For a given set of consumption bundles, how do we know there are bundles that the consumer is indifferent between?

We can obtain the result that Indiference Sets are not singletons, i.e. that consumption bundles that are equivalent under the preference relation must exist, by using a Consumption Set $X$ defined over $\mathbb R_+^n$, i.e. over real vectors, together with the property of Continuity of preferences (alternative-weaker prerequisites may exist though).

An equivalent way to state the property of Continuity of preferences (see for example Mas-Collel et al, ch 3.C), is that $\forall x \in X$ the upper contour set $UCS$ and the lower contour set $LCS$ are both closed, i.e they include their boundaries. The contour sets are sets that include consumptions bundles "at least as preferred as $x$" (upper) and "at most as preferred as $x$" (lower). In order for them to "include" their boundary", a boundary must exist in the first place. But their boundary is the Indifference Set, the set containing all bundles as preferred as $x$.

So your question essentially boils down to Is it possible that the boundary of the upper or lower contour sets of $x$, sets that are defined over real vectors, is a single point?

I believe that envisioning this in two-dimensions will show that it is impossible, but let's prove it.

Ad absurdum, assume that this holds for some $x \in X$. Then for any other consumption bundle in $X$, say $x'$, we will have either $x'>_{pr} x$ or $x'<_{pr} x$. In other words, any other point in $X$ will belong either in $UCS_x$ or in $LCS_x$, but not in both.

So consider two such points , $x''>_{pr}x>_{pr}x'$. Assume that $x'=\{\mathbf y_{(n-1)}, y_n\}$, $x''=\{\mathbf y_{(n-1)}, y_n+\epsilon(k)\}$, i.e. that they differ in only one of the goods in Goods Space, $\epsilon(k)>0$.

Transform now these two points into two sequences of points, $\{x''_k\}$ and $\{x'_k\}$, indexed by $k= 1,...$ and by constructing the sequence $\{\epsilon_k\}$ such that $\{\epsilon_k\} >0\; \forall k$ but also $\lim_{k\rightarrow \infty} \epsilon_k = 0$. The consumption space as defined permits these constructions.  We note of course that $\{x'_k\}$ is a constant sequence since its value does not change  as $k$ changes, but that's perfectly legal.

Consider now the sequence of pairs, for each $k$, $\{(x''_k, x'_k)\}_{k=1}^{\infty}$, and it holds that

$$ x''_k >_{pr} x'_k\;\;\;\forall k  \tag{1}$$
$$\lim_{k\rightarrow \infty} x''_k = x'$$
$$\lim_{k\rightarrow \infty} x'_k = x'$$

Under Continuity of preferences we should obtain
$$ (1) \implies \lim_{k\rightarrow \infty} x'' >_{pr} \lim_{k\rightarrow \infty} x'$$
but this is obviously not possible since these limits are identical.
So by assuming that there exists an $x$ in the Consumption Set to which no other bundle is equivalent in terms of the preference relation, we have violated the Continuity property.

Alternatively, since the bundles are defined over real vectors, and both $UCS$ and $LCS$ are closed due to the Continuity property, then every Cauchy sequence in each of them that converges has its limit in them. So the sequence $\{x''_k\}$ has its limit in $UCS_x$, while the sequence $\{x'_k\}$ must have its limit in $LCS_x$. But this limit is the same as shown above, and equal to $x'$. But the only common point of these two sets is $x$. So it must be the case that $x'=x$. But this cannot hold since we have started by assuming $x>_{pr} x'$.

An example where indifference sets are singletons is the case of Lexicographic Preferences. And Lexicographic Preferences are not continuous.


Robot taxation is a bit like corporate income taxation. Like corporations, robots don't pay taxes, people pay taxes. In the words of Herb Stein:

I remember that in addressing the issue in the 1980s, the late Herb
Stein said that it's as if people think that if the government imposed
a tax on cows, the tax would be paid by the cows.
Are Taxes on Corporations Taxes on People? by David Henderson

There really are only three entities to tax. You can tax the people who own the robots, you can tax the people who buy what the robots make, or you can tax the income of people who work with robots. If you try to tax the robots directly, you'll end up with an effective combination of these other taxes.
In general, it is thought that the providers of capital are unlikely to bear very much of the incidence of taxation. Instead, it is thought that generally expected capital returns will adjust to keep the expected returns similar before and after the tax. This happens because capital is mobile. If the US tries to tax capital heavily investors can invest elsewhere, and will do so until after tax rates of return are equated. Robots are just another sort of capital, suggesting that it will be as difficult to tax them as any other sort of capital.
Taxing the buyers of robot produced goods is also problematic. As long as consumers can purchase imports without a special robot tariff then robotic produced imports will force domestic produced goods with a robotic tax to have a lower prices to leave consumers indifferent. Or they may not produce at all. Either way, consumers likely won't bear the burden.
Which leaves labor, the relatively immobile factor of production, to bear the burden.


Does the WTO monitor the trade of players in association football? Could they?

No, for several reasons.

WTO deals with trade disputes between nations when one nation creates legislature that breaks WTO rules or some pre-existing trade agreement. For example, one of the most important WTO rules is the most favored nation principle which says that (WTO, 2020):

countries cannot normally discriminate between their trading partners. Grant someone a special favour (such as a lower customs duty rate for one of their products) and you have to do the same for all other WTO members.

But again this all applies to countries not companies. Companies can charge different prices to people in different countries if they want to.

Trading players would likely not even qualify as a trade. If one company releases player from a contract and that player then signs contract with another company that is no more trade in economic terms than when your company fires you and other company in another country hires you. For this to qualify as a trade the players would still have to be employees of the UK company and just provide service to the German company  (when this would become trade in services).

Even if there would be trade in services and $1\%$ government imposed tax this would not be violation of WTO rules as long as all countries Germany trades with are subject to this $1\%$ tax or if it would be part of some of the WTO exceptions (for example, WTO allows trading blocks to set among themselves lower barriers than just under pure most favored nation principle).

If these aren't under the purview of the WTO, could they be in a decade?

I doubt this would ever be under the purview of the WTO given that WTO's objective is to promote free trade by removing tariffs and non-tariff barriers to trade and trade cooperation amongst nations. WTO's objective is not to regulate/police trade among private firms. In addition depending on how the 'trading of players' is set up it does not even qualifies as a trade (e.g. if players change employer).


Utility function that generates a demand curve which will have an U shaped MR curve

Let $v(x) = \int  1 − x + 3x^2 − 3x^3 \text{d}x$.
The quasi linear utility function
$$
U(x,y) = v(x) + y
$$
generates the inverse demand curve given by Ubiquitous if income $I$ is large enough and $p_y = 1$, as
\begin{align*}
|MRS(x,y)| & = p_x \\
\\
v'(x) & = p_x \\
\\
1 − x + 3x^2 − 3x^3 & = p_x.
\end{align*}
Hence it will also result in the same $MR$ for $x$.

P.s.: I have not checked if $U(x,y)$ is convex, so second order conditions may need to be examined here. Note that $U(x,y)$ need not be convex everywhere, only above the $MR$'s U shape. Everywhere else it can be altered as we see fit.


Is inequality a micro or macro economics issue?

Inequality and public policy has actually its own field (Public Economics) so it does not narrowly fit in either of those categories, but given that most research and work on inequality is of microeconomic character (see Stiglitz Economics of the Public Sector 3rd ed) I think it is fair to put inequality under microeconomics if you have to make such binary choice as it fits there better.
First let us get the definitions of micro and macroeconomics straight. Following Mankiw Principles of Economics pp 27:

Microeconomics is the study of how households and firms make decisions
and how they interact in specific markets.
Macroeconomics is the study of economy-wide phenomena.

Neither of them is top-down or bottom-up, in fact most macroeconomics models are build upon micro-foundations (in fact even you do research in pure macro you need to learn a lot of microeconomics), so by your descriptions in your question they would both be 'bottom-up' - although I do not think it is sensible to talk about these fields as top-down or bottom-up. The distinction is based rather on the scale at which you are looking.
When it comes to inequality, often the appropriate level of analysis is microeconomic analysis. For example, do you worry about income inequality? Well income inequality arises from the fact that people are heterogeneous in their ability. Hence you need to study how this affects rewards they get for their labor supply on a labor market. Such analysis does not require tracking of inflation and other macro factors that would likely just 'muddy the waters'. To study that you want some micro models not macro models where everyone is aggregated to an 'representative agent' (some sort of average human). Similar arguments would apply to most topics to be considered in the realm of income inequality.
Macroeconomics, also touches bit on inequality as there can be interplays between inequality and economic growth (and growth is definitely macro topic). Or unemployment benefits or redistribution from people who have high marginal propensity to save (usually rich people) to ones with low propensity to save (usually poor) can be considered macroeconomic stabilizers. But macroeconomics does not really concern itself with inequality per se. If it would empirically turn out that poor people have higher propensity to save than rich people from pure macro perspective there should be redistribution from poor to the rich.
This is not to say that there is absolutely no research into inequality where inequality would be in the main focus and the research is just using macro models/tools. But this strand of literature is miniscule compared to microeconomic literature devoted to this topic.
For example, consider this. In Romer Macroeconomics, which is the standard widely used graduate macro textbook the word income tax is printed only 8 times. In the Varian Microeconomic Analysis you have separate sub chapter on optimal taxation, separate chapter for welfare analysis, separate chapter for public goods etc.
This is not because macroeconomists would want to ignore inequality but it is simply not considered to be part of the subject matter. This primarily comes to the fact that macroeconomists often have to aggregate when they try to describe whole economy. In order to aggregate you need to introduce average households, average individuals, average firms etc. Once you aggregate whole population to set of average individuals then income/wealth inequality amongst those averages is zero.
There are macro models that also look heterogeneous agents but even there you will usually have just some small limited number of options. For example, having two households that are different and each is supposed to be an avatar of their own category. It is possible to still introduce more heterogeneity into macro models but the more heterogeneity you include the complexity of the model raises disproportionately. This is less problematic in microeconomic models and thus they are preferred.
Lastly, it is possible that in the future this will change and people studying inequality and policy economics in general will start focusing more on macro models and treating inequality as a macro phenomenon. From my own experience work on inequality from macro side is nowadays more common then when I was undergraduate. Nonetheless, as of now I don't think anyone could reasonably disagree with statement that most inequality work focuses solely on micro  (for example, almost whole literature on optimal income taxes and transfers is microeconomics) or uses micro models. Hence, if you have only two options I think it makes more sense to put it under microeconomics and I think this was why the author did that.


U.S. Treasury defaulting on its bond obligations to China

the administration would threaten to selectively not pay its Treasury
  Bond obligations to China

  What exactly would a default trigger?

Selective default would be ineffective and it could backfire.

It would be ineffective because, at any point in time, a treasury bond has the same value regardless of who holds it. If the USA disavows its obligations toward China, clearly the top holder of US Treasury securities, China might opt to get rid of its holdings. China's move would predictably disrupt the bonds market due to the arbitrage opportunities that will ensue.

Selective default would backfire in two ways. Most certainly, the Treasury would attract fewer funds in the next round(s) of US bonds because the market would be "flooded" with the bonds of which China is getting rid.

The other way in which it could backfire is that selective default against China will make other countries wary of they themselves becoming the target of a subsequent selective default. Since no creditor likes the prospect of its holdings being singled out for default, there is a realistic chance that each country would decide to dump its holdings as well, and lead to a domino effect. That reaction is likelier now with the increasing efforts to replace the US as the "official" currency for international settlements (such as in oil or currency markets).


Is any aspect of the cryptocurrency adoption violating a widely held economic tenet?

There is quite a bit of research on it and a good argument as to why central banks should not issue cryptocurrencies, but should issue electronic money.  I would recommend reading https://research.stlouisfed.org/publications/review/2018/02/13/the-case-for-central-bank-electronic-money-and-the-non-case-for-central-bank-cryptocurrencies/

and

https://research.stlouisfed.org/publications/review/2018/02/13/blockchain-what-it-is-what-it-does-and-why-you-probably-dont-need-one/

https://research.stlouisfed.org/publications/review/2018/02/13/is-bitcoin-a-waste-of-resources/

Nothing about cryptocurrencies is surprising to me.  Indeed, the separate parts have been in use, in some cases, for hundreds of years.  Bitcoin is a clever combination of things that have long existed.  The other cryptocurrencies, such as Etherium, also offer self-executing contracts.  Those are also very old.  The simplest that you may physically see are sight drafts, which are common in the southeastern United States.  It is a draft printed on an envelope and is payable only when the envelope contains a properly executed document.

There is nothing surprising to me about how cryptocurrencies are behaving, but they cannot continue like this and they won't.  You should read "Good Money," by George Selgin, who writes about the creation of private money in Britain by button makers to meet an unfilled demand.

What you are seeing is a process by which people learn.  If you think this whole process through, you will find that you will not want to participate in this little revolution until supply and demand intersect.  By limiting the number of coins the various firms have created artificial scarcity.  However, we now have 759 cryptocurrencies and growing.  Each one makes the others just a little less valuable.  Although there is an argument against a central bank entering the market, if one did, the private money would vanish if the official money could be exchanged at a 1 to 1 rate.

Imagine the impact on Bitcoin prices if the Chinese or British central bank allowed people to walk in the door and exchange their cyptocurrency directly for cash?  No trading costs?  Think about how easy it would be to pop this bubble.  What if Bank of America were to offer one instead of an unknown, small, private company?  What if they allowed direct deposits and withdrawals?  What if they paid the mining costs to their own computers?

You are looking at the modern version of the South Sea Bubble or the Tulip Bubble.  When people are learning and making money, they do funny things.


Interest is a key mechanism to ensure that resources are properly allocated within an economy. There are lots of uses to which money can be put: consuming goods, building highways, constructing a new factory, investing to develop a new drug, etc. Just like goods, the allocation of money is governed by the price system—where the interest rate is the price (because the interest rate is how much you have to pay to get access to money). If the price is too low (i.e. if there is no interest) then few people will be willing to lend money and valuable investments (in, for example, infrastructure or new business ventures) will not take place because the funds aren't available to pay for them. A positive interest rate ensures that people with money they don't need right now have an incentive to lend it to people with valuable projects to fund.

We know that it is possible for economies to function without interest, though, because the charging/payment of interest is prohibited by Sharia law, giving rise to so-called Islamic banking.


Will the monopolist inevitably increase the price if demand grows?

In this post, we find that the optimal monopoly price can be written as

$$P^* =  \frac {|\eta|}{|\eta|-1} MC $$

where $\eta$ is the elasticity of demand with respect to price, and so optimal price will necessarily be at a point where $\eta$ is higher than unity in absolute terms.

From this one can deduce some unambiguous and some ambiguous scenarios when the demand schedule shifts outwards, e.g.

SCENARIO A: constant price elasticity of demand, convex costs: $\implies$ Price and quantity will increase.

SCENARIO B: constant price elasticity of demand, linear costs: $\implies$ Price will stay put, quantity will increase.

The floor is yours.

PS: Note that to have constant price elasticity of demand, the demand function will necessarily be non-linear.


noise traders in neo-classical economics vs behavioural economics

The claim that financial markets are efficient is backed by an implicit argument that misinformed "noise traders" can have little influence on asset prices in equilibrium. If noise traders' beliefs are sufficiently different from those of rational agents to significantly affect prices, then noise traders will buy high and sell low. They will then lose money relative to rational investors and eventually be eliminated from the market. But De Long et al, 1987 (https://www.nber.org/papers/w2395) presented a simple overlapping-generations model of the stock market in which noise traders with erroneous and stochastic beliefs (a) significantly affect prices and (b) earn higher returns than do rational investors. Noise traders earn high returns because they bear a large amount of the market risk which the presence of noise traders creates in the assets that they hold: their presence raises expected returns because sophisticated investors dislike bearing the risk that noise traders may be irrationally pessimistic and push asset prices down in the future. The journal version of that paper is De Long, J. B., Shleifer, A., Summers, L. H., & Waldmann, R. J. (1990). Noise trader risk in financial markets. Journal of political Economy, 98(4), 703-738.
According to google there is more than 7000 citations for that article  https://scholar.google.com.br/scholar?q=Noise+Trader+Risk+in+Financial+Markets&hl=pt-BR&as_sdt=0&as_vis=1&oi=scholart


Questions of the type "What do you think other people would think?"

Epistemic game theory would be the closest (sub-)field that deals with questions involving higher order beliefs among interacting agents.

The introductory article by Dekel and Siniscalchi is a good entry point to the literature. From its introduction:

  Epistemic game theory formalizes assumptions about rationality and mutual beliefs in a formal language, then studies their behavioral implications in games. Specifically, it asks: what do different notions of rationality and different assumptions about what players believe about ... what others believe about the rationality of players imply regarding play in a game?

"Epistemic Foundations of Game Theory" on the Stanford Encyclopedia of Philosophy has a less technical introduction.

There is also a strand of literature in behavioral economics that studies cognitive hierarchies. The theories there are developed mostly to explain behaviors in the lab settings. Crawford, Costa-Gomes, and Iriberri (2013) provide a good summary.


Why is consumer surplus the area under the curve?

This is quite fundamental, so I'd encourage you to look up your textbook as well... but here's a short explanation.

Imagine that there are $n$ consumers in the market. You sort them by their willingness to pay from high to low. So the first consumer is willing to pay very high, and so on. Let's say the willingness to pay (sorted) is 12, 10, 9, 7, 5, 4, 2. This is represented by the downward-sloping demand curve.

Let's say the price is \$8.

The first consumer's surplus is $12-8 = 4$.
The second consumer's surplus is $10-8 = 2$.
The third consumer's surplus is $9-8 = 1$.

The fourth consumer onward wouldn't buy the item, since their willingness to pay is lower than the price. Their surpluses are all zero. They are the ones on the right of the equilibrium quantity.

The sum of the consumers' surplus is 4 + 2 + 1 = 7, which is given by the area under the demand curve and above the price the consumer pays.

Hope that helps!


Debt service cost is felt by equity owners and debt holders but neutral to the firm. Taxation does not care for cash flow, it is levied on tax law specific accounting profit, not even reporting profit, which tends to treat most interest charges (not all and not uniformly either) as a deductible cost. Thus you make an adjustment to your WACC as you said. However, interest charge does not need to be uniform during the life of the debt. Unless you do your WACC adjustment for every period and apply a different WACC for each (you should always at least justify not doing so anyway), you will end up with a inflated cash flow and higher discount rates.


Replicate average risk-free rate in Wachter 2005, "Solving models with external habit"

I think the answer just comes down to rounding error. The reported values of $\delta$ are annualized. In the case of the CC1999 calibration, the annualized $\delta$ is $0.90$ and the implied monthly is $\delta = 0.991258$. Because of the compounding over 12 periods (4 in the Wachter 2005/2006 case), the average risk-free rate is very sensitive to small changes in this value. Taking the other parameters as given and solving (11a) for the appropriate $\delta$ in each calibration leads to the following annualized values that are needed to reproduce the desired risk-free rates:
\begin{align}
\delta &= 0.8957829809431154 \text{ (instead of 0.90 in CC calibration)} \\
\delta &= 0.9384246954559854 \text{ (instead of 0.93 in Wachter calibration)}
\end{align}
Below is the code to verify this.
# Campbell Cochrane parameters
ann_fact = 12 #annualization factor. Monthly
g = 1.89 / ann_fact / 100
sigma_v = 1.50 / np.sqrt(ann_fact) /100
gamma = 2.00
b = 0.00
phi = (0.87)**(1/ann_fact)
# delta = (0.90)**(1/ann_fact)
delta_CC = (np.exp(- 0.94/ann_fact/100 + (gamma * g - 1/2 * (gamma * (1-phi) - b)) ))**ann_fact
delta = (delta_CC)**(1/ann_fact)

# ## Wachter 2005 Parameters
# ann_fact = 4 #annualization factor. Quarterly
# parameterization_name = 'Wachter (2005)'
# ann_fact = ann_fact
# g = 2.20 / ann_fact / 100
# sigma_v = 0.86 / np.sqrt(ann_fact) / 100
# gamma = 2.00
# b = 0.011
# phi = (0.89)**(1/ann_fact)
# # delta = (0.93)**(1/ann_fact)
# delta_wachter = (np.exp(- 1.47/ann_fact/100 + (gamma * g - 1/2 * (gamma * (1-phi) - b)) ))**ann_fact
# delta = (delta_wachter)**(1/ann_fact)

bar_S = sigma_v * np.sqrt(gamma/(1 - phi - b/gamma))
bar_s = np.log(bar_S)
s_max = bar_s + 1/2 * (1 - bar_S**2)
Smax = np.exp(s_max)
print("Smax ", Smax) # CC1999 should be 0.0939
r_f = (-np.log(delta) + gamma * g - 1/2 * (gamma * (1-phi) - b))
print("Average risk-free rate", r_f * ann_fact * 100)
# CC1999 should be 0.94 %
# Wachter 2005 should be 1.47 %
```


Best strategy to incentivise third party investment

There are several possible routes to get conservation investment into national reserves, i.e. geographic areas that are designated for special protection and environmental conservation.

The government could set up a fund that rewarded the protection and enhancement of the reserves; by guaranteeing the returns on investment, then investors would be drawn in.

The government could specify that investment in national reserve conservation was a requirement for other rewarding activities: for example, it could specify that if a corporation wanted to win government contracts, it had to invest at least x% of its turnover in protecting national reserves. Or that planning permission was linked to such investment.

It could create special tax incentives for such investment: for example, the company's tax burden might get reduced by amount equivalent to a fixed percentage of the investment in the national reserve.

I suspect that in any case, an expert panel would be needed, to assess whether something would represent beneficial investment or not; and if it is, then what the suitable size of return on the investment should be.

And any system is prone to gaming, so the scheme would need to put in place measures to detect and curb such gaming.


Calculating growth rate of capital when not in steady state

Perhaps I misunderstood something but it seems to me you can calculate the growth rate by using all the parameters to calculate $K_{t+1}$. Then get the growth rate by taking $\frac{K_{t+1}}{K_t}$.

You will never reach the true steady state, as outlined in my answer here.


The typical answer is that it reduces money supply by incentivizing saving and increasing cost of loans.

This is backwards.  The Federal Reserve (the United States' central bank) sets a target for the federal funds rate.  The rate itself is a weighted average based on market activity.  The Fed encourages the average rate to match the target by buying and selling bonds in what are called open market operations.  Buying bonds puts money into circulation, increasing the money supply.  Selling bonds takes money out of circulation.  The Fed changes the money supply to change the interest rate, not the other way around.

  But couldn't market participants just react by adjusting FX rates, inflation, etc. by the appropriate amount?

Only if all (or even most) of the participants worked together.  If only one or a small portion of participants attempt this, then it creates arbitrage opportunities for the remainder.  Currency exchange rates and inflation are both set based on market actions.  They aren't so easy to adjust.


Ordinally Separable Utility Representation

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$  is a connected separable topological space by a theorem of Eilenberg.  A proof of Eilenberg's theorem is given in Debreu's book Theory of Value. Debreu assumes the domain there to be Euclidean, but the proof easily generalizes to the present setting. One can also use the assumption that each $X_i$ is a second-countable topological space (for metrizable spaces implied by separability). Debreu showed that continuous utility representations exist in that setting too, but the proof is considerably harder.
Define $\succeq_i$ on $X_i$ by $x_i\succeq_i y_i$ if $x_iz\succeq y_iz$ for some $z$. Clearly, this defines a continuous weak order and the $z$ used does not matter by singleton separability. There exists a continuous utility function $u_i:X_i\to\mathbb{R}$.
Next, we can show that $x_i~\sim_iy_i$ for $i=1,\ldots,N$ implies $x\sim y$. By repeated applications of singleton separability we get $(x_1,y_2,y_3,\ldots,y_N)\succeq (y_1,y_2,y_3,\ldots,y_N)$, $(x_1,x_2,y_3,\ldots,y_N)\succeq (x_1,y_2,y_3,\ldots,y_N)$, and so on, until we have $(x_1,x_2,\ldots,x_{N-1},x_N)\succeq(x_1,x_2,\ldots,x_{N-1},y_N)$. By transitivity, $x\succeq y$. Similarly, $y\succeq x$, and, therefore $x\sim y$.
Let $U=\prod_{i=1}^N u_i(X_i)$, a product of (possibly unbounded) intervals. Define $\succeq^*$ on $U$ by $u\succeq^* u'$ if for some $x,y\in X$ one has $u=\big(u_1(x_1),\dots,u_N(x_N)\big)$, $u'=\big(u_1(y_1),\dots,u_N(y_N)\big)$, and $x\succeq y$. By what we have just shown, $\succeq^*$ contains, together with $u_1,\ldots, u_N$, the same information as $\succeq$. It is also obvious that $\succeq^*$ is strictly monotone on $U$. If a utility representation $W:U\to\mathbb{R}$ of $\succeq^*$ exists, it must be strictly increasing. We are done if we can show a continuous such representation exists. Using any of the mentioned utility representation theorems, it suffices to show that $\succeq^*$ is continuous.
So let $u\succ^* u'.$ There exist $x,y\in X$ such that $u=\big(u_1(x_1),\dots,u_N(x_N)\big)$, $u'=\big(u_1(y_1),\dots,u_N(y_N)\big)$, and $x\succ y$. Since $\succeq$ is continuous and by the definition of the product topology, there exist open neighborhoods $V_i\ni x_i$ and $W_i\ni y_i$ such that $x'\in\prod_{i=1}^N V_i$ and $y'\in\prod_{i=1}^N W_i$ implies that $x'\succ y'$.  Let $V$ be the interior of $\prod_{i=1}^N u_i(V_i)$ and $W$ the interior of $\prod_{i=1}^N u_i(W_i)$. Then for $u''\in V$ and $u'''\in W$ one has $u''\succ^* u'''$, so $\succeq^*$ is, indeed, continuous.


Why can we finance government deficit by having a trade deficit?

To answer the question, I feel it is needed to correct one misunderstanding first: trade deficit does not mean the government buys imported products more than it sells the exported products. Rather, it means the imported products are more than exported products, which does not indicate whether the products are consumed or invested by the government or the private sector.

And this question relates the double deficit problem (also called twin deficit).
Let's start from the no tax situation, and there is:
$$Y-C-G=I+NX$$
Since the LHS represents the total saving of the country (denoted as $S_{T}$), we have:
$$S_{T}-I=NX$$
Recall that saving is the surplus of a country's production that is not consumed by private and government.That's why total saving equals its investment ($S_{T}=I$) if it is a closed economy ($NX=0$).

In the perspective of an open economy, the consumption of a country's production is not only domestic spending on domestic products from the private sector ($C_{d}$) plus domestic spending on domestic products from government ($G_{d}$), but also foreign spending on domestic products ($X$). The private and government spending on foreign goods, as well as investment on foreign goods and services, should be eliminated from GDP or $Y$, and we have $M=C_{f}+G_{f}+I_{f}$ ($Y=C+G+I+X-M$)

Come back to the equation $S_{T}-I=NX$. The RHS is the trade balance, and the LHS can be view as the balance of this country. If the LHS is negative, it means the country has less saving than it has invested, which means it is the borrower in the global capital market (there is a capital inflow) in order to purchase foreign goods and services. This is caused (or accompanied) by a trade deficit, because the country produces ($X$) less than it consumes ($M$), which requires it to borrow money in the global market (if not, say, in the case of no capital flow, its currency will depreciate to balance $NX$, due to an excessive supply of domestic currency caused by trade deficit).

Adding tax is simply separate total saving into private saving and tax, which is just slightly different from the original equation, that is,
$$S+(T-G)-I=NX$$
As government spending is always exogenous, and others factors are relatively stable, the double deficit hypothesis supposes that an increase in government deficit will result in an increase in trade deficit, accompanied by an increase in capital inflow to finance the extra government spending.


Given that the utility maximization problem is:
\begin{eqnarray*} \max_{\{y_j \geq 0 : 1\leq j\leq J\}} && \sum_{j=1}^{J} u(y_j) \\ \text{s.t.} && \sum_{j=1}^{J} p_jy_j \leq m\end{eqnarray*}
where $p_j > 0$ for every $j$, and $m > 0$.

If $u$ is convex and strictly increasing, then in optimum consumer will always spend all his money on the cheapest good. That is, buying $\frac{m}{p_1}$ units of commodity 1 when $p_1 \leq \min_j p_j$ will be utility maximizing for the consumer.

Here is the proof: First of all note that in optimum consumer will never spend less than his income because his utility is increasing. So, we are left with bundles on which he spends all his income. Now any consumption bundle that costs exactly equal to his income can be written as the convex combination of the bundles of the type in which the consumer spends all his money on a single good.

For example, in a two commodity case: Any $(y_1, y_2)$ on the budget line can be written as the convex combination of $\left(\frac{m}{p_1}, 0\right)$ and $\left(0,\frac{m}{p_2}\right)$.

Since, $u$ is convex, and the sum of convex functions is convex, so we also know that the utility $U$ is convex. By convexity of $U$, utility at the convex combination of the bundles of the type in which the consumer spends all his money on a single good is less than or equal to the convex combination of the utilities at those bundles, and that is in turn less than or equal to the maximum of the utilities at those bundles. This shows that the consumer can never lose by changing his choice to the bundle where he buys the cheapest bundle.

For example: In the two commodity case, we can write any bundle any bundle $(y_1, y_2)$ on the budget line as $\lambda\left(\frac{m}{p_1}, 0\right)+(1-\lambda)\left(0,\frac{m}{p_2}\right)$ where $0 \leq \lambda \leq 1$. Therefore, by convexity of $U$,

\begin{eqnarray*} U(y_1, y_2) = U\left(\lambda\left(\frac{m}{p_1}, 0\right)+(1-\lambda)\left(0,\frac{m}{p_2}\right)\right) & \leq & \lambda U\left(\frac{m}{p_1}, 0\right)+(1-\lambda)U\left(0,\frac{m}{p_2}\right) \\ & \leq & \max\left( U\left(\frac{m}{p_1}, 0\right),U\left(0,\frac{m}{p_2}\right)\right)\end{eqnarray*}
This implies that the utility is maximum at either $\left(\frac{m}{p_1}, 0\right)$ or $\left(0,\frac{m}{p_2}\right)$. Given that $U(y_1, y_2) = u(y_1)+u(y_2)$, it follows that it is maximum when one consumes the cheaper one.


After some more intensive (4 mins) googling I found it on the BLS website.


In order to start making this tractable, I'd start by stating in which order the two events – the player making their choice, and the machine deciding how much money to put into box B – happen. There's two options:

First, the machine deposits money into box B (based on a guess as to how the player will choose); the player chooses afterward.
First, the player chooses; the machine deposits money into box B afterward.

(Technically, there's a third option: both actions being performed at the same time. I'd consider this to be equivalent to the first, in that the machine cannot observe the player's actual choice, and the machine's choice is a given for the player.)

The first option is not interesting as a one-off game, as choosing both boxes is clearly a dominant strategy. However, it would be interesting as an iterated game where the player's goal is to maximize their total payout, while the machine's goal is to minimize it. Obviously the player's optimal strategy would depend on the strategy (programming) of the machine, and vice versa. In order to identify overall good strategies, one could perhaps come up with different strategies for both player and machine, pit them against each other over a certain number of rounds, and see which strategy is robust against a wide variety of adversary strategies, the way that Axelrod did for prisoner's dilemma strategies in the early 1980s.

For the second option, the machine's decision-making process needs to be clarified; what does it mean for the machine to be "almost always correct"? I'd implement this as follows: the machine has access to an oracle that will tell it the player's choice, but which, with probability $p > 0$, will give an incorrect answer. The machine always follows the oracle and puts \$1,000,000 into box B iff the oracle says the player did not choose both boxes. This scenario could then be analyzed using standard tools, and the player's optimal strategy determined for both one-off and iterated games, depending on the value of $p$.


Solution to the Bellman equation is a fixed point

I am by no means an expert on this, but maybe this helps. Here is a simple example for a bellman equation

$V(y) = \max_x u(x,y) + \beta V(y')$

$s.t. \, y' = f(x,y)$

This is a functional equation in an unknown function V. A solution to this problem is a function V that satisfies the equation above. If you look at the equation, it's pretty clear that the solution has to be a fixed point of the operator on the RHS of the bellman equation: if you take the correct V and an arbitrary y and calculate

$\max_x u(x,y) + \beta V(y')$

$s.t. \, y' = f(x,y)$

you will get $V(y)$. The operator that is the RHS of the Bellman equation operates on functions, and the solution is a fixed point in some space of functions.

It's a different question whether this fixed point exists and how to find it. Here, you appeal to the contraction mapping theorem: under typical assumptions on u and provided $\beta<1$, the maximization step above is a contraction mapping for any guess of V. This means that there exists a unique fixed point V, and you can find it by successive iteration.


Isn't it the case that figures on employment should equal the sum of employees plus the self-employed?

While you didn't share in which direction the equality is off, I would imagine that there is a double-counting issue. If someone is a self-employed person AND a wage earner (e.g. perhaps they have a salaried position but also do freelancing on the side as a registered independent contractor), this may lead to the sum of self-employed workers + wage earners to be higher than Total Employed. That's my thought.


Modeling job application process in labor market

Chapters 6-8 (especially 8) in Modern Labor Economics: Theory and Public Policy, a textbook by Ehrenberg and Smith, cover the topics concerning labor supply. I guess the parts in Ch. 8 that discuss hedonic wage theory would be most relevant to your inquiry.


How to choose between fixed and random effects using economic intuition?

Here is an example where just from an economic perspective fixed effects are better than random effects.
Suppose you have panel data and you want to regress earnings $y$ on some observable characteristics $X$ of an individual like education, tenure, experience, age, birthplace, etc. The regression you would estimate is
$$y_{it} = \alpha + X'_{it} \beta + \epsilon_{it}$$
where the error term $\epsilon_{it} = \alpha_i + \eta_{it}$, is a function of individual heterogeneity $\alpha_i$, which is not varying over time and some random shock $\eta_{it}$.
Pooled ordinary least squares and random effects assume that the observable characteristics and the individual heterogeneity component are uncorrelated, $Cov(\alpha_i,X_{it})=0$. As you know this does not hold when there is a correlation between your controls $X$ and the error term, which will bias your estimates - that's the standard omitted variables bias.
Does the assumption $Cov(\alpha_i,X_{it})=0$ hold in the earnings context?
In this context, your economic intuition will be useful. You may think of $\alpha_i$ as individual ability, which is unobserved by the econometrician but potentially correlated with some of the observed individual characteristics $X$, such as education or tenure. So, the $\alpha_i$ correlate with the regressors $X_{it}$, and the assumption $Cov(\alpha_i,X_{it})=0$, is violated. Then, a fixed effect approach, which effectively fits such intercepts will be more convincing.


Rent was one third of revenue. Revenue increases by 18%. Assuming this all goes to rent
$$
\frac{18}{100/3} = 54.
$$
Perhaps the 6 instead of 5 is a typo?


How does lowering a corporate tax rate benefit a nation?

I. The case for government revenue

The rationale behind this argument is that it it possible to increase the tax revenue by lowering the tax rate. A. Laffer draw a curve which represent this:

The idea is that if your tax rate is above $t^*$, then taxpayers try to evade taxes, develop the shadow economy, avoid taxes, or simply reduce the amount of work that they do because their is a lesser incentive to do more when the state takes a bigger share of the reward.

So the debate is therefore to know whether we are already beyond this optimal tax rate $t^*$ or below. Emmanuel Saez and Thomas Piketty argue that this optimal tax rate is about 70%. Others, such as Arthur Laffer, believe that it is much lower, and that the government revenue would actually increase if the tax rates were lower.

II. The case for growth

Lower tax rates can also be adopted in order to increase the attractiveness of the country to foreign investors, and hopefully develop the economic activity. Many countries have been known recently to struck tax deals in order to attract business: The UK with Google, Ireland with everybody, Luxembourg with Amazon and others, etc.

Similarly, developing countries tend to create special economic zones where foreign firms can develop their activities and build factories in return for lower taxation.


Is there any economic meaning to log returns?

Take the 1st order Taylor series the function $log(x)$. around one:
$$ log(x) \approx log(1) + \frac{x-1}{1} = x-1$$

Therefore, for a random variable r that is close to zero:
$$ log(1+r) \approx r$$

Log returns are typically something like $log(P_{new}/P_{old})$:
$$\Delta log(P_t) = log(P_{new}/P_{old}) = log((P_{old} + \Delta P)/P_{old}) = log(1 + \Delta P/P_{old}) \approx \Delta P/P_{old}$$
But $\Delta P/P_{old}$ is just the percent change in $P_{old}$. We've therefore shown that we can approximate the percent change in a variable $X$ with $\Delta log(X)$.

Why bother? Mostly because log differences can be summed but percent changes cannot be:

  Using logs, or summarizing changes in terms of continuous compounding,
  has a number of advantages over looking at simple percent changes. For
  example, if your portfolio goes up by 50% (say from \$100 to \$150) and
  then declines by 50% (say from \$150 to \$75), you’re not back where you
  started. If you calculate your average percentage return (in this
  case, 0%), that’s not a particularly useful summary of the fact that
  you actually ended up 25% below where you started. By contrast, if
  your portfolio goes up in logarithmic terms by 0.5, and then falls in
  logarithmic terms by 0.5, you are exactly back where you started. The
  average log return on your portfolio is exactly the same number as the
  change in log price between the time you bought it and the time you
  sold it, divided by the number of years that you held it.

Jame Hamilton: Use of logarithms in economics


Books and resources on how policies/regulations are made on exchanges like NYSE, NASDAQ, LSE, etc?

This document, published by the International Organization of Securities Commissions, states three objectives and describes thirty-eight principles of securities regulation:
https://www.iosco.org/library/pubdocs/pdf/IOSCOPD561.pdf
Objectives

Protect investors

ensuring that markets are fair, efficient and transparent;

reducing systemic risk.

The Securities and Exchange Commission (SEC) is the official federal regulator in the United States. Administrative law regulations are made in a context of procedures relating to the Federal Register and/or Code of Federal Regulations. This is a Guide to the Rule Making Process:
https://www.federalregister.gov/uploads/2011/01/the_rulemaking_process.pdf
North American Securities Administrators Association (NASAA) has much information about its efforts to solve problems under the objectives published on this site:
https://www.nasaa.org/
Rules established by government authority or customs with commercial authority are efforts to solve a particular social problem, procedures to solve those types of problems, and procedures to monitor behavior and enforce rules. Your research would be based on comparisons of these elements in unique case studies or contexts regardless of the information sources which should be available to the public.


Existing metric for personal productivity hours needed per life hour?

It seems to me this seminal paper has looked at the question

Gary Becker - A Theory of the Allocation of Time

You can use economic intuition to establish the value of one hour of work in terms of one hour of life. Suppose you have a consumer/worker that has to choose how much time to devote to work or leisure. His total daily hours allotment is

$24=h+\ell$

where $h$ is the number of hours worked and $\ell$ is the number of hours of leisure. He values leisure according to some utility function $u(\cdot)$ and his total utility also depends on the amount of goods $x$ consumed

$U(x,\ell)=x+u(\ell)$

he earns a hourly wage $w$. He spends all of his income in this single consumption good, which we assume has a nominal price of 1$. Therefore he has the following budget constraint

$x=wh$

Then, if the consumer/worker is choosing optimally how much to work the margin, it must be true that, on the margin, one hour of leisure has the same value as one hour of work

$\dfrac{\partial u}{\partial\ell}=w$

In other words, he must be indifferent on the margin between working
1 more minute or using it for leisure.

To know the value of every hour of leisure we need to know of course the function $u(\cdot)$. But at least on the margin, if you want to know what's the value of one unit of time not spent on work - there you have it: it's the hourly wage rate.

Becker suggested you could use that intuition to include leisure and household production in GDP. You should weight all hours not spent on work by their respective wage rate.

With regards to your initial question: how many hours of "life" do you gain for every hour of work? It should be 1, otherwise you're not choosing your leisure time optimally.

R.I.P. Gary Becker, Nobel Prize in Economics. I had the privilege of attending his last Price Theory course at UChicago before he died. He lectured on this.


There’s two prices for a bond:

Invoice or dirty price: what you actually pay; and
the clean price, which is the dirty price less accrued interest.

In market convention, the clean price is the quoted price.

Accrued interest starts off at \$0 at a coupon date, and then rises (roughly) linearly each day until it matches the value of a coupon at the coupon date. The exact formula is a fixed schedule that depends upon market convention.

If you buy a 1% annual coupon bond one day before maturity, you will probably have a clean price very close to \$100. The accrued interest will be very close to \$1, and so the dirty price would be close to \$101.

The deviation of the clean price from \$100 would depend on the deviation from 1% for the yield you bought it at. The exact accrued interest will depend on the daycount convention in the market.

In any event, people use the clean price precisely because there is a “pull to par,” we do not have to worry about the coupon when thinking about the bond price.


Proving consistency of quantile regression estimators

I can give you literature. An authoritative book on the subject is Koenker, R. (2005). Quantile regression. Cambridge university press.. In there we read (p.126)

  4.6 QUANTILE REGRESSION ASYMPTOTICS UNDER DEPENDENT CONDITIONS    Relaxation of the independence condition on the observations is
  obviously possible and offers considerable scope for continuing
  research. Bloomfield and Steiger (1983) showed the asymptotic
  normality of a median autoregression estimator for a model in which
  the observations were assumed to be stationary and ergodic martingale
  differences. Weiss (1991) considers models under α-mixing conditions.
  Portnoy (1991) considers a considerably more general class of models
  with “m-decomposible” errors, a condition that subsumes m-dependent
  cases with *m* tending to infinity sufficiently slowly... More
  recently, Koul and Mukherjee (1994) and Mukherjee (2000) have
  considered the asymptotic behavior of both the primal and dual
  quantile regression process under conditions of long-range dependence
  of the error process. This work relies heavily on the theory of
  weighted empirical processes developed by Koul (1992).

The section continues with a presentation of the properties of Quantile regression for AR, ARMA, and ARCH models.


What is the difference between a Start-Up and a small company which is not a Start-Up?

Probably, there's no clear-cut definition, however, there's an article of Paul Graham called Startup=Growth which represents the opinion many agree upon: startup is a company that plans to grow fast (... and scale). A quote:

  A startup is a company designed to grow fast.
      Being newly founded does not in itself make a company a startup. Nor is it necessary for a startup to work on technology, or take
  venture funding, or have some sort of "exit." The only essential thing
  is growth. The only essential thing is growth. Everything else we associate with startups follows from growth.

That's why a new barbershop isn't a startup since

  A barbershop doesn't scale.

This definition probably still leaves some space for interpretation, but not that much.


Why has China's integration into the global economy caused it to slow it's output?

Being integrated into the global economy means that country increases its trade openness and lifts restrictions on capital flows (you can look at this paper).
The article does not say that China's output slowed down. It says its rate of growth slowed down. Usually, a country that used to be closed and did not trade much before experiences a boost in the rate of economic growth. This is because trading allows the country to specialize in a good where the country has a comparative advantage. This boosts growth by a lot at the beginning but the effect eventually wears down as the country gets fully specialized and all low hanging fruit was already picked.


Is there a term for the indirect sponsorship of large corporates by the state?

In European Union's nomenclature this is called 'State aid' (see glossary on EU's legislation site for a quick summary).
The idea behind this term is to cover all direct and indirect aid, including economic policy. If the measure is selective and it distorts the competition, it constitutes 'incompatible State aid'.

Besides the EU, I think the word you are looking for is 'subsidy', which covers both direct and indirect aid, or more generally, 'aid measure'.

Hope this helps, if still relevant to you :)


At the end of the day, the firm seeks to maximize profits, which is a one-dimensional object. So the firm simply solves
\begin{equation}
\max_{\mathbf x}\;\mathbf{p}\cdot\mathbf f(\mathbf x)-c(\mathbf x)
\end{equation}
where $\mathbf x\in\mathbb R_+^\ell$ is a vector of $\ell$ inputs, $\mathbf p\in\mathbb R_+^n$ is a vector of $n$ prices for the final output, $\mathbf f:\mathbb R_+^\ell\to\mathbb R_+^n$ is the production function mapping $\ell$ inputs to $n$ outputs, and $c:\mathbb R_+^\ell\to \mathbb R_+$ is the cost function. This is a standard unconstrained optimization exercise that any first year grad student in economics should know how to solve. The standard textbooks --- MWG, Jehle and Reny, Simon and Blume --- have sections that cover the techniques.


Richard Musgrave (1957, 1959) invented the term. My interpretation of what he says (below) is that merit goods are simply goods that the "community" deems desirable, while demerit goods are those the "community" deems undesirable.

He (1987) writes that the concept (of merit goods) is "best applied where individual choice is restrained by community values" ... "Community values ... give rise to merit or demerit goods" ...

Such goods are where "community values" act "as a restraint on individual choice". The "evaluation of a good (its merit or demerit) derives not simply from the norm of consumer sovereignty but involves an alternative norm."

Examples of merit goods are "Concern for maintenance of historical sites,  respect for national holidays, regard for environment or for learning and the arts."
Examples of demerit goods are drug use and prostitution because they are "offences to human dignity (quite apart from potentially costly externalities)."

P.S. There is of course the difficult question of who or what exactly the "community" is. If I may add my humble opinion, all of this sounds very illiberal and even mildly fascist. Which is perhaps why this concept hasn't been much taken up by mainstream economics.

P.P.S. The above is merely Musgrave's interpretation. As Musgrave himself says:

  The concept of merit goods ... has been widely discussed and given divergent interpretations ... it is thus difficult to provide a unique definition.


How do you calculate the intrinsic value or fair value of real estate?

same as for a company, if you assume that you are buying it for the rent it will get you then the value of the asset is the sum of the discounted future cash flows (called FFO in RE)
you can also apply the comparables methods, in this field we usually apply it with a slight twist we read it in terms of yield (inverse of the multiple) which is called a cap rate and compare it to similar projects in similar areas for similar purposes.
this pdf is a good primer on all of this :
https://www.merrilledge.com/publish/content/application/pdf/gwmol/bofaml-reit-primer.pdf


A classic on public choice and taxation theory is

"The Power to Tax: Analytic Foundations of Fiscal Constitution" by Brennan and Buchanan

Really interesting book and relatively easy read.


Would Pilots and Air travel be considered complementary?

Consumers buy airline tickets but don't hire pilots, so I wouldn't call those two complementary in the usual sense of "complementary goods".


How can an economy pay for all the products it produces?

This is not really a riddle.
First, amount of money in economy is not constant. The amount of money economy has access to is typically growing thanks to policy of central bank/government. For example, you can see from Fred data that amount of money in the US measured by $M2$ increases over time (you can check similar statistics for almost any other country you want save for few that don’t publish it).
However, even if amount of money would be fixed that wouldn’t cause any issue. A price level in an economy is determined by the output and other factors so economy can’t really run out of money.
For example, following Mankiw Macroeconomics 8th Ed money market equilibrium can be described by:
$$MV=PY$$
Where $M$ is money stock, $V$ velocity of money (how often one banknote is used), $P$ is price level (aggregate prices) and $Y$ real output (this is only most simple representation of money market, in more complex models expectations of these quantities matter as well but for this explanation this simplified version will more than suffice).
Hence, if the amount of goods and services increases in the economy ($Y$ goes up), even if $M$ is fixed either $P$ can drop or $V$ can increase (or some combination of those two) to offset $Y$ and make sure that the amount of money in economy is sufficient to purchase or goods and services.
Consequently, economy will always be able to pay for what it produces. Heck, money is strictly speaking not even necessary - people could just barter with each other - money just helps transactions to run smoother.


Is loss in tax revenue a negative externality?

In my view yes, it would be considered an externality. An externality is said to occur when a production or consumption decision of an individual has an unintended consequence on the consumption or production of someone else.

The analysis of externalities usually does not include governments, considering only production and consumption, but one can think of the government as a firm that produces public goods like law and order and national defense.

In that case the decision of the individual to use online gambling rather than conventional gambling has an unintended effect on the production possibilities for the government. Now that the government has less revenues, it can produce less public goods.

When a person goes for online gambling rather than conventional gambling they do not intend to reduce the overall provision of public goods, they just went for online gambling because it was more convenient/cheaper etc.


...I'm confused if we can count the root of the contract curve (point M) as one of the efficient points since it's part of the contract curve?

The contract curve is, by definition, the set of all Pareto efficient allocations. Therefore, any point that you choose on this curve--even the end points--is also Pareto effecient.

  In this case the indifference curves (IC) in the box don't "touch each other", therefore is not a point of efficient trade (as if MRSa = MRSv).

In the example figure you included, they have only shown the pair of indifference curves for person A and person B at four points just to illustrate and to make the graph easy to read.  In reality, indifference curves can be found at any point (they're continuous), so there would be a pair of indifference curves that you could map on to the end point (point M) as well.

What would that look like?  From Person A's perspective, their indifference curve sits on point M and opens upwards. At this point, they have been allocated nothing, so they prefer the set of all allocations above their IC.

For Person B, their IC also sits on Point M and opens downward. Essentially, this is saying that have the entire allocation, and while they would prefer to have more, there is nothing available in the set of feasible allocations depicted in this Edgeworth Box (again, they have the entire allocation).

While this allocation may not be equitable (Person B has everything, Person A has nothing), from a Pareto perspective it is efficient in the sense that, you cannot move from this point (Point M), without making one of the people worse off--i.e. if you shift anywhere from point M, though it would make person A better off, it would make person B worse off.  So even here at the border, we can find a Pareto effecient allocation.

  Is my line of thinking correct? Can we also assume that every point in the contract curve are Pareto Optimal?

Correct.

  Is it also possible for the utility change from one to another (say from N to P)?

Not sure exactly what you mean here, but the second fundamental theorem of welfare economics says--broadly speaking--that you can get to any point on the contract curve in a competitive economy by changing the endowment allocations.


The net value of the two villages increases because value is in the eye of the beholder. If commodity S is more valuable to village A than to village B, and commodity T is more valuable to B than A, then, when they trade, both villages gained something of more value to them and the relative wealth of both villages increased.

Value comes from the valuator.

This is why--one of the reasons, anyway--Marxism was doomed. Value doesn't come from labor at all. Only cost comes from labor. For example, if I go stand in the middle of the road and labor all day pointing at the sky, I've worked hard but created no value, because no one else values my sky-pointing. Similarly, if I build two houses and put exactly the same amount of labor into both of them, but one is on a mountain with a beautiful view of the city, and the other is next to the city dump, the first will be far more valuable because people would value the location of the first, "envaluating" the house.


What's good about both exports and imports of a country rising at the same time?

There are plenty of issues that can be relevant here, which mean it is hard to give a precise answer. On the one hand, even though GDP is not growing, at least the balance of trade is roughly unchanged. Some countries could experience an equivalent unaffected change in GDP via a trade deficit (e.g. if part of the fall in net imports is compensated with, say, an increase in consumption), which can fuel a currency depreciation and create inflationary pressures, not only on imported goods but on locally produced good which use imported inputs.

On the other hand, it is interesting to compare two countries with the same GDP but different level of exports and imports. The country with greater trade levels is more likely engaged in global value chains, which spur specialisation, and learning for local producers. The literature recognises the role of trade as a mechanism to spur technology transfer (and even transfer of ideas), thereby fostering productivity and growth.


Multivariate cointegration methods in Econometrics

I think the main alternative to the Johansen (statistical) approach is the methodology propounded by Pesaran and Shin in their so-called long-run structural modelling (economic) approach. The main formal reference is Pesaran and Shin (2002). The methodology is also presented to a wider audience in Garratt et al. (2012).

Although referred to as the long-run structural modelling approach, you will also read about the type of models associated with the methodology, which are called VARX* models. The distinguishing feature is that the cointegrated models are estimated using reduced-rank regression and (over-)identifying restrictions are derived from economic a priori then tested. The sequencing is slightly different to Johansen since economic theory is given priority.

In the same school of thought is the cointegrated VAR approach associated with Juselius (2006). Again, Juselius propounds an economic approach to cointegration as opposed to the statistical approach by Johansen (and others like Phillips).

Pesaran's approach has been programmed and is available in the Microfit software.

Juselius' approach has also been programmed and is available in the RATS (CATS) software.

To my understanding, you can find MATLAB code in the GVAR toolbox, and this should give an indication of what's required for the Pesaran approach. In terms of programming, I haven't seen much difference when compared to Johansen (although I just quickly investigated this). Note that the GVAR approach can be thought of as an extension to the VARX approach (the difference lies in stacking individual country models, but estimation of the individual models is the same).

References

Pesaran, M and Shin, Yongcheol, (2002), LONG-RUN STRUCTURAL MODELLING, Econometric Reviews, 21, issue 1, p. 49-87.

Garratt, Anthony, Lee, Kevin, Pesaran, M and Shin, Yongcheol, (2012), Global and National Macroeconometric Modelling: A Long-Run Structural Approach, Oxford University Press.

Juselius, Katarina (2006) The Cointegrated VAR Model: Methodology and Applications (Advanced Texts in Econometrics).

GVAR Toolbox


Coefficient of Determination with Weighted Coefficients

If you want to explain the fraction of variance explained, use $R^2$. If you want to compare $R^2$'s but adjust (very weakly) for model size, use adjusted $R^2$'s. If you want to compare these models, use an $F$-test since Model 1 is nested in Model 2.


What is the reasoning behind the sharp fall of oil prices starting in mid 2014?

My understanding of the recent drop in oil price is due to recent changes in US drilling. They are now the worlds largest oil producer. This article by the economist does a good job explaining.

My answer (borrowed very heavily from the economist.)

Demand is low because of weak economic activity, increased efficiency, and a growing switch away from oil to other fuels.
America has become the world’s largest oil producer, creating a lot of spare supply.
The Saudis and their Gulf allies have decided not to sacrifice their own market share to restore the price. They could curb production sharply, but the main benefits would go to countries they detest such as Iran and Russia. Saudi Arabia can tolerate lower oil prices quite easily. It has \$900 billion in reserves. Its own oil costs very little (around $5-6 per barrel) to get out of the ground.

For further reading, the NY times has a decent article explaining it a little more.

edit:
I think it is safe to assume that OPEC has market power. They can get oil far cheaper than their competitors (as referenced above).

With that in mind,
The below reference provides a model which explains what we are experiencing here. Cournot competition seems a perfect model to use.

Naoum-Sawaya, J. and Elhedhli, S. (2010). Controlled predatory pricing in a multiperiod Stackelberg game: an MPEC approach. J Glob Optim, 50(2), pp.345-362.

A key quote from the above article: "After reducing or eliminating competition, the predatory firm
can raise its price and compensate the loss by earning monopoly profits in the long run".

This second provides a very brief overview of OPEC and their behaviour.

OPEC reflects on its history. Salisbury, Peter, Middle East Economic Digest.

I couldn't find an empirical study comparing a Stackleberg model to the OPEC cartel but I do not doubt that this has been done to death.


How to improve the underperforming construction productivity by correcting its market failures?

A great many things can be said on the question. But it might be worth pointing out that people have been building stuff for a very very long time, so maybe 1% productivity growth is not a bad result.

The most difficult is anti-corruption. You could compare the free press + independent judiciary + empowered investigative authorities within specified limits + free elections ... with the approach of President Xi, whose name is now entrenched in the Chinese constitution.

However, it seems that you're more interested to motivate a specific theory. So perhaps you could develop that and come back with a question that is more aligned to what you want to know. For example, you could be additionally descriptive about what it is about Tesla's operations that might have broader lessons in construction? (Note that Tesla makes cars, not ports, not bridges and not roads.)


Are opportunity costs synonymous with economic costs?

In economics, we argue that there is only one correct way to think of costs. And so, the adjectives economic or opportunity in front of the word cost are actually superfluous — the terms cost, true cost, economic cost, and opportunity cost are all exact synonyms.

Note. If they are superfluous, then why do we use these adjectives?

One reason is to strongly emphasize that other conceptions of "cost" that accountants and other non-economists use (e.g. accounting costs, historic costs) are not quite correct.

Another reason is that in practice, it may be difficult to figure out what the true cost or opportunity cost is. And so in practice, economists will — despite what they themselves teach in Econ 101 — often use the term cost in some narrower, incorrect sense.


What made The Wörgl Experiment successful?

It was successful because it combined three things at the same time.

Firstly, it increased available currency at a time when currency was scarce for any but the rich - so everyone was more solvent.

Secondly, it had a negative interest rate, so the encouragement was to spend quickly.

And thirdly, it was a local initiative, and belonged to the town, at a time that was particularly introspective - isolationist, even.


After doing some reading on the topic, it appears we would estimate this equation as we would with any other demand estimation.

the reason why the term $b(p)m$ is written this way is due to the assumption of homothetic preferences (or homogeneous of degree 1, this dosen't matter in demand analysis). Since Gorman polar form is a homothetic form this implies that:

$$v(p,m)=a_i(p)+b(p)m \equiv a_i(p)+b(p,m)$$

more specifically $$b(p,m) \Rightarrow b(p)m$$

if preferences are homothetic or homogeneous of degree 1.


Can money flow from Firms into the Financial Market in the Circular Flow?

Yes! I put together this sketch to show it

For example

Households can save money in banks, which inject money into the financial market
They are later compensated the form of a rate of return
Firms can borrow money
Later they need to pay an interest. This way the money flows from the firms back to the financial market


Income/Wealth Inequality; Why does it matter?

There could be some complicated models where high inequality could have effect on efficiency - the research is inconclusive (or to be more precise almost non existent) on the matter.

However, in Public Economics, depending on the selected welfare function (ralwsian, utilitarian, charitable conservative/libertarian etc.), inequality matters for optimal taxation. For example, even in competitive market without any imperfections someone’s marginal product of work and hence wage might be close to zero or not enough to afford food. From pure efficiency perspective that would be optimal, but under Ralwsian/utilitarian or also some other welfare function there should be some transfers to said person.

This being said economics does not provide any answers into which welfare function is the right one for that you need to study philosophy. Inequality as far as currently understood is a matter of taste.


In the theory of production (and similarly for consumption), a homothetic production function is compatible with the occurrence of fixed costs, while a homogeneous production function is not.
In both cases (in standard notations), the production can be written as $$y=F(h(x)),$$ with $h$ linearly homogeneous. It is also common to impose $h(0)=0$. Then, the cost function inherits the form $$c(w,y)=g(w)F^{-1}(y).$$
When the production function is homogeneous of degree $k$ it turns out that $F^{-1}(y)=y^{1/k}$ and so $c(w,0)=0$. For homothetic production functions however, $c(w,0)=g(w)F^{-1}(0) > 0$ in general.
A further advantage of the homothetic case, is that the degree of returns to scale can depend on $y$, while it is constant for the (degree $k$-) homogeneous technology.


Why are so many pharmaceutical drugs so expensive?

Prices are high because drug firms have monopoly power, granted to them by the patent system. In addition the demand for the drugs is fairly inelastic because once you fall seriously ill you're often willing to pay a high price for the cure. If your insurance covers it, you won't even directly notice the price. So the companies can charge almost any price.

Now the patent system is there for a good reason: developing drugs is expensive and if not patented others may use your hard-obtained knowledge to their advantage, without paying for it. Therefore, if your patent is granted you get monopoly power for a while to recapture your investment, on the condition that when the patent expires the knowledge can be used by anyone.

There is no doubt that developing drugs is expensive but whether it justifies the prices charged is a different matter. The pharmaceutical industry is generally unwilling to reveal its costs for good reasons, and that makes estimating profit margins hard.

Part of the question is normative anyway because what would we consider "excessive" profit? What costs are justified when it comes to someone's health?

A fairly old article in the J of Economic Perspectives on this interplay can be found here:
https://www.aeaweb.org/articles?id=10.1257/jep.7.3.97

As for tags: microeconomics would certainly be better suited as would monopoly


Do tax cuts to the rich lower interest rates?

You may want to look at the literature of the effects of taxation on savings. A survey is given in: Sandmo. A. (1985) The effects of taxation on savings and risk-taking, in: Auerbach & Feldstein, Handbook of Public Economics, Elsevier.

In this survey, the (simpler) problem of the overall effect of taxation on saving behavior is analyzed. A change in tax progression is much more subtle and harder to measure.

The survey states the problems associated with estimation of this simpler problem and the author comes to the conclusion that the empirical results are inconclusive. Since then there may have been some move in the literature, however.


Microeconomics: Question on Market for Factors

Nominal means measured in present day money. E.g. someone in South Africa may have a monthly salary of 100,000 South African Rands. You have no idea how much this is unless you are told what real goods can be purchased from this. If a TV costs 10,000 Rands then the real wage of this person would be 10 TVs. Now you see the difference between nominal wages and real wages.

If you reread your question focusing on this part: "The Wage(in terms of good we produce)" I hope all your confusion will go away.


The concept of information structure in incomplete information games

What you are defining is the complete information structure $\bar{S}$ with $\bar{T}_{i}=\Theta$ for all $i$ and  $$ \bar{\pi}(t|\theta) = \begin{cases} \ 1, & \text{if} \;\; t_{i}= \theta \text{ for all} \; i \\ 0, & \text{otherwise}, \end{cases} $$ for all $\theta \in \Theta$.

Note that this is not the same as the case in which $\Theta$ is a singleton, but it is rather an extreme information structure. The same authors define this here (Bayes Correlated Equilibrium and the Comparison of Information Structures - Bergemann & Morris), it beats me why they didn't include this detail in the paper you are looking at.


How can a country develop the skills of its population and have them stay?

The brain drain is a difficult issue facing many countries around the world. Simply, they need something to come back to. Laborers generally leave countries because there is insufficient economic activity for them domestically. If a country wants to retain workers, they need to build up educational infrastructure, encourage foreign direct investment, and open up border to free trade. However, building domestic economic infrastructure requires skilled labor - the very laborers who generally leave for foreign markets. Its a bit of a catch-22.


What actually went wrong in the economy as described by "The Big Short"?

The problem at the bottom of the 2008 crash is that the values of multiple instruments were dramatically over estimated because the risk of default in the various underlyings was dramatically underestimated since it was assumed that they were fully backed by the US government.

Although many commentaries focus on the securitization of debt that was not where the real problem lay, in fact the securitization of sub-prime mortgages into CDOs (collateralized debt obligations) covered by CDSs (credit default swaps - a type of default insurance) was as a result of firms wanting to divest themselves of risks surrounding unknown quantities.

In the (first?) Clinton and Bush administrations there was a political will (rightly or wrongly; it isn't important) to help people buy their own homes. This applied particularly to low income families who previously had not been able to buy, in many cases for generations. To do this bills were passed to offer subprime loans with the implicit backing of the US government. Since these people had never been mortgage borrowers before their ability to repay was not well understood or modelled and came with an assumption that they would continue to have jobs and that their homes would be worth at least what they paid for them. These extra buyers put upward pressure on the housing market (i.e. buyers exceeded sellers so prices rose) meaning that it looked like that latter assumption would prove true at least.

The mortgage companies then had a large tranches of debt on their books that, although apparently backed by Sam, were riskier than their previously core business. To mitigate this risk they rolled them up into CDOs which, given that the probability of any one loan defaulting was taken to be independent of any and all others defaulting, supposedly held lower risk than the individual loans. To further reduce default risk these were further rolled up with high credit rating corporate, municipal, and national bonds. In case there was a default in these wrappered securities insurance companies, such as AIG, wrote CDSs that would pay out a sum to cover losses in return for premia (as is usual with insurance).

Then the economy slowed down a little, house prices started to stall, and unemployment rose. As people who were the borrowers in these subprime mortgages had wages cut and lost jobs they began to default on loans and the value of the collateral (the house) fell, sometimes to less than the original value at purchase, as the bubble burst. The assumption that defaults were uncorrelated turned out to be unfounded and, as you can probably guess, that triggered CDS payouts and huge losses for many of the companies holding the CDOs and writing the CDSs. Drinking and strip clubs were basically what the borrowers preferred to spend their money on instead of their mortgages. Slightly less glibly the value of their house counted as an asset to them against which they could take out further debts (credit cards, personal loans etc.) so that they could afford to live outside their means under the assumption that they would pay it back using future months' salaries. When they lost their jobs they could no longer do so. Remember from earlier that these were people who had never been offered credit before and even professionals make mistakes in their finances when it comes to credit and loans so they were doubly cursed by a lack of understanding of credit and a lack of experience.

A major issue in all of this was hubris and people's belief that current trends would never end; the economy had been growing for the longest time in recorded history therefore it would continue to and everyone would always be employed and able to pay their bills. Add to this that the managers of the funds etc. that held these debt based derivatives believed that their models of the risks of holding them, underpinned by the fallacious assumption of the lack correlation of defaults, were accurate and would model potential profits and losses, and that, in any case, they had insurance if anything bad happened and you have the problem.

These were massively leveraged positions in little understood instruments that wrapped around very risky debts for which there were no good models of repayment and default rates, and those models that did exist did so under the fatefully incorrect assumption that defaults would be totally independent.


The position is both worse and better than you describe

If you look at Section 7 and Figure 6 of  https://www.ons.gov.uk/economy/nationalaccounts/balanceofpayments/bulletins/balanceofpayments/quarter1jantomar2018 you will find that UK liabilities (public and private) abroad were estimated to be £10,894.5 billion at the end of March 2018, about 5.30 times UK GDP in the 12 months to March

But at the same time UK assets abroad were estimated to be £10,632.2 billion, about 5.18 times UK GDP

So the net position here was a liability of £262.3 billion, about 0.13 times UK GDP (a minor rounding issue here), i.e. 13% of annual GDP.  In fact the net position is the difference between two enormous numbers largely resulting from financial activity in the City of London, both of which are estimated with errors, so large revisions to the estimated net position is common

A small UK net international liability represents the position for many years and so in that sense empirically seems to have been sustainable.  For Greece this net figure became large and unsustainable, contributing to its crisis, and not helped by the Eurozone restricting the possible solutions.  The answer to your question "When Does a Country's External Debt Become a Problem?" seems to be "when people start to worry that they will not be able to get their money back and so stop lending more money".  Turkey may be facing this issue at the moment.  But I doubt there is a fixed figure you can identify


Negative probabilities - Can we have negative payments in bonds?

I heard in a lecture at my university, that measure theory is applied in finance. This field also operates with negative probabilities a lot.

Anyways, I don't see the practical intuition behind using negative probabilities in a bond-pricing model. You can check this article by Burgin and Meissner: Negative Probabilities in Financial Modeling (Wilmott Magazine March 2012).

I think you should migrate your question to Quant SE.


Preference relations defined by $x_1^n + x_2^n$ converge to $\max\{x_1, x_2\}$

I am assuming you're familiar with the concept of monotonic transformations.

Let $f(z)=z^n$ be a monotonic transformation. $f(z)$ is an order preserving transformation for all positive $n$. Let $v:\mathbb{R_+^2} \to\mathbb{R}$ be an utility function, where $v(x_1,x_2)=(x_1^n + x_2^n)^{1/n}$. The function $f(v(x_1,x_2))=((x_1^n + x_2^n)^{1/n})^n=x_1^n + x_2^n$ will represent the same preferences as $v(x_1,x_2)$. Therefore, $(x_1^n + x_2^n)^{1/n}$ should also converge to $\max\{x_1,x_2\}$.

To prove the same, consider $v(x_1,x_2)=\Big(1+\cfrac{x_1^n }{x_2^n}\Big)^{1/n}x_2$. Assume $x_2 = \max\{x_1,x_2\}. $Now, as $n \to \infty$, two cases arise.

1) If $x_2>x_1$, then $\cfrac{x_1^n }{x_2^n} \to 0$ as $n \to \infty$. In that case, $v(x_1,x_2)=x_2=\max\{x_1,x_2\}$.

2) If $x_2=x_1$, the limit can be calculated by taking log on both sides as follows-
\begin{align*}
\ln(v(x_1,x_2)) = \cfrac{\ln(1+\cfrac{x_1^n }{x_2^n})}{n} +\ln(x_2) \,\,\,\,-(1)
\end{align*}
Since $x_1/x_2=1$,we have $(x_1/x_2)^n=1^n$. Using this in equation $(1)$, we get
\begin{align*}
\ln(v(x_1,x_2)) = \cfrac{\ln(1+1^n)}{n} +\ln(x_2)
\end{align*}
As $n\to \infty$, the term $\cfrac{\ln(1+1^n)}{n}$ becomes $\infty/\infty$. This limit can be solved using L'Hospital's rule. The value of the limit(you may check it for yourself) comes out to be $0$.
Now we have $\ln(v(x_1,x_2)) = 0 +\ln(x_2)$, or, $v(x_1,x_2)=x_2=\max\{x_1,x_2\}$.


Double counting because of gross mixed income

According to the scenario Mr. X is an employee of XYZ and not the owner of the business (sole proprietorship).  So his $50 dollars won’t get count towards his own earnings.  It should get counted toward Gross Operating Surplus.

If he was the owner of the business then it still wouldn’t be counted under compensation of employee for Mr. X because at that point he would not be an employee.  It would be counted toward Dr. Y's earnings.


What are the fundamental theorems of welfare economics?

I presume you meant The Fundamental Theorems:

The first fundamental theorem of welfare economics states that any Walrasian equilibrium is Pareto-efficient.
The second fundamental theorem of welfare economics states that, under the assumptions that every production set $Y_j$ is convex and every preference relation $\geq _i$ is convex and locally nonsatiated, any desired Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers. Further assumptions are needed to prove this statement for price equilibriums with transfers.

Or you could read Stiglitz's Paper yourself if you were so inclined.


Examples of firms having private information about negative event

Banks that were not bailed out in the 2008 recession and went insolvent i.e. Washington Mutual, IndyMac, Franklin Bank, First National Bank of Nevada e.t.c. You could also look into firms that were sanctioned by the Securities and Exchange Commission (SEC). I do not know the names of the top of my head but a simple google search should suffice.


Research Design: Indifference curves and budget lines

In order to derive a preference indifference curve between different quantities of two goods, one possible questioning protocol would consist in:

First choosing any arbitrary combination of quantities for the bundle of goods (e.g. 20 units of A and 5 units of B);
Then choosing any arbitrary quantity of one of the goods (e.g. 15 units of good A);
Asking the consumer whether he (or she) is willing to exchange the first bundle of goods (20 of A and 5 of B) by a bundle of goods comprising 15 units of A (or any other quantity as specified in the previous paragraph) and 10 units of B (or any other quantity provided that is greater than 5);
If the consumer chooses the first bundle then the previous question should be asked again provided that that the previous 10 units of B be replaced by a greater quantity (e.g. 15 units of B); otherwise, if the consumer chooses the second bundle then the question of the previous paragraph should be asked again provided that that the previous 10 units of B be replaced by a smaller quantity (e.g. 8 units of B);
When the consumer considers that he (or she) is indifferent between two bundles, it means that the two bundles are indifferent and therefore the corresponding points should lie over the same indifference curve.
Following the previous procedure for a significant set of other combinations of quantities (bundles) would allow to draw the corresponding indifference curve(s).


Showing that the elasticity of arrival rate for workers wrt $\theta$ is between $0$ and $1$

Let $m$ cobb-douglas. This means $m(u,v) = A u^\alpha v^{1-\alpha}$ for $\alpha < 1$ and $q(\theta) = A \theta^{-\alpha}$.

Then

$$\frac{\theta q'(\theta)}{q(\theta)} = -\alpha > -1$$

To what extent this holds for other functional forms I'm not sure, but I haven't seen anything but Cobb-Douglas being used as the matching function.


Uniform bounds on rate of merging for Bayesian learners

This paper by Acemoglu, Chernozhukov and Yildiz (2016) and the references therein may be of interest.

The results they derive are in a much more limited environment, but I think they still gesture toward the direction you're looking. Otherwise, their literature review should also prove useful.


A question on the accuracy of economic forecasts

Regarding Forecasts in General

Economic forecasts as any forecasts can be sometimes wrong, however that does not mean they are always wrong and many times people just don’t really understand how to properly evaluate if forecast was wrong. For example if mean forecast of GDP growth is 4% but 95% confidence interval is 2%-6% you cannot say forecast is wrong unless it’s outside the bounds and unless further forecasts from the model give wrong predictions significantly more than 5% of the time. Normally in literature to evaluate forecasts people compute some forecast accuracy metric like root mean squared error (or even more fancier ones) which check how the actual forecasts performed against real observations and while still imperfect many forecasting models are good enough so that you can reasonably take them in account when making some (policy) decisions.

Furthermore, also many people confuse forecasts with predictions. For example, even if we wouldn’t be able to forecast how UK GDP changes after brexit we can make predictions based on our understanding of the economy. For example, based of our understanding of international trade a country seceding from a trade arrangement and not replacing it with a better one will experience some negative shock - we need forecasting to estimate the size of the shock, but just to predict there will be a shock requires just knowledge of general economic models and their implications.

Regarding Brexit Forecasts

Forecasting effect of something like brexit is very difficult because effects of brexit will depend on many contingencies like UK’s post brexit trade arrangements. If post brexit UK will become some beacon of free trade and somehow would manage to get the same access to EU market as it had before then the impact would be of course completely different than if UK will become some protectionist island.

Hence in case of brexit you get different forecasts depending on what assumptions you are willing to make about UK post brexit trade arrangements as well as its macroeconomic and monetary policy which can try to accommodate brexit shocks more or less successfully.

Because of that, adding to the uncertainty, most serious organizations or scholars making forecasts for brexit are working with different scenarios. For example, OECD which is well respected and non-partisan world organization estimated that depending on various scenarios brexit would in terms of GDP cost UK about 3% in short term and in long term between 3-7% (see here) However, those are just mean estimates taking some baseline scenarios. For a nice overview of brexit predictions by different organizations as well as further explanation why it’s hard to forecast effects of brexit you can look at this article from the economist.

However, the problem is that you can’t really evaluate how accurate the OECD or other organizations’ brexit forecasts are in advance. To really properly evaluate their accuracy we have to wait for brexit materializing and then making post mortem analysis because it’s a one-off event. A developed country seceding from economic union is totally unprecedented in modern history.

It would be possible to provide some measures of forecasting accuracy of the parts of brexit models in their previous non-brexit use but that would very laborious task. For example, the forecasts impact of brexit on UK GDP also depends on how brexit will affect UK trade openness which according to  estimation of Fournier et al. (2015) could be between 10-20%, then you have separate forecasts what would happen to labor supply etc. As you can imagine doing survey on forecasting accuracy of all these partial components would be very daunting task  (hence the reason why I did not do it here even though it was part of your question). If you really want to go that deep into weeds then you can just check all the sources for those individual models cited in the reports and then try to find on google scholar some evaluation or comparison of those models in different situations. Or if you are good with R or Python you can just check some basic forecasting evaluation criteria and then laboriously estimate all models and check their out of sample performance compared to some benchmark like random walk.


Strategic form: mixed strategy nash equilibria?

The probability vectors you gave in red describe pure strategy NE. (They assign probability one to an action.)

$\tau$ shows the probability that the column player plays action $l_2$.
If $\tau = 1$, then he will play $l_2$ with probability one. This is essentially another way of saying that he plays the pure strategy $l_2$.

One way to get these probability vectors is to find the PSNE as you have done. The PSNE $(u_2,l_2)$ means that the Row player plays $u_2$ with probability one, and the Column player plays $l_2$ with probability one.
According to your notation this means that $(\tau, \sigma) = (1,1)$.


Capital/Income Ratio in Pikketty's Capital

First, important thing to note here is that $K/Y$ is not the capital income to labor income ratio, but ratio of capital to total income (or output which is macro-economically equivalent to income). $K$ is not an income derived from capital it is the stock of capital.

Wars destroy the stock of capital so even though returns to capital $r$ indeed increase the fall in the stock of capital makes the capital-income ratio fall.

Furthermore, the share of income going to capital is according to Piketty given by $\alpha = r \frac{K}{Y}$, and hence even if the rate of return increases when capital becomes more scarce the actual share of income going to capital might decrease if the capital-income ratio falls sufficiently to offset an increase in $r$.


What is this economic theory? Cost of investment vs production accounting for time?

Sorry I don't have enough reputation yet in the Economics community to comment. But the answer provided by @Alecos Papadopoulos can be illustrated further. Please feel free to move my "answer" to comments of his answer.

With the assumption of only $F' > 0$ and $F > 0$ and $h' > 0$, then there is a simple case where the minimization problem has no solution. Consider when $F(q) = \sqrt{q}$ and $h(x) = x$. Then the objective function becomes $1 / \sqrt{q} - \sqrt{q} - c$, and if you're considering for any $q > 0$, then there is no global minimizer for this (i.e. globally decreasing for all $q > 0$ and hence the minimizer is $q \to \infty$. Thus, the FOC you'd derived is not true. Indeed, the condition you'd provided would render as $4 \le 1$ --- clearly impossible.

In general, an increasing function $F$ over a linear function (i.e. say $F(x) / x$ in your case) will not necessarily be convex. And furthermore, when you subtract off the form $h \circ F$, the resulting $-h \circ F$ could again depend on the form of $F$ (which may not have any particular geometric form you could exploit). I think what would fix this model is a form such that $F(q) / q$ is convex in $q$ and $h \circ F$ is concave, so that $-h \circ F$ is convex, then the whole problem will be convex.


Your conjecture seems to be contradicted, at least for small values of $\sigma$.
You can draw the function with the following R-code:

qq_f = function(x,k,h,sig){
-pnorm(-k, sd=sig)*( (dnorm(h*(1-x), sd=sig))^2 ) - 0.5*dnorm(-k^2, sd=sig)*( 2*pnorm(h*(1-x), sd=sig) -1 )^2
}
curve(qq_f(x,k=0,h=1,sig=0.5),col='blue',xlim=c(-1,3),type='l',main="A candidate quasi-concave function")


Dif in Dif on countries, is it necessary to cluster by country?

You should cluster on level where you believe the autocorrelation or heteroskedasticity occurs.

If you don’t have enough clusters at a given level, for example, if you believe the above mentioned problems occur on country level but you have only 3 countries, you are correct you should not be using clustered standard errors which by rule of thumb require at least about 40 clusters to be properly estimated. However, clustered errors are not the only type of error adjustment there is. For example, you can instead of clustered errors use Newey-West errors or bootstrapped errors (of course these have their own advantages and disadvantages as well). The above mentioned are just examples other ways of adjusting errors exist as well.


Why is an inverted yield curve NOT a good predictor for recessions except in the US?

I’m currently researching this topic, and I have not found any single reference that answers this (... besides my work). I will summarise what I have seen.

I would point out that from a research perspective, having the yield curve work is interesting, while a failure to predict recessions is not interesting. Very few journals publish articles on failed indicators, since the number of those is very large.

If we ignore term premia, an inverted curve normally indicates that the bond market is pricing a cut in interest rates. The way central banks react is that they normally cut rates when recessions hit. As such, we would typically expect inversions ahead of recessions. However, the logic can break down.

Bond market participants are wrong about the forecast.
The central bank cuts rates, but no recession. (U.S., 1998).
Proximity to 0% makes inversion difficult (Japan, euro area).
Central bank does not cut rates in response to recession (possible for euro periphery).
“Technical factors” in bond market lead to flat/inverted curves (U.K., Australia.) For the U.K., pensions were forced to match the liability of duration, and there was not enough supply of long-dated bonds. For Australia, the government had surpluses, and I believe that duration supply was extremely low. (Would need access to paid bond index data to validate that claim.)

The issue is therefore not U.S. versus non-U.S., it’s just that the U.S. managed to avoid most of these factors (except in 1998 in the recent history).


Understanding what's in the frequently reported and discussed high demand in IT specialists in Germany - is the IT Fachkräftemangel real?

is the IT Fachkräftemangel real?

It is not. Reports fail to filter out the abundance of fake job posts with which intermediaries saturate --and obfuscate-- the labor market. This is the actual reason why intermediaries hardly ever give candidates any concrete information about the alleged job positions. Consequently, reports headlining a "shortage of skilled professionals" are meaningless.

To a great extent this also happens in the U.S., Canada, and perhaps other regions. But the fact that one same position is advertised by several intermediaries (very often their list of requirements being the same, verbatim) obviously does not mean that there are several open positions.

The broad unawareness regarding this issue allows for air castles when it comes to discussing the pseudo "shortage" of labor supply in labor markets.