Why do many economists believe that money is neutral in the long run?

There are multiple ways that a central bank can resort to apply its designated monetary policy. The most common one is called "open market operations (omo)", which is basically selling/buying government securities to/from private individuals or organizations. Through these operations, central bank affects the level of cash reserves in the banking system since it deposits/withdraws payments for these government securities to/from the banking system. When cash reserves in the banks go up -by central bank buying government securities-, these banks decrease their interest rate to deplete the extra added cash reserves, and the vice-versa.

As you can see, in this mechanism the only price that is changed "directly" by central bank is the price of government securities. Once the cash reserves in the banking system reaches its targeted level, the probability that the general price level in a specific market will be affected is equal for almost all markets. Thus the operations of central bank does not aim at specific markets, alterations in the interest rates affect the whole economy equally by average.

There are tons of literature, both theoretical and empirical, focusing on the neutrality of money. You can also search them via Google Scholars.


Why does quantity supplied increase with price in economics?

I believe that I have found the problem in this line:

  I don't understand why price has to increase if quantity increases.

The concept is actually the other way around.

Think of it in the same way as the Law of Demand.
If you go to the grocery store and you see a food that you like selling for \$.25/lb, you would buy a whole lot of it before the price rises.
Conversely, if you go to the grocery store and you see a food that you like selling for \$100/lb, you would probably wait to buy this item until it is cheaper or at least buy a small amount of it.
In economics, the price drives the quantity demanded by the consumer.

Now let us look at the Law of Supply.
Imagine that you are the owner of a company.  You go to the store, and you see that the item you are producing and the similar items produced by your competitors is selling for \$.25.  You would not necessarily want to produce a lot of the product because the margin between the selling price and the production costs (profit) is small.
Conversely, imaging going to the store and seeing that the item you are producing and the similar items produced by your competitors is selling for \$100.  You would want to produce a lot of the product because the margin between the selling price and the production costs is (presumably) large.
In this case, as in the other case, the price drives the quantity produced by the supplier.


Why did creating the Eurozone cause the cost of labor to rise in Greece?

I think we should ask why didn't Germany's labor cost increase, while everyone else's did:

The eurozone had economic growth before 2008, so appreciating labor was expected, especially due to increasing labor mobility. Wages were converging to Germany's.

In addition, Southern Europe saw an influx of capital (this is how they caught the debt crisis). Capital triggered some speculative economic growth — 4% in Greece — and an increase in wages.

One more thing. When the crisis hit in 2008, the private sector fired a lot of people. But high-paid employee remained. Which lead to this 12% pike in the business sector alone, as the average wage of the remaining workers increased:

Unit labor cost, year-to-year change: Business sector services only

Data source

The public sector kept its employees, so on average you have no peaks in 2010 (the first figure). And after 2010, you see a decline — basically caused by the same determinants, now working in the opposite direction.


Formulas can be checked against infinite online resources, not here.

The most widespread use of $x$ and $X$ to distinguish something, is when it is needed to emphasize what is treated in theoretical derivations as a realized value (a fixed number, $x$), and what as a random variable, $X$. When using matrix algebra though bold uppercase denotes a matrix (but in panel data where everything is a matrix, we see often simple uppercase to denote a matrix).

Sometimes, authors don't bother, since it should be obvious from context.

Perhaps the most usual confounding is to use lowercase $x$ for either a realized value or a random variable, but use uppercase letters for random variables.

As for calculation, it is always done using data, so there is nothing to "convert" here. If you have the realized data on $X$ and $Y$, then plug them in the formula.

Any other contrasting use of the two is of course not "forbidden", but then one should read carefully how each author defines their use. Nevertheless, to use simply $x$ to denote "a variable centered on its sample mean" $x=X-\bar X$, is rather unusual and confusing I would say, and in any case it does not warrant a change from lowercase to uppercase. We usually write things like

$$x^* \equiv x-\bar x$$

or some other symbol instead of a star.


Why is machine built this year but not sold this year included in GDP?

The comment provided by @Kitsune Cavalry:

  It counts as investment. Inventory, capital, and structures are part of investment. Unsold stuff counts as inventory.

is correct as to the accounting; let me provide a little more intuition to it.

GDP measures production (output), not sales— it is in the name: Gross Domestic Product. And for many firms, production into inventory, making a thing with the explicit intent of building inventories so that the thing is available for purchase when people want that thing, is a core part of the business.

Imagine a firm that makes snow throwers— they build these machines year round, then basically sell all of them during a period from late fall through mid-winter (the exact seasonal pattern doesn't matter, only the fact that there is one).

For the purpose of counting that firm's output in GDP, it should be clear why we want to recognize that they produced snow throwers all year, rather than having all that production only show up during the few months when the machines are actually sold.

This approach isn't without its downsides— in the event that large volumes of production into inventory occurs because demand for the product in question has fallen off permanently, then the market price of the units sold will have been applied to a bunch of units that would never have sold at that price, and the value of output will be biased. This bias is of smaller magnitude than the mis-measurement that would be introduced by normal inventory fluctuations if we were only to count output when products are finally sold, though, so we live with it.


How likely are the Sonnenschein-Mantel-Debreu results?

The space of economies can certainly be modeled as a measure space. Perturbing an economy (the parameters of preferences and production functions) generally changes the excess demand function continuously, so if there is one economy with a "weird" excess demand function (however defined), then there is a positive measure of them (open ball around the one weird economy).

As the number of commodities grows, and preferences for each new commodity are randomly drawn from some space, then my intuition is that the probability that at least one commodity has upward sloping demand converges to 1. The probability that all commodities have "weird" demand likely converges to zero.

Empirically, upward sloping demand is rare (examples of inferior goods are rare).


Understanding the linear expenditure system

For every $x_i$ separately you get

$$\frac {\gamma_i}{x_i-a_i} = \lambda p_i \tag{1}$$

Re-arranging,

$$\frac {\gamma_i}{\lambda} = p_ix_i - p_ia_i \tag{2}$$

Sum over $i$

$$\frac 1{\lambda}\sum\gamma_i = \sum p_ix_i - \sum p_ia_i \tag{3}$$

Re-arrange to solve for the optimal $\lambda$ taking into account the restrictions

$$\sum p_ix_i = m ,\;\;\; \sum\gamma_i=1$$

$$\lambda^* = \frac {1}{m-\sum p_ia_i}\tag {4}$$

Insert into $(2)$, re-arrange and you 're done.


proof of uniqueness of Dominant strategy equilibrium

You are along the right track but are making certain errors in your proof,

Your notation is confusing. You haven't defined what $S^{'D}$ is. I would suggest sticking with the defined strategy space and not creating unnecessary notation in the proof.
When you write $v(s'_{i},s'_{-i}) \geq v(s_{i},s_{-i})$, you are not only changing player i's strategy but also other players' strategies. However, you need to keep other players' strategies constant and then compare player i's options to choose the best strategy. Also, note which player's payoff function is being compared. So, you should instead write $v_{i}(s'_{i},s'_{-i}) \geq v_{i}(s_{i},s'_{-i})$.

Now, let's move on to how a proof by contradiction could look like.

Suppose there exists another dominant strategy equilibrium, $s^{A} \in S$. (Note that we are talking about both weakly and strictly dominant strategy equilibria here)
This means that $\exists i \in n$ such that $s^{A}_{i} \neq  s^{D}_{i}$ and $v_{i}(s^{A}_{i}, s^{A}_{-i}) \geq v_{i}(s^{D}_{i}, s^{A}_{-i})$.
However, $s^{D}$ is a strictly dominant strategy equilibrium, and so, $s^{D}_{i}$ is a strictly dominant strategy for player i. Inter alia, this implies that $v_{i}(s^{D}_{i}, s^{A}_{-i}) > v_{i}(s^{A}_{i}, s^{A}_{-i})$.
The inequalities in (2) and (3) contradict each other. Hence, the strictly dominant strategy equilibrium must be unique.


Nash Equilibrium and Dominant Pure Strategy when payoffs identical

First, make sure you have a definition of strategy clear. Strategy describes the actions a player chooses conditional on information etc. In your case it's a one-shot game so perhaps the distinction between an action and a strategy is not that important.

In game theory there is the notion of a "weakly" or "strictly" dominant (or equivalently "weakly" or "strictly" dominated) action. Nash equilibrium does not allow for any strictly dominated action to be chosen. It can be, however that a weakly dominated action is chosen. Hence you cannot cross out a whole row or column because an action is weakly dominated.

Your thinking regarding to NEs is correct. In both AC and AD no player can be strictly better off by changing his/her action conditional on opponent's action being held constant.

Finally, game theory done properly is rigorous enough (at least at undergraduate and graduate level), that there is never grey area. The definitions you gave are not stated mathematically and not 100% correct either, hence they seem a bit vague.


How monetary policy affects bank reserves

It's a direct result of how monetary policy is normally conducted. In normal-times monetary policy, the Fed will increase its target rate (the Fed Funds rate) by selling Treasuries to dealers, who will pay for them by withdrawing deposits they hold at banks. The cash that formed these deposits can therefore no longer be held as excess reserves at the Fed, which decreases the supply of funds in the Fed Funds market, increasing the price of borrowing funds (i.e., the Fed Funds rate). The reduction in bank reserves is actually the mechanism by which rates are increased; the sale of bonds is just a way of reducing reserves.

Here's a reference.


Writing the core as the intersection of pareto efficient outcomes of all coalitions

Most of what you write is correct, but the definitions of the $F_i$ sets is imprecise. The problem is that in the core $A$ and $B$ may get goods that do not match their initial endowments. In this case it is not true that the core allocation $x$ is Pareto-efficient in the restricted 2-person economy of $A$ and $B$, because $x$ is not even an allocation in that game.

Edit: (An example)

Consider the initial endowments
$$
\omega_{A} = (1,1),  \omega_{B} = (1,1), \omega_{C} = (2,2)
$$
and an allocation $x$
$$
x_A = (2,2), x_B = (2,2), x_C = (0,0).
$$
$A$ and $B$ cannot Pareto-improve on $x$. But $x$ is not a Pareto-efficient allocation in their 2-person economy, because it is not a feasible allocation of their economy:
$$
x_A + x_B \neq \omega_{A} + \omega_{B}
$$

A better definition for the sets $F_i$  would be something like:

Let us denote the set of feasible allocations of the 2-person economy of $A$ and $B$ by $Y_{A,B} \subset \mathbb{R}^{2}$. Then
$$
F_{1} =\{ x \in \mathbb{R}^{2} : \nexists y \in Y_{A,B} \text{ such that } u_{A}(y_{A}) \geq u_{A}(x_{A}), u_{B}(y_{B}) > u_{B}(x_{B}) \}
$$
There are still some issues with cases when $A$ is better off and $B$ is not worse off, but if the utility functions are continuous then this should not cause a problem.

You can define $F_{2}$ and $F_{3}$ in a similar manner.

A remark: $E$ is not 'special', it is the set of allocations that cannot be improved upon by the three player coalition. This is equivalent to Pareto-efficiency.

Yes. Why not? This is exactly what the core is.
I would not call it solving, because usually you do not get a unique solution, and in extreme cases you may get no solution. But yes, every economy ($n$ players, $m$ goods) has a core, and it is defined in this way. (As indicated, unless some conditions are met the core may be empty. A competitive equilibrium is always an element of the core, so if that exists, the core is non-empty.)


How to explain these two discrepancies between BLS OES wage data and salaries of pro athletes?

This may not be the complete answer but I believe this is a major issue:

It is extremely difficult to get famous and rich people to participate in surveys. They have handlers that prevent surveyor access. They have multiple phones numbers and homes that makes it difficult to find them. They also have complex affairs that may make it difficult for them to easily summarize. For example, having a complete picture of sponsorship income can be challenging and almost no one knows what a fair value is for their accumulated pension.

In summary, I assume that most of professional athletes are not covered by the BLS survey. Instead, it is people of more modest income, like minor league athletes and golf pros that are actually getting picked up.


What's the name of this pairwise price-setting scheme?

This is known as the "split-the-difference" mechanism, originally due to Samuelson (1985). See Section 6 of Gibbons et al. (1987) for a discussion of this rule. In particular,

this kind of rule only efficiently dissolves the partnership between the two siblings if they have equal (or sufficiently close to equal) shares in the house. Precisely, each sibling should own no more than 57.7\% of the house.
such a mechanism does not typically induce an equilibrium with sincere bidding (i.e., the siblings will report less than their true value for the house).


Is there a pull of skilled workforce from rural to urban areas?

It somewhat depends what you mean by that. The process is more complicated with migrants to cities also acquiring skills/education there, e.g.

  Another important piece of work related to education-based motivation is Lucas (2004), who constructs a two-sector model where the urban sector pays a higher wage due to its high-skilled jobs which are not available to rural workers. Unskilled rural agents migrate to cities to accumulate human capital and invest in skills. Households’ preferences are isoelastic and they supply one unit of time endowment to either work or invest in human capital. Rural production uses labor and land; it has diminishing returns because land is fixed in quantity. The urban production technology is linear and requires only skilled labor input. Human capital investment can only take place in cities and labor is free to move between sectors. Rural–urban migration takes place once and for all with the growth rate of production of the urban (rural) sector changes from zero (one) to a constant positive level (zero).

This is somewhat of a no-brainer, you don't expect PhDs in AI to just pop up in rural areas and then migrate to cities...

There's a monster EU document (some 350 pages) published in 2018 on migration patterns inside the Union.  As far as I can tell from that (its summary is poor), already [mostly the younger] skilled workers prefer to move to another EU country (with better wages, less corruption, etc.)

For healthcare workers, which the EU study spends a lot of time on, within a given EU country, there's indeed a more acute shortage in the rural areas (p. 266). However across EU countries healthcare workers may migrate from an urban area e.g. in Greece to a rural area in the UK, all because of pay, prospects etc. (p. 269)


Extraction and royalty time path for a non-renewable resource under a monopolist assumption

I would say that (a) follows from a downward sloping demand curve that causes marginal revenue to decline faster than price. As you write a monopolist accounts for that and to maximize profits thus extracts less to drive up the price. That also explains why generally exploitation would take longer.

That this is not always the case is shown by Stiglitz's (1974) Monopoly and the Rate of Extraction of Exhaustible Resources  where he shows that the extraction path for the monopolist and perfect competition are the same for a constant elasticity demand curve and zero extraction costs. That in turn, has been shown by Gaudet and Lasserre in "On comparing monopoly and competition in exhaustible resource exploitation" to depend on the fact that the resource stock is fixed and hence it is sometimes hard to restrict output to increase the price.

Lewis et al. provide examples where the monopolist may be more excessive than socially optimal


What are your assumptions about a market dominated by long term investors?

For example if the market is dominated by investors does that mean demand outweighs supply so that yields are generally lower?
Does long term mean this level of demand for bonds will persist for a long time or that the investors are predisposed to buy maturities of durations longer than another?
Do your long term investors have the ability to buy short term maturities and roll them on an on going basis to allow them the possibility of investing for the long term via multiple short dated instruments?

  What is the liquidity premium/discount?

Is your market very large and liquid (like US or UK treasuries), i.e. you expect prices to be more transparent and have narrower bid-offer spread so that execution costs are small?
Is your market illiquid because your investors are buy-and-hold so that there is effectively no secondary market (like SEK or NOK corporate bonds).
what impact does each have on the supposed prices of securities.

  What about exogenous factors?

are you basing the answer to the question solely on this one aspect of knowledge and ignoring all other factors, such as current yield levels and view of the direction of interest rates?

I know this isn't an answer to the question but I have seen first hand how models do not reflect reality and this model seems incredibly simple for this type of analysis. I don't agree with the solution because I don't agree that the 'implications' will always be upheld, and there are 3 of them before the conclusion with scope for much argumentation. This is my own pragmatic view and not necessarily consistent with any of the academics that you are being taught so take it with a pinch of salt.


What is the danger of (China's) investment-led growth as opposed to consumption-led growth?

From the Chinese perspective, it's "dangerous" because high investment for so many years, particularly when driven by state incentives or by state-owned firms might induce mis-allocation of resources, high indebtness levels and excess capacity (some might say even voluntary, for example dumping in steel to hurt developed countries' industries). For instance, this paper states that:

  The main  objective  of  management  in  Chinese  state-owned  and  state-
  dominated  firms  has been  to  maximise  not  profits  but  the  growth  of  investment  and  output.  The  reward  to such  bureaucrats  takes  the  form  of  prestige,  power  and  the  accompanying  perks  of commanding an organisation; the reward being greater the larger the organisation.

Another reason is that investment is more volatile than consumption. However, if investment is supported importantly by state-owned enterprises, this might not apply well to China, which by the way have not had a recession in decades).

From a western perspective, high investment (and low consumption) contributes to a high trade balance surplus and therefore to a high current account surplus, which might be good for the central bank of China (accumulation of reserves) aiming to keep the Yuan relatively low to foster the competitiveness of the export sector, but is bad news for the rest of the world. The latter would benefit from higher consumption of the Chinese population (via Chinese imports and rest of the world's exports). The same story is told of Germany, which would help other countries (mainly European partners, because of same currency) by rebalancing its economy from exports toward more consumption.

Additionally, note that some economists have blamed China (among others) for the Great Recession (e.g. here or here), partly because of their voracious reserve accumulation policies, a core element in the export-led Chinese development strategy. Thus, a more rebalanced global economy might be good for the world economy as a whole, decreasing the likelihood of another recession.


Auctions and finding nash equilibrium of a dynamic game

You are only required to get Nash equilibria and not sequentially rational/subgame perfect equilibria. Hence Player 2's actions at information sets that do not occur (that do not reflect Player 1's actual strategy) do not need to be best responses. All you have to make sure is that no one is better off by deviating.
In case 4., regardless of what Player 2 does, Player 1 cannot get a positive surplus; she wins with a higher than 500 bid or loses. This is not better for her than bidding 500 and losing.


Inflation , Housing Prices, Is there some form of the CPI that can include Home Prices?

House price increases lead to increases in wealth for homeowners and increases in rental payments for renters. Thus, unlike for consumer products of the CPI like vegetables, milk, cheese, meat, or toothpaste for which the welfare impact of the increase of the price is clearly negative, the welfare impact of house price increases is ambiguous. In many countries the homeownership rate is greater than 50%, hence the median household tends to experience pretty significant increases in wealth in hot markets.


Kimball (1995) Specification of Final Good Production

Some input:
Kimball assumes a constant-returns-to-scale (CRS) production of the final good $Y$ from the intermediate goods (and no other inputs are involved in this function). Turn to discrete space for a moment, and this means that we would have something like

$$Y = F(y_1,...,y_l,...,y_m)$$

Since this is a CRS function we have

$$1 = F\left(\frac {y_1}{Y},...,\frac {y_l}{Y},...\frac {y_m}{Y}\right)$$

But also, from Euler's theorem for homogeneous functions we have

$$Y = \sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot y_i \implies 1 = \sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot\frac{ y_i}{Y}$$

Combining and manipulating the index into $[0,1]$-continuity we get something like

$$1 = F\left(\frac {y_1}{Y},...,\frac {y_l}{Y},...\frac {y_m}{Y}\right) =\sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot\frac{ y_i}{Y}\rightarrow \int_0^1\left[\frac {\partial F}{\partial y_l}\cdot\frac{ y_l}{Y}\right]{\rm d}l$$

In a sense, $G(y_l/Y)$ is the elasticity of final output with respect the the $l$-th intermediate good. Given the assumptions on $G()$, it rules out a Cobb-Douglas CRS production function, where the elasticities not only sum to unity but they are constant, and it looks, say, to a C.E.S. production function with constant returns to scale, where the elasticities are variable but always sum up to unity.


What is the fundamental problem with Pakistan's economy?

Let's split the problems into two categories, proximate and fundamental. It is easy to confuse the two categories or mix the problems of fundamental nature with those of proximate. Econoimc historians, in the pursuit of determining the causes of Britain's industrialisation have come up with several plausible explanations. I think we can use their framework to explain why Pakistan faces economic problems. I believe these factors are relevant in this case - Pakistan has not been able to take off from the low growth state for a while now. If I remember, it was a success story in the 1990s, but it could not sustain its high growth. In that sense, the country faces Malthusian straightjack.

Quality of Institutions. Every since North and Weingast (1989) claimed that Britain's take off in the eighteenth century was possile thanks to its Glorios Revolution when the Crown transferred much of its power to the Parliament. With this transformation, Britain was able to lift itsel from centuries of low growth state, first in the world. Does this give any explanation to why Pakistan faces the problems it has had for years. I would say it does. First. The quality of institutions is poor in Pakistan - rules of the game are not enforced properly. The political and social setting is not condcuive to growth - property rights are not secure, corruption is rampant, it is, possibly, institutionalised, and the presence of the rent seeking political elite obstructs the enforcement of rules. The necessary institutions have eveloved there, but they do not function in a way that permits growth. Daron Acemoglu (MIT) has successfully built on North's work, Acemoglu has a nice book on why nations fail. I would recommend you to read it.
Culture. Joel Mokyr has been putting the idea of 'cultural supremacy' in centrestage to explain the Industrial Revoltuion. I think his ideas could as well provide some explanation why Pakistan is facing problems. Historically, Mokyr believes, the beliefs, values, and preferences in society that are capable of changing behavior were a deciding factor in societal transformations. He goes on explaning what these transformations were and how they lead to the take off of Britain. Does Pakistan have the type of culture Mokyr is referring to, possible yes, but it is suppressed due to the presence of the issues outlined above in point 1. (will continue later)
Ideology/Religion. Pakistan is a hotspot of religious conflicts. The conflicts are not between religions but within religions. Tribes fighting tribes and depriving the young generation from education (girls and women especially). Such a culture suppresses innovation and discourages innovative behaviour. There is a growing literature on the role of religion in economic growth. Consensus has not been achieved as in other areas, but the papers look into catholicism vs. protestant vs. anglican vs. bhuddism vs. islam vs. judaism etc. vs. growth. It is probably not possible to quanitify religion's role in growth precisely. There is a misconception that religion suppresses innovation. Some papers point at the events of the past, there have been many including the locking up of Galileo for his claims about solar system which we today know were true. Pakistan is an Islamic country (in fact, it is Islamic Replublic of Pakistan). Does Islam hinder growth there? The facts point at No. Islam, like any other religions, promotes peace, grwoth and wellbeing but people adhering to Islam do not follow the teachings correctly. Economic historians agree that during the time of Abbasid Caliphate, Muslims had their golden age. It would not be an exaggeration if I claim that muslims were the only people making progress in all aspects of life at the time. Much of the World was in turmoil (Europeans had not seen a long lasting peace until nineteenth century, spurts of wars hindered growth for a long time). Exceptions could be given to the nations of the Far East (China). So, what is the issue here, it is surely the individuals and groups wages conflicts between various religios factions. These acts do not promote growth.

The other factors (i.e. low investment, human capital, terrorism, etc.) causing the problems are generally proximate.

(I will add more as soon as I have time)


ARIMA modelling and Economic interpretation

So we can think of an AR(1) process as:

$y_t = a*y_{t-1} +e_t$

where $e_t$ is some shock / noise process that we can't explain. Note that we can rewrite an AR process as an MA process where by tracing through we get

$y_t = a^2*y_{t-2}+a*e_{t-1}+e_t\approx \sum_{i=0}^{t-1}a^i*e_{t-i}$

So that an AR process is just the weighted sum of all previous shocks where $a$ is what determines the persistence of past shocks.

So then what does it mean to say we have an AR(1) process? Well it means that the outcome today is to some extent dependent on the outcome yesterday (as you yourself suggested). In an extreme case we can set $a=1$ so that we have a random walk process where

$\Delta y_t=y_t-y_{t-1}=e_t$

This implies that the outcome today depends entirely on the outcome yesterday and any additional change in the series is purely due to random noise. Alternatively if $a\geq1$ we would say the process is non-stationary or explosive while if $a<1$ then the process is stationary around the mean of the process or of the shocks themselves.

An MA process has a similar interpretation except we think of it more in terms of unexplainable shocks or noise with some degree of persistence. The important thing is that it is not really something we have a variable for in the MA case but rather this notion that the the process just depends on this noise component. It is more common to find these types of models for forecasting in finance where there's alot of movement but a less clear sense of what might be driving it.

In terms of population forecasting, the AR component could definitely be interpreted as you suggested, while the MA component is less easy to interpret. It's really just a sum of shocks that are otherwise unexplained. If you had an explanation for it (i.e. immigration etc.) then you would likely want to include that as an explanatory variable to model it explicitly rather than approximately through the MA process.


Answering your question involves the various sub-fields of Regional Economics, such as Spatial Competition theory (Hotelling, 1929), Economic Geography, Urban
Economics (Alonso, 1964), New Economic Geography (Krugman, 1991), preferably in an integrated way.

Indeed, each of these subfields relate to considerations that complement each other. As stated by Thisse (2010), integrating space in economic analyses relates to (i) the use of a computable general equibrium (CGE) framework to endogenize the formation of incomes and relative prices, (ii) the realism of the modeled geographical space, the necessity of being multidimensional, asymmetric and empirically grounded and (iii) the number of encompassed spatial scales. Otherwise, it is difficult to figure out how spatial forces shape the so-called Space-economy (Isard, 1949) and, a fortiori, what policy or what institutional change can influence them if they are judged to go in an undesired direction.

I would reformulate your question as

  What to do to model a city, a priori itself within a system of cities in interaction.

Incorporating CGE theory

The modeling architecture to develop in order to spatialize a GE is implicitly specified by Thisse (2010), who, instead of soft-linking compact descriptions of the economy and spatial formalisms, insist on the necessity of their full entanglement. This intellectual movement is the same as the movement toward hybrid models approach to couple GE models and engineering-based energy models (Hourcade et al., 2006). In the case of spatial economics, this hybridization passes through the use of New Economic Geography (NEG), which intrinsically implies GE theory. Indeed, GE concerns are what makes Economic Geography New (Krugman, 1998).

Involving geographical realism

At the scale (implicitly) involved in Urban Economics (UE), i.e. the urban scale, you can read Anas and Xu (1999), Nitzsche
and Tscharaktschiew (2013), Choi et al. (2015), Anas and Liu
(2007). If we collapse the internal structure of a metropolitan area into a single point, we conventionally jump to the field and the scale of analysis of New Economic Geography (NEG). Because at its early stages the analytic tractability of the spatial processes governing the interactions of areas was a key objective, one NEG’s notable feature is its lack of realism with
respect to the geographical distribution of areas, e.g. distributed in a mono-dimensional world as in Krugman (1993)’s Racetrack economy or in a part of it as in Fujita et al. (1999b)’s Line economy, as well as in a bi-dimensional world as in Puga (1999)’s side-by-side Equidistant Economy (EE). Since then, adding more real spaces components to NEG has been thought of as the main step for future development (Behrens and Thisse, 2007; Fujita and Krugman, 2003). You may also want to read Behrens et al. (2007), Bosker et al. (2007), Stelder (2005), Brakman et al. (2006), Sheng et al. (2016), Mercenier et al. (2016).

Involving multiple spatial scales

Both UE- or NEG-related models rely on exogenous signals coming from upper geographical scales, be them national, regional or global. But it is harder to find modeling approaches in which these links are explicitly involved. Attempts to bridge this gap can be found in Waisman et al. (2013), Allio (2015, 2016) and Faucheux (2018).

Waisman, Allio and Faucheux integrate (i), (ii) and (iii) in their modeling exercices.


What properties does a stable economy possess?

I’ll cover two of the possible policies that can be stable: interest rates and fiscal policy.

I will assume that a “stable” interest rate policy is to keep the policy rate (e.g. Fed Funds in the United States) unchanged. This is is a case discussed in “mainstream” economics, and the argument is that such a policy leads to inflation drifting at random. (As in Section 2.4.1 of “Monetary Policy, Inflation, and the Business Cycle” by Jordi Galí.) In other words, if you lose monetary policy, you lose control of inflation.

Of course, the policy rate has been at the same level for years in many countries, and not much happened. The reason why the policy rate was stable was usually that the economy was growing slowly, and no inflation risk was seen. That is, the economy was in a sense (temporarily) “stable”, and policy rate reflectd that.

Even if the central bank says it will not change short rates, bond yields would likely still move around on the basis that the policy will not be sustained. So things like mortgage rates would still move.

Some critics of “mainstream” economics think that the models that lead to this inflation instability are incorrect. In particular, some proponents of Modern Monetary Theory argue in favour of locking the policy rate at 0% forever. That is very much a minority position.

If we turn to fiscal, I am unaware of any formal theory. In basic “mainstream” models, fiscal policy is often left unchanged. However, this was not viewed as having much if an effect, since monetary policy is believed to be the main policy variable.

We have seen unchanged tax rates for years in many countries, and I do not believe anything unusual was associated with that. However, taxe rates are only part of fiscal policy. Spending changes from year-to-year, and in the real world, that is unlikely to change. For example, the Canadian Federal Government is helping pay for a bridge near where I live. That spending will hopefully cease when that construction is complete.

To top it off, the dollar amounts of deficits vary based on the business cycle. If people lose jobs, they pay less taxes and end up receiving more governmental assistance. So if someone defines fiscal policy “stability” as an unchanged fiscal deficit, that is almost guaranteed to never happen. However, these changes to the deficit are believed by some economists to help dampen the business cycle (the “automatic stabilisers”). (I see little discussion of automatic stabilisers in modern “mainstream” theory, but it was discussed more by the old Keynesians.)

In summary, “stable” policy settings imply that the government cannot lean against fluctuations like recessions, or control inflation. Economists of the Keynesian persuasion would likely argue that this will result in deep depressions, such as in the 1800s. Many non-Keynesians would dispute that.


Nominal Values or Real Value & stationary

My question clearly is ; It is possible to use a nominal variable in a time series regression? (regardless of stationary issues)

Is it possible? Yes.
Should you actually do it? Probably not.
Nominal variables are combination of real variables and prices. For example, nominal consumption $C_n$ will be given by:
$$C_n = P C_r$$
where $C_r$ is real consumption. If you will run regression (changes are included to get rid of non-stationarity) with
$$ y = \beta_1 + \beta_2 \Delta C_n + \epsilon \Longleftrightarrow  y = \beta_1 + \beta_2 \Delta P C_r + \epsilon $$
$\beta_2$ will not tell you what is the effect of change in consumption on change in dependent variable. This is because you cannot know, without actually examining real consumption, whether variation in $C_n$ is due to $P$ or also due to $C_r$. It is very well possible that all variation in  $C_n$ is just due to prices where you would be estimating effect of prices on your dependent variable and not consumption. Of course, in reality it is even worse because likely $C_r$ and $P$ both change at the same time, meaning $\beta_2$ will be combined effect of both of those and will not have any stright forward interpretation.
This being said if you are using AR model for forecasting and you just want to forecast nominal consumption based on past nominal consumption where you don't care about above issue it would be fine to use nominal consumption.


Does the current situation of excess reserve imply failure of QE?

You ask a few questions. I'll try to generally answer them:

  Is this the failure of QE? The money being lent and given frivolously circles right back into the Fed in the form of excess reserve it seems.

No, the existence of excess reserves is not a reflection on bank lending. A significant fraction of excess reserves originate in money market mutual fund holdings. This is very clearly explained in Singh (2015):

  The interest rates in most of these markets are in the range of zero and 25 basis points (bps). The Federal Funds (FF) rate or policy rate has been around 10–15 bps since the crisis, and is largely a negotiated rate stemming from the excess cash balances of nonbanks such as GSEs (i.e., Fannie Mae, Freddie Mac etc.), and banks. Only banks have access to the 25 bps interest on excess reserves (IOER) and thus arbitrage by depositing nonbanks cash at the Fed.

The reason for nonbanks' extra cash showing up as the excess reserves of banks is that it flows through money markets to the Fed because it allows banks and nonbanks to split the 25bps IOER. Singh again:

  The Fed’s balance sheet increased from roughly \$1 trillion (end-2007) to over \$4 trillion by end-2014, owing mainly to some \$3.4 trillion of asset purchases that sit on its asset side. The approximate corresponding entry are excess reserves of \$2.9 trillion on the liabilities side— these are deposits of nonbanks (who sold assets to the Fed) at banks, who then place them as deposits at the Fed. Since October 2008, the Fed offers banks 25 basis points per annum for their deposits (including excess deposits over the required reserves), but pays zero interest on deposits from nonbanks, especially GSEs.

As I note in the discussion in the comments, zero dollars of QE were "lent" or "given"— QE consisted entirely of purchases of government and agency securities on the open market, at market prices. What QE does is change the mix of "safe assets" versus cash. There's nothing "frivolous" about it.

  Is this making it categorically impossible to raise real interest rates?

Due to the payment of interest on excess reserves (IOER) and the creation of the Reverse Repurchase Program (RRP), the existence of excess reserves does not prevent the Fed from raising real interest rates. As noted above, IOER effectively provides a reason for banks to continue to borrow funds from money markets by subsidizing them at a fixed rate, while RRP provides a floor under risk-free rates (by providing a place to park excess collateral at a positive rate, independently of the supply of collateral at dealers banks) that is under the control of the Fed.

Real interest rates are low because the FOMC has decided not to raise them yet for policy reasons.


Does economic growth always implies investments in infrastructure?

In economics, we commonly use the formula for GDP as a Indicator of "economic growth"

The GDP formula is: GDP=C+G+I+NX

Where C is equal to all private consumption, or consumer spending, in a nation's economy, G is the sum of government spending, I is the sum of all the country's investment, including businesses capital expenditures and NX is the nation's total net exports, calculated as total exports minus total imports (NX = Exports - Imports).

Notice how we can have an increase in GDP by increased consumption or government spending. So technically we can see economic growth without any  investment in structure (which would be represented by an increase in I) from a mathematical perspective.

Note how this is a static measurement of economic activity, which we cannot make any predictions regarding the future. However you may want to look at the Solow-Swan Model regarding economic growth with capital accumulation and technological change.

Hope this helps.


Interpretation of a 2SLS Coefficient - Civil War Determinants

You have a model
$$(1)\  y = x \beta + e,$$
where $x$ is endogenous $\mathbb E[xe] \not =0$. This is the structural model in the sense that the parameter you are interested in is $\beta$ interpreted by its appearance in (1). In your case

y is civil conflict onset
x economic growth

suggesting that you are interested in examining how economic growth affects propensity for civil conflict. So positive $\beta$ means economic growth increase propensity for civil conflict.
Using 2SLS you get an IV-estimate of $\beta$ which we can denote $\hat \beta_{IV}$. When $\hat \beta_{IV}$ is positive it suggests that $\beta$ is positive and hence  economic growth increase propensity for civil conflict.
Interpretation is independent of $Cov(z,x)$ where $z$ is the instrument. Which in this case is

z is natural disaster.

For instrumental regression instruments must be relevant which in the simple univariate case reduces to $Cov(z,x) \not = 0$, but the sign is irrelevant (you can try to use $w := -z$ as instrument then $Cov(z,x) < 0 \Leftrightarrow Cov(w,x) > 0$ but the 2SLS estimate $\hat \beta_{IV}$ will be the same).


Is the Federal Reserve issuing money in a fair way?

Normative questions require a standard of fairness in order to then be analyzed and answered with some degree of "objectiveness" so that the answer is not just a declaration of philosophical-ideological positions.

The OP starts by writing

  "Basic logic says that, once a central bank is issuing new money for increasing the money supply, the money should be distributed equally
  between all the citizens."

Why "basic logic" says that? Whose "basic logic"? I note that throughout human history and especially in the western civilization in the last 400 years, "fairness" is almost exclusively interpreted as "proportionality", not flat equality (i.e. as "per capita tax" is considered not in accord with current societal values, so are "per capita transfers").

Anyway, under the given fairness standard, certainly "new money is not fairly distributed".

Regarding how the creation of new money works, and how it affects the domestic or global economy (or not), which is an additional question, I believe there are many threads here that provide a lot of information on the matter. Selecting the tag "inflation" or "money supply" will bring up most of them.


Let's assume our agent purchases positive quantities of $x$ and $y$. The utility maximization problem (without constraints on purchasing non-negative quantities) is:

\begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{maximize (over $x, y$)} & u(x,y) \\
 \mbox{subject to} & p_x x + p_y y \leq w
 \end{array}
\end{equation}

The Lagrangian is:

$$ \mathcal{L}(x, y, \lambda) = u(x,y) - \lambda(p_x x + p_yy - w)$$

Assuming $u$ is concave and differentiable, the problem is a convex optimization problem (in the jargon of optimization problems), and Slater's condition holds, hence the first order conditions are necessary and sufficient for an optimum. The first order conditions are:
$$ \frac{\partial \mathcal{L}}{ \partial x} = \frac{\partial u}{\partial x} - \lambda p_x  \quad \quad  \frac{\partial \mathcal{L}}{ \partial y} = \frac{\partial u}{\partial y} - \lambda p_x = 0 $$

Combining both equations (taking out lambda) we have:

$$\frac{\partial u / \partial x}{p_x} = \frac{\partial u / \partial y}{p_y} $$

This gives you the classic microeconomic theory result that the marginal rate of substitution is the ratio of prices. From here can you answer the question?

(Note: as you may have noticed, I slipped into your question a few more assumptions: eg. an assumption that the agent actually buys positive quantities of Option  A and Option B.)


Max of a profit function: partial derivative of an integral function?

You are really asking about the marginal product of a CES production function.
The LHS of your second equation is $P * dY/dY(i)$ where $Y_t$ is a CES aggregator of the $Y_t(i)$.
Let's define $\rho = \frac{\epsilon-1}{\epsilon}$ so that $Y=[\int Y(i)^\rho]^ {\frac{1}{\rho}}$.
Now apply the chain rule to get the derivative (also available for lookup in any micro text book):
$\partial Y/\partial Y(i) = \frac{1}{\rho} (\int Y(i) di) ^ \left(\frac{1}{\rho}-1\right) \rho Y(i)^{(\rho-1)}$
Cancel a few terms and you have your answer.


Market with changing number of goods and services

I think the best candidate would be monopolistic competition as introduced by

Dixit and Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity,

in which two models are introduced. One central theme was product variety and the endogenous determination of the number of product varieties.
There are many models formulated within the monopolistic competition framework in diverse areas of economic theory including

International Trade (Krugman)
Economic Geography (Henderson,Fujita,Krugman,Venables)
Economic Growth (Smulders, De Groot)
Macroeconomics (see for example Blanchard and Kiyotaki 1987)

The welfare analysis has centered around the question whether there is too much or too little product variety in equilibrium. The problem is that there are counteracting forces in equilibrium. Because of setup costs, revenues may fail to cover the costs of a socially desirable product. As a result some products may be produced at a loss at an optimum. This force tends towards too few products. On the other hand firms hold back output making price higher than marginal costs, which gives an incentive to entry, which creates a tendency towards too many products.
The different models therefore vary with respect to whether the first best social optimum is a case of too many or too few varieties depending on the specifics of the model and how the influence the balance of the counterveiling forces.
A representative selection of the literature on monopolitistic competition is collected in book

Brakman and Heijda (2003) The Monopolistic Competition Revolution in Retrospect,

chapter one of which has a textbook example of a general equilibrium monopolistic competition model ala Dixit and Stiglitz.


What is the best observable proxy for the instantaneous risk-free market interest rate?

SOFR is becoming more and more popular
Also there are other measures of repo rates, and also so futures on sofr


Interpreting the q-intercept of the demand curve

There isn't usually a name for this, but we could borrow from the theory of the individual consumer the term satiation point.


How to calculate average buy price when you buy, sell and rebuy

I am a Canadian, and under Canadian tax laws, I think there are at least two ways of determining your cost basis for shares. I am not going to get into the details of tax accounting; please see a qualified tax professional for advice.

However, if you just want to track your profit and loss for analysis purposes, the standard way of determining the book value of your positions is to track the average cost. You track the number of shares you own, and their total cost.

When you buy more shares,

the total cost goes up by the total you paid in the transaction (=(# shares in the transaction) * (transaction price)),
the number of shares increases by the amount in the transaction

You get the average cost by dividing the total cost by the number of shares.

When you sell, the price you sell at does not matter for the determination of your average cost. You reduce the number of shares by the number of shares in the transaction, and you reduce the total cost by the (average price)*(number of shares in the transaction). This means that selling does not change the average price, just the number of shares.

Your profit on selling is based on comparing the selling price to the average cost. This would be the “cost of goods sold” in inventory accounting. (If you want more details on this subject, you could look for primers on inventory accounting on the internet.)

What you appear to want to do is to roll the profits or losses from a sale into the total cost. That is, if you made a profit, use that to lower the “total cost” of the shares so that you can compare that average of that “total cost” to the market price, and quickly tell if you are profitable.

Although that might be convenient, it probably would not be acceptable under most accounting conventions (so no textbook will tell you to do it that way). It also would give some strange-looking results. If you made \$1 million profit on some shares, and then sold all but one of those shares, you would have a “cost” of about -\$1 million for that one share. Most people would look at you funny if you showed them that number on a spreadsheet.


Is GDP the market value of goods consumed or goods produced in a time period?

Goods produced.
In your example, a good is produced in Q1 but sold only in Q2. This adds to Q1 GDP (addition to inventories). For Q2 GDP, there's (i) a subtraction from inventories; and (ii) an addition to consumption—so (i) and (ii) cancel out and there's no net change to Q2 GDP.


Difference between pigovian tax and sin tax?

A pigouvian tax is on goods or services that are known to have negative externalities, and for which some estimation of the economic value of those negative externalities can be made, and is there to correct the market failure. Its goal is to improve economic efficiency.

A sin tax is on goods or services judged to be morally negative. Its goal is to reduce consumption and/or appease the moralists and/or raise revenue in a way that is less likely to be successfully opposed.


Restrictions on goods and capital, but freedom of labour movement

The proposed policy you are describing is akin to a basic International Microeconomics model with free immigration but immovable capital/goods. Assume two economies, $A$ and $B$, where in the one wages are higher, say $w_A > w_B$.

The basic results will be:

1) Immigration flows from economy $B$ to economy $A$ will continue until the wage is equated in both economies.

2) This common wage $w^*$ will be in between the previous two: $w_A > w^* > w_B$. This is because, as labor is concentrated in host Economy $A$ its marginal product falls and so is the wage, while as labor becomes more scarce in economy $B$ its marginal product and so the wage there increases.

The basic distributional consequences will be

1) Workers from economy $B$, both those that emigrated to economy $A$ as well as those that "stayed behind", will have higher wages than before.

2) Workers in the host economy $A$ will see their wage get lower compared to previously.

3) Capital holders in the host economy $A$ will see their returns increase

4) Capital holders in the origin economy $B$ will see their returns decrease.

Introducing a dynamic aspect (capital accumulation, technological progress), would modify the above conclusions in the following way: immigration will partially "hold back" any increases in the host-economy wage, resulting in lower wage increases than what would happen without immigration.

So from the point of view of a host country like the UK is, under such a policy foreign workers (immigrants or not) and local capitalists will benefit, while local workers and foreign capitalists will get hurt.

In my eyes, the above constitute an interesting framework to contemplate the ideological principles alluded to by the OP, but this is a discourse that does not belong to this site.


Understanding the construction of stochastic processes

This construction you describe is not fully general. In fact it characterizes strictly stationary time series. You see that it's shift-invariant. This operator $S$ is essentially a shift operator.

For comparison, here's the usual definition of, let's say discrete-time, processes:

Definition A stochastic process is a sequence $\{ X_t \}$ of Borel measurable maps on a probability space $( \Omega, \mathcal{F}, \mu )$.

Now for what you're describing, you have a fixed Borel measurable map $X: \Omega \rightarrow \mathbb{R}^n$. It's the underlying measure that is evolving according to $S$. The map $S$ induces a new "push-forward measure" (in measure-theoretic parlance) on $\Omega$ by just taking preimages: define a measure $\mu_S$ by

$$
A \in \mathcal{F} \stackrel{\mu_S}{\mapsto} Pr(S^{-1}(A)).
$$

So the random vector $X: ( \Omega, \mathcal{F}, \mu_S) \rightarrow \mathbb{R}^n$ is $X \circ S$ by construction. They induce the same push-forward measure on $\mathbb{R}^n$. Do this with $S^t$ for each $t$ and you have your time series.

As for your question about $\omega$, inspecting the proof for the other direction should clarify this---i.e. any strictly stationary time series must necessarily take this form for some $( \Omega, \mathcal{F}, Pr)$, $X$, and $S$.

The basic point is that, from a general point of view, a stochastic process is a probability measure on the set of its possible realizations. This is seen in, for example, Wiener's construction of Brownian motion; he constructed a probability measure on $C[0, \infty)$. So in general, an $\omega$ is a sample path and $\Omega$ consists of all possible sample paths.

For example, take the two processes you named above. They are strictly stationary, if let's say the innovations are Gaussian. (Any covariance-stationary time series driven by Gaussian innovations is strictly stationary.) The construction would then start by taking $\Omega$ to be the set of all sequences, $\mathcal{F}$ the $\sigma$-algebra generated by coordinate maps, and $Pr$ the appropriate measure. For the white noise process (2), $Pr$ is just a product measure on an infinite product.

Reference This characterization/construction by shift of strictly stationary time series is mentioned in White's Asymptotic Theory for Econometricians.


You are not very clear on whether your model is uniequational or a system of equations (e.g. a VAR). I will assume the former is your case.

There are several approaches to estimating Real GDP. One approach is via estimating a production function (i.e. a supply side approach). Here, you need information on factors of production, like hours worked, education level, physical capital, etc. It would make little sense to use inflation as a regressor in this approach.

An alternative method is to focus on the demand side, by which you are interested in explaining the determinants of GDP sub-components like consumption, government spending, etc. This approach is in my opinion the most common among specialised institutions like central banks and etc. For example, Atlanta Fed, OECD, NIESR UK, researchers at the Bank of Canada.

The latter type of models are normally very complex, and go beyond the one equation system. As such, some of them do include inflation as one of the variables (both as independent and dependent variable), because inflation is expected to affect real wages and thereby consumption (for example, see the documentation of the Atlanta Fed model here.

Now, if you only have one equation, and are using the demand side approach, I would include inflation as long as other included variables are also nominal (like retail sales). Alternatively, you can redefine every variable as real, and not include inflation. This is a more restrictive scenario though, as you are imposing extra restrictions (which, in fact, you can test).


Is there a term for the personal wealth that is not used to generate income?

I do not believe that your suggested definitions will hold up. Since this is a forum for questions about economics, I do not see the point of proposing a new definition here. Who is going to see it?

What you call “income generating wealth” is captured by “wealth” in its standard usages. However, what you exclude (such as bank deposits) will be included in “wealth” or “claims on wealth.” From a portfolio perspective, it makes no sense to exclude bank deposits. Furthermore, “income” is well defined, and there are assets that generate no income for the holder, and yet are wealth (for example, a personal residence). Rephrasing to include future income does not help, since the sale of a primary residence is not considered income.

Everything else would be classified as “consumer goods.” If you want to only include things that “make you look like a rich person,” you would end up with the fuzzy category of “positional goods.”

Why not include consumer goods in wealth? One of the regularities of consumer goods is that they have almost no resale value. So there’s no good way to measure its value.

The lines are blurry. For a collector that buys and sells particular goods, those goods are actually assets that are expected to make a profit. However, no statistician could hope to measure the value of these exceptions.


There is a very nice survey by Aviv Nevo on Empirical Models of Consumer Behavior. He surveys some of the models used in the literature. In particular, he discusses two commonly used demand systems: multi-stage budgeting approaches and discrete choice models.

We may also find interesting Recent Developments in Empirical IO: Dynamic Demand and Dynamic Games by Victor Aguirregabiria and Aviv Nevo.


How to measure home value distribution in a city?

When it comes to constructing indices, the axiomatic approach is the standard approach in the economics literature.

An axiom is a property you would like your index to satisfy. For instance, in your example, you would probably wan your index to increase when the distribution in b) changes from $X$ to $Y$ and $Y$ first order stochastically dominates $X$.

You can require your index to satisfy the above property, as well as others, and hope that as you impose more and more desirable properties, you will narrow down the class of acceptable indices to a single index.

Now, such axiomatic characterization (i.e. proving that index $I$ is the only index that satisfies a set of properties) is typically not an easy task. As a matter of fact, some very fine researchers have made their whole career out of these kinds of characterization results.

These characterizations are also very much context dependent. Although there are some general results out there characterizing price indices (see https://scholar.google.be/scholar?q=axiomatic+price+index&btnG=&hl=nl&as_sdt=0%2C43) they might not be directly amenable to your problem .

(I don’t know enough about the topic to tell you whether something like an axiomatic theory of housing price indices already exists. It’s not impossible that it does, but I just don’t know.)

Depending on how much time you want to put into this, here is what I would do if I were you:

Learn about axiomatic price index theory ( again see https://scholar.google.be/scholar?q=axiomatic+price+index&btnG=&hl=nl&as_sdt=0%2C43 as well as Afriat, S. (2014). The index number problem: construction theorems (1st ed). Oxford ; New York: Oxford University Press. and
Diewert, W. (1988). The early history of price index research. Retrieved from http://www.nber.org/papers/w2713)
If you learn about an already existing theory of housing price indices, just pick the index you like best among the ones in the literature.
Otherwise, try to see if you can adapt some of the general price indices in the literature to your housing case.
In last resort, develop an axiomatic theory of your own.

If you have the time and ability, you can try to obtain characterization and publish nice research papers on the topic.
Otherwise, you can still use the axiomatic approach to craft an index that satisfies some of the properties you think are the most important. Just list those properties, and start looking for an index that does the job.
If you can’t find one, don’t disappear. It may be due to a lack of creativity, but it may also be that you have imposed so many properties that there does not even exist an index satisfying all of them (this is called an “impossibility result” in the axiomatic literature).
Just start relaxing or dropping properties one at the time up to the point where you find an index that works. Then if you still have the time and energy, start re-adding some of the properties you dropped, one at the time, and see if you can find a new index that works.


How can governments seemingly run permanent budget deficits?

Because they have low interest rates and growing tax revenue.

As long as your income is growing faster than your debt, you can keep borrowing.


Financial investment in the composition of GDP

What you're describing is a change in the capital account, not in GDP. They're related through the balance of payments, in that if a country is running a current account deficit (usually arising through being a net importer), they'll have a capital account surplus (i.e., foreigners will on net buy more domestic assets or domestic owners will be net sellers of foreign assets, or both), and vice-versa. So if you take foreign investment in domestic assets for example, it'll be associated with being a net importer, but it does not appear directly in the national accounting identity $Y=C+I+G+NX$.


Indirect utility from expenditure function

Mas-Colell, Whinston, Green chapter 3, especially p. 75 explains this. One possible way is $v(p,w)=e(p,v(p,w))$ where $v$ is the indirect utility function, $p$ the price vector and $w$ the wealth.


What determines a new cryptocurrency's initial value?

A cryptocurrency's initial value is set at the same way as an established cryptocurrency.

It's the price that people are willing to pay, when they believe the hype told to them by the crypto's promoters.

As such, the price is a measure of the promoters' marketing skills, and the gullibility of the buyers.

I completely understand that you feel it ought to be different. But for an intrinsically worthless thing, what I've written above, applies both to a new crypto and an existing one. It's worthless. Therefore its price is whatever the owner can fool someone into paying for it.

As for what number the promoters actually pick, they will play a guessing game. There's a story about how opticians price the pair of a new glasses. When they tell a new customer the price, they say, it will be £60. And they pause, and watch the customer's reaction. If they don't blink, they add "per lens". Pause, and if no blink again, "and £120 for the frame". If the customer looks scared, they can then add "but with your discount, the frame will be half price".

Pricing worthless assets is similar, but far worse. The seller make a pretty fairy story about the total value of the market, works out how many tokens there will be, throws in a big "get in at the ground floor" discount, and then plucks the asking price out of thin air.


What empirical evidence is there of distortions when capital gains taxes are introduced or raised?

Correcting Misconceptions in the Question:
Before providing an answer it is worth noting the premise in your question is simply incorrect. You state:

I have heard that introducing capital gains taxes would create distortions in investments, and that the optimal tax rate for capital gains is actually 0.

But this is not rationale for the famous 0 capital income tax Chamley-Judd result. Virtually all taxes create distortions, and any modern government, due the amount that modern governments spend, has to resort to distortionary taxation at some point. However, the famous Chamley-Judd result showed that (Chamley, 1986; Judd, 1985) the burden of capital income tax is in long run shifted to labor, so there is no point to those distortions and one should just levy those taxes on labor incomes instead (although more recent papers showed that the result does not always hold and that only some of the burden of capital income taxes gets shifted not all of it - e.g. see Straub & Werning 2014 or Diamond & Saez 2011).
So the 0 capital income tax result had nothing to do with distortions directly. Indeed taxing labor causes non-trivial amount of distortions as well and in spite of that those economists recommended taxing it directly.
Answer:
Now to your main question on whether empirically capital taxation causes distortion, the answer is clearly yes. There are numerous studies showing that, and I think it is a fair to say that the question is not if there are distortions but how much and how economically significant they are.
The deadweight loss and distortions caused by capital income taxes will vary from country to country and from time period to time period, and it is beyond scope of a single answer to cover every country on a planet and every time period but here are examples of some studies that found capital taxes create distortions:
Devereux MP (2004) Measuring taxes on income from capital. In: Sørensen PB (ed) Measuring the tax burden on capital and labour. MIT Press, Cambridge
Becker J, Fuest C (2004) A backward-looking measure of the effective marginal tax burden on investment. CESifo Working Paper No. 1342
Goolsbee, A. (1998). Taxes, organizational form, and the deadweight loss of the corporate income tax. Journal of Public Economics, 69(1), 143-152.
Shan, H. (2011). The effect of capital gains taxation on home sales: Evidence from the Taxpayer Relief Act of 1997. Journal of Public Economics, 95(1-2), 177-188.
These are just few examples, but in public economics there is no question that capital taxes create some distortions (generally the only non-distortionary taxes are Pigouvian taxes and lump-sum taxes plus some special cases), so the question is not if but how much and answer to that will differ in time and place. Moreover, this has nothing to do with the zero capital tax result even pro-capital income tax economists would not deny taxing capital is distortionary.
For example, Goolsbee (1998) founds that corporate taxes (which are economically taxes primarily on capital incomes) in the  US between years  1900–1939 caused distortions that resulted in deadweight loss of 5–10% of their revenue. Again that number will vary by place and time.
The positive effects of elimination of capital income tax on economy are elimination of the above mentioned distortions. Hence, if capital taxes caused in the US between 1900-1939 deadweight loss of 5-10% of revenue, then eliminating them would mean that this loss would not occur in the first place.


In the link one can very clearly see that the company has no contractually short term debt, and in the short-term (i.e. in the next 12 months) has to pay part of its long term debt. Also, that the debt amounts are not included in the line "Accounts payable"

Also, one can see its long term debts

And no, debt is not only bonds.


Can individual consumers ever be price-makers, and can they thereby have major effects on welfare?

Answer to the Question on Welfare
The welfare analysis is not as simple as that.  First, let us set aside for a second any inequality considerations (we can add them on later but there are some misconceptions about welfare analysis that have to be corrected first).
Welfare in supply-demand analysis is conventionally measured by amount of consumer and producer surplus. These are defined as the difference between maximum price consumer is willing to pay and price consumer actually gets to pay in case of consumer surplus and in case of producer surplus as the difference between the price consumers actually pay, in both cases summed across all consumers and producers. Graphically, the consumer surplus is given by the blue area and producer surplus the red area on the picture below:

Given this the expansion of demand can very well lead to increase of not just overall welfare but consumer welfare as well, even if price increases. Consider a trivial example:
Suppose demand is given as: $D=100-10P$ and supply as $S=6P$. Given this assumptions market price and quantity will be $P_e = 6.25, Q_e = 37.5$ which will gives us the following producer and consumer surplus: $PS = 117.1875, CS = 70.3125$.
Now suppose the demand increases (i.e. shifts to the right) which can be represented by new demand function $D = 120 -10P$ and supply remains the same. Now the market price and quantity will be  $P_e = 7.5, Q_e = 45$ which will gives us the following producer and consumer surplus: $PS = 168.75, CS = 101.25$.
As clearly demonstrated by the simple stylized example above increase in demand even when it leads to increase in price can very well leave consumers better off even measured in just consumer welfare terms.  Note the above is just example, the size of both consumer and producer surpluses crucially depend on shape of supply and demand, so in some situation consumers could be worse off. The point is that already from the start the assertions made about consumer welfare from shocks to demand are simply invalid.
Next, if we would want to start discussing inequality and similar issues then yes the above welfare analysis would change. If we would for example denote all producers poor and all consumers rich (or vice versa), assume that interpersonal welfare comparisons are possible etc., we would additionally have to scale the consumer and producer surplus accordingly to reflect that. Nonetheless, this does not necessarily change outcome of the above welfare analysis, the inequality would have to be severe enough so that scaled CS/PS would be lower in the new equilibrium.
However, the simple partial equilibrium generic supply-demand graph/model is not suited for analysis of inequality. For example, the stylized supply-demand model above does not explicitly model factor prices (e.g. wages, capital etc). The reason why supply is upward sloping is that the more producers produce the more factors they need to employ (e.g. more workers) which puts upward pressure on wages. Just because factory owner is rich that does not mean workers are so in more involved analysis we would try to model also supply demand changes on labor markets or some alternative partial equilibrium analysis that focuses specifically on people's incomes/wealth whether they are consumers or producers etc. Since this heterogeneity is not captured by simple stylized supply and demand you simply cannot do any serious welfare analysis that takes into account inequality simply from  supply and demand shifts on a given goods market. You need more complex model for that (see discussion in any public economics textbook for example Stiglitz, Economics of the Public Sector, 3rd ed. is good place to start).
Answer to the Question on Individual Consumers Changing Prices
Yes individual consumer demand can change market price unless we have special situations (e.g. situation where supply is completely elastic so increase in demand wont affect equilibrium price). I mean this is on some level trivial. The demand function is nothing more than an aggregation of individual demands, hence if demand and supply function are such that increase in demand leads to  change in price, then individual consumers can affect the market price.
You can of course have special situations such as 'step-like' supply because it might be simply too costly for suppliers to adjust prices just because one extra person demands one extra good (your example of shopper in general store).  An exaggerated case of such situation is plotted below.
In the case as shown below the demand will affect price only if it shifts sufficiently to point where supply is elastic. The probability of some individual consumer being the proverbial 'last straw' will depend upon parameters of the model. For example, if we assume that quantity demanded must increase by $100$ to create change in price, and if we assume this increase in demand is caused by $100$ individual consumers each raising their demand exactly by 1, then the probability of any single randomly selected consumer being the one that caused the price change would be $\frac{1}{100}=1\%$.    This is of course oversimplified because in reality one consumer my increase their demand by 10 another just by 1, this would make the probability calculation more complex but not impossible in principle.
This being said, I do not know of any research that would try to calculate the above. Normally when it comes to demand researchers do not have data on individual demand of every single customer. Even the demand curve has to be estimated as only equilibrium demand can be observed, but this is doable and can be done with reasonable precision. So I would be surprised if there would be some situation where there is enough data to realistically calculate this. However, we know that demand as a whole affects prices - there is whole literature on estimation of supply-demand systems (see  MacKay & Miller; 2019 for review and sources cited therein for particular examples) which shows that changes in demand change the prices and since demand is just an aggregation of individual demands individuals have an effect on this. Since demand curve can shift only when individual demands shifts there will always be someone on the margin responsible for triggering price change.


Utility maximimization for unusual Leontief utility function

Above is a plot of the indifference curves for $U=1,2,...,8$ and also for $U=8.5$ that I made in MATLAB. For $P_x=P_y=1$ and $m=4$, the budget constraint is simply the the line $y=4-x$ (not pictured here, but easy to visualize).

Notice this budget constraint overlaps the $U=4$ indifference curve (fourth contour from the origin) on the line segment connecting $(1,3)$ and $(3,1)$. Any other point on the budget line would yield $U<4$, hence consuming on the line segment connecting $(1,3)$ and $(3,1)$ is optimal. But since all of these bundles correspond to $U=4$, the optimal bundle is not unique, but could be any bundle on this line segment.

Here is the MATLAB code that generated the plot:

u=zeros(60,60);
for i=1:60;
    for j=1:60
        u(i,j)=min([(i+j)/10 4*sqrt(i/10) 4*sqrt(j/10)]);
    end
end
figure
contour(u,[1 2 3 4 5 6 7 8 8.5]);
ylabel('Good Y')
xlabel('Good X')
grid on
grid minor
xt = get(gca, 'XTick');                                 % 'XTick' Values
set(gca, 'XTick', xt, 'XTickLabel', xt/10)
yt = get(gca, 'YTick');                                 % 'XTick' Values
set(gca, 'YTick', yt, 'YTickLabel', yt/10)


What are the factors that drive openness in developed vs developing countries

I assume you mean openness to international trade, and that you're interested in both policy choices (does the country want to be open) and physical constraints (can they be open).  The main factors that come to mind are:

Size: the smaller you are, the more you are forced to rely on international trade, as it simply isn't feasible to complete many economic operations entirely within your borders.  Lichtenstein and Andorra are much more open than the US or Russia primarily for this reason.
Remoteness and transport costs: if you're tiny, but in the middle of nowhere, like Tuvalu, the cost of trading internationally means that anything you can feasibly produce domestically will be cheaper than imported goods.  There is little point in trying to protect your domestic market because transport costs already provide more protection than you're likely to want.  This also applies to a lot of small landlocked countries, like Burundi.  Some dependencies of more advanced countries overcome this by massively subsidising transport, often linked to military bases.  E.g. consider the money the UK has spent on an airport in St Helena.
Taxation capacity: this matters very little to rich countries, but developing countries often have a small tax base, large informal sector, and find it hard to replace tariffs with alternatives (VAT, income tax, etc).  Some would like to encourage international trade, but are constrained in how quickly they can remove tariffs.
Proximity to the technological frontier: Ha Joon Chang would argue that the history of protectionism in countries like the UK and US make this point very clearly.  If you lag behind your competitors, you're more likely to protect industries you want to catch up.  If you're ahead, you are more likely to advocate for free trade, knowing that the more unconstrained international competition is, the more you will gain.
Relatedly, supply capacity is perhaps the equivalent for developing countries far from the technological frontier.  Some countries have very weak technology but so much cheap labour that they ought to be able to compete in global markets for simple manufactured goods, but lack critical elements necessary to develop successful firms in these areas.  These constraints come in many forms, a couple of the most common are poor physical infrastructure and weak financial services.  If you have supply capacity you will be keener to promote international trade.  This is not an issue for rich countries.
Comparative advantage will tell you what your country is likely to produce and export if very open.  As I explained in this answer, you will be more keen to rely on your comparative advantage if it is in an industry that is experiencing rapid improvements in productivity.  If your advantages are all in stagnant industries, you'll be more tempted to use protection and other state intervention to shift your comparative advantage, and then open your markets.  This is a bigger issue for poorer countries.
Political culture: many countries have attitudes to openness that are not entirely or directly driven by economics.  For instance, some would argue that European economic integration in the late 20th Century was primarily an attempt to reduce continental violent conflict.


Welcome.
Do you mean: if $x^*_{k,it}$ are the panel variables, then $x_{1,i} = \min_t x^*_{1,it}$, $x_{2,i} = \max_t x^*_{2,it}$, etc.? Usually people use averages over time or values at certain points of time, while you mean the smallest (or largest) value during a period of time.
Computationally I do not see any differences. They are just numbers.
I am more curious (or doubtful) about interpretation, though. SCM balances on $\mathbf{x}_i$, where $\mathbf{x}_i = (x_{1,i}, x_{2,i}, ..., x_{k,i})$, i.e., SCM tries to find $w_2, \ldots, w_{J+1}$ such that $\mathbf{x}_1 - \sum_{j=2}^{J+1} w_j \mathbf{x}_j$ is close to zero (in terms of $v_j$-weighted sum of squares). But then what's the meaning of balancing on $x_{1,i}$? The treatment group may attain the minimum value in year 1960, while one donor in year 2000 and another in year 1990. Then what would it mean that $x^*_{1,1,1960}$ is close to $w_1 x^*_{1,2,2000} + w_2 x^*_{1,3,1990}$?
On the other hand, if it's the smallest during one business cycle, for example, that can make sense. If you have good reasons to use those min/max values, I think that's fine. I am very curious about what kind of min/max variables you have in mind.
Edit
If the treatment group's $x_1$ value is the smallest of $i=1,2,...,J+1$, then it is not possible to have $w_j$ such that $w_j\ge 0$, $\sum_j w_j=1$, and $x_{1,1} = \sum_{j=2}^{J+1} w_j x_{1,j}$. The convexity assumption (nonnegativity & add-up) is violated and you cannot use AG (2003) or ADH's (2010 and after) SCM. If you still use their SCM, you will be harshly criticized, I am afraid.


The Fed pays 1.50% interest on reserves; so why are interest rates so low?

There are several issues here:

There are banks offering close to $1.5\%$, some well-known such as Goldman Sachs and American Express
Most banks currently have excess reserves deposited in the Federal Reserve System which they could use for lending if they wanted to, so they are not missing profitable commercial lending opportunities due to lack of deposits
Many customers are leaving their money in very low interest accounts, possibly through inertia or the attraction of other banking services, and so banks have less incentive to push up interest rates.  If customers want more, they can go for longer-term certificates of deposit yielding over $2\%$
In eras of "normal interest rates", banks took a margin.  For example the old anecdotal 3-6-3 rule for bankers (pay $3\%$ percent interest on deposits, lend money out at $6\%$, and tee off at the golf course at $3$ p.m.) might have been associated with a base prime rate of around $4-5\%$ percent and a default rate of around $1-2\%$.  So with a base rate of $1.5\%$, short-term savings rates should not be expected to be close to this.  Earlier in the cycle when Fed interest on excess reserves rates were $0.25\%$, this margin was not compatible with positive savings rates; only now are banks able to restore their margins


Are monotonic and continuous preferences necessarily rational?

Consider a preference relation in $\mathbb{R}^2$ such that $x=(x_1,x_2)\succsim (y_1,y_2)=y$ $\iff$ $x_1\geq y_1$ and $x_2\geq y_2$.

1) You might like to argue whether this preference relation is strictly monotonic and continuous.

2) Is the relation defined above complete?

Then, as a side dish, you might also reconsider your claim that continuity is the cause of transitivity.

Note: I just wrote this particular one with the purpose of providing a thought experiment. More in a way to challenge your understanding. I am not sure whether this example provides an answer to your question or not.


The chart you are looking at is off by a year: it should have shown about £10 billion in 2007 and almost £100 billion in 2009.  See the ONS figures for the "Public sector current budget deficit, excluding public sector banks" and click on "table" and "year"

It is more common to look at "Public sector net borrowing, excluding public sector banks" but the story is similar: about £37 billion in 2007 and about £153 billion in 2009

There was a reduction of about £25 billion in central government current receipts  and an increase of about £60 billion in central government current expenditure  plus some other things happening.  Falling receipts included lower corporation and income taxes and part of a 12 month cut in VAT, while increasing expenditure included support for banks and increased numbers of people claiming unemployment and other benefits.  Interest payments (part of expenditure) actually fell by about £4 billion due to lower interest rates, though they rose the following year due to the increased debt

You can find more data in the ONS Public Sector Finances time series dataset spreadsheet


Prove if $\succsim$ is rational then: if $x \succ y \succsim z$, then $x \succ z$

By contradiction, assume $z \succsim x$. However, we already know that $y \succsim z$, which is a contradiction (after a couple more steps). Can you now see this?


$\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dd}{\, \mathrm{d}}$
How many arguments does $C$ have? If the only argument to $C$ is $Y_D$, then $C_Y$ and $C_\tau$ might not make sense.
I think it's easy to get yourself in trouble here.

Suppose that $C$'s only argument is $Y_D$. Yet, we want to impose the restriction that $Y_D = Y(1-\tau)$. Well, as you wrote
\begin{align*}
\dd C = C_{Y_D} \dd Y = C_{Y_d} (1-\tau) \dd Y - C_{Y_D} Y \dd \tau
\end{align*}
is correct. But what is $C_Y$ and $C_\tau$? Well, technically,
$$
\pd C Y = \pd C \tau = 0.
$$
This is because $Y$ and $\tau$ do not directly affect $C$.
I guess what you have in mind is something similar to the following.
Suppose you define a function $f$ such that
$$
f(Y, \tau) = C(Y(1-\tau)).
$$
Then
\begin{align*}
f_Y &= \pd f Y = \pd C {Y_D} (1-\tau) \\
f_\tau &= \pd f \tau = -\pd C {Y_D} Y.
\end{align*}
So, really, $C_Y \neq f_Y$ and $C_\tau \neq f_\tau$, because $C_Y$ and $C_\tau$ are both equal to zero.


What are some applications of Akerlof's 'lemons' in goods and services markets?

Health insurance. If an insurance company must charge a single premium because they cannot distinguish between low risk and high risk individuals, they will charge a premium based on average risk. As a result the proportion of high risk individuals in the pool of insured individuals increases, whilst the proportion of low risk individuals decreases (healthy individuals are less willing to pay a high price of insurance), which pushes up the average price of premiums, further reduces the proportion of low risk individuals, etc. The market fails.


How is reducing government expenditure supposed to reduce aggregate demand?

Aggregate demand is the total domestic demand for goods and services in an economy. Cutting government spending could reduce aggregate two ways.

Direct. Government demand is obviously lower, and is a component of domestic demand.
Indirect. The recipients of income from government spending have less money to spend.

It is theoretically possible that other sectors can raise spending to offset those effects. (If taxes were cut at the same time, this is obviously easier. In that case, there can just be a shrinkage of the government’s share of the economy.) There is a large debate about the empirical effects of spendng cuts. However, since the question just asked for a mechanism, that literature is somewhat out of scope.


In the benchmark hedonic price analysis, we assume a utility function of the general form

$$U = U(x, z_1,...,z_n)$$

where "$x$" stands for the composite good, and $(z_1,...,z_n)$ are the characteristics of good $y$ that are valued by the consumer. Assume for simplicity (as is usually done in the literature, and as is the OP case), that the consumer will only purchase just one unit of good $y$ (one house). The budget constraint in this case is

$$x+p(z_1,...,z_n) = M$$

where $M$ is the available income and $p(z_1,...,z_n)$ is the price of the good, expressed as a function of those same characteristics. This is an important departure from the standard consumer theory, because here, $p$ is not market price, but it is the "willingness to pay" of the consumer, decomposed into a function of many characteristics.

The lagrangean here is

$$\Lambda = U(x, z_1,...,z_n) + \lambda[M-x-p(z_1,...,z_n)]$$ and the f.o.c's are

$$\partial U / \partial x = \lambda$$

$$\partial U / \partial z_i = \lambda \partial p / \partial z_i, \;\;\; \forall i$$

So

$$\partial p / \partial z_i = \frac {\partial U / \partial z_i}{\partial U / \partial x}$$

Note that we have differentiated the price also, since again, this is not "the market price faced by the consumer" but his/her willingness-to-pay.

The hedonic-price econometric estimation is based on actual transactions, and so on data related to prices that consumers have actually agreed to pay.
So the specification

$$p = f(z_1,...,z_n, e)$$

estimates $$\partial p / \partial z_i$$ (and/or their binary-characteristic analogues), which is seen to be the ratio of the marginal utility of the characteristic scaled by the marginal utility of income.

If one assumes quasi-linear preferences of the form

$$U = v( z_1,...,z_n) +x$$

then $\lambda =1$ and so the marginal utility of the characteristic is the same as the marginal monetary willingness-to-pay for that characteristic.

On the basis of these remarks, I think you can conclude whether the econometric specification is a "reduced form" or not.


Resources to improve my economic intuition

While textbooks are the best way to learn the material (MC=MR etc), here are some suggestions for improving your intuitive understanding of economics.

Books

The Undercover Economist by Tim Harford

The Armchair Economist by Steven E. Landesburg

Blogs

Marginal Revolution

Noahpinion

Conversable Economist

The Enlightened Economist (Great for book recommendations)

Podcasts

Econtalk with Russ Roberts

FT Alphachat

This is just a start. Hopefully these will send you to other excellent sources.


Does estimating OIRFs from an SVAR (with uncorrelated errors) mean we are orthogonalizing the error structure twice?

Usually if you are estimating a SVAR model you would use structural impulse response functions (SIRF) not orthogonalized impulse response functions.  SIRF are not exactly the same as just IRF or OIRFs. SIRF already takes the identification problem into account when the SVAR is estimated.
To learn more about this you can have look at this blog about impulse response functions in R - the blog is brief but explains difference between simple impulse response, orthogonalized impulse response, structural impulse response and generalized impulse response. More info can be found in sources cited therein.


Derivation of Surplus in Paul Romer's paper on "mathiness"

I think you have a typo in your  $q_0$: the exponent of $N$ should be  $-\frac{b}{a+b}$. I did the whole calculus with this corrected type of $q_0$ and I was able to replicate your results (this is why assume $q_0$ has only a typo and you actually did the algebra with the correct $q_0$).
I suggest that the discrepancies to the paper are indeed connected to the Surplus function. But i cannot come up with any idea why this surplus function might be wrong.

Edit: Outside from the typo there was no error, @actuarialboi9 and my humble self calculated the individual surplus. If one multiplies that Surplus function with $N$ the number of people in the market results will be equal:
$\begin{equation} S= C(a,b,m) N^{\frac{b(a-1)}{a+b}} \;  \cdot \; N \;  = \; C(a,b,m)N^{\frac{a(1+b)}{a+b}} \end{equation}$


Minimalist example of voter profiles yielding different outcomes

First, let me state that this is a beautiful problem!

Here is a proof that $n=4$. To prove that $n \leq 4$, consider the following example:

$m+4$ voters have preference $a \succ d \succ b \succ c$
$m+3$ voters have preference $b \succ d \succ c \succ a$
$m+2$ voters have preference $c \succ d \succ b \succ a$
$m$ voters have preference $d \succ c \succ a \succ b$

where $m \geq 2$. I leave it to you to check that $a$ is selected by the plurality rule, $b$ by the simple run-off, $c$ by the elimination rule and $d$ by the Borda procedure.

Now, let us prove that $n>3$. The cases $n=1$ and $n=2$ are trivial, so let us assume $n=3$ and write $v_1,v_2,v_3$ for the number of voters in  the three voter profiles. Without loss of generality, let us assume that $v_1 \geq v_2 \geq v_3$, that $a$ is selected by the plurality rule, that $b$ is selected by the run-off election, that $c$ is selected by the elimination rule and that $d$ is selected by the Borda count.

The outcome of the plurality rule shows that $a$ is most preferred by voters of group 1 (since it wins with no ties). If $a$ were also preferred by voters of group 2, $a$ would win the run-off election, which is a contradiction. Thus $a$ is most preferred by voters of group 1, $b$ is most preferred by voters of group 2, and $v_1>v_2 \geq v_3$, the latter inequality being strict if $b$ is not most preferred by voters of group 3. But then it is clear that both $a$ and $b$ survive the first two rounds of the elimination rule, i.e. that $c$ and $d$ are eliminated, which is a contradiction.


How to find opportunity cost if each choice has more than one important effect?

Note that as a rational agent you will have preferences on the set of (wage, free time) bundles. Your own answer suggests that these two "goods" are perfect substitutes, but this is not the general case. If your preferences are given by a utility function you can calculate the utilities for the three bundles in your choice set and then proceed in the way you indicated, measuring opportunity costs by utils foregone.

Alternatively, you can for each of the three bundles find a respective indifferent point at the maximal-free-time line in the space of bundles (corresponding to zero traveling time) and compare the wage levels of these points to get a monetary measure of opportunity costs. This corresponds to converting the traveling time contained in each bundle to some wage loss which you would be prepared to face if traveling time could be eliminated from the respective bundle. (If your preferences happen to be quasi-linear in money, this amounts to calculating your willingness-to-pay (WTP) to avoid traveling time, but in the general case this WTP need not be independent of the wage level.)


Common Knowledge and the Red Hats Puzzle

Without the teacher, everyone knows that there is at least a red hat, but nobody knows that everyone knows - the fact is not common knowledge.

With the introduction of the teacher,

Girl 1 doesn't answer. Due to common knowledge, 2 and 3 can reason: "1 knows there is at least one red hat, and since she doesn't know her hat color, 2 and/or 3 must have a red hat.

Without the introduction of the teacher,

Girl 1 doesn't answer. Without common knowledge, there is nothing 2 and 3 can reason on top of their prior knowledge: 2 will keep knowing that 3 has a red hat, and 3 will keep knowing that 2 has a red hat. Nothing more.

In other words: Without the teacher, the set of knowledge is:

1: 2+3 have red hats
2: 1+3 have red hats
3: 1+2 have red hats

The teacher works as an injector of additional knowledge:

1: 2+3 both know that there is at least one red hat
2: 1+3 both know that there is at least one red hat
3: 1+2 both know that there is at least one red hat

And, common knowledge means that in the next level, everyone knows that everyone knows

1: 2+3 both know that I know that there is at least one red hat

etc, ad infinitum. This additional information is required to solve the puzzle.


Are there some examples of theoretical models that discuss pandemics?

Many links at
https://marginalrevolution.com/?s=pandemic+model
(MR is Tyler Cohen's blog. There isn't much else to say, but my reply needs to be at least 30 characters long.)


AGV mechanism and Individual rationality

The simplest example I could cook up. I might have messed up somewhere.

Consider the provision of a public good with two agents, $n=2$. Let the type space be such that $\theta \in \{0,1\}$, and both types are equally likely. Either the good is provided, $k=1$, and total cost $c=2/3$ occurs, or the good is not provided, $k=0$, which is costless.
Let
$$v_i (k; \theta_i) = k  (\theta_i - \frac{c}{2}).$$
That is, either the good is not provided and nobody pays or it is provided and both share the cost. Of course transfers can be used to compensate.
It is efficient to provide if at least one agent values the good, i.e., has a type of 1. AGV leads to this efficient rule $k^* (1,1)=k^* (0,1)=k^* (1,0)=1$ and $k^* (0,0)=0$.

Then,
$$\xi_i (1) = \frac{1}{2} \left( v_j(k^*(1,0);0) + v_j(k^*(1,1);1) \right)
= \frac{1}{2} (0+1 - c) = \frac{1}{6},$$
$$\xi_i (0) = \frac{1}{2} \left( v_j(k^*(0,0);0) + v_j(k^*(0,1);1) \right)
= \frac{1}{2} (0+1- c/2) = \frac{1}{3},$$

Now, suppose $(\theta_i,\theta_j)=(0,1)$. Then,
$$v_i(1;0)+\xi_i(0) = 0 - \frac{1}{3}  + \frac{1}{3}= 0 < \xi_j (1) = \frac{1}{6}.$$
So, we don't have ex-post IC in that case.


What was economics like as a field before Adam Smith, the father of *modern* economics?

There exists the... "Adam Smith of the Arabs": Ibn Khaldun. He lived during the 14th century. The work of his that focuses on Economics is Muqaddimah translated in English in 1958 as "The Muqaddimah: An Introduction to History".

He advocated less involvement of the state in production and trade activities, which should focus instead on stabilization policy. He also argued in favor of a reduction in military expenses, in order to invest more on education and technology. He also pointed to the increased efficiency that comes from specialization of labor.

Don't all these sound eerily familiar? (400 years before Adam Smith). For a summary of his economic theories, see here.


Fiscal budget: spending in real vs nominal terms

If there is no inflation then the two phrases would mean the same

But if there is positive inflation, then "freezing spending in nominal terms" would involve spending the same amount of money, and since prices are rising, would involve buying less.  Meanwhile "freezing spending in real terms" would involve spending an increased amount of money corresponding to the increase in prices, with the effect after taking price changes into account of buying the same amount.


This site https://www.statisticshowto.com/granger-causality/  shows about conducting the F-test for granger-causality.     It also talks about  alternative Tests.

"If you have a large number of variables and lags, your F-test can lose power.

An alternative would be to run a chi-square test, constructed with likelihood ratio or Wald tests. Although both versions give practically the same result, the F-test is much easier to run."


What does 'continuation' as in continuation games, strategies, plays, etc. exactly mean?

Consider a game with private information such as a privately known willingness-to-pay or any other type. We usually model this as a game in which at first "Nature" draws the type and then players make their moves. Such games do not have proper subgames because a proper subgame never splits up an information set and Nature's first move connects the entire game that follows into a subgame. That is why in such games we do not look for subgame perfect equilibrium, but a perfect Bayesian or sequential equilibrium.
A continuation game is kind of like a subgame that can start at an information set larger that just one decision node and it also assigns beliefs (probabilities) about at which decision node the player is.