compared to the other types of losses). The losses that were associated with interest rate swap trading was the result of needing to replace hedges that were lost when Lehman (and others) defaulted. (Not all of the problems that occurred during the Financial Crisis were documented, but I am unaware of issues in the interest rates swap market being a trigger. Once the crisis was underway, the swaps markets were dislocated, like most other ones.)

Otherwise, the movements in interest rates were not enough to trigger a crisis in 2007-2008. The source of the crisis was funding/credit risks.

You would need to go back to the 1994 bond bear market, or the LTCM Crisis of 1998 to have a historical event where interest rate swaps were a factor in the crisis.

In the case of 1994, there were very large movements in interest rates, coupled with risk management systems that did not really know how to measure interest rate risk. Since 1994, interest rate risk measurement is much better understood at any credibly managed investment firm or bank. (The comments about risk management are based on my knowledge of interest rate risk management that I gained from working in fixed income from 1998-2013. They are the result of my discussions with market participants who were involved with investment management during 1994. You would need to track down a paper on the evolution of interest rate risk management to get a formal reference.)

Whether a situation similar to 1998 or 1994 could happen again is an open question that could not expected to be answered here. Regulators are certainly aware of what happened then, so they could be expected to take actions that would prevent an exact repeat.


Network goods in competitive markets vs. socially optimal network size

There are two sorts of network effects a network good might have. It might have consumption externalities (such as with traffic) or it might have positive network externalities (such as with operating systems). The latter case is the case that your link discusses. I don't see a discussion of perfect competition in your link, but if you imagine a model in which consumers choose a level of some good with a positive network externality to buy at some price, equilibrium production should be less than socially optimal for the same reason that there is underproduction in perfect competition with any positive externality -- at the margin, consumers don't take into account the value buying the network good brings to other consumers.


Euler equation in Continuous time VS Discrete time

One way to see (intuitively) the connection between the left hand sides is to write the discrete case as:
$$
\frac{u'(c(t + \tau))}{u'(c(t))},
$$
for $\tau = 1$. Now if we generalise this to a setting where $\tau$ is now a variable in $\mathbb{R}$, this becomes a function of $\tau$. Taking the derivative with respect to $\tau$ and evaluating at $\tau = 0$, gives:
$$
\frac{u''(c(t))}{u'(c(t))} \dot c(t).
$$
which is the left hand side for the continuous case. As such the left hand side for the continuous case can be seen as the "derivative" of the discrete setting (when allowing the time interval to go to zero).


The law of demand is a microeconomic law that states, all other factors being equal, as the price of a good or service increases, consumer demand for the good or service will decrease, and vice versa.

Now, when you say that "if demand increases then the price of the good will increase"", you aren't changing the price and based on the change in demand you are now predicting that the price would rise which is clearly against the law of demand as your "increase" in demand is obviously due to more user engagement, advertising, or some other external factors.


A very nice survey from a now well-established journal is

Olken, Benjamin A., and Rohini Pande. "Corruption in developing countries." Annual Review of Economics 4, no. 1 (2012): 479-509. Here is the ungated NBER version.
It is not new and focuses on developing countries, but it is very influential. They try to answer three questions:

what is the degree of corruption?
What are the consequences of corruption in terms of efficiency?
and what determines the level of corruption?


Using the Federal Reserve's Implicit Price Deflator to Convert to 2020 Dollars

If you want to change the base year of a deflator just divide all values by the value in base year and multiply by 100. For example, if you have deflator with base year 2012:
           2011  2012 2013 ... 2020
Def_2012:    87   100  101      112

Then deflator with base 2020 will be given as:
               2011             2012           2013     ...     2020
Def_2020:    87/112*100   100/112*100      101/112*100      112/112*100

This way you you rescale the implicit deflator to 2020. Afterwards you can just apply the deflator as you would normally do.
Edit:
In order to convert all values to 2020 you can use the following formula:
$$X_{2020} = \frac{X_{t}}{\frac{\text{deflator}_{2020}}{100}}$$
where $X$ is the nominal variable you want to adjust and  subscripts indicate the years with $t$ being an arbitrary year.


I'm going to stick my neck out and say that the answer is no, not because I have a specific reference (people tend not to publish papers stating that something doesn't exist) but monetary systems has been a keen interest of mine for the past decade and if free banking was currently being practised somewhere I would almost certainly know about it.

It has however been practised historically, perhaps most notably in Scotland 1716 - 1845.

You can find many proponents of free banking in the Austrian School of economics, for example George Selgin.


Production Set: Not satisfying Free Disposal Assumption

This is something you can figure out yourself. If you have not yet tried, I encourage you to do so.

Draw any set $H \in \mathbb{R}^n $. Select any point $x$ of set. Is the set of all points $y$ "under" this point $x$, that is
$$
\left\{y\in \mathbb{R}^n | y << x \right\},
$$
a subset of $H$?
If no, free disposal is violated.
If yes, remove any of these points from $H$. The reduced set now violates free disposal.
For a figure of such a set, you can consider the Mercedes logo:


When a relevant variable unnamed on either axis changes, can the curve NOT shift?

The scenario of counteracting changes in relevant variables is theoretically possible, but unlikely in practice.

As an example that might come fairly close, consider the supply curve of oil in a small developing country which imports all its oil and subsidizes its sale, the subsidy being at a flat rate per unit.  The country will be a price taker for oil, so its domestic supply curve will be horizontal at the world price less the subsidy.  Now suppose the world price falls, and the government takes the opportunity to reduce the subsidy by a similar amount.  Either change on its own would shift the supply curve (the former down, the latter up), but together they could largely counteract each other.  Even in this case, however, it is implausible that the supply curve would never shift at all.  For one thing, the world price of oil is constantly changing, and it would be impracticable to adjust the subsidy so frequently.  For another, there would inevitably be time-lags in adjustments.

So although you correctly identify a theoretical exception to the statement by Mankiw, it is unlikely to be of much practical significance.


connection between foreign assets and trade surplus / deficits

Piketty  is correct, this is well known mechanism thats discussed in any international economics 101 textbook.
I think the best way how to see how this work is to forget about money for a second. Ultimately all economic transactions  are not about money but about exchanges of real goods and services. Money are just facilitators of transactions, they solve the double coincidence of wants. For the purposes of this discussion you can think of money as just technology that makes transactions easier and go faster like internet which allows you to send letters faster and easier compared to post office, but what really matters is the content of a message.
So lets forget about the money for now and consider a barter economy where you have your country X that does not want anything from country Y but country Y wants coffee from country X. In such situation country X would either:

export the coffee for nothing in return essentially making the coffee foreign aid - a noble act for sure but alas most people/countries will not just generously give away their  products or services for nothing.
the country X exports the coffee in exchange for country Y promising to pay them some goods and services back at some time in the future (i.e. for debt). Now debt is an asset for country X (for the creditor) and liability for country Y (borrower) so their net foreign assets position changes.

If you introduce money the same thing occurs except the debt is denominated in money terms rather than in some goods and services. In addition if we introduce not just money but also assume both country X and Y have their own separate currencies, dollars and pounds lets say then also some important exchange rate effects come in the play.
For example, if country X is the exporter country Y must always go through forex market to exchange their pounds for dollars to pay for X exports as people in X want dollars (since they dont want anything from Y and you can’t buy things in X with pounds). This will increase the demand for dollars at the same time as increasing supply of pounds. Hence the value of dollar rises (ceteris paribus) and value of pound decreases (ceteris paribus). This change of relative value of currency makes all foreign denominated assets and liabilities worth more so if the country is a net debtor currency depreciation increases its foreign currency debt burden making the net asset position even worse.
In addition this goes little bit beyond the scope of the question but eventually running trade account deficit leads to higher interest rates relative to those that run surpluses which encourages people to actually invest (i.e. purchase assets) in the country running trade deficit to the point where the flow of foreign investment is high enough to force exchange rate to equate the real returns in home and foreign country. An important caveat is that there are also other factors at play for example Batra & Belad (2013) show that thanks to USD being reserve currency US managed to keep low interest rates even though running huge trade deficit. This being said this is how, in general, the mechanism works in the long run and then some special circumstances can create exceptions.


I would say people usually use log-returns for continuous data (although no data is really continuous, not even tick data). And discrete returns when your data is discrete. In the case of GDP, you only get the data every 3 months, so that is as discrete as it gets.


Impact of election outcomes on individual's mood? Behavioral Finance

I can only provide an australian perspective, but we have a variety of consumer confidence sources tracking weekly results. Somewhere in

Here is one analysing elections. It shows a very sharp drop in the lead up but it is short lived. There is evidence that this impacts the stock market.

Shane Oliver wrote about the 2013 australian election, the 2016 Australian election and the Italian election. The most interesting is probably the one published around the 2013 Australian election as it looks at the impact on stock markets. Those sources should be a good starting point. There's a good amount of research going around. The basic gist is that mood is impacted for sure but how important this mood swing is is unclear.


Can debt to GDP ratio be reduced transfering debt to local autorities?

To answer your second question: a country could probably do that, but it would achieve not much more thereby than clean up that particular statistic. In Europe, the Treaty of Maastricht that specifies the maximum debt-to-gdp ratio a country can have pertains to the whole government sector, which is defined so as to include local authorities (see this explanation on the site of the Belgian government https://www.debtagency.be/en/datagovernmentdebtdebt).

I don't know the situation in the U.S., but I think it can be assumed that in general any state or international institution that wants to know about another state's debt levels will not only look at the debt-to-gdp ratio.

In an analogous case, most E.U. officials are aware that the United States unemployment rate is on average lower than the European version partly because of differences in labour force participation, which works 'behind the scenes' of the official statistic (http://bruegel.org/2017/09/employment-in-europe-and-the-us-the-eus-remarkable-strength/).


Randomisation and balance in experiments

Ok, I am far from an econometrician, but my train of thought would be as follows:

By using random assignment we have two groups that are on average equal to one another on all aspects. As you rightly point out due to sampling variation there will be differences.

My worry would be that if I start "correcting" those after randomization (as per your example with gender distribution) I introduce other differences between the groups in say age or even an unobserved but important characteristic.

Stratification is different as it is done upfront. You have a characteristic that you know is different for different groups so you take the random samples within those groups. As a rather naive example: we know that the effect of taking anticonception pills is rather different for men and women, so if you wanted to test the effect of the pill on fertility you could stratify by sex and then do a random assignment within the sexes.


Is it theoretically possible for a singular national currency to exist under a free banking system?

Is it possible for India to have a Free Banking arrangement with one singular national currency?

Under free banking you would only end up with single currency if one bank manages to become a monopoly on issuing the currency. That is possible, although at least historically it did not occurred under free banking systems. Also this would be in essence equivalent to central banking system with only difference that the central bank would not be government institution.
Also, for full disclosure,  I am mainstream economist and not fan of idea of free banking (which is a fringe idea that very few economist support, and it would likely be very inefficient monetary arrangement)  but to my best knowledge of the position of free banking advocates, they advocate free banking because they don't like monopoly on money so ideally the system should be operated with multiple currencies.
However, If the free banking currencies are backed by gold (which was true for most historical free banking systems but does not necessarily need to be part of the system as you can have fiduciary free banking) then even if they technically use different currency they are all in some sort of 'monetary union'. This is because even if they can still vary the number of banknotes redeemable per ounce of gold (and hence the value of individual notes) the exchange rates between the banks is essentially fixed by value of gold which is in practical ways virtually equivalent of being in monetary union (see Eichengreen & Temin, 2010).

Also, if several competing private currencies are to exist in one country, would each bank be willing to accept the currency of the other?

If the free banking banks operate under gold standard then they would have to because in essence what gives value to the money is gold. Lets say hypothetically that under such arrangement bank A would refuse to take money of bank B. Well all you would have to do is just to convert your bank B banknotes to gold and get bank A banknotes for that gold so it is not like bank A can stop you from converting the money.  In addition why would bank that wants to attract customers (and their gold) impose this extra administrative burden on its customers?
If there would some fiduciary free banking then there likely would be floating exchange rates for each banks money and when you would be going to another bank you would just have to pass through exchange rate like when you want to put money in foreign bank. Here I suppose it would be more doable for bank to impose restrictions on other banks money if they would want to - but again question is why would they want to restrict new customers coming to them?

How did historical examples of Free Banking function regarding this issue?

You can find nice accounts of how the system worked in writings of 18th century economists such as for example in Adam Smiths Wealth of Nations. I will only give very brief summary here.
The system developed as an extension of commodity money. Before paper bank notes people used gold/silver (or other metals) coins across Europe. However, travelling with coins is dangerous so people would put them in banks and get 'bank note' of their gold balance that they could get at any other branch of the bank on demand. Since these banknotes were as good as gold/silver coins people started using them instead of the coins.
Under free banking every bank issued their own notes but since the value of note dependent on the gold you can get they could be easily exchanged with each other. If banks A note entitled you to 1 ounce of gold but bank B note entitled you to 2 ounces of gold you would only accept 2 banks A notes for 1 banks B note. Later one as the system developed there were also dedicated exchanges for money issued by various banks.
However, the above is only very rough and crude and simplistic description. The system worked bit differently across different countries since back then there was much less standardization and uniformity when it comes to ways how financial institutions operate (at least relative to present day). For example, the US free banking system was relatively more regulated than lets say the Scottish or Swiss one (see Kenneth (1988) or Kroszner & Randy (1995)).


Questions regarding foreign exchange online trading

"Short selling" is a method of trading that includes selling a security (in this case currency) that the trader doesn't own, by borrowing it from a broker.
Prices on forex trading websites are quoted at the current market price, meaning that people around the world are buying and selling at that price at that time. Brokers buy for slightly under market price and sell for slightly over, and they are the ones who actually trade with the average currency trader.
In the scenario where there are more sellers than buyers at a current price, all buyers at that price level will make their purchases and then the price will drop so the market includes more buyers. Demand < Supply so price falls.
People trade currencies for many reasons. Forex trading as an investment does not make up the majority of currency exchange. Businesses that are involved in international trade need different currencies to pay for operations in other countries, to import goods from other countries, etc. and visitors to foreign countries have to buy the currency of those countries. Therefore price changes are not primarily affected by investor activity and many people/businesses do not win or lose on purchasing a currency because they do not hold it as an investment but just spend it.

Hope this helps.

http://www.dailyfx.com/forex/education/trading_tips/trend_of_the_day/2012/02/07/Short_Sell_Basics.html

http://www.econedlink.org/lessons/index.php?lid=342&type=student


Extensions of Nash equilibria to games with infinite strategies

Yes, there is such a setting. The result is that

  If each player's strategy space is

  convex
  compact

  and if payoffs are continuous then there exists at least one Nash equilibrium (possibly in mixed strategies).

This holds even when the set of possible actions is uncountably infinite. If one additionally assumes that payoffs are quasiconcave then the best-response correspondence will be convex even when we restrict attention to pure strategies so that we are then guaranteed to have at least one equilibrium in pure strategies in such a game.

I believe the original reference here is

I.L. Glicksberg. "A further generalization of the Kakutani fixed
point theorem, with application to Nash equilibrium points."
Proceedings of the American Mathematical Society, 3(1):170–174, 1952.

The treatment in Glicksberg's paper, though, does not  seem very accessible. A good starting reference is more likely to be section 1.3 of Fudenberg & Tirole's book "Game Theory".


Why is GDP growth so important that central banks are trying to impose negative interest rate?

Central banks do not try to increase growth. Instead, they commonly have two targets: 1) to maintain price stability and 2) to minimize the output gap. Let us ignore the first goal for a moment. The minimization of the output gap implies that if output falls short of what the central bank believes to be the natural level of output, then it will try to use economic policies which promote growth.

In your example, 10 units of bread being produced seems to be the natural level of output. If the central bank sees that the economy produces only 9 units of bread, it will try to stimulate the economy with low interest rates. If it observes 11 units being produced, it will raise interest rates to reduce the output. (Note that this is still possible even if firms produce 9 units of bread using 10 (or even 100) units of labor, i.e. if firms are not cutting employees efficiently.). Thus, if the central bank of Japan currently has low interest rates, it believes that current output falls short of the natural level of output, despite low unemployment rates.

You may still ask why the central bank tries to minimize the output gap in the first place. Why should it not just focus on price stability or why not even do something random. It turns out that up to second order approximations, an interest rate policy which minimizes a combination of price volatility and output gap volatility maximizes people's welfare in so-called new-Keynesian models. This was first shown by Woodford and Rotemberg in several articles:

  Woodford, Michael. "Inflation stabilization and welfare." B.E. Journal of Macroeconomics, 2, (2002).

  Woodford, M. (1999). Optimal Monetary Policy Inertia. Retrieved from http://www.nber.org/papers/w7261

  Rotemberg, J., & Woodford, M. (1999). Interest rate rules in an estimated sticky price model. Monetary Policy Rules. Retrieved from http://www.nber.org/chapters/c7414.pdf


How much is an asset worth that produces a constant income?

An asset that gives you a constant sum of nominal income every year is called a perpetuity. (The UK government used to issue such a product and called it a consol.)

Everyone "knows" that £1,000 received next year is worth less than £1,000 received this year. But how much less?

To try to formalize this notion, economists use the concepts of discount rate and present value. The discount rate is the rate at which you discount future returns.

If for example your discount rate is $5\%$, then this means you treat £1,000 received next year as equivalent to £950 received this year. (And an economist would say that the present value of £1,000 received next year is £950.)

We can now answer your specific numerical question:

For simplicity, suppose your discount rate is fixed as $\delta$.

Then the present value of this asset is: $$1000+(1 -\delta)1000+(1 -\delta)^21000+\dots = \frac{1000}{\delta}.$$

More generally, suppose your discount rate each year is $\delta_t$ ($t=0,1,2,\dots$).

Then the present value of this asset is: $$1000+(1 -\delta_1)1000+(1 -\delta_1)(1 -\delta_2)1000+\dots.$$

The brief discussion above is usually covered in greater depth in a course on Intermediate Microeconomics. See e.g. Varian (2014, 9e) Ch. 10.


The relation between econometrics and machine learning

I.)  2 Principles of econometrics can potentially be useful compared to Machine Learning.
(see  Hal R Varian 2014 Paper : https://pubs.aeaweb.org/doi/pdf/10.1257/jep.28.2.3)

A.) As you suggest the search of causality is one advantage but unlike what you think, even if causality sometimes could be tricky to measure it remains very useful and functional.

But first,  when you suggest that Instrumental variable is the only available tool which work for most of the cases and allow causal inference, i think there is a few more techniques that could  still be applicable  to measure causality in response to a treatment, a manipulation, or an intervention and still be relevant (unlike natural experiment as you precise because of its restricted direct applicability to most of the cases) in most of the situations such as:

● explicit experiments
● regression discontinuity
● difference in differences
● structural estimation

For instance you can investigate causality with theses techniques or instrumental variables because even if those techniques are subject to bias and correlation issues (all the more since big data because the increasing size of datasets limits the usefulness of instrumental variable methods, depending on instrument strength and level of confounding), it still give a hint to investigate causality.
In a way that, for instance, it enables the detection of conditional counterfactual (an if-clause which is contrary to fact) while also check the existence of potential selection bias.

What is a Conditional counterfactual:

If it is raining, then he is inside.  : Rain = Indicative Variable

If it were raining, then he would be inside. : Rain = Conditional Counterfactual variable

Hence, the better predictive model you have for the counterfactual, the better you will be able to estimate the causal effect.
Thus, even though a predictive model will not necessarily allow one to conclude anything about causality by itself, such models may help in estimating the causal impact of an intervention when it occurs.
Because it can highlights a bunch of conditional counterfactual which could be use as potential causal Variables to run the test on, in order to  eventually determined causality in a given dataset to do good decision making (clean up from confounding issues). If not, at least, it gives you some clues to conduct a deeper investigation to understand the problem : anything wrong with the theory ? anything wrong with my econometric model ? anything wrong with my data ?.

For more insight about the pro cons of differents causal inferences techniques in econometrics :

https://www.jstor.org/stable/pdf/44234997.pdf?refreqid=excelsior%3A18bacfd86299dcf19e7f1f13d9c52022

B.) Except causality, model uncertainty is another advantage of Econometrics compared to ML.

Probabilistic foundation of econometrics is a strenght in a way that it allows interpretability of most of the models and their parameters (avoiding black box phenomenon) and give a quantisation of uncertainty (with confident intervals ) .
The goal is usually to show that the estimate of some interesting parameter is not very sensitive to the exact specification used : how an estimated parameter varies as different models are used.This question illustrate a simple form of model uncertainty.
In this period of “big data,” it seems strange to focus on sampling uncertainty, which tends to be small with large datasets, while completely ignoring model uncertainty, which may be quite large.
One way to address this is to be explicit about examining how parameter estimates vary with respect to choices of control variables and instruments.

II.) On the contrary Machine Learning techniques could  also be useful for data analytics in social Science.

A.) Parameters Selection, Model validation methods in ML can improve traditional econometrics models

Researchers in machine learning have developed ways to deal with large datasets and economists interested in dealing with such data would be well advised to invest in learning these techniques.
For instance, Web Mining methods could discover new usuable explanatory variable.
Cross validation should identify a non-linear effect or a forgotten cross effect.
Model Validation should detect when a model is mispecified and thus allows a better specification of an econometric model and on the whole reduce omitted variable bias and error.
As an example, recent litterature in finance focus on Garch models (traditional times series models) improved with Neural Network to better predict volatility and asset prices.

For more insight about the usefulness of ML in econometrics check out this paper by Arthur Charpentier : https://arxiv.org/pdf/1708.06992.pdf

B.) Causal Modelling began to be a concern and a field of research in ML. Which means that in the long run ML could potentially overcome its own flaws ( like being exclusively focus on the fit ) and outperform Econometrics.

Some Theoretical computer scientists, such as Pearl (2009a, b) have
made significant contributions to causal modelling in computer science (by extension machine learning):

(see:  Causal Inference in Statistics: A Primer Wiley, 2016  Judea Pearl et Al)

Pearl defines counterfactuals directly in terms of a "structural equation model" – a set of equations, in which each variable is assigned a value that is an explicit function of other variables in the system. Given such a model, the sentence "Y would be y had X been x" (formally, X = x > Y = y ) is defined as the assertion: If we replace the equation currently determining X with a constant X = x, and solve the set of equations for variable Y, the solution obtained will be Y = y. This definition has been shown to be compatible with the axioms of possible world semantics and forms the basis for causal inference in the natural and social sciences, since each structural equation in those domains corresponds to a familiar causal mechanism that can be meaningfully reasoned about by investigators.

However, it appears that these theoretical advances have not as yet been incorporated into machine learning practice. Except in some recent research paper :

http://www.nasonline.org/programs/sackler-colloquia/documents/athey.pdf

As a Conclusion, in my opinion and as things now stand, Econometrics as more valuable input in social science than what machine learning could bring. So it is still adequate to use econometrics to conduct data analysis instead of ML techniques.


The game has positive expected value, so you should play (assuming constant utility of each dollar, which tends to break down somewhat for very large sums of money). If you start with \$X, after 1 round you have a 50% chance of having \$1.5X, and a 50% chance of having \$0.6X, for a total expected value of \$1.05X. That is, for every dollar you bet, on average, you'll win \$1.05. The amount of money and number of trials doesn't matter at all for this analysis - every round is effectively identical, so you should always bet.
In addition, there is absolutely no risk of "gambler's ruin" here, since you can never go bankrupt. In games where you can lose your entire wager, playing even a positive expected value game can lead to bankruptcy over time, since you'll eventually run into a losing streak that outstrips your bankroll. But here, since you will always have some amount of money on hand, you can keep playing forever no matter how bad of an unlucky streak you get.


Should one remove trend from time series before testing for cointegration?

It is not necessary to detrend the series although the trend has to be dealt with and detrending is one of the options but there are also other options.
Other options include explicitly modeling the trend. For example, many tests for cointegration including Johansen cointegration test, it is possible to estimate the test together with a linear or quadratic or other deterministic trend terms. The only issue there is that with some cointegration tests the test statistics with and without trend and critical values are different, but most modern packages/programs already provide adjustments for some common deterministic trends - for example linear or quadratic trends.  Hence in such cases this might be preferable to detrending.
However, detrending the series is still an option. The problem with detrending is that you are modifying the series itself and there is always a risk that when you remove trend you are also throwing away some other information that the series had. Nonetheless, it is still a valid option.


How to derive utility function from indirect utility in this exercise

Use Roy's identity to find demand functions:
$$\displaystyle x_1=-\frac{\frac{\partial v}{\partial p_1}}{\frac{\partial v}{\partial y}} =-\frac{\alpha yp_1^{\alpha-1} p_2^\beta}{p_1^\alpha p_2^\beta}=-\frac{\alpha y }{p_1} $$

$$\displaystyle x_2=-\frac{\frac{\partial v}{\partial p_2}}{\frac{\partial v}{\partial y}} =-\frac{\beta yp_1^{\alpha} p_2^{\beta-1}}{p_1^\alpha p_2^\beta}=-\frac{\beta y }{p_2} $$

Since $p_1x_1 + p_2x_2 = y$, from the above demand expressions we get $-\alpha -\beta = 1$.
Demand equations above can also be rewritten as:

$$x_1^{-\alpha}  = (-\alpha y)^{-\alpha}p_1^\alpha$$
$$x_2^{-\beta}  = (-\beta y)^{-\beta}p_2^\beta$$

Multiplying them we obtain the following:
$$x_1^{-\alpha}x_2^{-\beta} = (-\alpha y)^{-\alpha}p_1^\alpha (-\beta y)^{-\beta}p_1^\beta = (-\alpha)^{-\alpha}(-\beta)^{-\beta}yp_1^\alpha p_2^\beta $$

Comparing it with the indirect utility function, we get the direct utility function as:

$$u(x_1, x_2) = \frac{x_1^{-\alpha}x_2^{-\beta}}{(-\alpha)^{-\alpha}(-\beta)^{-\beta}}$$


What happened to the dollar in Nov 1978?

Well, the Fed began raising interest rates to record highs in late 1978. I thought it was mainly to quell inflation. Maybe Burns - who was Fed chairman up to Jan. 1978 - thought it was mainly to strengthen the value of the dollar.


Can a Discount Rate be Inferred within a Solow-Swan Growth Model?

In the Solow-Swan model, the interest rate will be determined by market-equilibrium: under the baseline assumption of competitive markets (and constant-returns-to-scale technology), we get the familiar per-period equilibrium relation

$$f'(k) = r + \delta \tag{1}$$

Regarding the rate of pure time preference, think of the following: in this model, savings is a fixed percentage of output. So consumption is also a fixed percentage of output:

$$c = (1-s)f(k) \implies \dot c = (1-s)f'(k)\dot k \tag{2}$$

while also we have

$$\dot k = sf(k) - (n+\delta)k \tag{3}$$

Combining, we have

$$\dot c = (1-s)f'(k)\cdot [sf(k) - (n+\delta)k]$$

$$= (1-s)f(k)f'(k)s - (1-s)f'(k)(n+\delta)k$$

and using the previous relations,

$$\dot c = s(r+\delta)c - (1-s)(r+\delta)(n+\delta)k \tag{4}$$

In the "Ramsey" model with consumer intertemporal utility maximization, we get the relation (for log-utility for simplicity)

$$\dot c = (r-\rho(t)) c \tag{5}$$

Note that I made the rate of pure time preference time-varying and this is indeed the case when we examine time-varying preference rates in the Ramsey model (see for example Barro, R. J. (1999). Ramsey meets Laibson in the neoclassical growth model. Quarterly Journal of Economics, 1125-1152).

For the consumers in the Solow-Swan model to behave like the consumer's that optimize intertemporaly equation $(4)$ must be equivalent to equation $(5)$ so (using now the time variable to clearly indicate what is time-varying and what is not), we get after eliminating and re-arranging

$$s(r(t)+\delta)c(t) - (1-s)(r(t)+\delta)(n+\delta)k(t) =[r(t)-\rho(t)] c(t)$$

$$\implies \big[s(r(t)+\delta) - r(t)+\rho(t)]c(t) = (1-s)(r(t)+\delta)(n+\delta)k(t)$$

$$\implies \rho(t) = (1-s)(r(t)+\delta)(n+\delta)\frac {k(t)}{c(t)} +r(t) - s[r(t)+\delta]$$

$$\implies \rho(t) = (r(t)+\delta)(n+\delta)\frac {k(t)}{f(k(t))} +r(t) - s[r(t)+\delta]$$

The consistent rate of pure time preference is endogenous, and time-varying under diminishing marginal product of capital, and becomes constant only in the steady state. The relation tells us to what $\rho(t)$ must be equal so that the optimizing consumer finds optimal to maintain a constant savings rate.

If the production function has constant output elasticities (say, Cobb-Douglas $f(k) = Ak^a$), then the above simplifies to

$$\rho(t) = (n+\delta)a - s\delta + (1-s)r(t) $$

or

$$\rho(t) = (n+\delta)a  - \delta + (1-s)f'(k(t)) $$

so it is still time varying (it would be constant in a model with constant marginal product of capital). If capital increases to its steady state, we see that the consistent rate of pure time preference should fall over time (which is also the result we get in the Ramsey model with time varying $\rho$).

At the steady-state we have $f'(k^*) = a(n+\delta)/s$ so

$$\implies \rho^* = (n+\delta)a  - \delta + \frac{(1-s)a(n+\delta)}{s}$$

$$\implies \rho^* = \frac {an+ (a-s)\delta}{s}$$

That's a nice one since $a$ is the golden-rule savings rate. To mimic the Ramsey model we should set therefore $s<a$ so we will be to the left of the golden rule (and even in the Solow context, setting $s >a$ is dynamically inefficient). Then the steady-state consistent $\rho$ will certainly be positive.

Just for a taste, set $a=0.45, s=0.35, n=0.01, \delta =0.05$, to get

$$\rho^* \approx 0.027$$

Not far from the benchmark value $\rho =0.02$.


Relation between incomplete contracting and the principal agent problem?

Not all principal-agent problems are the result of incomplete contracts, no. In fact, in most principal-agent problems, complete contracts are assumed.

An incomplete contract is one that cannot be contingent on every possible outcome that could occur after the contract is signed. You can't write a contract that gives out a different payment to each party under every possible circumstance, because you can't imagine every possible circumstance. For example, an American agricultural firm's profits might be affected in miniscule ways by weather in Peru. But a contract related to that firm might offer the same payouts regardless of Peru's weather.

A principal-agent problem arises when the incentives of the principal and of the agent are not aligned, and so the agent may not act as the principal desires. For example, putting in less effort than the principal would find optimal because the agent will be the primary benefactor of additional effort. A lot of the problem arises here because the principal cannot perfectly and costlessly observe what the agent does, and so must write a contract contingent on the outcome of the project, rather than contingent on the agent's actions, which cannot be observed.

You could, colloquially, say that the principal-agent problem is an example of incomplete contracts in the sense that you can't contract on the agent's actions. But that's not what economists mean by incomplete contracts. Incomplete contracts refers to the fact that your contract is not sensitive to every possible contingency, even though everyone will know what that contingency is after it happens, as opposed to principal-agent problems, which refer to the fact that you can't include unobservable actions of the agent in your contract, which are hidden to the principal.

As for how they're related, they're both examples of problems that can make it difficult to write a contract. There are certainly models that combine the two (for example this), but one doesn't subsume the other.


Is utility in neoclassical economics a circular argument/concept?

Let's be clear--the concept of utility is both unfalsifiable (as a singular proposition) and useful.

Joan Robinson is right, of course: Utility as a definition is circular. If you read a textbook that discusses the foundations of economic theory (such as the popular graduate text by Mas-Colell, Whinston, and Green), then you start not with utility, but with preferences. You would characterize an individual as being defined by a "preference relation" which describes, for every pair of potential allocations (i.e., sets of goods), which they prefer.

Some (not all!) preference relations are rational, which means that they are both complete (you have a well-defined preference over any pair of allocations), and transitive (if you like A more than B and B more than C, you like A more than C). A subset of those preference relations can be represented by a utility function (again, not all--the classic example is lexicographic preferences).

So a utility function is really just a thing that represents some underlying preferences. The statement "Steve's preferences are accurately modeled by a utility function" are not restrictive enough to be tested in any practical way. Rationality in general is hard to test as well, as you can always explain away seeming rationality violations with unusual preferences.

Instead, you should think about utility as a framework, not a theory. It's a language for developing theories. For example, you can describe Friedman's Permanent Income Hypothesis without invoking utility at all, but describing it in terms of utility is a lot clearer and simpler!

As an aside, the blog post you linked to is utter tripe, from a guy who seems to love bashing economics while not actually knowing all that much about it. I would avoid him like the plague.


What statistical concepts are used in introductory econometrics?

I'll list the statistics-related topics from my basic econometrics course.

Expected value, mean, variance and covariance as well as conditional expectation and conditional variance.
Basic probability and probability density functions (I'd suggest getting somewhat familiar at least with Normal, Student's T, and F distribution and maybe Bernoulli, Chi-Square and Poisson) and what random variables mean.
The central limit theorem (closely related with PDFs).
Correlation. Also understanding why correlation does not imply causation is useful, but you'll probably discuss this in class.
Methods of regression/estimation. You'll probably be introduced to Ordinary Least Squares early on and a basic understanding of what regression models are for might be useful although probably not assumed.
Statistical hypothesis testing. This will probably be a main subject in any applied econometrics course, so you might take a sneak peek on the basics.

You might have to understand some matrix and equation algebra depending on how theoretical the lecturer makes the course, but these were probably included in your calculus course.


Data for the distribution of US household annual income by year and state

Have you checked the US census bureau? Income related website here.

Furthermore there is a kaggle training competition about the census data. Have a look here for the available data tables. Data straight from the census bureau.


Is a lack of any barriers to entry a necessary condition for ensuring pareto optimality is reached?

Let us quickly define a competitive market. A competitive market satisfies the following:

1) All firms sell an identical product

2) All firms and buyers are price takers (have no control over price)

3) Buyers and firms have complete information about the market

4) The market is characterized by free entry (no barriers to entry)

If this is the market structure, we can have the existence of pareto optimality, so I am assuming this is the competitive market to which you are referring in your question. What happens when we try to relax assumption 4?

We can show this using your example of startup capital:

Suppose we have a competitive market and the amount of startup capital needed to enter this market is $A$. When deciding to enter this market the firm would subtract $A$ from the long term profit possibility $\Pi$ (because it would be another cost). By definition $A\geq 0$. Because the situation where $A=0$ is trivial (because that means there is no startup capital needed), we know that $A>0$. Suppose at a point in time there is opportunity for profit in this market. As long as $\Pi\geq 0$ firms will decide to enter the market, but if $\Pi <0$, a firm would decide not to enter the market. However, note that firms who are already in the market are earning positive profit. This is a contradiction because if positive profits persist, we do not have a competitive market.

As you noted in your question, a competitive market is a necessary condition for the existence of pareto optimality. If the nonexistence of barriers to entry is a necessary condition for a competitive market, nonexistence of barriers to entry is also a necessary condition for pareto optimality.


Osborne and Rubinstein has a book called "Bargaining and Markets" where they have a detailed exposition on bargaining models. I think its free to download from Ariel's website. Apart from that, Abhinay Muthoo has a very nice book on bargaining: "Bargaining Theory with Applications".
You can always look up chapters on bargaining in Tirole's Game Theory or Myerson's Game Theory books. They're short, but quite detailed to develop the basic toolkits.


If I have a £25,000 loan, in a balance sheet do I include it in the Cash as well as a liability?

Yes, you are correct: add the 25k both in cash and in liabilities.

The balance sheet represents what you have (assets) and what you owe (liabilities). So when you get the loan you do have 25,000 in cash under assets, but you also owe it and will need to repay it at some point (which appear as 25,000 in bank loans, under liabilities). As you repay it, the value in liabilities decreases. If you need to learn more there are several free online courses, like this one.


Is this an an endogeneity/simultaneity problem?

Let us consider Situation 1.
Let us assume that $\rho$ is observed. If it does not work when $\rho$ is observed, there is no reason why it (using a proxy of $\rho$ as instrument) should work when $\rho$ is not observed.
$\rho$ is endogenous so we can't just include it as a regressor in an equation. Thus, let us consider IV estimation.
Obvious instruments for the first equation are $\mathbf{x}_1$, $\mathbf{x}_2$, and $\mathbf{x}_3$. Can we use $\rho$ as an extra instrument? It depends on whether $\rho$ is relevant and whether $\rho$ is exogenous in the first equation (i.e., the $y_1$ equation). First, $\rho$ is relevant as it is correlated with $y_2$ (unless the last equation is degenerate). Next, is it exogenous?
To check it, let's go maths and consider the following simple model (without intercepts and exogenous regressors, for simplicity):
$$y_1=\alpha y_2 + e_1,\;\; \rho = \beta y_1+e_2,\;\; y_2=\gamma \rho + e_3.$$
Write them in matrix form:
$$
\begin{pmatrix}
1 & 0 & -\alpha\\
-\beta & 1 & 0\\
0 & -\gamma & 1
\end{pmatrix}
\begin{pmatrix}
y_1\\ \rho\\ y_2
\end{pmatrix}
=
\begin{pmatrix}
e_1\\ e_2\\ e_3
\end{pmatrix}
.
$$
Using Cramer's rule, we get
$$\rho = (\beta e_1 + e_2 + \alpha \beta e_3 ) / (1-\alpha\beta\gamma).$$
$\rho$ is correlated with $e_1$ unless $\beta=0$, so $\rho$ can't be used as an instrument.
Edit
What happens if $y_1$ is regressed on $y_2$ and $\rho$? Then will $\alpha$ be consistently estimated? For simplicity, suppose that $e_1, e_2, e_3$ are $iid$ standard normal. Then the OLS estimator vector converges in probability to
$$\begin{bmatrix}
 \alpha\\ 0
\end{bmatrix}
+
\begin{bmatrix}
 E(y_2^2) & E(y_2 \rho)\\
 E(y_2\rho) & E(\rho^2)
\end{bmatrix}
^{-1}
\begin{bmatrix}
 E(y_2 e_1)\\ E(\rho e_1)
\end{bmatrix}
.$$
Letting $e=(e_1,e_2,e_3)'$, write $\rho = e'g$ and $y_2 = e'h$ for some $3\times 1$ nonrandom vectors $g$ and $h$. Then the second term is
$$
\begin{pmatrix}
 h'h & h'g\\ g'h & g'g
\end{pmatrix}
^{-1}
\begin{pmatrix}
h_1\\ g_1
\end{pmatrix}
.
$$
The first element of the above is the determinant inverse times $g'gh_1 - g'h g_1 = g'(gh_1 - hg_1)$. Let us calculate it. We have already obtained $g$. We now need $h$:
$$y_2 = (-\beta\gamma e_1 - \gamma e_2 + e_3) / (1-\alpha\beta\gamma).$$
Let us ignore the determinant part (common). We can work out with $g=(\beta, 1, \alpha\beta)'$ and $h=(-\beta\gamma, -\gamma, 1)'$, ignoring the determinant. Then
$$ gh_1 - hg_1 = [0, 0, -\beta (1+\alpha \beta\gamma) ]' \text{ times a constant}.$$
Thus,
$$ g'(gh_1 - hg_1) = -\alpha \beta^2 (1+\alpha\beta\gamma) \text{ times another constant}.$$
The OLS regression of $y_1$ on $y_2$ and $\rho$ gives an inconsistent estimator in general, but gives a consistent estimator of $\alpha$ if $\alpha=0$ or $\beta=0$. Especially, if $\alpha=0$, then the OLS estimator is consistent. That is, we can regress $y_1$ on $y_2$ and $\rho$ if we want to test $H_0: \alpha=0$. This is unexpected. Not sure if my algebra is correct.


Does a negative interest decrease the money supply?

Definition of Money Supply:
The money supply is the entire stock of currency and other liquid instruments circulating in a country's economy as of a particular time. Because of it's definition, the money supply can only be affected if a central bank prints or burns currency. Thus, technically, rates don't affect it per-se.

Velocity of money and money in circulation:
What you may be thinking of is related more with how much money is in circulation and how fast it changes hands.

Effect of negative rates:
In theory, when rates go up, money flows into bonds and "leaves" circulation. By the same logic when rates go down, money flows out of bonds and "enters" circulation.


The change in base year keeps the dollar values meaningful to contemporary readers.  If we still used, say, 1900 as the base year for US GDP, the dollar values for today would be so tiny they would essentially be meaningless 'GDP units', which would make it harder for people to visualize what the numbers really mean.

Note that, when the base year changes, historical GDPs go up, not just going-forward GDPs.  So the 1900 US GDP, in 'constant' 2010 dollars, is about 10x as high as the NGDP for that year, to reflect inflation over the 20th century; but that number does a much better job of conveying to readers living in 2016 what conditions in the US in 1900 were really like than using 1900 dollars would.


Separate from their academic qualifications, no there are not.  The reason is that certifications generally serve two functions, either to show a legal status or where there are no academic qualifications required to show some learning process has happened that is similar to an academic designation.

To provide an example, the MD, the CPA and the PE designations have legal meaning.  Their existence is postulated in state and federal law as necessary to engage in certain forms of practice, though medical doctors at the state level also have additional licensing.  Federal physicians do not always have to have a state license, such a military doctors.  Physicians also have higher level certifications to show skill in a narrow area to convince customers they can be trusted in that narrow area.

Beyond this, some fields are heavily populated with people who have no academic degree and for those narrower certifications are available such as CFA or licensing such as the barber's license.  The barber's license is a strange throwback to when barbers did surgery, we still license barbers but the need is less apparent than it was when it started.

Economists have no legal standing.  There is no field of practice that the legislators of the world have felt necessary to impose additional qualifications for beyond a doctorate.  Indeed, you could have a doctorate in another field and engage in economic research and consulting.  This certainly happens with finance professors, mathematicians, sociologists, psychologists and political scientists.  Psychologists and mathematicians have won the Nobel for economics before.

It also is unlikely there are certifications that would be meaningful.  The simplest way to determine the technical skill of an economist is to read their doctoral dissertation, or masters thesis, and their research body of work.  It is an open source process, so unlike craftsmen whose work is only visible to their prior customers the work of economists is very public and very visible.

Of all professions, it should be easier to evaluate an economist than maybe any other.  Certifications would do what?  For whom?


Why is the graph of unitary elastic demand a hyperbola?

The price elasticity of demand is defined as:

$$E_P=\frac{dQ}{dP} \frac{P}{Q}$$

Although generally elasticity depends on price there is a special type of functions (isoelastic functions) for which elasticity remains the same along the whole function.

For example consider demand given by:

$$P=AQ^{1/e}$$

This demand function will always have the same price elasticity $E_P=e$ which you can verify by first solving the function above for Q and then applying the formula mentioned above. For $|e|=1$ the elasticity would always be unity. This is because the elasticity does not depend on anything else for example for a linear demand function the elasticity would depend on price as for $Q=a-P$ the $E_P=-P/(a-P)$ and hence as you correctly pointed out along line the elasticity constantly changes with price.

When you plot the function above you will get a picture that looks like hyperbola. See the picture below. However, not all functions that describe hyperbola have such property I don’t think your teacher meant to say all hyperbolas have constant elasticity.

.

Also the demonstration project from wolfram alpha where I got the picture has amazing utility that graphically shows how the elasticity remains same along every point of the curve. The utility even allows you to play with the parameters of the model. Here is the link for it if you want to explore it.

——————————————————-

Per request in the comment this is how you show that the elasticity is $e$ for $P=AQ^{1/e}$

Solve for $Q$ which gives you:

$$Q= \frac{1}{A} P^e$$

Apply the formula for elasticity which gives you:

$$E_p=e\frac{1}{A}P^{e-1} \frac{P}{Q}$$

Substitute for Q the original expression $Q= \frac{1}{A} P^e$

$$E_p=e\frac{1}{A}P^{e-1} \frac{P}{ \frac{1}{A} P^e}$$

Simplify the above expression and you will get:

$$E_P=e$$


When is expenditure function non-decreasing?

One possibility is to reparameterize your expenditure as follow: $$e(\mathbf{p},u) = E(f(\mathbf{p}),g(\mathbf{p})u),$$ with $f,g:\mathbb{R}^n\rightarrow \mathbb{R}$. Then use the chain rule to find $\partial{e}/\partial{p_j}$, $\partial{e}/\partial{u}$ and to prove that $e$ is homogeneous of degree one in $\mathbf{p}$ iff ...


A basic question about supply and demand graph

Yes, typically we would like estimate these curves’ functional forms. If we estimate a demand function $P=10-2Q$ we can rearrange it such that $Q=5-\frac{1}{2}P$.

This, just changes the way we interpret the relationship. For a given quantity how does price change? Compared to, for a given price how does quantity change? This begs even further questions however because in the market we typically just observe equilibriums and “shocks” that determine prices typically also affect quantity. This makes estimating these functional forms difficult from an econometrics point of view.


How to derive firm's cost function from production function?

"I am not given wealth $w$ although I suppose I could assume any firm who
  is purchasing has some budget."

No. This is exactly where the fundamental microeconomic theory of the firm differs from the microeconomic Consumer Theory: the firm is not constrained by a budget. The reason is that this fundamental theory deals most and foremost with the "long-term" view, or even better, with the "planning view". So we assume that the amount necessary to cover expenses, will come from sales, since the firm won't enter production at a loss (remember also, this is a deterministic set-up, there is no uncertainty). Working capital considerations (the fact that usually first you have to actually pay expenses and then to actually collect revenues), does not enter the long-term view, justifiably, it is a short-phenomenon. Also, in the long-term or planning approach, there are no fixed costs, all factors are variable.

Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the given level output, as determined by demand, in the context of efficient use of the always scarce resources.

But also, the case of a price-taking firm that is too small compared to its market, is closer to a cost-minimizing behavior rather than profit maximizing, since the firm has not really control over its production (except downward by direct decision).

In both of the above cases, an exogenous variable appears: the level of output itself. So we solve the problem by treating the level of output as a "constant" or better, we solve it for any given level of output, and the solution we obtain has the level of output as one of its components.

So

$$\min_{K,L} C\equiv rK + wL \\
s.t. F(K,L) = \bar Q$$

with the Lagrangean

$$\Lambda = rK + wL +\lambda[\bar Q - F(K,L)]$$

The first order conditions are

$$r = \lambda F_K,\;\;\; w=\lambda F_L  \tag{1}$$

which gives, at the optimum,

$$rK + wL = C = \lambda\big(F_KK + F_L L) \tag{2}$$

Now assume that the production function is homogeneous of some degree $h$ (not necessarily homogeneous of degree one, i.e. exhibiting "constant returns to scale", but homogeneous -and yours is, of degree $h=1/2$.). From Euler's theorem for homogeneous functions of degree $h$ we have that

$$F_KK + F_L L = hF(K,L) = h\bar Q \tag{3}$$

the last equality holding given the constraint of the initial problem. Inserting $(3)$ into $(2)$ we obtain

$$C = \lambda h \bar Q$$

The multiplier $\lambda$ is optimal marginal Cost, denote it $C'(\bar Q)$, so we arrive at

$$C = C'(\bar Q)\cdot (h\bar Q) \implies C'(\bar Q) + [(-1/h\bar Q)]\cdot C =0$$

This is a simple homogeneous differential equation with solution

$$C = A\cdot \exp\left\{-\int(-1/h\bar Q) {\rm d}\bar Q \right\} = A\cdot \exp\left\{(1/h)\ln \bar Q\right\}$$

$$\implies C^* = A\cdot (\bar Q)^{1/h} \tag{4}$$

for some constant $A >0$. To complete the solution, we need to express the object of interest, $C^*$, in terms of the exogenous entities: $r,w,\bar Q$.
To do that derive the optimal marginal cost (which is equal to the multiplier)

$$(4) \implies \lambda^* = (1/h)A(\bar Q)^{1/h-1} \tag{5}$$

Inserting $(5)$ into the first-order conditions we have

$$r = (1/h)A(\bar Q)^{1/h-1} F_K,\;\;\; w=(1/h)A(\bar Q)^{1/h-1} F_L  \tag{6}$$

It is time to use the specific functional form of the production function

$$F(K,L) = K^{1/2} + L^{1/2} \implies, F_K = \frac 12 K^{-1/2},\;\; F_L = \frac 12 L^{-1/2} \tag{7}$$

Inerting $(7)$ into $(6)$ together with $h=1/2$ we obtain, after manipulation,

$$rK = \frac {A^2}{r}(\bar Q)^2,\;\; wL = \frac {A^2}{w}(\bar Q)^2 \tag{8}$$

Sum the two to obtain an alternative expression for the Cost function

$$rK+wL = C^* = A(\bar Q)^2\cdot \left[\frac Ar + \frac Aw\right] \tag{9}$$

But inserting $h=1/2$ into $(4)$, we also have that

$$C^* = A(\bar Q)^2 \tag{10}$$
So

$$ (9),(10) \implies   A(\bar Q)^2\cdot \left[\frac Ar + \frac Aw\right] = A(\bar Q)^2$$

$$\implies \frac Ar + \frac Aw = 1 \implies A = \frac {wr}{w+r} \tag {11}$$

Inserting $(11)$ into $(4)$ we conclude obtaining

$$C^* = \frac {wr}{w+r}\cdot (\bar Q)^2 \tag{12}$$

Three things:
A) Verify that the second-order-conditions hold for all this to indeed lead to the optimal cost-function.

B) Solve the unconstrained profit-maximization problem with the same production function, normalizing the price of output to $p=1$ (i.e. treating the exogenous prices, $w,r$ as expressed in real terms), to verify that it will lead to a cost level that it is consistent with $(12)$.

C) If you are interested in the theory of the firm under a budget constraint, a related paper is
Lee, H., & Chambers, R. G. (1986). Expenditure constraints and profit maximization in US agriculture. American Journal of Agricultural Economics, 68(4), 857-865.


Balance of Payment (BPM6) - Direct Investment: how to record

Assuming country EXT builds a factory in DOMES:

For country DOMES: this is direct investment liability
For country EXT: this is direct investment asset