Why do some businesses choose scarcity as their business model instead of raising the price to meet the demand?

Your question is somewhat ill-posed as stated, because none of your examples are good instances of businesses creating artificial scarcity.  Each of them are businesses that face real capacity constraints, whether it be building space and fire-code restrictions or an attempt to manage a sort of tragedy-of-the-commons demeaning of the service.

But you are right in wondering why these businesses don't run their prices up until demand meets capacity. It's basically down to a choice by the business whether to ration through pricing, or ration through queueing.

Queueing is viewed, rightly or wrongly, as a more "fair" means of rationing, since (in the case of non-essential goods) everyone has the same endowment of hours in the day. To the extent that this is true, it's a means to distribute the good or service equitably in a cross-class sense.

It is also a more robust business model in terms of revenue stability, as it leaves much of the "clearing house" work in the respective markets to consumers themselves.  One alternative would be to continually monitor the market and adjust prices to keep patronage at or near existing capacity, which is a tricky balancing act that entails potentially high information and menu costs.

Another, obviously, would be to expand the enterprise - but economies of scale must be present.  Even then there are associated costs and risks - expansion can serve to dilute a brand, and very few restaurant franchises receive Michelin stars.  The market for event space is extremely fraught politically, and the subject of a lot of recent academic interest.

Scalping is an example of arbitrage that takes advantage of the inherent inefficiency of queueing.  One can argue that tickets sold in advance of an event must be priced below their value, with the difference being a combination of a risk premium the venue-owner pays to be assured of a full house well in advance and a risk discount that event-goers demand because of the non-zero possibility of their attendance being disrupted at the last minute.  Flesh-and-blood scalpers make money off of the rapid market shifts that arise due to the resolving of this uncertainty for both parties, and it could be argued that this leads to a more efficient market.

Of course, software breaks this model entirely, and so I wouldn't make that argument today.  But it's certainly true that whatever value scalpers have captured historically was money that even large monopolies like Ticketmaster were not especially inclined to capture for themselves, suggesting that the costs of doing so exceeded the expected benefits.


Was Haiti the richest society in the world in the late 18th century?

It has certainly been argued in various places that Haiti was the "richest" colony.  It's obviously difficult to reconstruct modern measures of wealth such as per capita income for historical territories, and the metric of "rich" that mattered in the day was how much revenue could be extracted by the colonial government, so assessments may be skewed towards this.  My books are boxed at the moment, but this is what I could find based on the footnotes from Understanding Power (ch 10 p62).

Citing Hans Schmidt, The United States Occupation of Haiti, 1915-1934, New Brunswick, NJ: Rutgers University Press, 1971, pp. 19-20:

  At the time of the French Revolution, Saint Domingue [Haiti] was the
  wealthiest European colonial possession in the Americas. As early as
  1742 the sugar production of Saint Domingue exceeded that of all the
  British West Indies, and on the eve of the revolution the colony
  accounted for more than one-third of the foreign commerce of France.
  In 1789 French trade with Saint Domingue amounted to £11 million,
  while the whole of England's colonial trade totaled only £5 million.
  In the same year the ports of Saint Domingue received 1,587 ships, a
  greater number than Marseilles, and France employed 750 ships
  exclusively for the Saint Domingue trade. The chief exports of the
  colony were sugar, coffee, cotton, indigo, molasses, and dyewoods. The
  French built an elaborate network of roads, irrigation systems, and
  magnificent plantations.

Citing Paul Farmer, The Uses of Haiti, Monroe, ME: Common Courage, 1994, p. 63:

  At one time or another, the colony was first in world production of
  coffee, rum, cotton, and indigo. On the eve of the American
  Revolution, Saint-Domingue [Haiti] -- roughly the size of the modern
  state of Maryland -- generated more revenue than all thirteen North
  American colonies combined. By 1789, the colony supplied three-fourths
  of the world's sugar. Saint Domingue was, in fact, the world's richest
  colony and the busiest trade center in the New World.

And citing Patrick Bellegarde-Smith, Haiti: the Breached Citadel, Boulder, CO: Westview, 1990, pp. xviii:

  In the second half of the eighteenth century... [Haiti] created more
  wealth than any other colony in the world. At that time, 50 percent of
  France's transatlantic commerce involved Haiti, and nearly 20 percent
  of the French population depended on trade with Haiti for its
  livelihood... Overall investments in the Haitian economic
  infrastructure were meager compared to the enormous profits and trade
  advantage gained by France. Economically, Saint-Domingue [Haiti] was
  the world's most profitable colony. It soon became known as "the pearl
  of the Antilles" and was the standard by which the profitability of
  other colonies was judged.


I'm assuming pollution is a bad in consumer utility (as is typically seen) or otherwise relevant to some agent's payoffs -- if it's not, then $P_t$ is irrelevant and you might as well ignore it.
$P_t$ is a state variable: it's something which describes the history of the system, and is necessary to compute utility. See page 117, Definition 3.1 in these notes for a more detailed definition of state variables ("utility function" = "contribution function"), the section has some more interesting discussion as well.
If pollution appeared in utility and you were solving a planner's problem, I think there would be no confusion since you would need $P_t$ to calculate the decision rule. Where your problem gets a bit different is that the agents don't use it to compute their decision rules. Mechanically, that makes your calculations a bit simpler: optimize agents' objectives over potential values of $K_t$ only, since the decision rule is going to be "flat" over $P_t$.
That an eigenvalue is bigger than one suggests that pollution or capital in this economy will "explode". But that doesn't bear on whether $P_t$ is a state variable or not.


TL;DR version: "the tunnel" and D+A+B have exactly the same area.

You are right to say that post-subsidy producer surplus is equal to the blue area in the following figure:

However, it turns out that The Tunnel (i.e. the dark blue area) is exactly equal in size to D+A+B.

Intuitively, there are two ways to think of a unit subsidy:

Paying the seller a subsidy of $S=p^s-p^d$ per unit sold has an effect equivalent to reducing his marginal cost by $S$. The seller's marginal cost curve shifts down in the same way as the supply curve in your figure. From the seller's point of view, its as if he is paid price $p^d$ for each unit he sells but had a low marginal cost. When we calculate producer surplus (by looking at the area between the marginal cost curve and the price) this creates some new producer surplus equal to 'the tunnel' (which represents the seller's lower perceived cost). This way of thinking about things gives us the blue area above.
The subsidy does not reduce the seller's cost [so the seller's perceived marginal cost curve coincides with S(pre-subsidy)], but the subsidy is paid to 'top-up' the price received by the seller from $p^d$ to $p^s$. So from the seller's point of view it is as if he is being paid price $p^s$ with a high marginal cost. When we calculate producer surplus we look at the green area in the following figure:

Since these are two ways of thinking about the same subsidy, they must yield the same value for producer surplus. But you should be able to see by comparing the blue and green areas that this necessarily implies that the tunnel has the same area as D+A+B.

You can check this is true in an example by drawing the curves $Q_S=0.5+p$ and $Q_D=2-p$ and then applying a subsidy of $0.5$ to produce a new supply curve $Q_S'=p$. It looks like this:

Geometrically calculating the areas, you should find that 'the tunnel' and A+D+B both have area 3/8.


Heating oil more expensive than gasoline?

I would expect the gasoline market to be more competitive than heating. Transporting heating oil into homes requires large infrastructural investment and high fixed costs, so you'd expect there to be more market power concentrated in a fewer number of firms, driving the price of it up. Creating gasoline for sure requires a lot of investment too, but not for nearly a large area I would imagine.

So basically, consider heating oil an input for a "utility" sort of company, which acts as a natural oligopolist/monopolist. I'm not exactly sure what the market looks like here in the United States, but this is my best guess. Someone else can probably elaborate on this answer or provide an alternative suggestion.


Still don’t understand why a trade deficit must be financed by foreign borrowing (even after reading other answers)

The assertion that a trade deficit must be financed by foreign borrowing is technically incorrect (although it is often approximately true).

The accounting identity for international transactions has a few extra terms, one of which is grants between governments (foreign aid). A country could run a trade deficit financed by foreign aid.
Equity financing also shows up in international transactions. Instead of emitting debt, a country can emit equity.

The reason why the statement is approximately correct is that the  largest new flow across borders is the transfer of debt instruments, noting that currency notes are a liability of government. Since net imports had to be paid for with some instruments (possibly bank deposits), there has to have been an offsetting flow of instruments out of the country. Since those instruments are debt instruments, people loosely refer to this as “borrowing,” although it could be the transfer of a bond issued years earlier. Most people do not refer to selling a bond out of their portfolio as “borrowing,” so one needs to be careful of this terminology.


The supply of chips will probably decrease, although the situation is a bit more complicated than you describe.

The subsidy will tend to increase demand for corn by ethanol producers, both existing producers expanding production and new producers entering the industry. Increased demand for corn for ethanol production may have two effects:

More corn will be sold to ethanol producers.  If total corn production were unchanged, less corn would be sold to chip producers and the supply of chips would decrease.
However, total corn production will probably increase because the increased demand for corn for ethanol production will increase the profitability of growing corn relative to that of other crops or other land uses.  So the likely effect will be that corn is grown on land previously used for other crops, or possibly more land is brought into cultivation. In the medium term (since changes in land use take time) this could partly offset any decrease in the supply of chips.


Is NIIP a good measure of a country economical development?

For common definitions of the word development the NIIP would be a very bad measure. The NIIP is a measure of net foreign debt. I don't see any clear first order link between foreign debt or assets and economic development.

A popular measure of development is the HDI, which is constructed from data on live expectancy, adult literacy rates and GDP.


In law of DMU statement what is meant by "keeping other commodities"?

You have a beer. The beer is great. You have a second one. It's equally good, but it adds less to your utility (decreasing marginal returns). You have a third beer, which makes you also feel better, but the effect is relatively small. Since you're drinking only for a while, you become hungry and you order a spicy pizza. The pizza makes you thirsty and you have, you name it, another beer. This fourth one is likely to be better than the third one, as it kills the spices.

In this example, the pizza is the 'other commodity'. The term "keeping consumption of other commodities constant" is just a typical economics way of saying "I look at beer consumption in isolation, but I am aware that things are usually more complicated". The law of diminishing marginal utility is just not valid anymore if you consume lots of commodities because of complementarities in consumption.


Are there any surveys of economists on the determinants of healthcare price increases in the US, particularly over several decades?

This was actually already indirectly answered by the IGM forum.
In this question the forum asks economists the following:

The US spends roughly 17% of GDP on healthcare, according to the OECD; most European countries spend less than 12% of GDP.
Higher quality-adjusted US healthcare prices contribute relatively more to the extra US spending than does the combination of higher quantity and quality of US care (interpreting quantity and quality to reflect both greater American healthcare needs due to underlying population health and the delivery of more or better healthcare services to Americans).

With the following answers:
From European economists:

From US economists:

This answer (albeit only indirectly) suggests most top policy economists think there is some factor other than technology or quantity demanded that is leading to the increased cost of American healthcare. If you skim the survey some of the alternative reasons presented include increased barriers to entry for American physicians, a dysfunctional insurance system, and market power in the American hospital system.


A general drought in the world, raises the total income that farmers receive?

The demand for the world grain is relatively inelastic, while the demand for the Kansas grain is relatively more elastic. As a worldwide drought reduces the supply of the world grain, the world price of grain rises significantly and the quantity of grain falls insignificantly. Therefore, total revenue that world farmers receive rises. On the other hand, as a drought only in Kansas reduces the supply of Kansas grain, the price of grain rises insignificantly and the quantity falls significantly. Therefore, total revenue that Kansas farmers receive falls.


Type - I Error & Type - II Error: Pregnancy test analogy - is it legit?

Presumably here

the null hypothesis is $H_0:$ You are not pregnant
the alternative hypothesis is $H_1:$ You are pregnant

so being pregnant would be the positive result.

You take a pregnancy test

if the pregnancy test gives a positive result when you are not pregnant then this is a false positive, a Type I error when the null hypothesis $H_0$ is in fact true but has been rejected by the test
if the pregnancy test gives a negative result when you are pregnant then this is a false negative, a Type II error when the null hypothesis $H_0$ is in fact untrue but has not been rejected by the test

So in a statement of being a true/false positive/negative test, the true/false part is about the accuracy of the test while the positive/negative part is about the result of the test rather than being the real situation


Issues if insurance is not compulsory for population

If insurance is not compulsory for all drivers there can be a case where the affected parties are unable to be compensated when an accident happens. In that case those affected by the accident wont have the means by which they can have their property replaced.

Insurance provides a service which guarantees a certain level of wealth for a party who is insured. Without insurance sub optimal transactions can take place making everyone worse off. This can be illustrated via an abstracted example.

Example
 1. Party A has wealth of \$50 and Party B has a wealth of \$100 and all their wealth is stored in their cars.
 2. Party A and Party B both get into a car accident.
 3. Party A destroys Party B's Car and legally owes Party B \$100
 4. without insurance Party A is unable to compensate Party B and loses all his income.

This whole issue can be avoided via the purchase of car insurance. So the law requiring all drivers to have insurance actually leads to a net increase in welfare for all involved  because it allows legally required transactions to take place.


Why does firing employees appear to increase the share price of a company?

Is firing employees to raise stock prices a good strategy for a
  company?

For the range of scenarios whereby more productivity can be obtained from employees (units of workforce), it certainly is.

By opting for layoffs, a company seeks to get rid of redundant capacity and its associated costs, such as insurance, retirement pensions, or vacations. Since labor is the production factor which can be adjusted in the short-term, the benefit from cutting labor costs is reflected in the company's financial statements quicker than when the company envisions to sell its facilities, real estate, and equipment. Market agents are mindful of that difference, whence they foresee that the company's numbers are about (or are likelier) to improve.

That "strategy" (or, said more properly, tactic) starts to fail where the remaining workforce burns out and/or gets demoralized. These conditions are evidenced in accidents; production failures; product recalls; costs of litigation for unfair labor practices or faulty products; sickness; sudden resignations that cause bottlenecks; and when excessive layoffs render the company unable to meet the demand. It also fails when managers make their firing decisions influenced by personal motives (politics: favoritism, animosity) instead of business-minded criteria.


Why was Modigliani-Miller so relevant and innovative when it came out?

People, particularly business leaders, seem to remain confused about this issue even today. At the core of is the question Is equity finance expensive?. We certainly observe in the data that the realized returns on firm debt are much lower than the realized returns on firm equity. Does this mean that firms have too much equity?

If equity capital always costs 9 percent per year and debt always 3 percent then a firm would be worth more if management kept equity to a minimum and used as much debt as possible. But in the real world there are lots of complications that might make debt cheaper than equity. Monitoring costs are real, default is costly, tax wedges are large, equity is riskier, and contracts are incomplete. It may be that factoring in these costs, which are hard to measure, provides a complete explanation for the differences in return between debt and equity. Or maybe not. If observable characteristics fail to explain this difference does this mean that firms should take on more debt? Or does it mean that these characteristics are poorly measured?

Modigliani-Miller provide a clear way of thinking about this question. In the absence of frictions, as you shrink the amount of equity and offset it with more debt the required returns adjust to leave the firm's cost of capital unchanged. But even so, there is no problem with return on debt being lower than the return on equity. So in and of itself, there is nothing to learn from the lower cost of debt compared with equity. This is a surprisingly robust result:

  The two Modigliani-Miller theorems hold good, irrespective of
  individual differences between shareholders' valuations of risk,
  leverage effects, durability of loans, etc. The logic of the theorems
  rests in fact upon the assumption of perfect markets, namely that a
  shareholder can always, through his own borrowing or lending, compose
  his asset portfolio as he sees fit and that he can, without costs,
  give it the composition he desires with respect to risk, leverage,
  etc. If for instance the risk level of a firm's assets is increased,
  the shareholders can neutralize this by lowering the risk of other
  assets in their portfolios.

Press Release: 15 October 1985 THIS YEAR'S ECONOMICS PRIZE AWARDED FOR PIONEERING STUDIES OF SAVING AND OF FINANCIAL MARKETS

This is an extremely valuable insight. The explanation is also clear and powerful. So good and powerful that it is hard to see the world any different once you hear it. In one of the Sherlock Holmes stories, Holmes says "The world is full of obvious things which nobody by any chance ever observes." In TV tropes this is called Seinfeld Is Unfunny:

  It wasn't old or overdone when they did it. But the things it created
  were so brilliant and popular, they became woven into the fabric of
  that show's genre. They ended up being taken for granted, copied and
  endlessly repeated. Although they often began by saying something new,
  they in turn became the status quo.

Additionally, it is hard to remember from the perspective of the present how devoid of theory was corporate finance in the period before Modigliani and Miller. Here's a quote from an award ceremony speech related to the Nobel award:

  Until the latter part of the 1950s, no viable theory of corporate
  financing of investment, debt, taxes, and so forth had been developed.
  It was not till Modigliani and Miller presented their theorems that
  more stringent theorizing began to appear in this field. By treating
  financing decisions within the framework of a theory of
  financial-marketplace equilibrium, Modigliani and Miller provided the
  general guidelines for continued research in this area.

Award Ceremony Speech: Presentation Speech by Professor Ragnar Bentzel of the Royal Academy of Sciences


How does Perfect Elasticity imply infinite quantity for any higher price?

"Calculus at infinity" is a useful tool in Economics also, when we have managed to represent mathematically an economic situation. For example, when we model an intertemporal utility maximization problem with an "infinite" planning horizon, we only do it because it simplifies the mathematics considerably, while the approximation error is usually negligible for our purposes, since due to discounting, a finite-horizon setup would not produce any additional insight of value. We do not really argue that economic agents have "infinite" planning horizon.

The convention to talk about "infinite supply" belongs to the same category -but in this case it does not appear to help in anything.

The reason that the Economics discipline exists is the finiteness of resources. Supply of a good or service, or anything else related to Economics cannot be infinite - so whenever we use the term we usually mean "very large"...

...and the excerpt the OP quotes from the textbook does exactly that: it talks about "price elasticity approaching infinity... meaning that very small changes in price lead to very large changes in quantity supplied". This passage takes a mathematical concept, and correctly translates it in meaningful economic terms.

More formally, "any quantity" does not mean "infinite quantity" because "infinity" is not a number, so it cannot represent a quantity (and let's not confuse quantity with the concept of cardinality used in set theory). The expression $x\geq 0$ does not include "$\infty$". $x$ belongs to the real numbers, and $\infty$ is added to the real numbers to create the "extended real numbers" - it is not a real number itself.

Perhaps by "logical extension", if all non-zero real quantities are supplied at USD $4$, then what is left for higher prices is only "infinity". But this has no economic meaning -apart from the approximate one already mentioned in the passage quoted.


As you noted correctly, it has something to do with the costs.  An important point here is a cost of producing an additional unit, and not average cost.

Let me give the following example.  Suppose that you are an owner of a bakery, and you have a single employee who works $8$ hours and produces $8$ breads a day ($1$ bread an hour). Assume you have to pay your employee $\$1$ an hour for the first $8$ hours and $\$1.50$ (overtime) for each additional hour (after $8$ hours), and you sell each bread at $\$1.20$.

One day you think to yourself: why not to produce and sell $9$ breads (you come to this thought by the same reasoning as in your question)?  Then, you realize that in order to produce the $9$th bread, you need to keep your employee working for an additional hour and have to pay overtime, namely $\$1.5$ for the $9$th hour.  However, you will still sell the $9$th bread for $\$1.20$, which means you will lose $\$0.30$ from producing and selling the $9$th bread.

Suppose that the price of bread has increased to $\$1.60$.  Now, you surely want to keep your employee for the $9$th hour (or even more), since for each additional bread produced (after $8$ breads), you will make $\$0.10$ profit per bread.  So, you can see that an increase in price induced you to supply more.


Limits to Growth: Is environmental collapse by 2030 a likely scenerio?

We know one core element of the Guardian article is nonsense - the discussion of Peak Oil.

The Peak Oil lament, fashionable a decade ago, was that we didn't have enough oil, and that was going to become a major problem very soon.

However, we know from climate science that the reverse is true - we have far too much fossil fuel reserves (disclosure - that's a paper by my colleagues). Our challenge isn't that we won't extract enough to continue economic growth - it's that we'll extract too much and cause a major environmental collapse. Turner's report makes a significant show about the age of cheap and easy oil being over (pp 11,12) - and that's a prediction that was made to look very silly, quickly, as the report's publication in August 2014 happened just as the 18-month oil-price crash caused by huge global over-supply was beginning.

Is environmental collapse possible? Yes.

Are we causing huge ongoing damage to the environment? Yes.

Were all of the significant predictions of the Club of Rome correct? No. Not only has geometric population growth stopped, but even arithmetic growth is in decline - and the growth wasn't capped by disaster, but by better healthcare, better access to contraception, and more education and emancipation of women. Food production per capita has way exceeded predictions. And the renewable alternatives to fossil-fuel energy are already economic and scalable for static applications, and becoming so for transport too.

Are we still on a trajectory of yesterday's brown economy, that will require a trajectory change if we are to avoid environmental collapse?
Yes.
Do we have enough evidence to say that the date for that collapse is 2030?
No.
Do we have the technical means to transition to a modern, clean economy and avoid that collapse?
Yes.


Are there data for the evolution of global economic inequality

It is possible, but it involves consiberable work as you need to collect underlying income distributions across the world or to be able to estimate them.

This has been done, for example in a World Bank policy research working paper 6719, Global Income Distribution From the Fall of the Berlin Wall to the Great Recession by Christoph Lakner and Branko Milanovic in December 2013. It included this chart of global income distributions:

In fact the real story here is the reduction in the number of very-low-income people in absolute terms, especially in countries such as China over 20 years, plus the rise of middle incomes (in a global sense).

It is possible to use this to estimate Gini indices and the paper gives the following estimates:

1988: 72.2
1993: 71.9
1998: 71.5
2003: 71.9
2008: 70.5

Points to take from these numbers are

global inequality is wide with a Gini index higher than almost all countries or regions
the numbers suggest a global reduction in inequality even when inequality had been increasing within most major countries (especially China)
the reduction has been small and can easily reverse (either in the future or on re-estimation of the past)

Regional trends and other inequality measures are in Table 3 of the paper, while other people's estimates of a global Gini index are in the Appendix Table 1.


Where can I read more about formula (G-T)= -(S-I) -(X-M)? And how is this formula called?

Since you already answered yourself I am just filling the textbook recommendation:

You could  find more on this in Blanchard et al. Macroeconomics: an European Perspective. You will find this there (assuming based on the question you are looking something at undergraduate level).


As you pointed out the diagram is used to illustrate that Production (or more appropriately output) = Income. In this case it equates it to consumption but that is not generally true of all circular flow diagrams.  However, besides that it has many other uses as well as it is an easy way of provide a bird’s eye view of all important flows in the economy.

Also you cant judge a tool only based on its most simplified version. A more involved versions of circular flow diagram as for example the one taken from Krugman's macro textbook that you can find below the text also include financial markets (where people can save), government, international sector and so on.

The point of these is to provide birds eye view of the economy, to show students that in economics everything is interconnected (and you would be amazed by how many people do not realize that), and also it also highlights the specific interactions that occur between different sectors of an economy. The circular flow also is a good and easy way how to introduce 101 econ students to the idea that in macroeconomics income is endogenous, I seen it also be used as illustration of how money multiplier works.

  Thoughts: It seems to me that these circular flow diagrams are just some cutesy device that some textbook writer long ago first came up with, and which other textbook writers have simply and mindlessly stuck with.

In a sense yes (except for the mindless part) because circular flow is just a didactic tool - you compare it to a water cycle and thats actually a good comparison. You can also describe water cycle in terms of physical processes in a mathematical model and you can do the same with circular flow which is ultimately based on the national income identity and related macro general equilibrium models but that might be too intimidating for highschool/undergraduate 101 students. The tool does not get included in textbook because professors mindlessly copy it but because its genuinely useful classroom device.

Arguably the circular flow does not have much use outside classroom or self-study, but  that does not mean that it is not a useful or significant tool as it helps students who's math skills might not be yet ready for mathematical presentation understand important concepts in macroeconomics. In most fields you will find that there are tools that outside classroom have no much use yet I would argue thats enough significance in itself. I would also not say it has any less significance than water cycle which is again mostly just teaching/visualization tool. You can compare it to other tools, for example when I was studying math and we were covering topology our professor would bring us examples of Klein bottle, 3D printed tesseract and other 'cutsy toys' - ultimately they were literary just cool looking paperweights but I found them useful when learning topology and I am sure other students did as well.  The circular flow just serves similar purposes and makes teaching as well as learning easier.


Converting per capita growth rates for subgroups into contribution to total per capita growth rate

The problem is that you've got two inconsistent definitions of growth rate.

One is driven by per capita debt across the whole population.

The other by per capita debt within each subgroup.

These are inconsistent, because different subgroups have different baseline debts in Q2, and different population sizes. Because both of those things are different, and both are divisors in the calculation of your statistic of interest, then simply re-weighting by population size can't fix the problem.

So you can calculate a meaningful contribution of each to overall growth rate, so that together they satisfy  r1 + r2 +r3 + r4 = r0. But then the individual r1 to r4 won't mean much within their own groups - only in terms of their contribution to the whole. Or you can calculate growth rates for each sub group, as you've done in your question, but then they won't mean much in the overall context.


In this kind of cases, I think you can not use a simple OLS estimation because of endogeneity of your variables.

Probably, there would be causality between prices and sales data, this means that sales are influenced by prices but also prices are influenced by sales. So, in this case, in order to do a non-biaised estimation, you should use instrumental variables in order to avoid endogeneity issues between your variables.


(Why) was von Neumann not a fan of the Nash equillibrium concept?

Is there any more information about why von Neumann had this attitude?

There must be, but I have never seen it.

  Or can we infer a reasonable answer?

Here is one possibility based on hearsay among the Game Theory community.

Von Neumann thought that there was a sharp distinction between zero-sum games and other games.

In zero-sum games, there was no hope of cooperation. Therefore, it makes sense for players to engage in selfish, individualistic decision making. It is natural to assume that players choose what to do independently and with their best interests in mind. The resulting solution concept for such games is the concept of minimax equilibrium, which coincides with the concept of Nash equilibrium.
In other games, there is usually at least some room for mutually beneficial cooperation. It makes sense for rational people to talk to each other and try to find ways to cooperate. Hence, it is not evident that players must choose what to do independently and with only their best interests in mind. The way I've heard the story is that, for such games, Von Neumann favored the approach of Cooperative Game Theory in which decisions could be made collectively by groups of people.

I will add a further note. In the 1950s, there was no empirical evidence to support either hypothesis. In retrospect, we do see in practice that people often cooperate when they face prisoners' dilemmas. Also, the most widely cited field evidence supporting equilibrium behavior is from competitive settings that are close to zero-sum games (penalty kicks, tennis serves).


When were the First and Second Welfare Theorems proven?

Perhaps you have heard the competitive equilibrium referred to as the Arrow-Debreu equilibrium.

The idea of the theorems existed earlier but Arrow introduced the theorems formally in a way that is equivalent to what we today call the first and second welfare theorem in 1951 in his paper

AN EXTENSION OF THE BASIC
THEOREMS OF CLASSICAL
WELFARE ECONOMICS.

Debreu is more associated with proving that the equilibrium exists under certain conditions. If you think about it this makes the first welfare theorem much stronger. His seminal paper on the subject is

The Coefficient of Resource Utilization.


Difference-in-Difference (DID) Regression with Non-Stationary (but Cointegrated) Treatment and Control Groups

If your question is the broad: "How do I estimate the differences between these two groups under these problematic contexts?" I recommend taking a look at event study literature, which has a lot of similarities to the DiD literature, but is frequently applied in the context of stocks (which can be non-stationary).
MacKinlay, A. Craig. "Event studies in economics and finance." Journal of economic literature 35.1 (1997): 13-39.


Any research done on relationship between fertility rate and wealth?

Population changes slowly over time but has large effects in the long-run.
Here are some well-studied facts about fertility:

Before the industrial revolution, the rich had more surviving children than the poor.
The transition from stagnation to economic growth is accompanied by a demographic transition from high to low fertility.
Today, both within and across countries, rich and educated mothers have less children than poor and unskilled ones.
Most of the literature finds that the income of the father positively affects fertility, while the income of the mother negatively affects fertility.

Economists would stress that one explanation among others is Parents' income: Opportunity cost of child-rearing time is high for high income/education mothers. They are other reasons.

David de la Croix works on this topic and offers interesting lectures on Fertility, Education, Growth and Sustainability here. You will find references for the above facts and other reasons for the decline in fertility.

An academic reference is: De la Croix D. and M. Doepke, Inequality and growth: why differential fertility matters, American Economic Review, 93, 1091-1113, 2003.


http://econsmalaysia.blogspot.my/2013/05/gdp-deflator-inflation-consumer.html. Here read this. Its almost a non-related type of measurement
From the article:

The differences in calculation are quite substantial and meaningful – they’re not looking at the same thing. In fact, calculating the GDP deflator is fairly convoluted as these things go – first tabulate all goods and services in current prices, then measure the same goods and services in prices prevailing in the base year, then calculate the ratio of current price production to constant price production. This ratio (multiplied by 100) is the GDP Deflator index, from which growth rates (inflation) can be calculated.
The CPI on the other hand, is based on changes in prices only, as the volume of goods consumed by a “representative” (average) household is taken to be fixed. The index is just a weighted average of the changes in prices across the same basket of goods.

As an example, below are the GDP deflator and CPI for Malaysia, which show weak correlation

The key thing to understand is that while both can be affected by broad macro policy (like loose money), they do not neccessarily always drive each other, as the GDP deflator could be driven by a surge in export commodity pricing that is largely captured by foreign firms with little corresponding increase in pay to local workers and consequently little increase in CPI.
The relationship would have to be examined in terms of a single economy because the weight of outside factors confounds the relationship between the CPI and GDP


Cross price elasticity of demand vs Price elasticity of demand

Not necessarily.

For example if the utility function is $U=\min\{x,y\}$, then the demand function for x is given by $D=\frac{M}{P_x+P_y}$.

The own price elasticity of demand for x is: $-\frac{M}{(P_x+P_y)^2} \frac{P_x}{x}$.

The cross price elasticity of demand for x is: $-\frac{M}{(P_x+P_y)^2} \frac{P_y}{x}$.

If these elasticities are evaluated at $P_x=P_y=p$, then for any value of x, you have the same elasticities.


Is poverty rightly measured internationally?

This measure of poverty depends on your definition of basic needs and local prices. Argentina is a good example for how local prices matter. It is widely accepted that official price and inflation statistics published by the Argentinian government are cooked and inflation rates in the last few years are have been substantially bigger than official numbers suggest. I assume that the world bank has to rely on those statistics and it is therefore likely that its  estimates of poverty in Argentina are too low, simply because the price level adjustment is not done with good data.

Alberto Cavallo has a nice paper showing the discrepancies of official statistics with other measures of Inflation in Argentina, see here.

Living on 90$ a month is obviously not nice, but around the world many people do. This measure is intended to document the extent of  extreme poverty in developing countries, i.e. the circumstances where you are literally close to starving. Many more developed countries use a much higher poverty line in their domestic statistics. The world bank as an international body publishes statistics on development that are internationally comparable, and hence uses the same threshold for each country.

The World Bank's extreme poverty line used to be much lower - I believe it started out at 1$ per day around 1990. You can read up on the intention of publishing such a statistic, and how the threshold is chosen in this interesting blog post by a World Bank economist.


Prove all cost functions are concave in input prices and demand for inputs is downward

Let $x(w, q)$ denote the solution to the cost minimization problem :

\begin{eqnarray*} \min_{x} & \ w\cdot x \\ \text{s.t.} & \ \ f(x) \geq q \end{eqnarray*}
where $f$ is the production function.
Since $x(w, q)$ minimizes cost at $(w, q)$, following holds for all $w$ and for all $q$ :
\begin{eqnarray*} w\cdot x(w, q) \leq w\cdot x(w', q) \ \ \  \forall w' \end{eqnarray*}

Also we know that cost function is cost of cost minimizing input choice :
\begin{eqnarray*} C(w, q) = w\cdot x(w, q) \end{eqnarray*}

First, we'll show that $C$ is concave in $w$. Consider any arbitrary $w'$, $w''$ and an arbitrary $\lambda \in [0, 1]$.
\begin{eqnarray*} C(\lambda w' + (1-\lambda) w'', q) & = & (\lambda w' + (1-\lambda) w'')\cdot x(\lambda w' + (1-\lambda) w'', q) \\  & = & \lambda w' \cdot x(\lambda w' + (1-\lambda) w'', q) + (1-\lambda) w''\cdot x(\lambda w' + (1-\lambda) w'', q) \\  & \geq & \lambda w' \cdot x(w', q) + (1-\lambda) w''\cdot x(w'', q)  \\  & = & \lambda C(w' , q) + (1-\lambda) C(w'', q) \end{eqnarray*}
Therefore, $C$ is concave in $w$.

To show that input demand is downward sloping, consider any arbitrary $w'$, and $w''$,
\begin{eqnarray*} w'\cdot x(w', q) & \leq & w'\cdot x(w'', q) \\ w''\cdot x(w'', q) & \leq & w''\cdot x(w', q) \end{eqnarray*}
Adding them we get
\begin{eqnarray*} (w'-w'')\cdot (x(w', q) - x(w'', q)) & \leq & 0 \end{eqnarray*}
We now consider a special case where $w'$ and $w''$ differ only in the $j$th input's price. Consequently,
\begin{eqnarray*} (w'-w'')\cdot (x(w', q) - x(w'', q)) = (w_j'-w_j'') (x_j(w', q) - x_j(w'', q)) & \leq & 0 \end{eqnarray*}
Therefore, demand for $j$th input is inversely related to its price.


How do you benefit from a fall in interest rates in general as a creditor of public bonds?

This is because bond prices increase when interest rate falls.
A general bond pricing formula can be expressed as as a discounted stream of future cash flows from the coupon and the value of the bond itself. Hence we have:
$$ P = \sum^T_{t=1} \frac{C_t}{(1+i)^t}+\frac{M}{(1+i)^T}$$
Where $P$ is the price of the bond, $C$ is the coupon $i$ the interest rate, and $M$ the value at maturity (i.e. the face value). You can also learn more about the formula here if you are interested.
As you can see from the above formula when $i$ increases $P$ decreases and vice versa.  Hence fall in the interest rate is good for you if you are a bond holder as now you can sell your bond on secondary market for more than before.


Suppose you write some software that you can then freely sell at practically no cost per unit (the wonders of the internet). You want to make as much profit as possible. Since you have almost no per unit cost, maximizing your profit will amount to maximizing the revenue, the price per unit times the number of units sold.
Note what did not enter the calculation: The cost of developing the software. How much money you might eventually get might influence whether you put in some effort to create the software, but it will not influence your pricing decision once you made the software.
The companies bidding for spectrum are essentially in the same situation. Once they have gotten their spectrum rights, how much they paid will not influence how much they charge..
Of course someone has to pay, but these are companies bidding for monopoly rights. Apart from the cost, the license gives them a huge profit and this profit is what the cost is paid from.


Why would a commercial bank borrow from another commercial bank instead of the central bank?(more questions inside)

Yes, the “overnight rate” in the inter-bank market would be 4.75%, based on that description. The idea is that banks borrow from each other, not the central bank; the central bank is just providing back up facilities to cap the interbank rate. The objective is that banks need to convince other private sector entities that they are solvent, and not just the central bank.
The most important inter-bank market for domestic USD transactions is the Fed Funds market, but there is also the London inter-bank market.
This question was vague. Keeping the lending facility at a higher rate than the inter-bank rate keeps banks borrowing from themselves, for the reason noted earlier. The spread matters, not the absolute level (which was possibly implied by the question).


Why are some economists remarkably productive?

A short answer to your question is both yes and no. You have to first define productivity: productivity, in my opinion, cannot simply mean publishing the number of articles, but rather, their impact. A typical way of modelling the impact of authors is by looking at their h-index. There are many other ways of measuring productivity as well- where the articles are published is important as well. The top 5 journals in economics are (in no particular order): Econometrica, AER, QJE, JPE and ReStud. Articles published in these journals are generally of high quality. However, there are many articles that are published in these journals that do not always have the desired impact. Many a time, articles ranked in lower journals have a high impact. Consequently, it is extremely difficult to measure productivity. To me, the h-index is a good measure.


The Fed has a $4.5 trillion balance sheet: where did they get that money?

"Money" in this case is a liability of the Federal Reserve - mainly reserve balances held at the Fed, but also curency in circulation (which was largely stable).

Like any other entity, the Fed creates liabilities by issuing them. It did this by buying bonds in the open market (mainly Treasury and mortgage securities).

The seller of the bond (more likely, its bank) will get a deposit ("money").
The Fed gets an asset (the bond).

As people have found out, there is little to stop the Fed from doing this. They get matched assets and liabilities, and the assets it buys pays a higher rate of interest than its liabilities.

Other entities have to worry about funding risk when conducting such trades -- others may fear bankruptcy, and not lend them money. Very few people worry about the central bank running out of money.


Does the GDP contribution of a service sector represent the gross profits of that sector?

The second way of estimating GDP is to use "the sum of primary incomes distributed by resident producer units".
Gross domestic product on Wikipedia (emphasis mine)

If GDP is calculated this way it is sometimes called gross domestic income (GDI), or GDP (I). GDI should provide the same amount as the expenditure method described later. (By definition, GDI = GDP. In practice, however, measurement errors will make the two figures slightly off when reported by national statistical agencies.)

This method measures GDP by adding incomes that firms pay households for factors of production they hire - wages for labour, interest for capital, rent for land and profits for entrepreneurship.

So no, it is missing (at the very least) the wages and interest paid on the debt of the financial industry.


Market Clearing with Dividend and Investment

Look at textbook definitions of equilibrium in macro models of General Equilibrium with production and investment. Basically, the Equation $Y_t=C_t+I_t$ that says that the amount produced at a certain period must equal the consumption plus investment in that period is the relevant market-clearing condition in the market for goods.


Weak axiom of Revealed Preference application

Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set.

This is not true. Instead WARP would say that it is not possible for Andrew to be chosen whenever both Andrew and Barbara are in the choice set, which is different than your claim above.

Just because $b$ is chosen from the set $\{a,b\}$ doesn't mean that $b$ should always be chosen, e.g. from the set $\{b,c\}$. It just means that $b$ is revealed-preferred to $a$, but it doesn't mean that $b$ is preferred to all other possible alternatives (in particular, $c$). For example, the preference $c\succ b\succ a$ is perfectly consistent with the revealed choice pattern of the firm.

Write out the two conditions explicitly:
$$
\begin{align}
C(\color{red}{\varnothing},\color{red}{\{a\}},\{b\},\color{red}{\{a,b\}})&=\{b\}\tag{1}\\
C(\color{red}{\varnothing},\color{red}{\{b\}},\{c\},\color{red}{\{b,c\}})&=\{c\}\tag{2}
\end{align}
$$
$(1)$ just says that whenever the four choices --- hire no one, hire $a$ only, hire $b$ only, hire both --- are present, the firm hires $b$ only.
Likewise, $(2)$ says whenever the four choices --- hire no one, hire $b$ only, hire $c$ only, hire both --- are present, the firm hires $c$ only.
These two are consistent with WARP (for $(1)$, the x in your quoted definition is $a$ and y is $b$; for $(2)$, x is $b$ and y is $c$).

For restrictions on $C(2^{\{a,b,c\}})$, writing it out explicitly, we get
$$
C(\color{red}{\varnothing},\color{red}{\{a\}},\color{red}{\{b\}},\color{black}{\{c\}},\color{red}{\{a,b\}},\color{black}{\{a,c\}},\color{red}{\{b,c\}},\color{black}{\{a,b,c\}})=\{\{a\},\{a,c\},\{a,b,c\}\}
$$
where the elements colored in red are revealed to be inferior according to $(1)$ and $(2)$.


why not use $Ap - p$ instead of $p - Ap$?

$p$ represents total production.

$Ap$ represents the intermediate goods and services used in production, i.e. intermediate consumption.

So $p-Ap=(I-A)p$ represents net production, i.e. output minus input.


What is the point of monetary policy if most monetary variables are determined by non-monetary factors?

I do not know of any serious economist who would think that monetary variables are not to great extent determined by the decisions of central banks.
Monetary quantities depend on what the price level is which is in turn determined by the money market equilibrium. The money market equilibrium, in its simplest form is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing (so $\%$ changes in right hand side variables give us the $\%$ change in $P$) we get:
$$\ln P = \ln M + \ln V - \ln Y. $$
First observation that you can clearly see from the above is that technology growth or trade do not generally cause inflation. Both trade and technological growth expand production possibilities of our economies and hence they generally lead to higher real output $Y$. This causes deflation not inflation.  Inflation is positive change in price level but increase in output causes decrease in price level.
The velocity depends on what nominal interest rate is so this can be affected by central bank, although velocity can depend also on other factors so it is not fully under its control.
What the money supply is, is completely exogenously given by central bank. Some countries can have institutional restrictions/caps on how and when central bank can increase money supply but these are not economic restrictions and in principle central bank can set $M$ to any value it wants on interval $[0,\infty)$.
Setting the theory aside empirically there is strong relationship between inflation and changes in the money supply. For example, consider the two scatterplot below (from Mankiw Macroeconomics pp 91-92). The first plot shows the relationship between money supply growth in the US and inflation in the US at different points in time (figure 1) and the second one does that internationally for averages for period 1999 to 2007 (figure 2). As you can clearly see data indicate very strong positive relationship. In addition you can see the same strong relationship between nominal interest rates and inflation. These data support also the theory from the above. Of course, correlation does not immediately tells you what the direction of causality is (but these are differenced data so the correlation is definitely not spurious and indicative of some relationship), but there are further academic studies that show that there is without doubt causal relationship between money supply and inflation, and also interest rates and inflation (for example see Maune,  Matanda, Mundonde, 2020;; Barsky, 1987; Mankiw 2019; Romer: Advanced Macroeconomics 4th ed; Grauwe & Polan, 2005; and sources cited therein).
Appendix:
Figure 1:Historical Data on U.S. Inflation and Money Growth

Figure 2: International Data on Inflation and Money Growth

Figure 3: Inflation and Nominal Interest Rates Across Countries


Would a society of cannibals be self-sustaining?

No, an animal population that subsist solely by consuming its own members is not sustainable.  This is easy to see because there's no external energy input into the population (all energy comes from other members of the same population), yet the individuals will necessarily expend some energy in their metabolism (turning it into heat).  Eventually, if the population cannot find any other food sources, the individuals will die (and be consumed) one by one, until the last one will run out of food and starve.

That said, there are animal populations in nature where some of the individuals may subsist almost entirely on cannibalism.  This is typical of e.g. fish, which are very tiny when born, grow gradually over their lifetime, and tend to consume prey smaller than themselves.  Thus, in small lakes that happen to host only one species of fish, it's not uncommon for the large adult fish to mostly eat smaller juvenile fish of the same species (which in turn eat plankton and invertebrates, providing an external source of energy to the system).


The two Banks of the United States (the First and Second) were nothing like the modern Federal Reserve system.  For example, the First was prohibited from buying government bonds (one of the main roles of the Federal Reserve system is to buy and sell government bonds).  Further, neither of the national banks had any role in regulating the banking system.  They were simply the only interstate banks allowed.  All other banks had to be confined to one state.  The national banks primary impact on the money supply was to require smaller banks to pay them in silver and gold to redeem checks deposited in the national banks.

By contrast, the Federal Reserve system actively regulates banks, requiring them to maintain a percentage of their deposits as reserves.  It actively sets two rates:  the rate at which it loans money to banks to cover reserve shortages (called the discount rate) and the rate it pays on excess reserves.  The latter power is relatively new.  Previously the Federal Reserve did not pay interest on excess reserves.  The Federal Reserve also sets a target for the funds rate, which is the average rate at which banks loan money to each other.  The funds rate target is what most people mean when they refer to the Federal Reserve setting interest rates.

The Federal Reserve has two main ways to affect the funds rate.  First, it publishes a target rate for the average.  Second, it buys and sells government bonds in what are called open market operations.  When it buys bonds, it adds money to the system, reducing the number of banks that need to borrow and increasing the number of those with money to lend.  This pushes the average rate to fall, as there are more lenders and fewer borrowers.  When it sells bonds, it pulls money from the system, reducing the number of banks with money to lend.

The Federal Reserve can also compete with lenders by loaning money to banks at the discount rate.  This decreases the number of banks looking to borrow from other banks, which would push rates down.  It rarely does this however.  Most banks prefer to borrow from other banks, as there are fewer restrictions.

The Federal Reserve also has two ways to influence how much banks keep in reserve.  First, it sets an explicit reserve requirement.  Every bank must keep that much in reserve.  Second, it can pay interest on excess reserves.  The higher that rate, the more likely banks are to keep excess reserves instead of loaning them to individuals, companies, or other banks.  This would push up the funds rate average, as fewer banks loan.

Again, the two Banks of the United States were not central banks the way that the Federal Reserve is.  Yes, they could print banknotes, but so could other banks of the time.  They had no special banking privileges.  They were simply the only interstate banks allowed during their charter.


Here is what I think should be the answer based on your problem description. Otherwise this is hopefully still helpful for you to solve the problem.
I star by considering the model equations
$$(1) \ \ \ y = \beta_0 + \beta_1 x + e_y $$
$$(2) \ \ \ w = \lambda_0 + \lambda_1 x + e_w, $$
and then I manipulate equation (1) to get
$$(1b) \ \ \ y = \beta_0 + \frac{\beta_1}{\lambda_1} ( \lambda_1 x) + e_y, $$
from equation (2) I then find
$$(2b) \ \ \ w -\lambda_0 - e_w=  \lambda_1 x,$$
which I insert in (1b) to get
$$y = \beta_0 - \frac{\beta_1}{\lambda_1}\lambda_0 + \frac{\beta_1}{\lambda_1} w - \frac{\beta_1}{\lambda_1} e_w+ e_y,$$
I then define $b_0 :=\beta_0 - \frac{\beta_1}{\lambda_1}\lambda_0$ and $b_1:=\frac{\beta_1}{\lambda_1}$  and $u :=  - \frac{\beta_1}{\lambda_1} e_w+ e_y$ to get the estimation model
$$y = b_0 + b_1 w + u$$
where I know the probability limit of $\hat b_1$ is
$$var(w)^{-1}cov(wy) = b_1 + var(w)^{-1}cov(wu),$$
where I note that $cov(wu) = cov(w,-(\beta_1/\lambda_1) e_w+ e_y) = -\frac{\beta_1}{\lambda_1} var(e_w)$.
I assume that $cov(x,e_w) = cov(x,e_y) = cov(e_w,e_y) = 0$.
Here is a small test code in R
N <- 1000
beta <- 1
lambda <- 2
x <- rnorm(N)
e_y <- rnorm(N)
e_w <- rnorm(N)
y <- 1 + beta*x + e_y
w <- 1 + lambda*x + e_w

model <- lm(y~w)

W <- cbind(rep(1,N),w)
bias <- -(beta/lambda)*solve(t(W)%*%W)%*%t(W)%*%e_w
coef(model)[1] - bias[1,1]
1 - (beta/lambda)*1
coef(model)[2] - bias[2,1]
beta/lambda


The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form:
$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$
This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the utility function that leads to the Linear expenditure system.
To see that it is linear homothetic notice that:
$$
\begin{align*}
u(\beta \tilde x + a, \beta \tilde y + b) &= (\beta \tilde x + a - a)^\alpha (\beta \tilde y + b - b)^{1-\alpha},\\
&=\beta (\tilde x)^\alpha (\tilde y)^{1-\alpha},\\
&=\beta ((\tilde x + a) - a)^\alpha ((\tilde y + b) - b)^{1-\alpha},\\
&= \beta u(\tilde x + a, \tilde y + b).
\end{align*}
$$
It could be that there is more work on utility functions with subsistence levels that lead to other preferences that are also linear homothetic.
For example you could define a CES utility function with subsistence levels:
$$
u(x,y) = (\alpha_x(x -a)^\sigma + \alpha_y(y-b)^\sigma)^{1/\sigma}
$$
This will also satisfy linear homogeneity. This paper of Baumgärtner, Drupp & Quaas does something like this.
In general, if you take any homothetic utility function $u(x,y)$ then the modified 'subsistence-augmented' function:
$$
\tilde u(x, y) \equiv u(x - a, y - b),
$$
will be linear homothetic.


What do you call a game that is both simultaneous and sequential?

A sequential game with imperfect information?


How can we incentivize volunteer rescue forces like the Cajuan Navy which assisted in Louisiana?

The United States Code provides for such a force and I believe Louisiana has one.  They are usually called "state defense forces."  They are a parallel state force with the state components of the National Guard.  They are not subject to call up by the President and serve in policing and emergency management roles in the states.  Not all states have them.  In some states they are both voluntary and require you to provide your own equipment, in others, they are fully funded.

I believe the Military Department of Louisiana has or had such a structure.  Nothing prevents the State of Louisiana from providing either pay or tax incentives to organize and manage the membership and to permit it to exist broadly.  Considering the frequency of natural disasters in Louisiana adding formal training and incentives to such a force would benefit all parties involved.

Imagine the effectiveness of the Cajun Navy if it had received formal training and worked within a planned response.  In most states, these are token forces primarily made up of ex-military who wish to continue service, but no longer wish to be in a potential combatant role.  They serve a small, but often important role, in mitigating the immediate effect of a disaster, but I don't think the states actually think of organizing any large number of people.  Most likely this is because they exist, but are so small that legislators forget they exist and don't expect much from them.


How exactly did Venezuela's economy fail?

I think you summarized most of it quite well. Few things that you missed:

There was a host of bad macro and microeconomic policies. For example, nationalization of many firms, price controls, subsidizing many uneconomical activities etc. Also actually the investments into oil industry and productivity declined in Venezuela after the industry got nationalized so that part was not fully correct, and Hugo Chavez was appointing people who were making it worse (see  thisforeign policy article).

Also when it comes to social policies, it’s definitely possible and legitimate to want to have Ralwsian welfare state - this is matter of philosophy and value judgements not economics. However, this does not mean that a country can just arbitrarily sets its social policies and ignore resource constraints. Venezuela was running large fiscal deficits even in their best times when oil prices were high save for few years with surpluses.

Moreover, one thing that many states that are running out of money resort to is printing more of them which causes inflation. Now moderate inflation is fine but if you are running large deficits that you can’t really hope to repay with future economic output you need to start printing so much that it will eventually lead to very large inflation which will in turn only accelerate collapse. Once money is worthless people start resorting to barter which is less efficient and just exacerbates the problems further.


Can a position at a hedge fund be securitized?

Actual securitization would be impractical for a number of reasons, but secondary market transactions of shares in hedge funds do occur with the consent of the funds in question (though it’s not a particularly liquid market). Using these shares as collateral would not generally be practical, as they’re not assignable interests.


Caveat: my familiarity with the international agreement literature is limited to that of environmental and resource economics problems.

For those cases instead of a utility function the usual approach is to formulate the problem in terms of costs and benefits for countries' economies as a whole.

When it comes to environmental problems this usually boils down to individual benefits of pollution versus collective damages, or its inverse individual abatement costs versus collective benefits. For resource problems it is individual profits (or consumption) from harvest versus the collective diminishing returns in terms of stock growth or increasing harvest costs.

Active authors in this field are a.o. Scott Barrett, Carlo Carraro, Michael Finus in terms of environment and Marco Lindroos and Pedro Pintassilgo for fisheries.


How to treat different frequencies of a time series data?

CPI is a stock while GDP is a flow. Re-sampling of stocks to higher frequencies can be approximated with a number of choices, but probably the most common is linear interpolation. In some contexts filling forward and filling backward are common. But in principle you could fit any function you want through all the points you have and use the values from the function instead. Flows are a bit more complicated because annual flows are the sum of monthly flows. Dividing the annual flow into twelfths across the entire year is analogous to filling forward or filling backward (depending on what month of the year is the date of annual measurement). For example, if GDP is 144 and measured in December of 2018, assigning every month in 2018 a value of 12 is similar to filling backwards. But again, as long as you remember that the flows are supposed to add up across months to equal or approximate annual flows you can you a number of functional forms to approximate them.

This is a big topic. I recommend searching for papers related to "interpolated time series". Resampling and Subsampling for Financial Time Series by E Paparoditis, DN Politis might be a good, if advanced, place to start. Modeling and Forecasting Time Series Sampled at Different Frequencies by Casals, Jerez, and Sotoca is another useful reference. In comparing the effectiveness different methods, Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment by Lepot, Aubin, and Clemens provides a nice list of methods but doesn't focus on the stock versus flow distinction.

Deterministic methods (fill forward or back)
Nearest-neighbor
Polynomial interpolation
Distance weighting methods
Fourier transformation based methods
Regression models
Auto-regressive models
Machine learning based methods
Kernel methods
K-nearest neighbors
Box-Jenkins models
Kriging-based methods


There are several issues here, but the three most significant are:

The National Accounts treatment of the public sector, which is set out in international standards and controlled by the Office for National Statistics.  This is what you might naturally look to if you wanted to compare the size of the UK's public sector with those of other countries, or its impact on GDP or other macroeconomic indicators.  Examples of ONS data include https://www.ons.gov.uk/economy/governmentpublicsectorandtaxes and https://www.ons.gov.uk/economy/governmentpublicsectorandtaxes/publicspending/datasets/esatable2mainaggregatesofgeneralgovernment
Her Majesty's Treasury's control of public expenditure for budget management purposes, which includes passing on money voted by Parliament.  The Treasury has a natural tendency to try to minimise this, and so often encourages parts of government to obtain resources in other ways such as levies and fees, and sometimes allows these to be deducted from the control totals it applies in a way not deemed economically meaningful by the Office for National Statistics. Total Managed Expenditure is part of these control totals.  Sources of information include https://www.gov.uk/government/statistics/public-expenditure-statistical-analyses-2016
Individual parts of government managing their own budgets, primarily ensuring that money coming in  balances money going out.  This can include receiving money from or giving money to other parts of government, which as internal transfers would not count as public expenditure from an economic point of view, but do matter to an individual body

The Treasury paper you link to is largely trying to describe some of the differences of treatment between (1) and (2), explaining phases in its analyses such as "Receipts treated as negative DEL [departmental expenditure] but revenue in National Accounts".  Meanwhile, in the case of the Agriculture and Horticulture Development Board, it gets its money from the industry either in the forms of levies or commercial income and is an example of (3); in this case the Treasury interest is largely in ensuring the board does not become a burden on public funds and that it operates efficiently, while the ONS is interested because it how it is funded (it is similar to a tax as the levies are compulsory, but it is different in that the board uses this money solely to provide a explicit service to the levy payers)

If you are looking to measure the size of the UK public sector as a whole, you may find the ONS figures in (1) most useful, though they may not give the detail you can find from (2) or (3)


Unable to determine price ratio for a pure exchange economy

Your analysis seems to be spot on, except for some minor typos:
$$
x_A^* = 100 \frac{p_x + p_y}{p_x + p_y} = 100,
$$
because the value of A's endowment is $100 \cdot (p_x + p_y)$. And you could include the case of
$$
x_B^* \in [0,50]  \mbox{ if } p_x = p_y.
$$

As you state at the end the equilibrium price vector is not unique, there is equilibrium for all $\frac{p_x}{p_y} < 1$, because with this price vector we have
$$
(x_A^*,y_A^*) = (100,100), \hskip 20pt (x_B^*,y_B^*) = (50,0)
$$
and the market is then indeed in equilibrium as
$$
x_A^* + x_B^* = x_A + x_B, \hskip 20pt y_A^* + y_B^* = y_A + y_B.
$$
The price ratio $\frac{p_x}{p_y} = 1$ also results in equilibrium, because one of the optimal consumptions of B is still $(x_B^*,y_B^*) = (50,0)$.


A maximization program expressed by per capita variales - Ramsey model

Take the differential equation for the natural resource. I will write $S \equiv sL$ where lowercase is the per capita magnitude. So

$$\dot{S}=\left(1-S\right)S-\gamma AK$$

$$\implies \frac {d(sL)}{dt} = (1-sL)sL - \gamma AkL$$

$$\implies \dot sL + s\dot L = sL - s^2L^2 -\gamma AkL$$

Divide by $L$ to get

$$ \dot s + sn = s - s^2L -\gamma Ak$$

and using $L=e^{nt}$ and re-arranging we get

$$ \dot s = (1-n) s - s^2e^{nt} -\gamma Ak$$

Setting this equal to zero, we have a quadratic polynomial in $s$

$$\dot s = 0 \implies e^{nt}s^2 - (1-n) s + \gamma Ak=0$$

We see that the leading term increases as $t$ increases. The only way to maintain equality with zero, and constant $s$, is if $k$ decreases as time passes (and this will be only temporary since $k$ cannot turn negative).

So it appears that you cannot have a "steady state in per capita terms" for all variables here, consumption, capital and the natural resource.
Most probably you can maintain consumption and capital at constant steady state values while the per capita, at least, natural resource depletes to extinction (but you have to show that this is optimal with respect to intertemporal utility maximization).


Does control system engineering have a place in economics?

Economists have been exploring control theory applications to macro economics for decades. For example, here is a 40 year-old research paper written in 1976 on the topic.

top of page 2 (also numbered 171)

  In the past decade, a number of engineers and economists have asked the question: "If modern control theory can improve the guidance of airplanes and spacecraft, can it also help in the control of inflation and unemployment?"

Non-linear adaptive human behavior, uncertainty and governmental decentralization of control (i.e., democracy) are the main reasons I believe control theory has not become a more common tool for policy makers. Economics is simply not an exact science. Whereas engineering is more so. So control theory is more useful in the engineering discipline.


Rothschild-Stiglitz (RS) Separating Equilibrium

I recommend reading 'Informational Equilibrium' by John Riley (1979). It discusses this setting more generally. How the "insurance model" of Rotschild and Stiglitz is a special case of this is presented on pg. 335, and then Theorem 3 on pg. 343 is what you need. The proof is not trivial, but - in my opinion - well presented.


Intuition for why $EV>CV$ for a normal good

I think the magnitude of EV is less than the magnitude of CV for a normal good which increases in price (and the reverse for a normal good which reduces in price, so there may be a sign issue)

Intuitively, you should be able to argue that for a small price increase for an income-inelastic good (so neither normal nor inferior) the increase in income based on the new price you would need to take you back to your old utility (CV: the Compensating Variation) should be equal to the amount based on the old price you would be prepared to pay to avoid the price change (EV: the Equivalent Variation). The shift along the utility curves caused by the price change is similar in both cases as the unchanged-priced goods substitute for the increased-price good.  Graphically the two utility curves are in a sense almost equally spaced with a vertical translation

Meanwhile with a normal good, you now face a increased price for this good, so in the CV calculation not only do you need an increased income to compensate for the loss of utility, but it has to be even higher because you will want more of this increased-price normal good when you get your compensation than you would if it were income-inelastic, so CV is even higher for a normal good than for an income-inelastic good for a given level of EV calculated on the old prices.  For a decrease in prices you would get the opposite effect

Some Oxford lecture notes (pages 5 and 6) use a similar argument:

  For a normal good, for an increase in the price of good $x$, the compensating variation must be greater than or equal to the equivalent variation because once $p_x$ has increased, with $p_y$ remaining constant at $1$, it must cost more to compensate the consumer in get them back to their original indifference curve than would be required to take from them at the original price level to take them from the old to the new indifference curve, because both goods are at least or more expensive than before (another way of saying this is that the marginal utility that the consumer gets from each additional unit of cash income must be lower at the new price level, so more cash income must be given to get the consumer back to their original utility level). For a decrease in the price of a normal good, this result is reversed.


Income Offer Curve In Basic Microeconomics

It is the change in the demand of $x_1$ and $x_2$ for different levels of income $m$ (when we shift the budget curve out). Intuitively, how will your consumption of the two goods change when income changes. If we let $x_1$ be bus tickets and $x_2$ taxi rides, then if your income increases, we might expect that you will takes the bus less and the taxi more often.


How should I interpret (BTC/CNY) / (BTC/USD)?

Regarding your question before the update:

1) Yes, they have the same units ("how many units of CNY you need to buy one USD", either directly or through a bitcoin intermediary). That is the whole point of arbitrage.

2) It seems that "the global forex market"  and "the bitcoin market" seem separate to you. If by the former you mean big investment banks, then maybe you are right, as there seem to be no indication that big investment banks are systematically engaging in Bitcoin trading (at least I found no evidence of it; please correct me if I'm wrong). However, there are tons of bitcoin-oriented banks and brokers doing transactions on currency pairs, and I'm not sure these should not be counted as part of the global forex market.

Having said this, I think the simplest interpretation of the lines is that of a currency pair, and that of the currency pair being arbitraged through bitcoin.

3) The green line refers to how many units of CNY you need to buy one unit of USD. Or equivalently, how many units of CNY you get for one unit of USD. If the purple line is above (below) the green line, then you can get more (less) units of CNY by (1) selling USD through bitcoin and (2) buying CNY with those bitcoins. This means your interpretation in the update is correct. If you have USD, sell them to BTC, and buy CNY (purple line), and then exchange these directly to USD. As the green line is below the purple one, you need less CNY to buy one USD, which is the same to say that you get more USD for the same amount of CNY.

So, there seem to be arbitrage opportunities, at least in a zero transaction cost market.


What does it mean when the natural rate of unemployment is greater than the actual rate?

Natural unemployment is the 'expected equilibrium' rate. In the same way unemployment can go above the natural rate (usually when the economy is in a downturn) it can go below (usually when an economy is doing well).

An economy could report actual unemployment below the natural rate in multiple circumstances, one such being a shortage of labour across the economy. (usually due to an economic upturn). In that case employers may try to reduce the time workers take to switch jobs and they might be temporarily less stringent in their hiring requirements causing some 'structurally unemployed' workers to get picked up. To cope with the labour shortage, firms may also encourage retired or serially unemployed workers who have stopped looking for jobs (who in both cases are excluded from the labour force) to gain employment thus these people may go from not being counted in unemployment statistics to increasing the population of employed workers and therefore influencing the unemployment rate downward.

Another effect that should probably be considered is that real unemployment can be directly measured via surveys and the natural rate is estimated indirectly based on assumptions of what the equilibrium rate of unemployment 'should' be. There is no direct way to validate whether or not a particular methodology is 'the most correct' and when we are presented with only a graph like this we are trusting that the person compiling the graph did their work correctly. If their assumptions were incorrect then the graph is wrong and there isn't much point pulling our hair out to figure out an explanation for the results of the graph. In your case it looks like the graph is coming from a textbook so you're probably expected to believe that the calculation is reasonable and the answer is something lile the above. In real life graphs tend to have hidden assumptions that very well may be wrong (either intentionally or incompetently) so if something doesn't look right it's almost never a bad idea to start asking questions to validate if the result is truly different from your intuition or if somebody made a bad assumption.


When can we expect China to become a developed country?

Just because a country has a lot of resources (including human capital), it does not necessarily form a path for a developed nation. China is also very involved in military conflicts. It just takes one war for a country to drift back decades. Also, you can't ignore the brain drain problem.

With that, "developed" nation is a fluid definition. What does that mean? Is the US developed? Thousands of US citizens don't have electricity yet. No clean water. Massive health care and student loan crisis. Not to forget the trillions of dollars of debt. But many would consider the US a developed nation. If you are basing "developed" as in militarily-wise, then China will soon be a developed nation, if not already.

However, if you account for freedom of speech and all, then China is still decades behind from its neighbors like India.

Hope that helps!


You are at a decision point and you don't know what strategy the other player has taken. But you still know that which game you are in, you still know the other players' strategy sets etc.
Under incomplete information you might not even know what game you are into.

For example: Two firms are taking up R&D projects sequentially. One is the first mover. It decides whether to take up R&D or not. Now the second firm has to decide but does not know what first firm has decided. This is imperfect information.

Now imagine ypu are playing a card game and need to throw a card but you don't know whether you are playing poker or sweep. This is incomplete information.


The question was clarified in the comments as

  How to calculate $\min\limits_q \max\limits_p E(p,q)$ for a given function $E(p,q)$?

What this notation means is that $p$ is chosen first, and $q$ is chosen afterwards.

Assume that $p$ is a given parameter. Then $E(p,\cdot)$ is a function over $q$ only. So you take the minimum of the function $E(p,\cdot)$. In economics this usually means taking the first derivative with respect to $q$. From this you get the optimal value of $q$, given $p$. This is basically a function $q^*(p)$. So you know how $q$ will be chosen given the choice of $p$. This gives you the function $E(q^*(p),p)$. You choose $p$ to maximize this function. Again in economics this usually means taking the first derivative w.r.t. $p$.

With $\max\limits_p \min\limits_q$ the procedure is essentially the same, but there you first derive the maximum for $p$ given $q$. This yields $p^*(q)$. Using this, you minimize $E(p^*(q),q)$ w.r.t. $q$.

A sidenote:
If $E(p,q)$ is linear in both $p$ and $q$ the functions $q^*(p)$ and $p^*(q)$ are usually not continuous. To be more precise they are not functions but mappings. As such, they upper hemicontinuous but that is very technical and you may not need to know what it means.


Effect of a depreciation on foreign interest rate?

According to the interest rate parity condition , if there is a depreciation of the domestic exchange rate, then the domestic interest rate will increase and the foreign interest rate will decrease. Am i Correct?

This verbal formulation does not match what interest rate parity states. Note that there are two versions of interest rate parity: covered interest rate parity (an arbitrage condition), and uncovered interest rate parity (an equilibrium condition that appears questionable).

See this question for a description of interest rate parity: link to question I will not repeat the description of interest rate parity, as it is done in that answer.

In both versions of interest rate parity, what matters is expected depreciation, not a historical event of depreciation.

The statement that interest rates adjust in reaction to expected exchange rate movements is also questionable. Short-term rates are largely under control of central banks in floating currencies, and so a more plausible description is that expected currency movements conform to interest rate differentials.


This a highly disputed question! There is a new and interesting paper by Òscar Jordà, Moritz Schularick and Alan M. Taylor about "Leveraged bubbles". They study bubbles in housing and equity markets in 17 countries over the past 140 years.


If you measure how well off he is by his utility, then no. His utility is 10 in both scenarios.

The fact that career A is open to him increases his utility more in the second scenario than in the first scenario, this is what the "net utility benefit" shows you.


A speculative bid in a Vickrey auction means that it is not what the bidder really thinks that the product is worth. Hence, a speculative bid is anything different from the true valuation of this product by the bidder (either a smaller or a larger price than the true one).

In a Vickrey auction, the bidder has incentive to reveal his/her true valuation. If he/she does not do so, he/she risks losing (either by winning the bid and paying more than what the product is worth for them - if price was higher than true valuation -, or by losing the bid - if price was lower than true valuation).


Economics of Education: Private School Behaviour

I actually found the answer in a paper called  Subsidies, Hierarchy and Peers:The Akward Economics of Higher Education. by Gordon C. Winston.

He goes on at length discussing the objectives of colleges, and concludes we cannot use the same methods by which we predict firm behaviour to predict the behaviour of colleges both public and private.

In his conclusion he writes:

  This paper suggests the standard economic intuition and analogies, built on an understanding of profit-making firms and the economic theory that supports it are likely to be a poor guide to understanding higher education and to making predictions and public policy. One who thinks a college is like any other business will look in all the wrong places...

Don't completely understand of all of his arguments but he addresses my question.


Optimal price function: application of calculus of variations

Not really an answer, but too long for comment.

The $P$ in your
$$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$$
expression from the insider's problem and the $P$ in the expression
$$
\min_{P(\cdot)} \cdots
$$
from the market maker's problem should not be the same. That is not the definition of rational expectation equilibrium in this context.

The insider takes the pricing rule as given, per definition of rational expectation equilibrium. This means insider has optimal demand
$$
y(u,v, P_1(\cdot))^*= \frac{v-P_1(y^*)}{P_1'(y^*)} +u
$$ for any pricing rule $P_1(\cdot)$. Then market maker minimizes
$$
E[ ( y(u,v, P_1(\cdot))^* - u) \cdot ( v - P_2(y(u,v, P_1(\cdot))^*) )  ]
$$ over $P_2$---taking $P_1$ as fixed. It's a simultaneous game.
If the minimizing $P_2$ coincides with $P_1$, then $P= P_1 = P_2$ is an equilibrium pricing rule. Doing this in the one-period Kyle setting recovers Kyle's lambda, whereas your calculation would not.

The optimization problem you're solving is therefore not the correct one.

The insider optimizes with respect to a conjectured pricing rule $P_1$. In equilibrium, the market maker confirms the insider's conjecture, making the latter's expectation rational.

Your calculation assumes that the market maker acts after the insider. This is not the case in Kyle and I don't believe it's the case in Rochet and Vila either.


(Microeconomics) How to classify Public Goods and Common Resources in Externalities?

First let's clarify some terminology: I assume that with "common resources" you mean goods that are non-excludable but rivalrous. The term that applies in those cases is open access. In common resources there are shared property rights so some excludability (this is one of the mistakes in Hardin's tragedy of the commons).

The answer to your two related questions then is: it depends on the type of good. Public goods and externalities are related but distinct problems.

Take a classic public good: national defense. That is not a problem where production or consumption have unintended effects and hence there are no externalities. It is also a very clear example of a public good: one cannot be excluded from national defense and one person's protection does not diminish someone else's protection.

That is not to say that the two never go together. Air pollution is a classic example of a production consumption externality (the companies do not intentionally pollute the air, but reduce consumer health nonetheless) and often a public bad: if one lives in the area one cannot be excluded and breathing the polluted air by one doesn't reduce the overall pollution level.

The classic Gordon-Schaefer fishing model is both an open access resource problem and a production-production externality: the catch of an additional fish by one producer raises the costs of other producers.


the US Government instead deducted the same amount from the US debt that the Chinese Government owns

That would be equivalent to a partial sovereign default of the US: we don't like you, therefore our bonds you own are now worth less/crap.

More technically, it would be a repudiation of (some) sovereign debt:

  Repudiation happens when the sovereign rejects its obligation to pay, which could
  happen before or after any payment is due. Governments typically avoid questioning
  the validity of their debts, or announcing their intention not to pay before missing a
  payment, since in practice, repudiation carries all the traumatic consequences of
  payment default [...]. Repudiation may go hand in hand with
  governments questioning the legitimacy of one or more obligations. The literature on
  “Odious Debt” includes a handful of examples (King 2016, Lienau 2014). In 2008,
  Ecuador claimed that two bonds issued by a previous government were illegitimate and
  pledged not to pay them. It launched a buyback offer the following year in the shadow
  of the illegitimacy claim, and ultimately secured nearly 2/3 debt relief with more than
  90 percent of the creditors participating.

  [...]

  Repudiation [...] is rare in modern times. Repudiations most often
  occur after a regime change, and examples of repudiations include those in the wake of
  communist revolutions—such as Russia in 1917, China in 1949, and Cuba in 1960; Rhodesia
  in 1965, after its unilateral declaration of independence from the United Kingdom; Zaire in
  1979; Ghana in 1979 and 1982; and North Korea in 1976.

  Repudiation can go hand-in-hand with the concept of “odious debt,” which posits that a
  government is not obligated to pay for those debts incurred by a previous government contrary
  to the interests of the public. Though, the concept has strong moral appeal, it is difficult to
  define odious debt in a sufficiently limited way to allow its practical application (Reinhart and
  Rogoff 2009). The doctrine has not gained traction with arbitrators, courts, or credit rating agencies (Gelpern 2007; Blair 2014) and, the earlier example of Ecuador aside, states tend not to assert it explicitly.

As for effects, that's harder to predict. China would probably accelerate its sales of US treasuries, since the US move you propose is not going to impact all their holdings at once. The sooner they would sell, the fewer losses they would incur over time from the proposed US tax-like measure. On the other hand, selling too much too soon would hurt them too by dropping the price too much.

Russia substantially reduced its holding of US treasuries in 2018, presumably just based on fears:

  Between March and May, Russia's holdings of US Treasury bonds plummeted by $81 billion, representing 84% of its total US debt holdings. [...]

  "One theory is that this was Russia's revenge for US sanctions," said [Jason ] Bush [an analyst at consulting firm Eurasia Group].

  Another theory is that Moscow feared further US sanctions that could cause its holdings of US debt to be frozen or even seized.

The Russian side didn't acknowledge this explicitly, but their central baker said something like

  Russia assesses "all the risks: financial economic and geopolitical."

Russia however held a lot fewer US treasuries, only around 1/12 of China's even at their peak holdings. So Russia could sell the bulk of theirs relatively quickly.


Compound interest problem using PMT in Excel

The reason of this difference is that you do not fully understand what Compound interest is.

Actually, if you want to get the same amount as the site, while using your monthly payment, which is almost correct (more about this "almost" thereafter), what you must do is less straightforward than simply multiplying it by $120$ (calculation which would return the correct value in the $0$ interest rate case.)

What you must understand is that, the first monthly payment, let call it $P_0$, is going to capitalize (at a $\frac{5}{12}\%$ rate) for $120$ months, $P_1$ during $119$ months, $...$ and $P_{119}$ during $1$ month. And this is the sum of all these differently capitalized identical payments that you must compute to get $8235.05$.

That being said, if you do so, you will get $8263.36$ instead of $8235.05$, since the payment is actually due at the beginning of the period (whence the term almost above), i.e. you must set a $1$ instead of $0$ for the argument Type, which will return a monthly flow of $(\$52.81)$.

Thus, to summarize

$8235.05 = 52.8127046833568 \times \sum_{i=0}^{119} \left( 1+\frac{.05}{12} \right)^{120-i} \approx 52.813 \times 155.9292889
$

Where $155.9292889$ can be interpreted as an implicit capitalization factor.


Slutsky equation with marshallian demand

The first term on the RHS of the Slutsky equation (as you write it) should be $\frac{\partial h_i(p_i,p_j,I)}{\partial p_j}$, not $\frac{\partial h_i(p_i,p_j,I)}{\partial p_i}$.

So your subsequent algebra is fine but you're conflating cross-price substitution with the own-price substitution: when you let $i \neq j$ in the Slutsky equation, the first term on the RHS $\left(\text{again, }\frac{\partial h_i(p_i,p_j,I)}{\partial p_j}\right)$ measures the cross-price substitution effect of an increase in the price of $j$ on the (compensated) demand of $i$. You found it to be positive, so goods $i$ and $j$ are net substitutes (the cross-price partials of $h_i$ and $h_j$ cannot be oppositely signed -- this follows from Shephard's Lemma).

If you let $i = j$, however, you will then find that the Slutsky equation implies that the own-price substitution effect is negative for both goods, which you correctly note should be negative:

$$
\begin{align}
\frac{\partial h_i(p_i,I)}{\partial p_i} &= \frac{\partial x_i}{\partial p_i} + x_i \frac{\partial x_i}{\partial I} \\
&= -\frac{I}{2p_i^2} + \frac{I}{2p_i} \left(\frac{1}{2p_i}\right) \\
&= -\frac{I}{4p_i^2} < 0.
\end{align}
$$


What happens to effective consumption and capital in the Ramsey/Cass-Koopmans(RCK) model when technological growth decreases?

When the (exogenous) rate of technological change/efficiency is smaller, the corresponding steady-state levels of consumption and capital per unit of effective labor, increase.

For capital, we have

$$f'(\hat k^*) = \rho + \theta g_x \implies \frac {\partial}{\partial g_x}f'(\hat k^*) = \theta >0$$

So if $g_x \downarrow \implies f'(\hat k^*) \downarrow \implies \hat k^* \uparrow$ due to decreasing marginal product of capital.

For steady-state consumption per unit of effective labor we have

$$\hat c^* = f(\hat k^*) - (n+g_x)\hat k^*$$

$$\implies \frac {\partial \hat c^*}{\partial g_x}= f'(\hat k^*) \frac {\partial \hat k^*}{\partial g_x} - \hat k^* - (n+g_x) \frac {\partial \hat k^*}{\partial g_x}$$

$$=[f'(\hat k^*) - n - g_x]\cdot \frac {\partial \hat k^*}{\partial g_x} - \hat k^* $$

The term in brackets is assumed positive, i.e. we have already assumed

$$f'(\hat k^*) = \rho + \theta g_x > n + g_x \implies \rho > n + (1-\theta)g_x$$

in order to exclude infinite utility.

Moreover, evidently $\frac {\partial \hat k^*}{\partial g_x} <0$ so in all

$$\frac {\partial \hat c^*}{\partial g_x} <0$$

Therefore, if $g_x \downarrow \implies \hat c^* \uparrow $.

See Barro & Martin, ch. 2 page 102, where they discuss the  $g_x \uparrow$ case (in p. 101 they discuss the constraint on the parameters).

Comment: This result may appear counter-intuitive, but a deeper examination of the model shows that if $g_x$ is lower, utility per capita is lower. Using "consumption per unit of effective labor" is a modeling tactic, what interests us is what happens per individual. So no, the model does not argue in favor of lower productivity/technology.


What links can be made between the marginal productivities of the factors of production and the costs of production

Assume the following profit function: $\pi = pY - wL -rK$ and a two factor economy $Y(L,K)$.
Optimal labour and capital inputs for the firm can be found by taking the respective derivatives and setting them to zero:
$\frac{\delta \pi}{\delta L} = p \frac{\delta Y}{\delta L} -w = 0  \iff \; \frac{\delta Y}{\delta L} = \frac{w}{p} $
$\frac{\delta \pi}{\delta K} = p \frac{\delta Y}{\delta K} -r = 0  \iff \; \frac{\delta Y}{\delta K} = \frac{r}{p} $
Now you can see that marginal productivities are equal marginal costs. The derivative with respect to the input factor tells you what the next unit of capital or labour will produce and the right hand side tells you what they will cost (real wage or real interest rate). If this condition does not hold firms could be better off by adjusting their production towards this condition. For example assume marginal product of labour is greater than the real wage $(\delta Y / \delta L > w / p)$. The firm should hire an additional worker since she will produce more than she costs. This logic holds true until the conditions are fulfilled.


Why don't Governments do away with the optics of taking on debt against new currency, and instead issue a limited currency every year (say 5% of GDP)?

Well first of all we have to make it clear that we are talking about government debt held by central banks. Non-trivial amount of government debt will be borrowing from households firms and other governments where it is not valid to argue that the debt is an illusion.
When it comes to debt held by central banks not much would change although it would make reducing amount of money in circulation bit difficult for a central bank. An advantage of creating new money by purchasing government bonds is that the money can be easily destroyed just by central bank asking for government to pay back the money. In principle government has other options of how to reduce money supply even in such case but doing it through repaying bonds is the most direct way how central bank can do it without help of rest of the government.
This brings me to the final reason why such institutional arrangement is justifiable in the first place. It is virtually agreed by most economists that central banks work best when they are independent technocratic institutions rather than when they are directly controlled by executive branch of the government (and consequently under political influence). You can read about all the benefits of central bank independence in this ECB working paper, but as a non-economists you can view it analogously to courts independence. You want law to be administered by experts in impartial manner that will help society run more efficiently than political courts and similarly independent monetary policy helps more efficient management of money supply and other things that are under purview of monetary policy.
Having central bank and rest of the government operate at arms length, like increasing money supply through purchase of bonds, or by not even buying bonds directly at primary market but only at secondary market helps to build a wall between government and central bank.