What is the difference between a loan and an investment?

A loan is an agent lending funds to another agent. This money can be used for investment spending, or it can be used for personal consumption expenditures. It can be used to buy fixed assets like real estate, which may or may not be "investment" depending on how you use the terminology. In the case of the IMF, the role of the institution is to ensure financial stability by providing funds to countries in an emergency; for example, a country which becomes insolvent and is unable to service its sovereign debt is likely to seek help from the IMF. The IMF would then provide a bailout loan along with a stabilization program aimed at improving the country's public finances. This is not investment, in the sense that it does not grow the country's capital stock; it just allows it to meet its obligations to past creditors.

Investment is an expenditure which will yield revenue in the future, and hopefully amortize itself through that revenue. In the case of a household, investment can take the form of acquiring financial assets; in the case of an economy, investment often refers to actions which improve the country's productivity. This can be the construction of new factories, higher wages at state subsidized education institutions, higher research and development spending by the private sector, etc.

The reason the two are related is that quite a lot of investment is done through borrowing. A corporation can borrow to ramp up capital expenditures, a country can borrow on international capital markets to invest more in infrastructure, an individual can take out a mortgage to acquire a house. All of these can be legitimately described as "investment".

China is said to be investing in Pakistan (and quite a lot of other countries) under its Belt and Road Initiative because of two reasons: these expenditures are aimed at boosting domestic productivity in those countries, and China expects to reap benefits from its program. The potential benefits for China are plentiful. China is still very much an export-oriented economy and a global manufacturing hub. Strong growth in underdeveloped countries is likely to increase demand for Chinese goods, allowing the Chinese economy to experience faster growth. China also hopes to get direct revenue from its investment operations under this program, and they have the additional goal of expanding their political clout. Contrast this with the case of the IMF, whose founding goal is to ensure financial stability and to prevent crisis contagion and panic. The two forms of lending are evidently quite different in their desired goals.


A private person will almost never have an access to borrowing at risk free rate. However, governments such as Germany or Switzerland can borrow at essentially for all practical purposes at risk free rate by issuing government bonds.

As a private person you might get access to risk free loan if you are rich enough to be able to negotiate the rate with bank and if your project is extremely safe (think of Jeff Bezos asking for small loan to build a private parking lot).


How to calculate change in GDP using valued added?

You're dismissing the value added by the companies producing the leather and the zippers. The value added method calculates the total value added to goods and services at each step of production.

The value added by Roots is 9-5-1=3 (total value of final products - value of resources) and then you add the value of produced leather in Alberta (5) and zippers in Toronto (1).

3 + 5 + 1 = 9.

All produced goods and services are accounted for with this method, not just the ones sold to consumers (the jackets were produced even if they weren't sold), so the 7 million figure is irrelevant in this.


What is the opportunity cost of going to a football match?

Note that the opportunity cost is what you gave up to get it, not necessarily the price of what you gave up to get it. Your opportunity cost is not what you perceive as the value of the rock concert since you haven’t even bought the tickets to the concert yet.
If you had bought the tickets to the rock concert already, then the opportunity cost of going to the football game would be $50 since that’s the value you placed on the concert and you’ve already bought the tickets.
But in this case, you haven’t purchased the tickets yet so your opportunity cost should be 10 dollars since that is the net gain of that activity: the tickets cost 40 dollars, you value them at $50. You can kind of picture opportunity cost as potential profit. Your ‘potential profit’ of the concert is 10 dollars since the value you think it’s worth outweighs the cost by 10 dollars.
tl;dr
You haven’t bought the tickets to the rock concert yet so you have to account for the cost of the tickets as well.


Elasticity, link between demand and supply

Example

if p = -1%

and the production increase is +2%

elasticity is -2

[2/-1 = -2]

Our exercise

if this is true

when p = -20%

and production is 60% more

elasticity is -3

[60/-20 = -3]

To sell the 60% more the price has to decrease, if it decreases of 20%, the ratio among products sold (+60%) and price decrease (-20%) is equal to -3.

So, elasticity is the ratio among variation in quantity (60%) and variation in price (-20%) = -3.


Domestic Demand, Private Consumption, Public Consumption, Gross Fixed Investment Relationship

I assume your numbers come from the 2012 numbers on page 4 or 5 of this document, which are rather different to the 2012 domestic demand numbers the IMF published a year earlier on page 28 or 29 of this document (the earlier version had domestic demand increasing in 2012 by 0.3% rather than decreasing by 0.6%)

They do look curious, for the reason you give.

Digging into Statistics Sweden's website, there are figures for various selected components of the expenditure measure of GDP.  It is particularly noticeable that in 2012, gross fixed capital formation fell by 0.2% while gross capital formation fell by 5.0%.

So an explanation could be a dramatic in the difference between gross capital formation and gross fixed capital formation.  This difference could include changes in inventories and changes in valuables: there seems to have been a fall in inventories in Sweden in 2012 after a large rise in 2011 and it is the combination of these two which leads to the sharp fall in gross capital formation in 2012 and so the fall in domestic demand.


To obtain $\frac{\partial y}{\partial z_n}=0$, you implicitly assume that each consumer takes its own skill level as given. And in such a case, in reality the consumer optimizes with respect to labor supplied only. Perhaps it would be better to optimaze with respect to $l$ given $n$.

Regarding aggregation the discrete analogous would not be what you write but

$$\sum_{s=0}^{\infty}p(s) T(z_s)$$

where $p(s)$ would be the proportion of the population having income $z_s$.

Note that $y = E[T(z_s)]$, the expected value of individual tax burden w.r.t skill level. The density of $s$ plays in the continuous case the role of "population proportions" in the discrete setting


Why does Iceland have 5% interest but Sweden 0% interest with the same GDP growth?

Here are some elements:

Considering the two countries independently (like a closed economy), it depends on the real interest rate in the domestic country. If the inflation is zero in both countries, the real interest interest is probably higher in Iceland than in Sweden. To go deeper on this path, look at the Taylor rule which is a good proxy for the monetary policy followed in practice.
In an open economy, it depends also on the "confidence" foreign agents have in your currency. If they are afraid of a future inflation (so their returns will shrink) they "ask" for a better remuneration. For example the ECB had for years higher interest rate than the FED, and some argued the lack of confidence in the euro was one motive. The interest rate parity is the key word for deeper research.


Confused about relation of inflation to a currency's depreciation?

Your confusion might be mitigated if you thing about these as short- and long-term effects.

That is, the more immediate effect of inflation is that exports fall, demand for currency falls, and so the value of currency falls.

This causes imports to fall, domestic spending to increase, and foreign demand for domestic goods to increase. That is, exports rise. This increase in exports causes an increase in demand for the home currency and this puts upward pressure on the currency.

The short-term effect of inflation is that we have currency depreciation. The long-term effect is less clear cut, since the depreciation should trigger an increase in foreign demand which should counteract the initial effect of inflation.


Build a balance sheet:

Assets = Liabilities + Equity

Assume a simplified government budget constraint:

Deficit Spending = New Money + New Debt - Taxes

Assume that spending is to accumulate assets on the balance sheet. So to accumulate assets without spending new money or new debt into an economy over a period the government must run a tax surplus. The government would accumulate equity claims instead of matching debt to assets with no change of equity.
Taxes that transfer money from non-government to government savings accounts kept on the books of the aggregate bank amount to a zero-sum transfer of savings from non-government to government. What public purpose is served if the government accumulates significant financial savings in the aggregate bank sector? The benefit from taxes or borrowing or money issue is the government spending and using assets wisely for public purposes. There is no benefit from government financial saving because society discounts nature at the interest rate and creates financial relations as we go along from past to present to future.
One famous case study for debt financed purchase of asset is the Louisiana purchase where USA bought large territory claimed by France:
https://www.monticello.org/thomas-jefferson/louisiana-lewis-clark/the-louisiana-purchase/
Then the public benefit of debt or tax financed purchase could be debated in the political process whether one is a voter, citizen, or economist because no one had a definitive answer to the ethical question what is the public good and how should the government use finance to cause the public good?
However if the United States government issues debt just to accumulate a financial savings account, that is, a debt claim against some other non-government sector, then there could be adverse consequences which are associated with government corruption and bad public policies.
Currently in the United States the American families owe over $1 trillion in student loans which are financed by the issue of Treasury securities held as assets by some other non-government units. So the federal government acts like a bank or financial intermediary when it subsidizes the university system via direct federal loan policy in this way and the working class families with student debt may or may not be able to service and pay down debt via income flows in aggregate.
In terms of the financial float in a banking region the bankruptcy rate and net repayment of debt destroys financial assets (and liabilities) while the credit dealer systems generate new financial assets and matching liabilities in some other sector or unit. Funding government via taxes either enables the private credit systems to function better or may impair the function of the private systems depending on your theory for how financial systems operate in a case study. When the United States ran the so-called Clinton surplus there is a Congressional report studying the problems caused in the financial sectors if the persistent surplus retires the Treasury securities. The Treasury securities are the backbone of credit dealer markets in the global economy.


Calculating elasticity for $y^2e^{x+\frac{1}{y}}=3$


How does the Fed's stimulus push stocks markets higher?

If there were economic laws that could be used to determine the level of the stock market, why would people post them here, and not use these rules to make money themselves? (This leads ti the Efficient Markets Hypothesis).
Stock prices are what investors are willing to pay. If investors believe that practically anything causes the price to go up, they will bid up prices. It does not matter whether these beliefs make sense.
In this case, the premise is that Fed stimulus will cause the economy to grow faster, and thus profits are likely to be higher.


Textbook for industrial organization without calculus

Luis M Cabral : Introduction to Industrial Organization :- This book has a combo of game-theory and Cost curves. It explains strategic concepts using no calculus.
However, if you want more theoretical perspective , I would suggest reading the research papers (I have a course named Strategy , most of my readings are the journal papers on JSTOR or elsewhere).

If you could just expand on what your course comprises of , etc ;  or give a link to the university website , may be I can tell you more books I know of.


Where does overseas money from exports go?

There are two scenarios depending on the invoice currency of the trade.

If the trade is invoiced in USD, then the exporter receive USD from buyer, deposit this into the local bank. The local bank increase the amount in the exporter's bank account (in local currency), and put put this into central bank's reserve.

If the trade is invoiced in local currency, then the buyer has to exchange USD into local currency (similar process as above), then pay the exporter with local currency.


How are robocallers able to make international calls profitable?

The short answer is yes

In general, technology today allows to transfer calls between telephone switches, over the internet, at zero cost (with only the fixed broadband cost), using SIP protocol

So one of the main reasons for international calls to cost differently between countries is the internal market structure of each country

Internal market structure includes in many cases operators interconnect costs or wholesale costs, regulated or not. It can also include taxes.

Calls to the US are extremely cheap
So in theory, one can download a SIP program and connect to any other person with a SIP program. All that is left is connecting to an international calls provider, and make calls...


What effect would converting the lowest denomination note to a coin be?

The usual rationale for replacing notes with coins is that small-denomination notes and coins circulate heavily, and notes need to be replaced much more often than coins. However, as an example, the costs of transitioning to a dollar coin (think vending machines, etc.) in the U.S. would be significant enough to make the change not worthwhile, according to a FRB staff working paper. See "Costs and Benefits of Replacing the \$1 Federal Reserve Note with a $1 U.S. Coin" for details on their approach, which covers the issues pretty well.


How to graph a monopoly that does not exhibit the properties of a contestable market?

The diagram above describes a Natural Monopoly. These economies have the characteristic that the long run average costs are decreasing (and hence the LRMC is below the LRAC curve). That is, costs reduce as the firm keeps producing more. *Firms with network effects (e.g. telecom industry) is an example of such an industry.

Lets look at 2 cases: Competitive market and a single firm.

If the market is competitive, the firms will produce at a price-quantity pair such that $LRMC =P$. But at this price, the total revenue is negative (since the firm earns 0 profit at $(Q_C,P_C)$).
Thus the firms will be better off shutting down and earning 0 profits. Clearly, a competitive market won't work in such cases.

For a monopoly, the firm will charge price $P_M$ and produce $Q_M$ (the point where $LRMC=MR$). This firm earns a positive profit ($Q_MP_M>LRAC*Q_M$).


What are good books about monopolies and market failures?

Your question is quite broad in the sense that various forms of market failure cover a significant portion of all of microeconomics. I presume you have already looked in a general undergraduate micro book such as Varian's Intermediate Microeconmics, which provide coverage of many of these topics.

More detailed coverage of market power and monopoly can be found in Church and Ware's Industrial Organization: A Strategic Approach. Discussion of market failures caused by public goods and externalities can be found in  Hindriks and Myles' Intermediate Public Economics. For market failoures originating in information asymmetries, a good source might be Bolton and Dewatripont's Contract Theory.


The impact of a devaluation of the national currency under a fixed exchange rate system

This looks good assuming that your country is a small country (i.e., the change in the domestic demand of foreign assets does not change the interest rate in the foreign country). To be clear on the full story, at point A, we have that the foreigners want domestic assets, and so that's why foreigners demand more domestic currency, and that's why (as you said) the Central Bank has to counteract by buying up foreign currency.


When estimating VAR(1) by OLS, I get transpose of AR matrix. Why?

I see that $n=2$ and $\mathbf \mu = \mathbb0$.
At single (2-dimensional) observation level, we usually write (ignoring the disturbance vector)

$$\mathbf y_t = \Phi \mathbf y_{t,-1}$$

where dimensions are for your case $(2 \times 1) = (2 \times 2) \times (2 \times 1)$.

But notice that the standard regression formulation has the coefficient vector to be "to the right" of the regressor matrix... so in order to be able to apply the standard OLS estimation formulas, we transpose and obtain

$$\mathbf y'_t = \mathbf y_{t,-1}'\Phi'$$

and at full sample level we get

$$\mathbf Y' = \mathbf Y_{-1}'\Phi'$$

where with $T+1$ 2-dimensional observations available, the dimensions are ($T \times 2$) = $(T \times 2) \times  (2 \times 2)$

So, when applying the standard OLS formulas as your code does, the estimator estimates
$\Phi'$, not $\Phi$.


How Does Land Value Tax (e.g. Georgism) Decouple Lease Transfer From Improvement Transfer?

In practical terms the leaseholder will need a certain degree of certainty over her future rent payments on the lease to be able to make the investment. That’s why leaseholds often have long durations (in Germany typically 99 years). At the end of the lease the buildings become property of the freeholder. Depending on the jurisdiction the freeholder may have to compensate the leaseholder for the building. That is why leaseholds typically specify which type of building is permitted. Also the limited lease duration limits the ability of the leaseholder to actually benefit from the positive externality because her investment is already made; neither can she liquidate the increased land value at the end of the lease.

If the lease were permanent and subjected to real time land value taxes the leaseholder would have little incentive to invest. This would be equivalent to a one-year rental contract in which the tenant has a perpetual option to renew but the landlord can adjust the rent each year. It would be hard to convince someone to make a significant investment under these conditions. A solution may be for the leaseholder to buy insurance against excessive land value tax increases. But I have not heard of this in practice.


Can all the regressors be endogenous in a given model?

In principle, there is nothing to exclude the case where all regressors are endogenous.

My issue is that the Hurlin-Dumitrescu test has to do with Granger-causality, and Granger-causality examines the relation between the regressors and the dependent variable, not the relation between the regressors and the error term. On the other hand, endogeneity refers to the existence of correlation of the regressors with the error term.

So it appears that the Hurlin-Dumitrescu test is of no use on testing for endogeneity. Do I miss something here?


Any interior solution for $u(x,y) = min\left \{ x,y \right \}^{2} + max\left \{ x,y \right \}$?

Because the two segments of the IC bend away from the origin, the only possible solutions are at the corners or at $x=y$.

Write $p_x$ and $p_y$ for the prices, and $M$ for the budget.

Suppose we try to set up a corner solution in which only one of the two goods is consumed. Since utility is symmetric, it makes sense to spend all of your money on whichever good is cheaper. Suppose this is good $x$. We then have $x=\frac{M}{p_x}>0=y$ and $u=x$.

The best $x=y$ bundle that can be afforded is such that $x p_x+x p_y=M\iff x=y=\frac{M}{p_x+p_y}$. Utility is then
$$\frac{M^2}{(p_x+p_y)^2}+\frac{M}{p_x+p_y}.$$

You should easily be able to compute a condition on $M$ such that the bundle with $x=y$ is optimal: $$M>\frac{p_y(p_x+p_y)}{p_x}.$$

You can see this dependence on M if we look at a contour plot of the indifference curves:


Are social goods and public goods the same thing?

I guess it might depend on the author. Here's another classification:

  The classifications of public goods associated to agriculture identify two main categories: environmental goods and non-environmental goods (or social goods). In the first category are placed those public goods closely related to
  environmental externalities, such as farmland biodiversity, water availability and
  quality, resilience to flooding and fire, climate stability (mainly carbon storage and
  reducing greenhouse gas emission), agricultural landscape. In the second category
  are placed those public goods more related to the social dimension of agricultural
  activities, such as farm animal welfare and health, rural vitality and food security.

So under this classification social goods are non-environmental public goods.

And for some other authors, public and social goods are entirely synonymous:

  Turning to allocative problems and policies we note first, that the category of good and services provided by government are of the nature of public (or social goods). In the case of private goods, we found that efficiency in the competitive economic sense meant the maximization of individual utility alone [...]

  Public goods are those the consumption of which jointly by all individuals could make any one individual better off but no other individual worse off. Examples of public or social goods are public education, hightways, water and power, the postal system and so on.

There also exists some economic dictionaries, which might reflect the wider usage of the terms.

The Oxford one, in its first edition only defines public good:

  public goods Goods or services which, if they are provided at all, are open
  to use by all members of society. Examples include defence, law and order,
  and public parks and monuments. As nobody can be excluded from using
  them, public goods cannot be provided for private profit. Public goods can
  be and frequently are provided privately, by individuals and voluntary
  organizations, from motives of altruism or ostentation. Really expensive
  public goods such as defence are necessarily provided by government bodies,
  which alone have the power to raise the taxes needed to pay for them.

"Social good" is not defined in that one. There's a newer 3rd ed online of that dictionary, which also doesn't seem to define social good.

Likewise, The Economist defines only public (but not "social") good

  Public goods

  Things that can be consumed by everybody in a society, or nobody at all. They have three characteristics. They are:

  non-rival - one person consuming them does not stop another person consuming them;

  non-excludable - if one person can consume them, it is impossible to stop another person consuming them;

  non-rejectable - people cannot choose not to consume them even if they want to.

  Examples include clean air, a national defence system and the judiciary. The combination of non-rivalry and non-excludability means that it can be hard to get people to pay to consume them, so they might not be provided at all if left to MARKET FORCES. Thus public goods are regarded as an example of MARKET FAILURE, and in most countries they are provided at least in part by GOVERNMENT and paid for through compulsory TAXATION. (See also global public goods.)

They also define

  Global public goods

  Public goods that cannot be provided by one country acting alone but only by the joint efforts of many (strictly, all) countries. Some economists, along with global institutions such as the UN, reckon that such goods include international law and law enforcement, a stable global financial system, an open trading system, health, peace and enviromental sustainability.

There's also a Routledge dictionary of economics, which defines (quoting 2nd ed. of that):

  public good (H4)
  A commodity or service which is available
  to everyone in a particular catchment area,
  cannot be withheld from non-payers and
  is ‘non-rival’, i.e. one person’s consumption
  does not diminish that of others. The
  main examples are national defence, sewerage,
  street lighting, lighthouses, public
  health measures such as mass vaccination,
  and scientific research. As individuals
  cannot be charged according to consumption
  for goods and services collectively
  provided, it is usual to finance their
  production by taxation. Although the
  provision of public goods was small until
  the twentieth century, writers as early as
  PETTY, SMITH and John Stuart MILL argued
  for their existence. It is not always easy to
  distinguish between a public and a private
  good as some private goods gratuitously
  benefit third parties, e.g. maintaining a
  garden in a beautiful condition confers
  pleasure on many people in the vicinity.
  See also: club good; local public good;
  mixed good; private good

  social good (H4) see public good

The MIT dictionary of modern economics has quite long entries, so I'll only reproduce snippets here (from 4th ed.). For public good is says among other things:

  The extreme, or
  'polar', case of a 'pure' public good has been
  defined by Paul A. SAMUELSON as a good
  which is:
  1. non-rival in consumption
  2. has the characteristic of NONEXCLUDABILITY
  - that is, if the good is provided
  the producer is unable to prevent anyone
  from consuming it.

This source also defines "collective goods" (defined by non-excludability alone), "local public good" (public goods may depend on the chosen community), "mixed good". The entry for the latter is quite long, and says in the end

  Mixed goods are certainly more common
  than pure public goods and possibly more
  common than private goods - thus their
  analysis is of great importance. The term
  impure public good is sometimes used to
  describe mixed goods.

The MIT dictionary does not define social good.

The New Palgrave dictionary (which is more like an encyclopedia, with articles/entries contributed by different authors) mentions (in its second edition) that

  The
  distinction between private and public or social goods arises from the mode in which benefits become
  available, i.e., rival in the one and non-rival in the other case (see Public Goods).

So that entry (which is actually on merit goods) considers the terms social and public goods synonymous. They further use the public/social goods interchangeably a few times in their entry on Musgrave. Musgrave himself wrote the "public finance" entry for that Palgrave dictionary, and it too uses social good to mean public good at one point.

Their entry for "public goods" (written by Agnar Sandmo) is basically dedicated to the Samuelsonian approach to the topic. It doesn't define or mention social goods, but it does define a "social welfare function", talks of "social marginal cost of the public good" etc. There's no entry for "social goods" in that Palgrave dictionary (even though it's 7,000+ pages long).

I think the trend from all of these is clear: "social goods" are much more idiosyncratically defined, if defined at all. Most authors who do use the term "social goods" seem to use it synonymously with public goods (unlike what Wikipedia claims, without a citation.)


2 firms production decision for one agent

Seems like you are assuming there will be an interior solution.
$$
(a-2b x-c^X)w-by = 0
$$
only needs to hold if the value of $x$ is positive. If it is not, but $x = 0$, the lowest possible amount, then
$$
(a-2b x-c^X)w-by < 0
$$
is not a contradiction. Further decreasing $x$ would increase utility, but it is not possible. Therefore in the case when $w = 0$ and
$$
(a-2b x-c^X)w-by = -by
$$
you do not necessarily have $y = 0$.


Empirical tests for symmetry of cross-price elasticities

Even if it is a very general result, its implications are not immediately intuitive and actually often quite difficult to accept. Why should the impact of an increase in the price for fresh strawberries on the demand for gasoline, be the same as the impact of a price increase for gasoline on the demand for fresh strawberries (while keeping $u$ constant) ?
One of the more convincing empirical investigation of Slutsky symmetry is experimental and provided in this paper which is also very closely related to the questions your ask:
Sippel, Reinhard, 1997, "An Experiment on the Pure Theory of Consumer's Behaviour," The Economic Journal, 107, 1431-1444.


European Commission getting rid of excess carbon emission permits

What were these excess permits of carbon emissions? Is it the amount of permits that are
still up for sale (or to be distributed for free); and hence, aren't used?
Page 92 of the ETS handbook states that since 2009, because of the economic crisis of 2008, a higher than expected imports of international carbon credits and a large uptake of renewables led to a significant surplus of allowances in the EU ETS.
So, yes, excess permits are the ones that aren't used by the holder in the near future. According to the ETS handbook, page 92, in early 2012 there were 1 billions surplus allowances. And 2 billions by the start of 2013.
From the handbook:

The presence of the surplus and resulting low price may discourage participants in the
EU ETS from taking actions now to reduce emissions, which may result in a deviation from the most
cost-efficient path to long-term emission reduction goals. The large surplus has led to persistently low
carbon prices in the EU ETS, reducing the price incentive for low-carbon investments.

One mechanism of tackling this surplus in allowances is the creation of a market stability reserve (MSR).
And how did the European Comission get rid of them? Is it by decreasing the cap of overall supply of permits within the EU?
The MSR acts to combat this imbalance of allowances. From the handbook, page 95:

Using the total number of allowances in the market as an indicator, market imbalances due to
unexpected shocks that impact demand, such as the economic crisis, can be addressed. This allows
the EU ETS to maintain its objective to reduce emissions in a cost-effective and economically efficient
manner, even under unexpected circumstances.

Hence, this way, each year the number of allowances injected into the market by auction will take the market demand into account.
So, yes, the amount of overall supply of permits, by auction, will be decreased if there is an excess. This will act as a carbon reduction incentive, by keeping prices competitive.
From this diagram of the ETS handbook, company B, with a shortage of allowances, has two options. Either buy from company A's surplus allowances or buy them at an ETS auction.

Also, what is the unit of an allowance?
The website of the European Commission states that each allowance gives the holder the right to emit one tonne of carbon dioxide (CO2), or
the equivalent amount of other powerful greenhouse gases, nitrous oxide (N2O) and perfluorocarbons (PFCs).
But isn't this ultimately limited by the overall EU cap that is decided politically?
Yes, indeed the total amount of allowances injected into the market by the ETS is determined politically. This overall cap will decrease accordingly, to realize the European Green Deal; no net emissions of green house gasses by 2050.


The social welfare maximizing outcome you computed (by equating the sum of marginal benefits to marginal cost) is just one of the Pareto efficient outcomes. Although maximization of joint utilities is a sufficient condition for Pareto efficiency, it is not a necessary condition. Pareto efficiency is defined by the lack of Pareto improvement -- a reallocation of resources to make at least someone better off without making any other worse off.

Option C is not Pareto efficient because a Pareto improvement exists. At $(0,80,20)$,
\begin{equation}
u_1(0,20)=0+6\sqrt{20}\approx 26.83 \qquad
u_2(80,20)=80+10\sqrt{20}\approx124.72.
\end{equation}
Observe that at this bundle, consumer 2's MU for the good $y$ is $5/\sqrt{20}\approx1.11$ and his MU for $x$ is $1$. Therefore, increasing one unit of good $y$ at the expense of one less unit of $x$ is utility enhancing for consumer 2. At the same time, one extra $y$ benefits consumer 1 as well. So we can easily verify that $(0,79,21)$ is a Pareto improvement over $(0,80,20)$:
\begin{equation}
u_1(0,21)=0+6\sqrt{21}\approx27.495 \qquad
u_2(79,21)=79+10\sqrt{21}\approx124.83
\end{equation}
where both consumers experience higher utilities.

In a similar way, you should be able to verify that no Pareto improvements exist for options A, B, and D.


Does Progressive Taxation Actually Follow The Decreasing Marginal Value of Wealth?

tl;dr
No they do not for several reasons:

Your post is based on misinformation, I know of no work on optimal income taxation that would aim to 'equalize sacrifice' (heck you would not find such philosophy even among most relevant moral/political philosophies). Consequently, the whole premise is false, and you would not even expect to see arguments in favor of implementation of and implementation itself of such tax system.
Even if your question would be rephrased to whether governments generally set marginal tax rates as dictated by optimal taxation literature the answer would be again negative. Generally speaking, politicians for better or worse are simply not interested in optimizing income tax rates (from societal perspective - of course, they likely do optimize it in a way to curry political favors).

Correction of Misinformation in The Question
Let me first address some misconceptions/misinformation in your question before answering the question:

From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: "Equal Sacrifice" - everyone's "sacrifice" should be the same, and the "Decreasing marginal value of wealth" - the actual value inherent in receiving an additional unit of money is less for someone with more money.

This is incorrect understanding of optimal taxation literature.

We are talking about optimal income taxes so marginal value of wealth does not even enter the calculation here. Optimal income taxation is as the name says about optimizing income taxation, and thus what enters optimal income taxation problems is marginal value of income.

From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: "Equal Sacrifice"

I am not sure where you (mis)heard the above, but it is extremely inaccurate statement.
They are based on the following optimal non-linear tax function which comes from the seminal works of Mirrlees (1971) (who in fact got Nobel Prize in Economics for this contribution), Diamond (1998) and Saez (2001). The optimal marginal income tax will be given by:
$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_n0)}$$,
with $b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $.
I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part $\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$ is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part $\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b_n$ which depends on both utility of consumers and the societal utility function,  and finally $\frac{1-F(z_n)}{z_nf(z_n0)}$ is the part that captures the relative magnitude of distortions created by this taxation. Again since we are talking about improvements in actual underlying utility what the aggregate price level is does not matter.
Now utility does enter the problem via $b$, but there is no principle of equal sacrifice or anything like that.

It is correct to say that people with concave utilities will have diminishing marginal utility of income (they will also have diminishing marginal utility of wealth but this is optimization of income taxes not wealth taxes). However, this has almost no effect on the progressiveness of optimal tax rates and actually does not affect the shape of marginal tax rate schedule that much.
What is far more important is what is the societal welfare function which assigns welfare weights to each individual. Here most commonly used societal welfare functions are (e.g. see Saez 2011):

A) Utilitarian social welfare function -  This social welfare function is based on the idea that poorer people value on a margin 1 dollar more than richer people - but by no means it implies 'equal sacrifice'. As a matter of fact if there would be no efficiency cost to taxation a utilitarian would make all incomes completely equal since if there is any person anywhere with higher income than someone else that means aggregate utility could be increased by moving that dollar from higher income person to poor person. Utilitarians  are not trying to make sure everyone sacrifices equal utility - they care about maximizing total utility.
B) Rawlsian social welfare function - This is based on Rawlsian MaxMin idea which states that it is moral to maximize the welfare of the poorest members of society (Rawls (1971), Theory of Justice p. 152). This welfare function gives welfare weight of 1 to the poorest member of society and everyone above zero (of course, in process of redistribution someone else becomes the poorest one - don't interpret this in a way that all redistribution goes to single person). Tax systems based on this function are as far from idea of equal sacrifice as possible. This social welfare function literally does not give a damn about sacrifice of anyone else than a poorest member of society and taxation and redistribution will occur to the point that it generates so much distortions that government is simply not able to make a positive redistribution (in terms of welfare) to the poor anymore.
C) Charitable conservative/Libertarian social welfare function - This one is typically also based on MaxMin criterion with a twist that although poorest still get welfare weight of 1 people who are not poorest also get some non-zero welfare weights (Atkinson 1995 and Diamond 1998). Again nothing to do with idea of equal sacrifice when it comes to utility.

What has the major impact on whether tax system is progressive or not actually does not depend that much on either of the above mentioned but primarily on the shape of income distribution.

For example, early Mirrlees (1971) and Tuomala (1984) simulations based on assumption of log-normal distribution yielded regressive (but approximately linear) marginal tax rates even under maxmin or utilitarian scenarios (e.g. see example below).

The true reason why modern optimal income taxation shows that tax system is progressive is that we later empirically discovered that log-normal distribution does not fit data nicely because income distributions seem to have pareto-tail (e.g. see Saez 2001). If we switch from log-normal income distribution to log-normal distribution with pareto tail suddenly we get progressive tax rates as you can see below under both utilitarian and maxmin scenario (the marginal tax rates are high for poor due to redistribution but average tax rates are progressive here).

So the premises of your question are prima facie wrong. Even though it is probably at least in principle possible to derive some new social welfare function that would be about  equal sacrifice, I know of no published research or even famous economist advocating such principle. The shape of an optimal marginal income taxes is primarily determined by shape of income distribution, different moral philosophies typically stretch or squeeze it or move it up and down (of course, one can create arbitrary moral philosophy and some would probably change a shape itself significantly but that is not what you will see with the usual ones).
Answer

My question is, then, to what extent do modern progressive income tax systems match money's "actual" quantity/value curve?

If I reinterpret this question about asking whether modern progressive income tax systems match the ones prescribed by optimal income tax calculations, the answer is that they simply generally don't. If they do it is often by accident rather than an intent.
For example, below you can see simulation of utilitarian and Rawlsian optimal marginal tax schedules in the Netherlands compared to the real observed ones from Zoutman et al (2012):

First as you can clearly see the utilitarian optimal marginal tax rate is completely different from the real observed one (although they intersect at 1 point).
Second, the optimal income taxation under Rawlsian scenario also intersects the real marginal tax rates only at a single point, although here above about 80000 euro income the optimal marginal income taxes are more close to the real observed ones, they still not match, and below 80000 euro they completely diverge.
The result that you see above for the Netherlands is not unique (it is not possible to go over all literature and all simulations in scope of SE answer but I never seen work that would show real life progressive marginal tax rates match the optimal ones), even though many countries have different tax codes in most countries you will get match between optimal marginal income tax schedule and real marginal income tax schedule at best at a single point (in case they cross) but often not even that. This is typically explained by the fact that taxes are determined via democratic processes so we should not even expect them to match anything what is predicted by optimal taxation literature. In fact, as shown and argued by Zoutman, Jacobs & Jongen (2016) real world tax systems are often set up to give some extra weights to middle/upper middle class and this is argued to be due to their political importance.


There is a lot to be said about this subject, and I will only give a partial insight.

First off, value is subjective. Everyone will place different values on different things, depending on such things as personal preferences, their economic status and income, and thoughts about the future such as expected future income and wealth, and the future worth of the item. Your reservation price is the highest price you would theoretically be ready to buy or sell something for. This may or may not correspond to the actual price you would be getting on the market. If the price of the product is lower than your reservation price ($ P > P_{r}$), then the difference is your surplus value, which is what you gain beyond what you think the actual value is. For example, on the market for goods, let's say that the price of a good is $ \$ 100 $. Your reservation price is $ \$ 110 $, so that means that you think the product is worth $ \$ 10 $ beyond what you pay. The total value to you, is $ \$ 110 $. On the other hand, the seller's reservation price is $ \$ 90 $ – i.e. the seller would be ready to let go of the product for that amount. On a very efficient market, that price might be what it cost to make the product. So $ \$ 90 $ is the value of the product for the seller. $ \$ 100 - \$90 = \$ 10 $ is the surplus value for the seller, if it sells for that price.
If your reservation price is $ \$ 100 $ to sell something, but the best offer you can find is $ \$ 90 $, then you should distinguish between the personal value that you attach to the product, and the market value. In the public world, we usually value things by its market value. So if $ \$ 90 $ is the best offer you can find, then that is its fair value.
This assumption means that a building that cost $ \$ 1\ million $ to build, can actually be worthless ($ \$ 0 $) on the market if nobody wants to buy it. This could of course change in the future, perhaps people will get around to see the value you see, with time. So how is it that a building can be considered completely worthless, even though it's "obviously not"? Well, it can also be said, from a more financial perspective, that a value of an asset is worth the future income that it will provide. So if the market doesn't believe that the asset can create future income, then it's basically of no use. In market crashes, fear may drive investors to undervalue assets, or simply not be willing to engage in transactions at the time, driving prices down.
We can create a measure of something's value by calculating its present value. Typically, the formula would look something like:

$ PV = \frac{FV}{(1+i)^{n}}$

where $ FV $ is the future value (what you can sell it for in the future), $ i $ is the best alternative rent you would be able to get per period during that time (e.g. by investing in a risk-free bond), and $ n $ is the time periods until payment. The formula discounts the future value to give a comparable present value, that you can use to compare it with other investments. $ (1+i)^{n} $ is the compounding effect of the rent over time.

So let's say that you own a nice painting, and you think that it will be worth $ \$ 500$ in 10 years. Given that you could sell the painting today, and place the money in a bank account which pays $ i = 1 \%$ per year in interest rate, you should be willing to sell it for $ \frac{\$ 500}{(1+0.1)^{10}} = \$ 193 $ today, meaning that you either can get $ \$ 500 $ by selling the painting in 10 years, or the same money in 10 years by selling it now and keeping the funds in a bank account until then. Again, market value is what should be used here to value the asset in transactional terms. But for you personally, the painting may be worth more than $ \$ 193 $ because it brings pleasure looking at it every day. In that case, you might actually value it in such a way that you would rather keep the painting for 10 years, rather than sell it now.


Devaluation of Venezuelan Bolívar - Will it increase or fight hyperinflation?

lopping 5 zeros out of its regular Bolívar

This will make the numbers used in daily shopping more manageable.

  However, if I remember well, won't the devaluation of a currency increase inflation by increasing prices of imports, and in some cases also increasing prices from domestic products by increasing exports?

Devaluing the currency might help make imports possible again.
The exchange rate vs the USD is held superficially high by the government. However, it is not possible to exchange Bolivars for USD because of the shortage of foreign currency caused by a drop in oil income. Getting paid in overvalued Bolivars is not very appealing, so imports have essentially ground to a halt causing massive shortages in products that aren't produced locally.

  What's the rationale here for this measure?

The government hopes that this will stabilize the currency and reduce inflation to manageable levels. The market usually requires real change for that to happen, so in my opinion not much will change. There will still be hyperinflation and shortages.

References: https://en.wikipedia.org/wiki/Shortages_in_Venezuela


Limiting the scope to the question of why the US encouraged the proliferation of many banks, one important contributor was the 1927 McFadden act, which placed significant restrictions on interstate banking (rolled back through state laws in the 1980s-90s, and eventually substantially repealed by the Riegle-Neal Interstate Banking and Branching Efficiency Act of 1994).

The reasons for this were primarily concerns about concentration in the financial services sector, and concerns about the ability of state banking supervisors to monitor across state lines the activities of institutions for which they were responsible.

The trend in the US since at least the 1980s has indeed been towards consolidation: http://research.stlouisfed.org/fred2/series/FREQ

Finally a (dated, but good) review of the effects of branch banking reform is available from the FRBNY: http://www.newyorkfed.org/research/current_issues/ci1-2.pdf


How do you keep up to date and find the most relevant papers?

Seminars and conferences are the best ways to get exposure to working papers, in addition to following specific authors. (Some of my colleagues also follow economists on Twitter, where they sometimes post new working papers.)

In addition, subscribe to the email alerts from the journals in your field/interest. This will get you up to date with what comes out in the latest issues. The AEA journals (as well as Econometrica) will also send you the titles and abstracts of the articles; the Elsevier journals will only send you the TOC with the article titles only.


In the exactly identified case, it is natural to assume that a unique $\theta$ satisfies $\bar{g}(\theta) = 0$, because the number of parameters is equal to the number of equations. Let $\hat\theta$ denote such $\theta$ value.

When $W$ is positive definite, $\bar{g}(\theta)' W \bar{g}(\theta) \ge 0$ (because $W$ is positive definite), and $\bar{g}(\theta)' W \bar{g}(\theta)$ attains zero if and only if $\bar{g}(\theta)=0$. That is, the global minimizer of $\bar{g}(\theta)' W \bar{g}(\theta)$ equals the solution to $\bar{g}(\theta)=0$, which is $\hat\theta$, for whatever positive definite $W$ matrix. Thus, $W$ is irrelevant.

Note that this argument does not hold if $W$ is not positive definite. For example, if $W$ is positive semi-definite but not positive definite, then the GMM estimator may not be unique.


Does it make sense to apply complex mathematics in economics given that the result can be off by substantial percentage points?

In the field of economics complex math is often used for

a) Economic modelling

There are usually tons of factors that influence even the simplest economic event. It is often necessary to resort to complex math to include all the variables that represent those factors in the right way.

b) Analysis of statistical data

It is often very difficult to support economic theories with good data, again because there are so many factors in play. On top of that, many simple mathematical instruments have drawbacks and should not be applied in certain situations. For example, to test relationship between two variables, you can use a simple correlation coefficient, but there are cases where outliers or certain patterns can make it unreasonably high or low. Then you can use simple linear regression, but if you spent months gathering data you may want to spend a few days trying to build a custom regression equation for a better fit. Finally, if you are preparing a paper for publication, you may want to use as many statistics as you can to try to convince reviewers and readers that your findings are actually true.

Drawbacks of using complex math in economics

1) There are not that many readers out there who are proficient in advanced mathematics. Unless you want to limit your target audience to a few math enthusiasts, keep it as simple as possible.

2) If you are preparing a paper for review at a journal, keep in mind that reviewers in economic journals are not necessarily mathematicians. It usually takes longer for math-heavy papers to get reviewed and the quality of those reviews may not be great, unless its a top journal.


Constant elasticity proof for log-linear demand curve

$\frac{dlnQ}{dp}=\frac{dlnQ}{dQ} \frac{dQ}{dp}$ thus:

$\frac{dQ}{dp}=\frac{dlnQ}{dp} \frac{dQ}{dlnQ}$

Since we know that if $f(x)=lnx \Rightarrow f'(x)=\frac{1}{x}\Rightarrow \frac{1}{f'(x)}=x$

We replace $\frac{dQ}{dlnQ}$
by $Q$. You get:

$\frac{dQ}{dp}=\frac{dlnQ}{dp} Q$

It is readily found that
$\frac{dlnQ}{dp}= \frac{e}{p}$

So our expression for the derivative of $Q$ wrt $p$ now reads:

$\frac{dQ}{dp}=\frac{e}{p} Q$

Divide both sides by $Q$ and you get $\frac{dQ/Q}{dp}=\frac{dQ/dp}{Q}=\frac{e}{p}$

Elasticity is now found by simply multiplying both sides by $p$, you get:

$\frac{dQ}{dp} \frac{p}{Q} \equiv \eta=e$


Why in the quantitative equation: $MV=PY$, $V$ and $Y$ can be taken as fixed?

There are multiple answers to this question.

In any model you can always make a thought experiment where you hold certain variables fixed.  So one answer, although not very satisfying one, is that you can view it as a thought experiment. For example, in physics distance traveled equals velocity times time or $D=tv$ and you can always make a thought experiment where you choose 1 or 2 variables that will be fixed and see how others behave.

If you are asking why it would make an economic/intuitive sense to take $Y$, $V$ fixed then:

Regarding $Y$: From economic theory we know that $Y$ in a long run depends on productive capacity of an economy that is independent of $M$, $P$ or $V$ (you can learn more about that in any 101 Macro/econ book such as Mankiw Macroeconomics or Mankiw Principles of Economics). So you can consider it to be exogenously given and holding it constant is just performing a thought experiment where you assume you have economy that is not growing so you can see how other variables behave.

Regarding $V$, in the past (before 2008), empirically speaking velocity of money was very stable (see the Fed graph below). As you can see between 60s and late 2000s it was always hovering around 1.9 so on empirical grounds many economists considered $V$ stable, although it is worth while pointing that recently it significantly declined. Nonetheless, given that it was historically very stable this might be just transitional, or it is possible there was a structural break and now velocity will remain approximately constant at some lower level. It is also possible that it was wrong to assume it is constant in the long run based on previous empirical observations.


Proving a result In Jones (1999) "Growth: With or Without Scale Effects"

We know that $g_A = \dot{A}/A$ and thus $g_A = \delta L_A A^{\phi-1}$.

Take logs of both sides of $g_A = \delta L_A A^{\phi-1}$ and differentiating with respect to time gives us

$$ \frac{\dot{g_A}}{g_A} = n + (\phi-1)g_A.$$

Multiply both sides of the equation by $g_A$:

$$ \dot{g_A} = n g_A + (\phi-1) g_A^2$$

Given that $\phi<1$, and in the steady-state BGP the growth of the growth rate $\dot{g_A} = 0$, we divide both sides by $g_A$ to get:

$$n+ (\phi -1) g_A = 0 $$

Rearrange to get the desired expression:

$$g_A^* = \frac{n}{1-\phi}$$

The second equation is just the result above multiplied by $\sigma$:

$$g_Y = g_A \sigma =  \frac{\sigma n}{1-\phi}$$


When a support function is found to be more profitable than the primary business

The Aerobie Sports company mostly makes odd-ball equipment for having a catch. But they made a coffee maker called the Aeropress as a side project and now it makes up more than half of their sales.

The Woolworth Company (of five and dime fame) founded the Foot Locker shoe store as a side business and now it is the core-surviving brand of that corporation when the main Woolworth business was closed.

Post it notes were a side business for 3M but now amount to \$1 billion in sales a year. But that's only about 3 percent of revenue.

The Michelin Guide books were developed to help promote the sale of tires through encouraging road touring. It survives today and though the business might be a small fraction of the revenue produced by the tire business it has significant cultural influence and an independent life.


Macroeconomic models with labor-replacing capital?

The Grundrisse’s “fragment on the machines” expounds on the tendency to universal automation rather famously, without Marx having stripped his more idealistic hopes from his exposition, and without the deadening jargon of Capital’s chapters on the organic composition of capital. It isn’t perfect on the automation of science itself, but that just needs to be reread with science posited as a commodity, AI as fixed capital etc. Nick Dyer-Witheford works in this area last time I checked.

The assumption, or rather observation, of the centrality of a diminishing living component of labour is central to Marxist political economy. Normally it is posited as the OCC critique where transcending this process only exists as a negative, but the fragment on the machines pictures the real possibility of social mastering of social production. In “fragment” Marx goes a bit further than UBI and instead poses universal leisure (relief from labour) due to hyper abundance. This is viewed as a fundamentally political achievement and one not achieved by parliament, treasury or finance ministries.

As far as formalisations, OCC seems to be an accepted category amongst the analytical Marxists (Basu 2018 …Ground-rent… [working paper] https://scholarworks.umass.edu/cgi/viewcontent.cgi?article=1242&context=econ_workingpaper). This means it ought to appear in heterodox Marxist economics, as opposed to just Marxist political economy.


Difference between the International Dollar and PPP

They are different stages of the same process. You convert a country's GDP into international dollars USING PPP.

PPP are rates. They are a relation between prices in different countries. International dollars is a unit of measurement.

The World Bank says "GDP per capita based on purchasing power parity (PPP). PPP GDP is gross domestic product converted to international dollars using purchasing power parity rates. An international dollar has the same purchasing power over GDP as the U.S. dollar has in the United States"


Crypto-tokens: how do they function in a latter stage of company?

The companies who engage in these sort of mechanisms expect the token to have a variable price which remains accessible and fair for the product in question, and also keep some tokens for themselves which can be used to influence the. In case the price rises beyond intended, some companies reserve large amounts of initial tokens to sell in the market (sell high volumes of tokens to keep the price low), or they even reserve the right to produce new tokens.
However, these companies also benefit from a high valuation of the token, which creates a speculative risk, benefiting the company but not necessarily consumers. See the long example/explanation below.
Example: Amusement park tokens
The situation can be illustrated as an amusement park who requires tokens to access and use the rides. Instead of selling the tokens themselves, the park sells lots of tokens before being built (similar to an ICO). Consider they sell it for €1 instead of the typical €2 or €3 on other amusement parks. You can then go to the park and try to sell the tokens to anyone. The logic is any visitor would still pay 2 or 3€ for the tokens, since he/she is a casual visitor. If it is a regular visitor, maybe he or she would try to save some money by buying the early tokens, but once those were used, the rides are still fairly priced at €2 or €3.
However, if the tokens are in too much demand, their price would climb to, for instance €10. When this happens, people would stop buying until the price lowers again until an acceptable value. If this doesn't happen due to whatever reason (e.g. speculation) the amusement park could sell some reserve tokens at the intended price (2€). Some amusement parks would even say they can produce new tokens at any time, to avoid this risk. However, remember that the amusement park benefits from selling reserve or new tokens for high prices at any time, and this would be a profit without delivering any value or product.
Research and further reading on ICOs
Please note that this is a very new area and very few studies have been performed yet on this field. I've recently read one paper on "Red Flags for ICOs", and another one is the economics of ICOs, if you want to learn more.


Find $\hat{\beta_2}$:

$$\hat{\beta_2}=\frac{\Sigma(Y_i-\bar{Y})(MALE - \overline{MALE})}{\Sigma(MALE - \overline{MALE})^2}$$

Assuming m males in a sample of n people, $$\overline{MALE}=\frac{m}{n}$$

We can say:

$$MALE - \overline{MALE}=\begin{cases}-\frac{m}{n} & \text{if }MALE=0\\\\\frac{n-m}{n} & \text{if }MALE=1\end{cases}$$

With some algebra, the numerator of $\hat{\beta_2}$ becomes:

$$\Sigma(Y_i-\bar{Y})(MALE - \overline{MALE})=\frac{m(n - m)}{n}(\bar{Y_M} - \bar{Y_F})$$

The numerator of $\hat{\beta_2}$ becomes:

$$\Sigma(MALE - \overline{MALE})^2=\frac{m(n - m)^2 + m^2(n - m)}{n^2}=\frac{m(n - m)}{n}$$

Dividing the two gives the correct answer:

$$\hat{\beta_2}=\bar{Y_M} - \bar{Y_F}$$

Find $\hat{\beta_1}$:

$$\hat{\beta_1}=\bar{Y}-\hat{\beta_2}\overline{MALE}=\frac{m\bar{Y_M}+(n - m)\bar{Y_F}}{n} - (\bar{Y_M} - \bar{Y_F})\frac{m}{n}=\bar{Y_F}$$


Why are the Fed's goals often described as the "dual mandate" and not the "triple mandate"?

The textbook answer is that the interest-rate objective is considered redundant (implied by) price stability, in the long run, i.e. low inflation is expected to bring down interest rates anyway.

It's interesting that Congress actually added the (3rd) interest rate objective in 1975-1977. Maybe there's some historical context that explains their insistence on it. I don't know more.

As for "some say the central bank can focus on price stability" that's an allusion to ECB's primary mandate (sometimes/unofficially called "single mandate"), no doubt.

And hat-tip to BKay for bringing this up, there are indeed more similarities between the Fed and ECB; even though not by law...

  As
  explained in the 9th edition of the Federal Reserve’s Purposes and Functions, published in
  2005: “Stable prices in the long run are a precondition for maximum sustainable output
  growth and employment as well as moderate long-term interest rates.” In 2004, Chairman
  Greenspan pointed to the primacy of price stability in explaining the success of Federal
  Reserve policy since 1979. As he explained, this resulted from “maximizing the probabilities
  of achieving our goals of price stability and the maximum sustainable growth that we
  associate with it” (Greenspan, 2004).
  With this interpretation of the Federal Reserve’s mandate, closer comparison with that of
  the ECB suggests more similarities than differences. This is because while the mandate of
  the ECB explicitly recognizes the primacy of price stability, it does not ignore other
  objectives.

Furthermore, analysis of actual macro models used by the ECB and the Fed (as they were in 2008 anyway) points to a number of "real" similarities, including the fact that neither is strictly IT (Inflation Targeting) in the sense of NCM (New Consensus Macroeconomics)... unlike that of the Bank of England, which was found to be IT.


Relation between demands of $x, y$ and $z$

Let $\min\{x,z\}=\Omega$, where $P_\Omega=P_x+P_z$. Now the problem becomes $U(y,\Omega)=y\Omega$, which is a standard Cobb-Douglas with degree 2 of homogeneity. Now in this case the choice for each good is:
$y^*=\frac{\alpha_y100}{P_y(\alpha_y+\alpha_\Omega)}\implies y^*=\frac{100}{2P_y}\;\;\;\;\;\;\;$in this case $\alpha_y=\alpha_\Omega=1$
For $\Omega$: $\;\;\;\;\;\Omega^*=\frac{\alpha_\Omega100}{P_\Omega(\alpha_y+\alpha_\Omega)}\implies \Omega^*=\frac{100}{2P_\Omega} \implies \Omega^*=\frac{100}{2(P_x+P_z)}$
Now, since $P_x=P_y=P_z$, let $P_x=P_y=P_z=P$ a general price, therefore substituting in our optimums:
$y^*=\frac{100}{2P}\;\;\;\;\;\Omega^*=\frac{100}{4P}$
Now it's straightforward (since we already know $x^*=z^*$ and as this is the optimum for $\min\{x,z\}$ which is $x$ OR $z$) that $y^*=\frac{100}{2P}>\Omega^*=\frac{100}{4P}$, so this implies that:
$y^*>x^*=z^*\;\;$****
Also I found this document, where this question is number 13.
Hope this helps.
Disclaimer: It would be helpful if other people could assess this approximation, since I hadn't never seen this problem before.


What does the vertical axis of the Kuznets curve depict (mathematically)?

Kuznet talked about income inequality. In this context it is often measured by the Gini coefficient or the Kuznet ratio, but basically any measure of income inequality would do, since Kuznet's hypothesis is merely a qualitative one.


What are the Impacts on an economy when a large mining venture begins to reduce production and lay off workers in a remote region?

It depends on what they mine there and how big are the layoffs. Boomtowns are usually formed when they set up mines in remote regions. After the mines are depleted or become economically non-viable, workers leave and the boomtown becomes a ghost town. Since the economy of such places heavily relies on income from mining, it cannot survive once the money stop flowing. You can read up on the history of boomtowns and ghost towns on wikipedia to see if that is what you are looking for.

Boomtowns:
https://en.wikipedia.org/wiki/Boomtown

Ghost towns:
https://en.wikipedia.org/wiki/Ghost_town


Shapley value with large number of players

The sampling approach rigorously would look like this. For each player $i$, we want to estimate the expected marginal contribution, where the expectation is taken over a the subset of players that precede $i$ in the permutation ordering. So for each $i$, we do the following.

Let $X_i$ be a random variable equal to the marginal contribution of $i$, when we draw the permutation of players randomly. Let $\mu_i$ be the true Shapley value of $i$ (which we do not know). Then $\mathbb{E} X_i = \mu_i$. Now, if we sample many independent copies of $X_i$ and average them, this average, call it $\bar{X}_i$, should be very close to $\mu_i$, and the closeness is given by e.g. Hoeffding's inequality, which says

  Let $K$ be an upper bound on any marginal contribution (so they are always between $0$ and $K$). Let $m$ be the number of independent copies of $X_i$ that we draw. Then
  $$ \Pr[ | \bar{X}_i - \mu_i | \geq \epsilon ] \leq 2e^{-2m \epsilon^2 / K^2} . $$

So for each $i$, we sample $m$ permutations randomly and calculate the average marginal contribution $\bar{X}_i$.

For example, if $K=100$ and we want an accuracy of $\epsilon = 0.01$ and a probability of failure of $2e^{-50}$, then we need $2m\epsilon^2/K^2 = 50$, so $m = 25K^2/\epsilon^2 = 2.5$ billion. So we need to sample $2.5$ billion permutations to "guarantee" accuracy of $0.01$, except with a miniscule $2e^{-50}$ chance of failure.

As comments have mentioned, there are big improvements for many special cases.


Principal Agent Question Private Benefit

If we try to directly optimize with respect to $t$, given this system (two unknwowns, one equation), we get a nonsensical result.

$$\frac{\partial P}{\partial t}: -s = 0$$
Which may not be true, and I'm assuming you are taking $s$ as given, not something to choose. You can only optimize with respect to $x$.

$$\frac{\partial P}{\partial x} = 1 - sx = 0 \implies \boxed{x^* = \frac{1}{s}}$$

What does this tell you about the optimal value of $t$ if anything?


Can the Income Consumption Curve have a non smooth shape?

An easy example:
When the utility function is
$$
U(x,y) = \ln x + y
$$
and the prices are
$p_x = p_y = 1$, the IOC is kinked at income level $I=1$:
$$
(x,y) = \left\{
\begin{array}{ll}
\left( I, 0 \right) & \text{ if } I \leq 1 \\
\left( 1, I-1 \right) & \text{ if } 1 < I
\end{array}
\right.
$$

A more complicated example:
When the utility function is
$$
U(x,y) = \min\left(2x+2y,x+10\right)
$$
and the prices are
$p_x = 8$, $p_y = 6$,
the good $y$ exhibits inferior (and Giffen) behavior. The IOC is kinked at income levels 30 and 80:
$$
(x,y) = \left\{
\begin{array}{ll}
\left( 0, \frac{I}{6} \right) & \text{ if } I \leq 30 \\
\left( \frac{2I}{10} - 6, 8 - \frac{I}{10} \right) & \text{ if } 30 < I \leq 80 \\
\left( \frac{I}{8}, 0 \right) & \text{ if } 80 < I
\end{array}
\right.
$$


What is the estimated propensity to consume out of wealth, and how does it depend on income levels?

This is the "wealth effect". It has been shown empirically that as one's assessment of their wealth increase, the more they spend. An important point to be made is that this may be disconnected from actual wealth. If you are counting on a big inheritance that may lead you to spend a higher % of your wealth now in expectation of the future payoff.

There's a good paper that empirically looks at the correlation between wealth and spending but it's important to understand that the factors that influence an individual to spend money are numerous and it's virtually impossible to completely remove the signal from the noise.


An equlibrium is a property of an entire market, not of any given investor's portfolio. The CAPM is an equilibrium theory.

In equilibrium, all (nontoxic!) assets must have non-negative prices. Hence, the weights of all assets in the market portfolio must be non-negative as well. The market portfolio weights will be strictly positive only if the assets have positive value, i.e., if their prices are strictly positive.

Within an equilibrium, it may be possible that some individual investors will find it optimal to be short some assets. But that does not imply that it's possible for market portfolio weights to be negative.


Where did Adam Smith write the following and in what context?

This (paraphrase) appears in Book III, Chapter IV of The Wealth of Nations. Smith is discussing how, in his opinion, towns engaged in commerce and manufacturing have contributed to the improvement and cultivation of their country.
Specifically, the quote appears in the context of a discussion about feudalism, and kings being unable to control the violence of feudal lords on page 448 (line breaks and emphasis mine for clarity):

But what all the violence of the feudal institutions could never have
effected, the silent and insensible operation of foreign commerce and
manufactures gradually brought about.
These gradually furnished the
great proprietors with something for which they could exchange the
whole surplus produce of their lands, and which they could consume
themselves, without sharing it either with tenants or retainers. All
for ourselves, and nothing for other people, seems, in every age of
the world, to have been the vile maxim of the masters of mankind.

Smith continues:

As
soon, therefore, as they could find a method of consuming the whole
value of their rents themselves, they had no disposition to share them
with any other persons. For a pair of diamond buckles, perhaps, or for
something as frivolous and useless, they exchanged the maintenance,
or, what is the same thing, the price of the maintenance of 1000 men
for a year, and with it the whole weight and authority which it could
give them. The buckles, however, were to be all their own, and no
other human creature was to have any share of them; whereas, in the
more ancient method of expense, they must have shared with at least
1000 people.
With the judges that were to determine the preference,
this difference was perfectly decisive; and thus, for the
gratification of the most childish, the meanest, and the most sordid
of all vanities they gradually bartered their whole power and
authority.

So, Smith seems to be making a greater historical observation about civilization while specifically discussing the concentration of power, greed, and wealth as it was in feudal society.  Hopefully this helps explain the context of the observation a bit.


Permanent Income Hypothesis - Changes in r

If you impose the condition $ \beta (1 + r) = 1 $ and then move around $ r $, you're assuming that consumer preferences are changing. In other words, you have introduced a third effect: increased interest rates lead to consumers decrease their preference for consuming in the future. In other words, there is no substitution effect in your model, there is only an income effect. The result you derived for consumption is fine, and the reason you see an ambiguity is because the consumer not borrowing today doesn't mean the consumer will not borrow at some point in the future, and if the consumer does want to borrow, the higher interest rates will induce them to borrow and consume less. This happens when the consumer expects their income to rise in the future. To illustrate, assume that $ Y_t = Y (1 + \gamma)^t $ for some $ \gamma < r $ (which could change the NPG condition you used to derive this equation, but whatever) so we get

$$ C = \frac{r}{1+r} A_0 + \frac{r}{r- \gamma} Y $$

If $ \gamma \leq 0 $ and $ A_0 > 0 $, i.e if the consumer expects his future income to stay constant or decline, then the impact of a rise in $ r $ is unambiguous: it leads to higher consumption in all periods. However, if the consumer expects his income to rise, i.e $ \gamma > 0 $, then the two effects work in opposite directions: higher rates mean that the consumer borrows less, and this decreases $ C $. The net effect is ambiguous, and depends on the values of the variables in the model.


Is the value of a product measured in money a reliable measure of the economic value of this product?

No, the economic value and monetary value (price) are different concepts in economics.
Value of a good to an individual is usually defined as the utility derived from that good. Unlike prices, it is not necessarily same across individuals or even for the same individual with varying quantity of consumption. In fact, an important assumption on which many of the results of economics depend is that the value of good to an individual decreases with abundance.
The difference of utility derived and price paid is termed as consumer surplus. It is considered a measure of welfare because it measures the net gain from consuming that good.
On how the price amd value correlate: well price is just an outcome of the market forces. Valuation of the good by different economic agents in this market is the primary force here so naturally there is some correlation. However, some markets are controlled by the state (also an economic agent in such markets) to ensure such that it's price is actually much lower than it's value. For example, water, oxygen.
So value in terms of price is the outcome of many, many market forces. How you value it is one them.


Can lower working hours beat unemployment?

In Germany in crisis times working shorter hours (short-time work, Kurzarbeit) can be used by firms to prevent layoffs. There are legal regulations how labor unions need to be consulted on how this is done. It has been argued that this has helped to overcome the 2008 crisis better than some other countries, e.g.: "What Explains the German Labor Market Miracle in the Great Recession? The Evolution of Inflation Dynamics and the Great Recession", Brookings Papers in Economic Activity, Spring 2011, Michael C. Burda & Jennifer Hunt; "Explaining the German Employment Miracle in the Great Recession – The
Crucial Role of Temporary Working Time Reductions", Alexander Herzog-Stein, Fabian Lindner, Simon Sturn, June 2013


Why is elasticity not defined simply as the slope of the graph?

A key feature of elasticity is that it is a unit-free measure.  Suppose we are considering the price-elasticity of demand for apples which is $\frac{{\triangle}y/y}{{\triangle}x/x}$ where $y$ is quantity demanded and $x$ is price.  The elasticity will be the same whether apples are measured in pounds (lb) or kilograms (kg) because both $\triangle y$ and $y$ will be in the same units.  Similarly it will not matter in which currency price is measured, because $\triangle x$ and $x$ will be in the same currency units.  This makes comparisons of elasticities, between goods measured in different units, or between countries with different currencies, much more meaningful than comparisons of slopes.


What methods - inspired by Haavelmo’s Structural Econometrics - can show that a partial equilibrium model is unreliable?

I can think of two big ways.

Formal tests of model fit: If you are using SMM, GMM or indirect influence check out the J-stat. If you are using maximum likelihood you want a likelihood ratio. Bayesian model testing is similar to likelihood ratios, but more complicated. Many of the DSGE models were rejected on these bases (you can see Sargent discussing it briefly in this interview). These tests are all very similar.
Out of sample performance: Estimate the model, then use it to predict out of sample data. This is a tough test, but it is super useful. They aren't so common in economics, but statisticians use them a lot (often times they leave out part of the data and try the model out there). You probably want your model to pass a formal test in sample before you do an out of sample test.

All these tests rely on the same, extremely basic, intuition: If the model is good then it should fit the data well. If the model does not fit the data well it is probably not good.


How is GDP counted between different countries?

So, from the link posted by @Brian Romanchuk, I got this paper and got the answer:

GDP = Consumption + Investment + Government spending + eXports – iMports

So in the example, country A GDP would end up to be:
GDPA_A = 3 000 - 2 000 = 1 000 and being correctly a smaller quantity than country B's GDP.

Thanks to contributors


Why can't countries immediately stop the deficit?

Some countries have very strict fiscal rules that basically force them to keep on spending more, or reduce the government's ability to cut spending (Brazil is a good example). Other countries have looser rules, but can't cut because of political pressures.

Remember acute spending cuts mean less money for hospitals, schools and many other services that rely on public money. It's often a very impopular measure.

That's why economists often favor clear, well-defined spending rules. Such as establishing a primary surplus target during times of economic growth, a maximum deficit during recessions and maybe a debt/GDP ratio ceiling.


Your attempt is almost correct.

Let the demand function be $D(p) = p^{-\epsilon}$ and the cost function some $C(D(p))$ so that $R(p) = D(p)p = p^{1-\epsilon}$, right?

Then, I'd rewrite your first and second order conditions - as you only provided the expressions but it would be great to write them in a form of equations (or inequalities). So that:

First order condition: $R'(p) - C'(D(p))D'(p) =0$

And the second order: $R''(p) - C''\left(D(p)\right)\cdot D'(p)^2 - C'(D(p))D''(p) < 0$

This is the moment when we kind of split our approaches. I also do agree that the second term in second order condition is negative. However, it is not necessarily $R''(p)$ that is negative - rather the whole relation $$R''(p) - C'(D(p))D''(p)<0 \;(*)$$ Now, we want the second order condition to hold for every $p$ that satisfies the first order condition. Thus, I would substitute the $C'(D(p))$ from the first condition to the $(*)$ and hence, I end up with: $$R''(p) - \frac{R'(p)D''(p)}{D'(p)} <0 \;(**)$$

Now it's time for the brute force. Calculate all derivatives in $(**)$.

$$ -\epsilon(1-\epsilon)p^{-1-\epsilon} - \frac{(1-\epsilon)p^{-\epsilon} \cdot -\epsilon(-1-\epsilon)p^{-2-\epsilon}}{-\epsilon p^{-1-\epsilon}} <0 \implies \epsilon > 1$$ for both, $p$ and $\epsilon$ positive.

P.S. If you think about it, letting $\epsilon<1$ would imply that $R'(p)>0$ and thus, the first order condition would have no solutions. Why?

P.P.S. I believe a similar exercise is in Industrial Organization: Markets and Strategies by Belleflamme and Peitz in their chapter about "monopoly pricing strategies" but with the inverse demands $P(q)$.


Find a Perfect Bayesian Equilibrium with a game that includes Nature

You've rightly identified A's optimal actions at his last decision nodes (shown as red arrows in the figure below). Next, you need to

conjecture a pair of actions at A's initial nodes,
use those actions to determine B's belief at her information set,
find B's best response(s) to that belief, and
check whether A's initial actions are optimal with respect to B's best response(s).

If the answer at step 4 is "yes", then you have a PBE; if not, repeat the steps with a different conjecture.
In your particular example, there is no pure strategy PBE. I will nevertheless go through an example just to demonstrate the above approach:

Suppose A chooses "pass" at both initial information sets (blue arrows).
Using Bayes' rule, B's belief must be $0.9$ at the top node and $0.1$ at the bottom node.
Based on this belief, B's expected payoff is $1$ from choosing "exit", and $0.9(3)+0.1(-5)=2.2$ from choosing "pass". So her best response is to choose "pass" (green arrows).
Given that B would choose "pass", A's best response after Nature chooses the top path would be "exit" (purple arrow), since $4>3$.

Thus we do not have a PBE involving A choosing "pass" at both initial nodes. You can go through the rest of A's possible action combinations at his initial nodes to verify that there is no PBE with pure strategies.


Is there something that does not change its value that can be used to measure currency inflation?

TL;DR -- No

Now for the fun part.  No!  No-no-no-no-no!  And here's why...

Conditions change.  Any commodity you can name will change in value over time, as various things happen:

People stop wanting it (I mean, you can buy myrrh these days, but it's not a hot-ticket item)
The availability changes (New mines show up, or old ones shut down)
People start wanting it (use of gallium really spiked when electronics came along)

But wait!  How about gold?  It's been in demand for like evah, right?  Well.  Let's look at the inflation-adjusted price of gold, just for the last century:

Maybe you could try something like "one day's rations", ie enough food to feed the metric standard guy for one day.  Of course, as agriculture gets more efficient...  Hmm.  The US gov't uses a "standard grocery basket" to measure inflation, that works ... okay, though politics occasionally sneak in.

Anyway.  Short story -- nope.


You are correct in stating that if MPS increases, then MPC decreases.  But the question doesn't actually say that MPS increases. What it says is that saving increases at each level of income. That's consistent with a scenario such as the following in which MPS does not change while autonomous consumption falls from $10$ to $5$:

$\qquad$Before: $S = -10 + 0.2Y$ and $C = 10 + 0.8Y$

$\qquad$After: $S = - 5 + 0.2Y$ and $C = 5 + 0.8Y$

In this scenario, consumption falls by $5$ at each level of income - a parallel shift of the consumption function as per option C.

Actually, the question is not as precisely worded as it could have been.  If C is given as the correct answer, the implication is that households decide to save the same amount more at each level of income.  That could have been made explicit.


The ability of markets and prices to direct resources to higher valued uses: Is this a short run or a long run attribute?

No its not necessary long run phenomenon it can hold in short run as well. Rather what matters is how competitive markets are and whether there are any market failures such as externalities etc.
To be even more specific under the first theorem of welfare economics if markets are competitive, complete, information is symmetric, there are no externalities and there are no transaction costs or in general there is an absence of any market failure, then any general equilibrium in competitive markets will be Pareto-efficient.
Pareto-efficiency requires that all resources are allocated to their highest valued uses as if there would be any unexhausted potential gain from trade (i.e. a situation where two individuals each have something they do not value more than what other individual has) two individuals could increase their utility further by actually doing the trade and so the outcome would not be Pareto-efficient.
However, in real life of course the first welfare theorem fails due to market failures and other reasons. In that case market might fail to allocate resources to its most valued uses. Some of the problems might have an effect mainly in the short run and in long run they might still be efficient, but in other cases they wont. In the case you describe the two firms are actually not perfectly competitive but engaged in some monopolistic competition. In monopolistic competition market indeed wont allocate resources to their most efficient uses always in the short-run as firms will have some excess capacity (see Mankiw Principles of Economcis for example) which may imply that not all resources are employed to their most valued uses. But even here I would have to add caveats on that the result would change depending on whether we assume competition occurs on price or quantity and so on.


A Special Case of Difference-in-Differences

Take a look at Ch 5 of Mastering Metrics (Angrist and Pischke) or their earlier "Mostly harmless econometrics".


Pareto efficient quantity of public good (quasi-linear preferences)

I don't think it is true in a standard public good economy the question is referring to. Consider the following counterexample:

Suppose $I  = \{1,2\}$ and utility of the individual $i$ depends on his consumption of public good $(G)$ and private good $x_i$: $u_1(G, x_1) = 2\sqrt{G} + x_1$ and $u_2(G, x_2) = 2\sqrt{G} + x_2$,

Also, the CRS technology used for production of public good uses private good as input:
$G = f(x_0) = x_0$.

If the society has only 4 units of private good in the beginning, then the set of feasible allocations can be written as

$\{(G, x_1, x_2)\in\mathbb{R}^3_+: G+x_1+x_2 = 4\}$.

Notice that the allocation $a_1 = (G, x_1, x_2) = (1,3,0)$ is Pareto efficient, but does not maximize the sum of utilities. The reason is that allocation $a_2 = (G, x_1, x_2) = (4,0,0)$ yields the higher sum.

$\color{blue}{u_1(1,3) + u_2(1,0)} = 5+2 =7 \color{blue}{<} 8 = 4 + 4 = \color{blue}{u_1(4,0) + u_2(4,0)}$.


Suppose you have a differentiable function $f(x)$, which you want to optimize by choosing $x$. If $f(x)$ is utility or profit, then you want to choose $x$ (i.e. consumption bundle or quantity produced) to make the value of $f$ as large as possible. If $f(x)$ is a cost function, then you want to choose $x$ to make $f$ as small as possible. FOC and SOC are conditions that determine whether a solution maximizes or minimizes a given function.

At the undergrad level, what is usually the case is that you need to choose $x^*$ such that the derivative of $f$ is equal to zero:
$$f'(x^*)=0.$$
This is the FOC. The intuition for this condition is that a function attains its extremum (either maximum or minimum) when its derivative is equal to zero (see picture below). [You should be aware that there are more subtleties involved: look up terms like "interior vs corner solutions", "global vs local maximum/minimum", and "saddle point" to learn more].

However, as the picture illustrates, simply finding $x^*$ where $f'(x^*)=0$ is not enough to conclude that $x^*$ is the solution that maximizes or minimizes the objective function. In both graphs, the function attains a zero slope at $x^*$, but $x^*$ is a maximizer in the left graph, but a minimizer in the right graph.

To check whether $x^*$ is a maximizer or a minimizer, you need the SOC. The SOC for maximizer is
$$f''(x^*)<0$$
and the SOC for minimizer is
$$f''(x^*)> 0.$$
Intuitively, if $x^*$ maximizes $f$, the slope of $f$ around $x^*$ is decreasing. Take the left graph, where $x^*$ is a maximizer. We see that the slope of $f$ is positive on the left of $x^*$ and negative on the right. Thus, around the neighborhood of $x^*$, as $x$ increases, $f'(x)$ decreases. The intuition for the case of minimizer is similar.


By buying stocks, am I as an individual measurably affecting the economy?

From an aggregate level, the only direct income effect is that you will have paid a brokerage fee, which is part of the financial sector’s revenue.

After that, all that has happened is a rearrangement of an existing financial asset. If the seller uses the proceeds to purchase goods and services for consumption, that could be attributed to the purchase, although that might be a stretch.

The seller may also pay capital gains tax, and since they need a buyer to sell, that could be pinned on the purchase.

The effect on the price is hard to judge. In the short term, there might be some effect (although it would be negligible for a large cap firm), it will not really matter in the long run, if we assume that prices drift towards the fair value determined by corporate profitability.


Interesting question. Indeed, Basel/I,II,III-type regulation changes the behavior of banks, that´s probably the goal of said regulation.

Its possible that their market behavior with respect to "notes" (which notes?) has changed.

bason on the typical Modigiliani/Miller intuition, it seems that there is no reason why Basel/type regulation will affect a sector or another. The main effects are in changing how firms get financed. If banks are expected to hold senior debt, then firms are more likely to issue senior debt. This changes how funds are intermediated from savers to investors, but in principle it should not change which sectors are financed. It should only affect how they are financed, with equity, preferred, junior debt, senior debt, senior collateralized debt, etc.


Why does the same parameter estimate change in multiple regressions?

Yes, it is because they are estimated jointly and the point estimates are conditional on other point estimates and also because of omitted variable bias. To be more specific, contrast an example of  univariate OLS with bivariate OLS, both of which are estimated by minimizing the sum of squared residuals, but resulting formulas for estimated coefficients are different.
In univariate OLS by minimizing squared residuals we get
$$\min_{b_0,b_1}\sum e^2 = \sum (y_i-b_0-b_1x_{1i})^2 \implies \hat{b_1} = \frac{\sum (x_{1i}-\bar{x_1})(y_i-\bar{y})}{\sum (x_{1i}-\bar{x_1})^2}$$
In a bivariate regression the result will be different
$$\min_{b_0,b_1, b_2}\sum e^2 = \sum (y_i-b_0-b_1x_{1i}-b_2x_{2i})^2 \implies \hat{b_1} = \frac{\sum (x_{2i}-\bar{x_2})^2\sum (x_{1i}-\bar{x_1})(y_i-\bar{y})- \sum  (x_{1i}-\bar{x_1})(x_{2i}-\bar{x_2})\sum  (x_{2i}-\bar{x_2})(y_i-\bar{y})}{\sum (x_{1i}-\bar{x_1})^2 \sum (x_{2i}-\bar{x_2})^2- \sum  (x_{1i}-\bar{x_1})(x_{2i}-\bar{x_2})}$$
Hence we can see that the difference is that in the second expression the calculation of $b_1$ critically depends on what the covariance of $x_1$ and $x_2$ is (i.e. $\sum  (x_{1i}-\bar{x_1})(x_{2i}-\bar{x_2})$), what variance of $x_2$ is (i.e. $\sum (x_{2i}-\bar{x_2})^2$ and also what covariance of $x_2$ and $y$ is (i.e. $\sum  (x_{2i}-\bar{x_2})(y_i-\bar{y})$) and hence $\hat{b_1}$, hence including additional variable will change the $b_1$ (and also other coefficient) formula. This will extend to multiple variables as well with formula getting progressively more complex.
Moreover, the coefficients will change substantially especially if there is some omitted variable bias present. If we assume that the true model is given by:
$$y= \beta_0 + \beta_1 x_1 +\beta_2 x_2 + u$$
but you will try to fit univariate model regardless of the above.
$$\hat{y} = \hat{\beta_0} + \hat{\beta_1} x_1$$
In such conditions it can be shown that even though we fit univariate regression the expected estimated beta will be (see Wooldridge Introduction to Econometrics):
$$E[\hat{\beta_1}]= \beta_1 + \beta_2 \frac{\sum  (x_{1i}-\bar{x_1})(x_{2i}-\bar{x_2})}{\sum (x_{1i}-\bar{x_1})^2}$$
Where the second term is the omitted variable bias. The intuition here is that if there is any covariance between $x_1$ and $x_2$ and if $y$  actually depends also on  $x_2$ (i.e. $\beta_2 \neq 0$) then if you apply univariate regression on bivariate relationship the estimate of $\beta_1$ will be biased as it partially captures the relationship of $y$ and $x_2$. For example,  if $\beta_1=0.1$, $\beta_2= -2$ and $cov(x_1,x_2)=0.5$ and $var(x_1)=2$ then if you estimate univariate regression the estimated $\hat{beta_1} = -0.4$ whereas in bivariate regression it would be $\hat{\beta_1}=0.1$ and $\hat{\beta_2}=-2$
Hence to sum up, the coefficients are different because they are in multivariate regression calculated conditional on other coefficients and also if you are omitting an relevant independent variable its effect will be 'hidden' in the estimates of non-omitted variable (if they are correlated).


Hint
Let $\{1,2,\dots,N\}$ be the set of outcomes. Without loss of generality, let $1\succsim 2\succsim\cdots\succsim N$. Let $\mathbf e_i$ denote the degenerate lottery that assigns probability $1$ on outcome $i$ and $0$ on the rest. Note that any lottery $L=(p_1,\dots,p_N)$ can be written as a compound lottery $p_1\mathbf e_1+\cdots+p_N\mathbf e_N$.
Since lottery $\mathbf e_i$ amounts to getting outcome $i$ for sure, we must have
\begin{equation}
\mathbf e_1\succsim \mathbf e_2\succsim \cdots \succsim \mathbf e_N.\tag{1}
\end{equation}
Successively applying the independence axiom to $(1)$, you'll obtain $\mathbf e_1$ as (one of) the best lottery and $\mathbf e_N$ as (one of) the worst.


Why can't we use $R^2$ for different dependent variables?

$R^2$ measures the fit of the model to the data.  Notice the ordering of that sentence, "model to the data," and not "the data to the model."  You want to use $R^2$ as a criterion to select among two possible equations.  The fact that you are posting here also probably means either someone told you that you cannot do that or you read online that this will not work.  The fact is, it can work, but it does not function well.

Several aspects of how $R^2$ is calculated make it an inadequate criterion.  First, and this is the classical reason, all you have to do to increase $R^2$ is add variables.  If you were predicting the impact of milk substitutes on babies' performance on a physical test and you added in the price of flawless, color D, three carat diamonds as traded in Hong Kong your $R^2$ may go up, but it will not go down.  In addition, your total sum of the squares will change under the transformation you propose.  Because of this, it is not well suited to using an F-test to compare models.

This brings you to the AIC and the BIC.  Philosophically, the AIC and the BIC are algorithmic approximations of the Bayes factor to choose between the two models.  Because you know your data and we do not, you should read a book on model selection, such as this one.

However, given the limited information that you did provide, it would appear, at least on the surface, that either the AIC or the BIC would produce equivalent results.  That is because you have only two models and they have an equal number of parameters.

If you have not used a Bayesian method before, they reverse the direction of probability.  Instead of assuming a model is true and determining whether or not the data is as extreme or more extreme than some standard, it assumes the data is fixed and not random and that the models are uncertain, and so selects the best parameters and models based on the data.  There is no null hypothesis.  So the AIC or BIC are approximations of odds transformed into an algorithmic rule.

They differ in two ways.  First, the BIC gives each model equal prior probability of being "true," while the AIC gives probabilities in proportion to their number of parameters so that complex models are penalized for being complex.  Second, they approximate the likelihood function differently so that the AIC ends up penalizing complex models less than the BIC unless the sample size is large.

The reason to use either tool, rather than a full-blown Bayesian method, is that they are faster, they are good approximations in most circumstances to the Bayesian solution, and they are less complex than a Bayesian model selection process.

You cannot formalize the idea of the logarithm reducing volatility and creating nicer models because it may not do so.  Consider a model whose true form is $y=5x+7$ and you then transform those variables.  A log-linear model would be expected to be a worse model than the untransformed model.  Although using logs would reduce the scale of all variables, $R^2$ is based on the relative scale.  Dividing all variables by two would also reduce variability, but not improve $R^2$ or model selection.


As they clearly say on pp 2267:

Alternative specifications were compared using the
specification test by Davidson and MacKinnon and
the Wald test of coefficient restrictions.

So they two tests Davidson and MacKinnon specification test and Wald test.
Wald test is a classic test for coefficient restrictions you can learn can learn more about it in any econometrics textbook, for example Verbeek (2008). Guide to modern econometrics. 4ed.
For Davidson and MacKinnon specification test you can have a look at the original Davidson and MacKinnon (1990) paper.


How does the stock market correlate with the economy?

This study, "Is There a Link between GDP Growth and Equity Returns?", compares a number of advanced economies over the period 1969-2009. On average, the growth in stock prices was just 0.3% higher than the growth in GDP. However this ratio varies considerably between individual countries. Stock price growth was 4.5% higher than GDP growth in Spain, but -3.5% in Sweden. Here's the relevant data table.


Take your two beginning expressions and cross multiply them:

$$p_1 c_1^F = p_2 c_2^F$$

$$p_1 c_1^H = p_2 c_2^H$$

Then subtract each right hand side for both equations. Multiply one of the expressions by $-1$ on each side:

$$p_1 c_1^F - p_2 c_2^F = 0$$

$$-p_1 c_1^H + p_2 c_2^H = 0$$

And set each expression equal to each other:

$$p_1 c_1^F - p_2 c_2^F = -p_1 c_1^H + p_2 c_2^H$$

Put the $p_1$ and $p_2$ terms together:

$$p_1 c_1^F + p_1 c_1^H = p_2 c_2^F + p_2 c_2^H$$

Factor the $p_1$ and $p_2$ terms and rearrange:

$$p_1 (c_1^F + c_1^H) = p_2 (c_2^F + c_2^H) \implies \frac{p_1}{p_2} = \frac{c_2^F + c_2^H}{c_1^F+ c_1^H}$$