What are the recent advancements in building a unified theory of bounded rationality?

The term bounded rationality was introduced by Herbert Simon. He wrote

  "The term, bounded rationality, is used to designate rational choice
  that takes into account the cognitive limitations of both knowledge
  and cognitive capacity. Bounded rationality is a central theme in
  behavioral economics. It is concerned with the ways in which the
  actual decision-making process influences decisions."

This passage seems to focus only on incomplete information, limited information processing time, limited knowledge -and not on any "psychological biases" like framing effects and the like. But the passage is complemented by a last phrase

  "Theories of bounded rationality relax one or more assumptions of
  standard expected utility theory"

which suddenly opens the content of the term to anything. So "bounded rationality" has come to signify "deviations from strict rationality in any way", making it rather impossible to arrive at a general, all-encompassing, or even most-encompassing, modeling approach, that at the same time, will be specific enough so as to be operational.


Does constant market cap imply reversion to the mean

Identically:
$$ Market\_Cap \equiv Number\_of\_Shares \cdot Price\_per\_Share$$
If a market cap is constant than stock prices only fluctuate because of changes in the number of shares. Therefore, this would imply mean reversion in share prices only if the fluctuations in the number of shares were also mean reverting.

As a general matter in the United States over the last hundred or so years, the average price per share has mostly been \$20-\$40 while market capitalization has increased substantially, implying a large increase in the number of shares. This has mostly been achieved by stock splits, stock based acquisitions, and secondary equity offerings.


Why is the USA not capable to pay its debt?

The US is in a privileged position of producing a currency that people in other countries want to use (as reserve). There's even a French name for that. Anyway, a result of that is cheap credit for the US government:

Many believe that this dollar dominance has allowed the United States to live beyond its means, running sizable current account deficits financed by borrowing from the rest of the world at cheap interest rates. Some other countries have chafed at this “exorbitant privilege” enjoyed by the United States. [...]
There are tangible and intangible benefits to a country whose currency serves as a reserve currency. In addition to the prestige conferred by this status, it also means access to cheap financing in the country’s domestic currency and the benefit of seigniorage revenue—the difference between the purchasing power of money and the cost of producing it—which can be extracted from both domestic and foreign holders of the currency.

In more mundane terms, the Chinese (and others) don't just hold on to their reserve dollars, but they then lend them back to the US gov't.
So the incentive to not be indebted is less than elsewhere.
In particular, the classical idea is that:

Economists say the US need not ever pay its debts back entirely, so long as the economy keeps growing enough to cover the payments.

But there are even more extreme ideas like the MMT, which says the US can pay even less than that.
Of course, you could ask, why don't the Chinese (and others) do something else with their dollars. The catch is that there's nothing better on the horizon, in a certain (risk averse) sense:

Moreover, there are no alternative currencies or investments that provide a similar degree of safety and liquidity in the quantities demanded by investors.
The reason the United States appears so special in global finance is not just because of the size of its economy, but also because of its institutions—democratic government, public institutions, financial markets, and legal framework—which, for all their flaws, still set the standard for the world. For instance, despite the Federal Reserve’s aggressive and protracted use of unconventional monetary policies, investors worldwide still seem to trust that the Fed will not allow inflation to get out of hand and diminish the value of the dollar.

(That was written in 2014, quantitative easing is even less controversial now.)

As for "can not": it is a matter of time frame and it is relative to other objectives. Here are some silly ways to pay the debt (quickly):

print money; it will causes substantial inflation, loss of confidence etc., and not really solve the problem because a lot of costs the government faces will rise with inflation.

mega-austerity, i.e. cut suddenly spending, a lot. The a-word is nearly unheard of in the US, unless it's news about Europe. Largely unacceptable politically. Also a shrinking economy means less tax revenue. Classical example nowadays:

Under the auspices of ‘bailouts’ from the IMF and the EU, Greece cut more than 20 per cent of GDP in spending. It lost nearly 30 per cent in final consumption.

There's of course debate whether a little austerity will do that; it depends what you'd cut.
But here's an example what Greece cut (and didn't):

The International Monetary Fund (IMF), which joined the rescue of Greece in 2010 alongside the EU and the European Central Bank (ECB), never asked for defence cuts, but insisted on reductions to wages. So while the defence equipment budget remained intact, soldiers suffered a near 40% drop in salaries.

And that's actually a little misleading too, because (except for defense) investment tanked as well, in fact more than the GDP did, to some 40% of the pre-crisis value.

Someone with better macroeconomics knowledge may be able to quickly estimate what paying the whole US debt at one would do. But since serious sources dismiss the idea off-hand, I can't find a published calculation.
There's no quick fix for the debt that won't impact some other long-term goal.


Universal Basic Income: Indexation and Inflation

Your real value column is wrong, but only slightly. You are decreasing it by 2% per year, but inflation of 2% doesn't cause real value to decrease by quite 2%. If there is 2% inflation, real value doesn't decrease by 2%; it decreases by 1 - 1/1.02 which is approximately 1.961% .

This is just the way percentages work. Since they are multiplicative, the inverse is different in magnitude:

If you lose 80% of an investment you have to gain 400% to break even.
If you lose 50% of an investment you have to gain 100% to break even.
If you lose 20% of an investment you have to gain 25% to break even.
If you lose about 1.961% of an investment you have to gain 2% to break even.

If you change your real value formula to =F6/(1+$F$4) you will get the correct values. You can also calculate the real value column by just calculating what portion of the buying power is left (1000/H26) and multiplying it by the nominal value: =1000*1000/H26. Both methods will give you the same result.


Is manufacturing automation really a bad thing for the economy?

As an answer let me offer a paradox that goes in your direction and is related to automation and the future of jobs:

  Despite a century of creating machines to do our work for us, the
  proportion of adults in the US with a job has consistently gone up for
  the past 125 years. Why hasn't human labor become redundant and our
  skills obsolete? In this talk about the future of work, economist
  David Autor addresses the question of why there are still so many jobs
  and comes up with a surprising, hopeful answer.

Here is David Autor's TEDx Why are there still so many jobs? and I will have added [despite automation]

Autor's answer is that technology magnifies the importance of our expertise and creativity and since we never get enough the wealth and the time saved by automation boost our endless inventiveness. So, automation will NOT replace EVERY job by robots. There's always new work to do.

The challenge that this automation phenomenon creates, what economists call job polarization, is that it knocks out rungs in the economic ladder, and shrinks the size of the middle class. The share of employment in occupations in the middle of the skill distribution has declined rapidly in the US and Europe. At the same time the share of employment at the upper and lower ends of the occupational skill distribution has increased substantially. A compelling explanation of this job polarization is that middle-skilled manufacturing and clerical occupations are characterized by a high intensity of procedural, rule-based activities which they call "routine tasks", which can relatively easily be coded into computer programs (see Autor, Levy, and Murnane, QJE, 2003).

Update

Daron Acemoglu and Pascual Restrepo have new working papers tackling this important question.

Robots and Jobs: Evidence from US Labor Markets
Automation and New Tasks: The Implications of the Task Content of
Technology for Labor Demand
Demographics and Automation

Abstract of Automation and New Tasks: The Implications of the Task Content of
   Technology for Labor Demand  [emphasis mine]

  We present a framework for understanding the effects of automation and
  other types of technological changes on labor demand, and use it for
  interpreting changes in US employment over the recent past. Automation
  enables capital to replace labor in tasks it was previously engaged
  in. Because of the displacement effect it generates, automation is
  qualitatively different from factor-augmenting  technological changes;
  it always reduces the labor share in value added (of an industry or
  economy) and may also reduce employment and wages even as it raises
  productivity. The effects of automation are counterbalanced by the
  creation of new tasks in which labor has a comparative advantage,
  which generates a reinstatement effect raising the labor share and
  labor demand by expanding the set of tasks allocated to labor. We show
  how the role of changes in the task content of production—due to
  automation and new tasks—can be inferred from industry- level data.
  Our empirical exercise suggests that the slower growth of employment
  over the last three decades is accounted for by an acceleration in the
  displacement effect, especially in manufacturing, a weaker
  reinstatement effect, and slower growth of productivity than in
  previous decades.


Are prices part of total factor productivity?

No actually by default when you use Cobb-Douglas function  the output is not even measured in monetary units but rather as output per unit of time.  This output per time can be still called income without assigning it any 'monetary' value. For example, in Robinson Crusoe economy if you catch 5 fish then those 5 fish are your income from economic perspective.
Moreover, let us suppose we measure $Y$ by GDP. In that case for calculation of TFP we will always use real GDP not nominal GDP. This is done precisely to get rid of any influences change in price level may have on the measurement.  Even though nominal GDP is calculated involving prices and we can express nominal GDP as $YP$ (where $P$ is the aggregate price level) when we calculate real GDP we deflate the nominal GDP by dividing it with the aggregate price level. Hence real GDP is $\frac{YP}{P}=Y$.
The above being said when it comes to TFP it is incredibly hard to do dimensional analysis (see this working paper) and generally $A$ does not have consistent units. To be more specific, of course in each specific case there will be some units but the units will depend on parameters of the model (unless we impose some restrictive assumptions as mentioned in that working paper). However, note this is not problematic from a statistical perspective as statistical analysis generally does not require dimensions to be consistent (like regression analysis for example). Consequently, in empirical work we usually just avoid assigning any particular units to TFP.


A few clarifications on Utility equations and indifference curves?

From a mathematical point of view, the indifference curve is an equation

$$I_{\bar u}:a^{2}b^{3} = \bar u$$

for every fixed $\bar u$. Put $b$ on the vertical axis and $a$ on the horizontal axis and write this as a function of $a$:

$$b^3 = \bar ua^{-2} \implies b = (\bar u)^{1/3} a^{-2/3}$$

This fully characterizes the indifference curve for $\bar u$. Now you have a one-dimensional function, and for it to be convex, its second derivative with respect to $a$ must be non-negative. In fact it is strictly positive (we assume goods are measured in positive quantities, and we ignore the uninteresting corner solution for $\bar u=0$ which is a single point).


If it is borrowed and budget deficits become a problem then the Fed will step in anyway.

The USA situation is the same as with the BoJ, which helps fund a Japanese 250% debt-gdp ratio: the Fed's profits are donated to the govt anyway. Purchasing govt debt with printed money essentially cancels the debt as any returns paid on the bonds are donated back to the govt.

The 2t coronavirus package, unless a vaccine is found, will not nearly be enough to stop a depression. Helicopter drops will be needed for at least the next 6 months. However, the US only has about 110% (?) debt-GDP so it can easily afford another 20t stimulus before it even approaches Japanese debt levels.


Consumer's Budget Constraint is

$$p_1x_1 + p_2x_2 + c\alpha x_1 = y$$

Holding $\alpha$ fixed, consumer's utility maximization problem is

\begin{eqnarray*} \max_{x_1, x_2} & \ (\alpha q + (1-\alpha))x_1 x_2 \\ \text{s.t.} & \ (p_1 + c\alpha)x_1 + p_2x_2  = y \end{eqnarray*}

Solving it we get the optimal bundle of goods as:
\begin{eqnarray*} (x_1, x_2) = \left(\dfrac{y}{2(p_1+c\alpha)},\dfrac{y}{2p_2}\right) \end{eqnarray*}

Corresponding utility as a function of $\alpha$ and other parameters is

\begin{eqnarray*} (\alpha q + (1-\alpha))\left(\dfrac{y^2}{4p_2(p_1+c\alpha)}\right)  \end{eqnarray*}

Now we find optimal value of $\alpha$ by solving
\begin{eqnarray*} \max_{0 \leq \alpha \leq 1} & \ (\alpha q + (1-\alpha))\left(\dfrac{y^2}{4p_2(p_1+c\alpha)}\right)  \end{eqnarray*}

Equivalently, we can find the optimal value of $\alpha$ by maximizing $\log$ of the objective
\begin{eqnarray*} \max_{0 \leq \alpha \leq 1} & \ \ln(\alpha q + (1-\alpha)) - \ln(p_1+c\alpha) + 2\ln y - \ln 4 - \ln p_2   \end{eqnarray*}

Differentiating the objective with respect to $\alpha$, we get
\begin{eqnarray*} \frac{p_1(q-1) - c}{(\alpha(q-1) + 1)(p_1+c\alpha)} \begin{cases} > 0 & \text{if } p_1(q-1) > c \\ = 0 & \text{if } p_1(q-1) = c  \\ < 0 & \text{if } p_1(q-1) < c \end{cases} \end{eqnarray*}

Therefore, optimal value of $\alpha$ is $1$ when $p_1(q-1) > c$, all values of $\alpha$ in the interval $[0, 1]$ are optimal when $p_1(q-1) = c$ and optimal value of $\alpha$ is $0$ when $p_1(q-1) < c$. In short, optimal value $\alpha^*$ of $\alpha$ satisfy

\begin{eqnarray*} \alpha^*  \in \begin{cases}  \{1\} & \text{if } p_1(q-1) > c \\  [0, 1] & \text{if } p_1(q-1) = c  \\  \{0\} & \text{if } p_1(q-1) < c \end{cases} \end{eqnarray*}

Consequently, final demand $(x_1^*, x_2^*)$ for $(x_1, x_2)$ satisfy
\begin{eqnarray*} (x_1^*, x_2^*, \alpha^*)  = \begin{cases}  \left(\dfrac{y}{2(p_1+c)},\dfrac{y}{2p_2}, 1\right) & \text{if } p_1(q-1) > c \\  \left(\dfrac{y}{2(p_1+c\alpha)},\dfrac{y}{2p_2}, \alpha\right) & \text{if } p_1(q-1) = c \text{ and } 0 \leq \alpha \leq 1 \\  \left(\dfrac{y}{2p_1},\dfrac{y}{2p_2}, 0\right) & \text{if } p_1(q-1) < c \end{cases} \end{eqnarray*}

Alternatively, we can write demand for $x_1$ as
\begin{eqnarray*} x_1^*  \in \begin{cases}  \left\{\dfrac{y}{2(p_1+c)}\right\} & \text{if } p_1 > \dfrac{c}{q-1} \\  \left[\dfrac{y}{2(p_1+c)}, \dfrac{y}{2p_1}\right] & \text{if } p_1= \dfrac{c}{q-1}  \\  \left\{\dfrac{y}{2p_1}\right\} & \text{if } p_1 < \dfrac{c}{q-1} \end{cases} \end{eqnarray*}

Here is the figure demonstrating the relationship between demand for $x_1$ and its price $p_1$, holding $q$, $y$ and $c$ fixed.

Now we consider Uno's revenue as a function of price $p_1$ :
\begin{eqnarray*} r(p_1) = \begin{cases} \dfrac{y}{2} & \text{if } p_1 \leq \dfrac{c}{q-1}  \\ \dfrac{yp_1}{2(p_1+c)} & \text{if }  \dfrac{c}{q-1} < p_1 \leq \overline{p}_1 \end{cases} \end{eqnarray*}

Consequently, profit as a function of price $p_1$ is
\begin{eqnarray*} \pi(p_1) = \begin{cases} \dfrac{y}{2} - \dfrac{ky}{2p_1} & \text{if } p_1 \leq \dfrac{c}{q-1}  \\ \dfrac{y(p_1-k)}{2(p_1+c)} & \text{if }  \dfrac{c}{q-1} < p_1 \leq \overline{p}_1 \end{cases} \end{eqnarray*}

Maximizing profit with respect to $p_1$, we get the optimal price as:

\begin{eqnarray*} p_1^* = \begin{cases} \dfrac{c}{q-1} & \text{if } 1 - \dfrac{k(q-1)}{c} \geq \dfrac{\overline{p}_1-k}{\overline{p}_1+c} \\ \overline{p}_1 & \text{if } 1 - \dfrac{k(q-1)}{c} < \dfrac{\overline{p}_1-k}{\overline{p}_1+c} \end{cases}\end{eqnarray*}

Now you can solve the remaining two parts in a similar way.


What is the real reason for china's stock market crash (August 2015)?

Stock prices predominantly are a quantification of the company's future growth and profit prospects. These are expectations, and are partly based on how the specific company performs currently and has performed in the recent past, but also on what the "general economic outlook" looks like (expectations again): if it is "gloomy", we expect that this will constrain even the best of companies.

A usual "stock market bubble" is a situation where there is a unabated inflow of new investors (so new money, available to buy stocks, flows in the market), and where the general tendency of the prices of the stocks is to rise, and at a fast rate.

Experience has taught us that even in a sample of a few hundred companies, there is no way that the majority of them will excel in economic performance at the same time. So the "short-term general tendency of stock prices to rise" must come either from positive expectations on the "general economic outlook" (rather than the assessment of each individual company separately), or from a "Ponzi-scheme" climate, which is often the case at least in new stock markets in emerging economies: investors demand to buy stocks simply because they believe they will be able to quickly sell them at a higher price to somebody else, irrespective of what the "economic fundamentals" are, as regards actual economic performance. This fuels stock prices which in turn represent a signal to outsiders that indeed, they too can come in and play this game.

The thing with Expectations is that they may have no inertia: they can swing abruptly. So if in the middle of such a rising-stock-prices trend (which already may not be based on the actual economic performance, as said), comes a change of Expectations for the gloomier, suddenly investors "see" that the prices are over-inflated. The inflow of willing investors to buy at even higher prices suddenly stops, and those with the papers in their hands (but not with the money) start to accept lower and lower prices in order to reduce the damage. And this too happens fast.  And this is when "the bubble bursts".


Suppose that A={a,b,c,....,z} is a finite set of social alternatives, and let P={>1,>2,....,>N} be a profile of strict preference orders on $A$ (where the set {1,2,...,N} indexes the voters).    We say that the profile P is single-peaked if there is some way to order the alternatives in A (e.g. in alphabetical order) such that, for each of the preference orders >n in P, there is some "ideal point" (say, h) such that

a<n b <n c <n ... <n g <n h >n i >n j >n .... >n  y >n z.

Note that the condition does not impose any particular preferences between, say, g and i, or indeed between any element of {a,b,...,g} and any element of {i,j,..,z}.  Note also that different voters could have different ideal points.  (So voter n has ideal point h in the above exampe, but maybe voter m has ideal point d....)  Note also that the question here is not whether a single preference ordering >n is "single-peaked" ---the question is that entire the profile P is single-peaked, meaning that we can find a single ordering of A such that all the preferences in P are single-peaked with respect to this ordering.  (In the above example, I used alphabetical order, but this was just for simplicity.  The question is only whether some order exists.)

Why is this condition useful?  Suppose you try to construct a social preference order >0 by determining the comparison between each pair of alternatives through a simple majority vote (as recommended by Condorcet).   So for example, for alternatives b and g, we will stipulate that b >0 g if and only if  b >n g for a majority of the voters {1,2,...,N}.  (For simplicity, suppose N is odd, so that majority vote never results in a tie.) We know that in general, this method does not result in a transitive preference relation.  (This is Condorcet's Paradox.)  However, if the profile P is single-peaked, then this method of "pairwise majority vote" always yields a transitive social preference order.  (This was discovered by Duncan Black.)   Furthermore, the "maximal" element of the resulting social preference order is the median of the ideal points of the voters.  (This is the famous Median voter theorem.)

In summary, if the profile P is single-peaked, then not only does majority rule induce a non-empty choice set (as you said), but this choice set is rationalized by a complete and transitive social preference order (obtained by pairwise majority votes), and furthermore we can say exactly what the choice set is (it is the median of the ideal points of the voters).


Will money still have any value in the fully automated world?

Your argument is built on this assumption:

  When there is no work that people can do, I imagine that there are no costs associated with it, and consequently, everything is free for everyone

That assumption is not true, and so your conclusion does not stand.

We usually classify production factors into labour, land, capital, enterprise. As you can see, labour is only one of the four factors. But all four factors incur costs: land still needs rent, enterprise still needs profit, capital still needs investment.

Another way of looking at it, is that even if the supply of labour is effectively unlimited (because so little of it is needed to produce anything), the supply of other things is still limited, i.e. there is still some scarcity. There is still a limited amount of iron in the world, or copper, or cabbages. And money is part of markets, which are the main mechanism for managing those limited supplies.


How Do I Calculate the After-Tax Equilibrium Quantity of a Supply and Demand Graph?

Implementing @dismalscience comment suggestion, the unit tax burdens the suppliers. So the demand schedule is not affected, only supply. How? Since the tax is fixed per unit sold (and not a percentage charge), then the slope of the supply curve should not change. Therefore what remains is an upwards shift, that will lead to increased equilibrium price-decreased equilibrium quantity. The algebra should lead one to

One could see this as a fixed shift in overall (not just production) marginal cost: the quantity has the same production marginal cost as before -but now "$2" is added as an obligation per unit, to cover the tax.


An explanation of the formula for the present value of an annuity

Take your second statement: $\dfrac{x}{r}$ is the present value of a perpetuity paying $x$ starting from now (first payment in year $1$)

Now consider the present value of a perpetuity paying $x$ with first payment in year $n+1$.  This is the value in year $n$ of $\dfrac{x}{r}$, discounted back to now so dividing by $(1+r)^n$ to give $\dfrac{x}{(1+r)^n r}$

So the present value of an annuity paying $x$ for the first $n$ years is the difference between these, namely $\dfrac{x}{r}-\dfrac{x}{(1+r)^n r}$

An alternative approach would be to consider the sum of the present values of the $n$ payments $$S= \dfrac{x}{(1+r)} + \dfrac{x}{(1+r)^2} + \cdots + \dfrac{x}{(1+r)^{n}}$$ and you might calculate this using the following to allow cancelation  $$(1+r)S - S = x -  \dfrac{x}{(1+r)^n}$$ which since $(1+r)S-S=rS$ becomes $$S=\dfrac{x}{r}-\dfrac{x}{(1+r)^n r}$$

It may be worth noting that the second term tends to zero if $n$ increases, so the value or cost of a long-term annuity approaches that of a perpetuity


Note 1.: It is rude to edit a question after it was answered; I had to make significant edits to make my answer consistent.

Note 2.: this is not a system of equations. There are two functions defined, but only one equation:
$$P_D(q) - P_S(q) = 0$$

What helps here is that inverse demand is decreasing in quantity while inverse supply is increasing. So given any $q$, if $P_D(q) > P_S(q)$ we know that $q$ is under the equilibrium value of $q^*$, and if $P_D(q) < P_S(q)$ then $q > q^*$.
You can also narrow down the range $q^*$ can be found in. Clearly $q^* \geq 0$.
Also $P_D(q)$ should not dip into negative territory, so you can find a very large $\bar{q}$ for which $P_D(q) \leq 0$. One can do this easily by starting from $\bar{q} = 1$, evaluating $P_D(\bar{q})$, and doubling $\bar{q} = 1$ iff $P_D(\bar{q}) > 0$.
Given the interval $I = [0, \bar{q}]$ which we know contains $q^*$, you can now apply interval halving.
An algorithm for approximating $q^*$ with a desired level of precision $\epsilon$:

Define the interval $I = [0, \bar{q}]$.
Start of LOOP
Select the midpoint of the interval as a "guess" for $q^*$. (In the first iteration this is $q = \bar{q}/2$.)
Evaluate the statement $P_D(q) > P_S(q)$.
If true, then

$q^*$ should be smaller, the solution is in the lower half of the interval, so in the next iteration we will use that as our new interval $I$.
If false, then

$q^*$ should be larger, the solution is in the upper half of the interval, so in the next iteration we will use that as our new interval $I$.
Evaluate the statement new interval is very small, i.e. has length less than $2\epsilon$.
If true, then
select its midpoint and say that you have approximated $q^*$ with reasonable precision, end program.
If false, then
Go to start of LOOP.


Nominal versus real wages in panel data modes

Since the dependent variable is log-wage, I presume this is some form of earnings equation that it is estimated.

To the degree that nominal wages may be correlated with the price level (or some appropriate delfator), using the nominal wages as dependent variable, immediately requires that the (log) deflator be included as an explanatory variable in the right hand side of the equation. This some times is useful, because it essentially tests whether the coefficient attached to the deflator is equal to unity. If it appears to be so, then using the nominal wage with log-deflator in the right hand side is statistically equivalent to use the real wage.

If this coefficient does not (statistically) appear to be equal to unity, it means that, in the presence of the other variables, nominal wages do not correct fully for inflation. In such a case, it is perhaps better to use real wages, and estimates the effect of the regressors on them.

Controlling for years is an indirect way to account for inflation, usually used in hedonic price analysis.


Purpose of a monotonic transformations in utility functions

Certain calculations using utility require utility functions to have some particular properties. The point of bringing up the invariance of preferences represented by different utility functions that are just monotonic transformations of each other is to show that these utility-based calculations are not entirely dependent on the utility functions that you happen to choose. The same results should at least hold up to order-preserving transformations.


Difference-in-differences in 2SLS regression

Well, if you believe that treatment is endogenous (which depends on the problem at hand here and is not an inherent feature of the model), then using eligibility as an instrumental variable will help you to get rid of the biases due to the safe selection in treatment. (Incidentally, DID is intended to do the same, but won't do as good a job as a well chosen instrument, so there is some doubts whether applying both of them is better then resorting to only one). However it is up to you to decide whether eligibility is exogenous, as it well may be, that those who are expecting higher return to treatment made sure to be eligible.

Taking that we believe that there are some biases that arenot eliminated by DID and that eligibility can help us, there is still considerations of efficiency. In many cases eligibility may happen to be a weak instrument and then the reduction is bias will come at a cost of significant efficiency loss.

And taking a look at the particular specification that you have sugested, it seems not very reasonable in general setting. You may choose when you believe that eligibility is changing quickly, or the interaction term in second equation will be generally unhelpful. Inclusion of time After in that equation can have even more drastic consequences, as it is likely to be endogenous and will weaken the bias reduction effect. If not endogenous, it is likeliy to be negligible as well as interaction, unless Treatment is rapidly changing on it's own.

So in this case I would recommend leaving only the eligibility as an instrument in the first equation and specifying the third one in a DID form.

With respect to interpretation, my specification does not allow for a nice interpretation of difference in changes in two subgroups and should be interpreted as a difference in changes in two hypothetical subgroup where each person is divided between them with some weights.

Your specification, however, loses all interpretation as DID, because you do not use the resulting interaction coefficient, but just employ more variables as instruments for treatment.

Unfortunately, probably due to the aformentioned reasons, I was unable to recall or find any appropriate paper, sorry about that.


Question about Locally non-satiated preferences

Consider a locally non-satiated preference represented by the utility function
\begin{equation}
u(x,y)=-(x-4)^2(y-2)^2
\end{equation}
It is easy to verify that $u(4,2)=0>-9=u(1,3)$. Therefore $B$ is strictly preferred to $A$.
Likewise, you can construct a locally non-satiated preference, one represented by $$u(x,y)=-(x-1)^2(y-3)^2,$$ under which $A$ is strictly preferred to $B$.

What local non-satiation does is that it rules out indifference "zones", so that we're left with indifference curves. Hence, with local non-satiation alone, it's impossible to tell whether one alternative is preferred to another.
The following figures from Jehle and Reny (2011) illustrate preferences with local non-satiation (Fig 1.3) and without it (Fig 1.2).

Axioms 1 through 4' are completeness, transitivity, continuity, and local non-satiation, respectively.


Bond market vs stock market during crisis

Short answer:
A bond issued by a company is less risky than a share of the same company.

Longer answer:
A company is in bankruptcy if its assets are worth less than its obligations. In case of bankruptcy the assets are sold off in the market or by individual agreements with the creditors. Direct creditors are payed first. If there is any money left, bond owners are payed. Since the assets were worth less than the obligations, no money is left, shareholders get nothing.

So even if things turn bad, bond owners can expect reimbursement. (Unless the change in asset price was drastic they can expect a large fraction of their money to be reimbursed.) Stock owners can expect no money if the company goes bankrupt. Even worse, stock prices already go down if asset prices decrease, because the company will have less profits after meeting its debt obligations, and hence it can pay less dividends. Basically a stock owner is betting on high profits while a bond owner is betting on medium profits. During crisis most companies cannot expect high profits anymore and their prices fall accordingly. Most bond issuing companies still don't go bankrupt so bonds are less affected.


The definition of cost-push inflation is an increase in aggregate prices driven by an increase in the cost of labour and/or production inputs.

A shortage of labour could put upward pressure on wages, yielding this type of inflation. So, yes, it is possible. However, it is not necessarily true that cost-push inflation is caused by a labour shortage and it is also not true that a labour shortage will always cause cost-push inflation.


If $X$ is a Giffen good then $Y$ must be a normal good

Reason: Both goods cannot be inferior.

Let's say originally you consume $x$ and $y$. So your budget constraint looks like

$$p_x x + p_y y = I.$$

If both X and Y are inferior, when income goes down from $I_0$ to $I'$, the quantity demanded for both has to go up (by definition) from $x$ to $x'$ and $y$ to $y'$. This implies

$$p_x x' + p_y y' = I' < I = p_x x + p_y y.$$

This is a contradiction, since $x' > x$ and $y' > y$.

Your question is a bit more specific. But since a Giffen good must be inferior, this answers your question as well.


What are the best books for developing a solid understanding of fundamentals of Economics?

Read Hal Varian's textbook "Microeconomic Analysis". It's brief, yet a standard text in a 4th year or graduate course in advanced micro.  Then grab a good book on mathematical economics that's actually well written, and includes suggestions on how to develop good mathematical intuition, not just a bunch of recipes on how to solve problems.  For example, you can learn how to use determinants in matrix algebra, but how many actually know what a determinant is, both algebraically and geometrically? That's the blood, sweat, and tears bit, I think.   I suggest the two volumes by Knut Sydsaeter (check Amazon)

You may also want to learn some advanced macroeconomic theory, and some spatial, urban and resource economics.

Take a good course in econometrics, and learn a programming language like Python or Julia so your are not always beholden to other people to provide you with data, which may be dirty or faulty, but you won't know it.

Watch lots of Youtube videos on economics, esp. controversial topics. Take some cool courses on Coursera or other MOOCS. They are often from some of the top universities in the world, and they are free.

OK, so now you have some basic tools; if you enjoy using them, and find that real work gets done, why not go to graduate school?

Then go practice for 20 years, and keep in mind that some aspects of economics may just be bunk anyway, so stay modest, don't get too overly seduced by beautiful models.


What would be the results of improved goods manufacturing efficiency?

Step 1) Improved manufacturing efficiency would lower the marginal cost of production for firms, thusly lowering the average total cost per unit.

Step 2) In a generic competitive market, lower ATC leads to profits in the short run, and increased number of producers in the long run.

Step 3) Increased number of producers moves the supply curve to the right (increasing supply), and the resulting equilibria has a lower price and a higher quantity. Profits for the individual firms decline as price falls, settling again at zero.


Difference between GDP and GNI in this exercise

GDP refers to income associated to particular geography (or residence).

GNI refers to income associated to a particular set of individuals (citizens), regardless of where they earn their income.

Since geography and citizenship do not fully overlap, they might be different:

I think this is enough to help you answering the question.

(code for reproducing the figure here)


Solving a problem using CAPM and capital structure theory

@chsk gives the correct answer if the firm has to raise funds using the equity market and the cost of those funds is the same as that for the firm. However, I would recommend that this firm do the project.

As indicated in the project, the risk free rate of return is 10%, and this project "...will earn a 13% rate of return for certain into the indefinite future." So this is a risk-less project with a return above that of the risk free rate. So, if you can borrow at the risk free rate, such a project will be profitable. That should be clear by introspection, but here are some numbers, even assuming that there are no cash flows during construction:
$$\mathrm{PV} = \$-1,000,000 + \sum_{t\ge 1} \frac{ \$130,000 }{ 1.1^t } \approx \$295,000$$
and if cash flows appear during the initial period the value is about \$425,000.

In general, the internal cost of capital should be tied to project risk and not overall firm cost of capital. Consider that we could start a new firm that does only the new project. What is the appropriate cost of capital to lend to this new firm? Well, by assumption, the new project has no risk. Therefore, the appropriate cost of funds should be the risk free rate (10% in this problem).

In the real world this might be tricky. You might have synergies which complicate considering the project as a stand alone firm. You might not actually be able to find a investor that will provide funds to you based on project risk. You might have other frictions which put you in a pecking order project financing setting. But, safely in the confines of a study problem, these frictions are not of first order importance.

The way that I would answer this problem is as follows:

  The appropriate cost of funds for this project should be the risk free
  rate because there is not uncertainty to the project's returns.
  Therefore, the NPV of this project should be positive (show math to
  that effect). However, there are a number of reasons (sketch some of
  these out) why it might be difficult to raise funds for a risk-less
  project at the risk free rate. It is likely that an upper bound on
  market financing on this project would be the firm's average cost of
  capital, which is 18%. At an 18% discount rate this is not an
  economical project (show math to that effect). This firm should
  review its financing choices to determine their marginal cost of
  capital for this project. The internal rate of return on this project
  is 13% (show math to that effect), so as long as the firm's cost of
  funds is $<13$% then they should fund this project.


We could distinguish between two kinds of "wage subsidies":

A) The government pays to the firm part of the wage cost, usually social security fees.

B) The government pays to the employee a markup on his wage.

In scenario A, labor supply is not affected but labor demand shifts outwards: the tendency should be higher employment and higher equilibrium wage (so eventually more disposable income for the workers). Usually, what is observed is that employment rises (after all, usually such programs are on  condition of net new jobs creation in a company), but wages do not increase (i.e. firms don't share the subsidy with the employees). So no betterment of standard of living of those employed.

In scenario B, take-home income will increase at no cost to the firms. So now, workers know that if the current wage is $w$, if they work they will receive $w+s$. This will shift the labor supply curve outwards (because now it responds to $w+s$), while the labor demand curve will remain unchanged (which still responds to $w$ only). This will tend to result in higher employment and lower equilibrium wage. But this contradicts (and partly or wholly offsets) the very purpose of the measure, which is not to increase employment, but to increase disposable worker income.

So in both cases, wage subsidies are more about employment rather than income, while minimum wage is clearly about income and standard of living of those employed, even though it may hurt employment.

...and one cannot avoid in these matters a dash of political economy and political science (as well as social psychology): subsidies bear always a risk of fraud, and are always suspect of being an instance of pork-barrel politics. Moreover, they are mostly seen as measures in time of macroeconomic (or regional) recession/depression, where the main concern is to boost employment.

On the other hand, minimum wage is not about a faltering economy as regards activity and level of production/income, but about market failure as regards distribution of income. Issues of distribution are inextricably linked to issues of fairness, and this creates a more passionate political agenda, occasionally  satisfying also the expectations of the public for a "strong government" which, "when markets create unjust misery", intervenes "decisively", imposing by decree "what's fair" (or fairer).


In a binary-dependent variable model (logic) will the inclusion of a scale introduce endogeneity?

Based on the clarification in your comment this is not because you use scale variables but because the scale variable that you use is self-reported.

It is well known in literature that self-reporting leads to endogeneity. See for example: Lindeboom, M., & Kerkhofs, M. (2004). Subjective health measures, reporting errors and endogeneity in the relationship between health and work.

The reason why that is when people self-report some outcomes those are often affected by other unobservables that you do not include in your model which is a form of endogeneity. For example, studies that measure happiness show that the self reported happiness can vary depending on which time of the day/week/season you make the measurement.

Moreover, you can have also endogeneity in more narrow sense unemployed people can self-report that they cannot work as a rationalization of their unemployment status and hence you would get reversed causality where the unemployment actually causes the self-reported inability to work even though the person actually would be able to work as measured by objective measurable criteria (in a model where employment is regressed on self-reported ability).

So it’s not that there is any problem with having scale variables. For example, having some 1-10 index but rather the problem is that in your case the scale was made from self reported data.


Am I correct that cyclical unemployment can be negative?

Cyclical unemployment $U^C$ is by definition difference between actual observed rate of employment $U$ and the natural rate of employment $U^N$ so we have: $U^C_t \equiv U_t-U_t^N$, see for example this paper.

Hence cyclical unemployment can be negative if actual observed rate of unemployment is lower than natural rate, because it is defined as a difference between the two.


What was the reason behind destorying or re-minting counterfeit gold coins of the same karats as the real ones?

I don't know about the story you refer to, but I'd say this was likely due to the intrinsic fungibility of gold as a commodity, before it serves as money.

Besides its natural scarcity and the significant amount of energy it requires to extract and mint, other important properties of gold (to serve as real money later on, anyway) include:

being stable (phys. / chem. speaking), portable, divisible, and fungible.

Fungibility: both as a store of value and as a medium of trade, any one ounce of gold is (relatively) easy to recognize and test for authenticity, and thus easily interchangeable with any other ounce of gold, of same purity -- if one excludes other considerations such as numismatic value or tastes in shape, etc.

Because the operation in the story you recall was fraudulent to begin with (extra-minting undeclared on the books) it was then important to find a way to differentiate unambiguously those ounces from the rest that were duly kept on the books -- i.e., to temporarily "deny" that gold its fungibility, so to speak, and at the physical level (by mangling it) -- if only temporarily.

Of course the rationale was to prevent a priori those coins from going into circulation on the market, after the bust -- and whether or not the decision would be made later on to reintroduce them (but this time in a controlled way).

Just my understanding anyway.

'Hope this helps.


Does quasi-concave utility function imply convex indifference curve?

No that is not true. Consider $u(x, y) = -x^2 - y^2$ defined on $\mathbb{R}^2_+$. Since $u$ is concave it is quasiconcave. Observing the graph of the indifference curves, we see that ICs of $u$ are not "convex".


Economic Effects of Presidential Candidate Proposal: Pay the same % in taxes no matter what you gain

Looking at the matter from a strictly positive and not normative or political view, then

1) Ignoring "Laffer-curve" effects, bleeding the Government treasury may not happen, if the flat tax rate is set to a suitable level (the specific politician has exactly in mind to bleed the federal government treasury, but that's discussing politics)

2) As regards income inequality, if indeed the current taxation system is effectively progressive and not just nominally, then it will indeed increase inequality, since this flat tax rate should be expected to lie above the current minimum tax rate implemented, and below the current maximum tax rate implemented.

A plausible anticipated effect is that lower incomes will reduce their saving (i.e. we should expect that the loss in disposable income will hurt savings more rather than consumption), while higher incomes, which will now have more disposable income, will increase savings. While the overall effect on the "consumption/savings" mix may be ambiguous, this will affect also the distribution of wealth, increasing also wealth inequality.

Is inequality bad for economic performance? Check some recent work by S. Turnovsky, that models links between inequality and growth.


Who paid the bill for cheap Chinese exports?

The United States was paying the bill. (This has now changed).


Why do there seem to be relatively few imports from India into the United States?

India has significantly worse physical/logistics infrastructure than China, Japan, Korea and Taiwan, so it is difficult to produce goods in India because it's difficult to move supplies and product around the country.  This, coupled with the comparatively higher complexity of dealing with government bureaucracy, makes it challenging to manufacture products in India for export.

Here is the World Bank's 2014 Logistics Performance Index for the countries, which illustrates the significant competitive disadvantage India has compared to China and others.

India still has a low-cost workforce, so they do export items where cost is very important (e.g. clothing, sneakers).  More importantly, India exports services (call centers, software offshoring) because it's much easier to lay down fiber optic cable than to build a railroad so certain cities (Hyderabad, Mumbai, Bangalore) have developed strong export services industries which do not need to depend on a lot of outside physical infrastructure to create viable exports.


Why isn't the Federal Reserve bothered by the drawbacks of its unlimited Quantitative Easing?

The Fed uses quantitative easing to provide ample liquidity and to prevent deflation.  Inflation has not been a problem in modern economies for over 12 years.

Deflation is much worse than inflation.

Technological advances, especially fin tech,  cause deflationary pressure.

I’ll say it again.  Deflation is the problem, not inflation.  Even the price of gold deflated this month.

There has not been a single reasonable economist worried about inflation in the u.s.  for 12 years.

A drawback to the Fed’s activity (and fintech) is that it would make it harder for banks to make money due to falling credit spreads.  Right now, though, the opposite is happening.  Credit spreads are skyrocketing due to stressed liquidity.

(My opinion) if the financial sector of the economy is making lower profit margins
 off the backs of people, then that should in theory be better for the economy as a whole.

As the Fed adds more government debt to its balance sheet at low rates, it makes it easier for the government to invest in infrastructure.  People have been complaining about national deficits since the 80’s.  Actually it was a lot worse in the 80’s than it is now.  At some point you can go too far with government spending (for example Greece).  Most other govt debt situations are far better than Greece.


Running ols when dependent variable is correlation coefficient

Welcome. Many things can be done. Some simple ones are:

Option 1: Do not care and just do OLS.

Option 2: Do some transformation such as $\ln (1+y) - \ln (1-y)$, logit of $\frac12 (y+1)$, to change the range from $(-1,1)$ to $(-\infty, \infty)$ and do OLS. I assumed that there are no $\pm 1$.

Option 3: Change your model. Let $y_{1,ij}$ and $y_{2,ij}$ be the firm-level ($j$=firm) variables from which you obtain the correlation coefficient $y_i$. As $y_i$ is also obtained by regressing $y_{1,ij}$ on $y_{2,ij}$ (up to a scaling factor), you might want to regress $y_{1,ij}$ on $y_{2,ij}$ including other industry-level variables $x_i$ as regressors. But in that case you need to think about which one of $y_{1,ij}$ and $y_{2,ij}$ to use as the dependent variable. If you are worried about endogeneity, you can also consider instrumental variable estimation, but that's a different story.

I would choose Option 1 unless many correlation coefficients in the sample are very close to $\pm 1$. Note that interpretation changes in Option 2.


This is simply the geometric distribution.

A job lasts 1 month with probability $s$.

A job lasts 2 months with probability $(1-s)s$.

A job lasts 3 months with probability $(1-s)^2s$.

$\vdots$

A job lasts $n$ months with probability $(1-s)^{n-1}s$.

Hence, the average length of a job is: $$\begin{align*}
 & 1\cdot s+2\left(1-s\right)s+3\left(1-s\right)^{2}s+\dots+n\left(1-s\right){}^{n-1}s+\dots\\
= & \left[s+\left(1-s\right)s+\left(1-s\right)^{2}s+\dots\right]+\left[\left(1-s\right)s+\left(1-s\right)^{2}s+\dots\right]+\dots\\
= & \frac{s}{1-\left(1-s\right)}+\frac{1-s}{1-\left(1-s\right)}+\frac{\left(1-s\right)^{2}}{1-\left(1-s\right)}+\dots=\frac{s}{s}+\frac{1-s}{s}+\frac{\left(1-s\right)^{2}}{s}+\dots\\
= & \frac{1}{s}\left(s+s\left(1-s\right)+s\left(1-s\right)^{2}+\dots\right)=\frac{1}{s}\frac{s}{1-\left(1-s\right)}=\frac{1}{s}.
\end{align*}$$

In the specific numerical example given by Mankiw, we have $s=0.01$ and hence the average length of a job is $1/s=100$.


You mostly got it, there are some minor mistakes.

You cannot have the moving index $t$ of the Cartesian product in the superscript of said product
$$S_i={\sf X}^{t-1}_{t=1} A_i^{A^t}.$$
At the start there have been no moves, so in round 1 you should map the empty history $H_0 = A^0$ to $A_i$, not $H_1 = A^1$ as you seem to have done.

Considering these I think the strategy set of $i$ is
$$
S_i={\sf X}^{T-1}_{t=0} A_i^{A^t}.
$$


If Y=K+C, why the value of a marginal unit of capital is the same of the value of a marginal unit of output?

This is a single-good model, that is, output is of one good which can be either consumed or added to capital.  The dynamic optimisation problem is to maximise the discounted present value of utility, utility being a function of consumption meeting the standard conditions $U_C > 0$ and $U_{CC} < 0$.

Perman et al identify the second of two state variables as capital $K$.  Now a state variable features in the statement of a dynamic optimisation problem in a limited number of ways:

An equation of state, in this case $dK_t/dt = Q(K_t,R_t) – C_t$.
An initial condition, in this case $K_{t=0} = K_0$.
A terminal condition, in this case by implication that (assuming an infinite
planning horizon) $K_t → 0$ as $t → \infty$.

Suppose that instead we were to specify the second state variable as the stock of the single good, say $G_t$.  Would this result in any essential changes to 1-3 above, which would change the solution of the problem?  It would of course if we allow for stocks of the good destined for consumption but not yet consumed, eg being packaged, transported or on shelves awaiting purchase.  But in a simple model such as this we will probably want to ignore such complications.  Suppose therefore we also assume that, for the part of output that is consumed, consumption is immediate.    Then the stock of the good would consist entirely of capital, and we could re-write 1-3 substituting $K$ by $G$ throughout.

This substitution would not affect the substance of the problem, so the same optimal solution would be obtained whether we make the substitution or not.  In particular, the time path of $\omega_t$ consistent with the optimal solution would be the same, whether $\omega_t$ is regarded as the shadow price of capital or the shadow price of the good.  Hence along an optimal path $\omega_t$ is also the value of a unit of output of the good.


Do rising inequalities create inflation?

This is not an answer that will state exactly which assumptions are valid/invalid but rather a guide to find answers to your question.

Here are various scientific articles about the topic:

These first two papers take what is called a Political Economy perspective on the matter. They assume that the central bank controls inflation, but the central bank is to some extent controlled by the population. If the population has higher income inequality, it prefers higher inflation and thus countries with higher inequality will face higher inflation.

Albanesi, Stefania. "Inflation and inequality." Journal of Monetary Economics 54.4 (2007): 1088-1114.

Dolmas, Jim, Gregory W. Huffman, and Mark A. Wynne. "Inequality, inflation, and central bank independence." Canadian Journal of Economics/Revue canadienne d'économique 33.1 (2000): 271-287.

This paper is also interesting, since it considers the different inflation faced by different income classes. It is somewhat orthogonal to your question, however.

Hobijn, Bart, and David Lagakos. "Inflation inequality in the United States." review of income and Wealth 51.4 (2005): 581-606.

However, to my knowledge, there are no papers written on income inequality as a direct cause of inflation without the involvement of the Political Economy argument above. This is mainly because in Economics our models usually tell us that the central bank has almost perfect control over inflation (except for negative inflation rates).

PS: If you find the readings above too difficult, consider reading a textbook on monetary economics:
Mishkin, Frederic S. The economics of money, banking, and financial markets. Pearson education, 2007.


How does the IS/LM model respond to organic GDP growth when the money supply is held constant?

We would move upward along the LM curve. However if there is growth because of investment or some technology shock that would cause the IS curve to also shift upward.

IS: When the growth "shock" you mention happens, there is now more demand for goods (Demand= C+I+G in a closed economy), without any change in interest rates. Since demand shifts, supply goes along with it in IS-LM and GDP increases

The IS captures the GDP-to-interest rate relationship (along the curve) since this relationship has now changed, we must shift the curve. Specifically we have more GDP now per unit of interest rate compared to before. Since interest rates didn't change in your example, but there is more GDP (caused not by an interest rate increase) the IS curve must represent more GDP for each interest rate, which - if you draw it now - is a shift to the right of the IS.

LM: Represents what happens to interest rates when GDP increases. This relationship has not changed, so we don't have a shift. The reason this relationship has not changed is because the only reason interest rates are increasing now, is because of GDP, something the current LM curve already captures. (If it would be for something else, like money supply change, there would have to be a shift.)

So there can be growth without money supply increases. However since there is now more demand for money (because there are more things that we can buy) but not more supply of money, then money increases in value, i.e. the "price of money" which in a way is the interest rate must go up. For that reason, part of the growth will be reduced (as investment goes down), but we will still have growth. If money supply also increases exactly enough to accomodate the GDP increase (typically is the case in reality) the interest rate will not increase and the LM would also shift and we do not have the mentioned negative effect.


Money will always be used for exchanges between nations. As a universal equivalent it will still be the best commodity for nation-to-nation exchanges and for measuring debts and credits between nations too.
Besides using alternative open source equivalents, such as cryptocurrencies, how would national banks manage to regulate inflation rates? This certainly is a critical point. Bitcoin for instance is a finite source of exchange: if it were to become a universal equivalent, virtually it would infinitely increase in its value, since the volume of the market exchange gets bigger and bigger.  Other cryptocurrencies tried solving this creating non finite sources, yet this still implies a arbitrary growth of the volume of available currency.


Is there an algorithm to find as many NE as possible in this generalized second price auction?

This is a pretty general algorithm, you can probably tailor a better one to your specific problem.

If you stick to this discreet strategy space it seems to me you would have to find the equilibria via brute force. Basically you would look at all bid profiles $(x_1,x_2,x_3)$ and check whether it is an equilibrium. To do this, you would have to check if player 1 would be better off by making another bid. Let us denote the payoff of player 1 by $\pi_1$. This is a function of the bids. Let's assume $x_2 > x_3 + 1$. Then you would have to check
\begin{eqnarray*}
\pi_1(x_1,x_2,x_3) & \geq & \pi_1(x_2+1,x_2,x_3) \\
\\
\pi_1(x_1,x_2,x_3) & \geq & \pi_1(x_2-1,x_2,x_3) \\
\\
\pi_1(x_1,x_2,x_3) & \geq & \pi_1(x_3-1,x_2,x_3) \\
\\
\end{eqnarray*}
If all these and the corresponding inequialites for other players hold then you have found an equilibrium.

If $x_2 = x_3 + 1$ you need to take the tie breaking rule into consideration.

You could limit the number of profiles $(x_1,x_2,x_3)$ you check by reasoning about the reservation prices. (E.g. for any N.E. where someone bids higher than their reservation price you have an N.E. where they bid just that.)


This is not a weird case, but a Leontief production function which is not homogeneous of degree one, but homogeneous of degree $b$. You can see this if you use the connection between a C.E.S. production function and the Leontief one.

Consider
$$Q_b=[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{b}{\rho}},\;\; b>0$$

$$\Rightarrow Q_b = \frac 1{[a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{b}{\rho}}}$$

Take the limit when $\rho \rightarrow \infty$. Since we are interested in the limit when $\rho\rightarrow \infty$ we can ignore the interval for which $\rho \leq0$, and treat $\rho$ as strictly positive.

Without loss of generality, assume $K\geq L \Rightarrow (1/K^{\rho})\leq (1/L^{\rho})$. We also have $K, L >0$. Then we verify that the following inequality holds:

$$(1-a)^{b/\rho}(1/L^{b})\leq  Q_b^{-1} \leq (1/L^{b}) $$

$$\rightarrow (1-a)^{b/\rho}(1/L^{b})\leq  [a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{b}{\rho}} \leq (1/L^{b}) \tag{1}$$

by raising throughout to the $\rho/b$ power to get

$$(1-a)(1/L^{\rho}) \leq a (1/K^{\rho}) +(1-a) (1/L^{\rho})  \leq (1/L^{\rho}) \tag {2}$$
which indeed holds, obviously, given the assumptions. Then go back to the first row of $(1)$ and

$$\lim_{\rho\rightarrow \infty} (1-a)^{b/\rho}(1/L^{b}) =(1/L^{b})$$

which sandwiches the middle term in $(1)$ to $(1/L^{b})$ , so

$$\lim_{\rho\rightarrow \infty}Q_b = \frac {1}{1/L^b} = L^b = \big[\min\{K,L\}\big]^{b} \tag{3}$$

For more, see this answer.


Why is Walras equilibrium inefficient when we are dealing with public goods?

In a competitive market for a private good (y) individuals may consume different quantities but the equilibrium condition requires that: $$ \frac{\frac{\delta u^{i}}{\delta y}}{\frac{\delta u^{i}}{\delta x}} = MRS^{i}_{yx} = MRT_{yx} \; \forall \; i $$

In the case of a public good (g) individuals may have different MRS but consume the same amount of the public good. Then for a given amount of a public good, the marginal benefit is the sum of individual marginal rates of substitution. Therefore, the optimal allocation must satisfy: $$ \sum_{i} \frac{\frac{\delta u^{i}}{\delta g}}{\frac{\delta u^{i}}{\delta x}} = \sum_{i} MRS^{i}_{gx} = MRT_{gx} $$

You can easily see that both conditions are not the same. In the case of an economy with only private goods, the benefit to
society of the last unit of a private good provided (expressed as the
willingness to forgo units of another good) is equal to the benefit of the
one person in society who receives this last unit. If there are some
people who receive a higher benefit from the last unit than others, we
don’t have Pareto efficiency. Hence, the marginal benefit of a private
good must be the same across all people for the allocation to be
efficient. Hindriks and Myles explain this thoroughly if you want to read further on the Samuelson condition versus competitive equilibrium (see Chapter 8 in particular).

Edit: There is no benefit that producers get here. Think of the MRT as the opportunity cost of producing an extra unit of a good and of the MRS as the marginal benefit of another unit of a good. The marginal cost of a unit of public good is one unit of private good (i.e. the MRT). Therefore the rule says that an efficient allocation is achieved when the total marginal benefit of another unit of the public good, which is the sum of the individual benefits, is equal to the marginal cost of another unit. If the good were private then the marginal benefit of an extra unit can only be attained by a single individual, so there is no sum.


How do I find out which form of utility function is being used?

I would call this a mean-variance utility function. The agent likes higher mean values, which is the first term, but trades that off against higher variance, which is the second term.

If the random variable of interest is normally distributed with mean $P_{t+1} + \delta_{t+1} - (1+r^f)P_t$ and covariance matrix $\Omega$, and if the agent has constant absolute risk aversion utility, such as $u(w) = 1-e^{-\gamma w}$, then maximizing that utility is equivalent to maximizing mean-variance utility. Here are some details I found with a quick google search.


I think one might use either vector auto-regression or perhaps factor-augmented vector auto-regression to make such predictions. The later will allow an economist to condition on hundreds of time series to predict future unemployment values without encountering the degrees-of-freedom issues that plague VAR whenever VAR attempts to condition on larger information sets.

Here is a paper detailing Kalman filtering and VAR for predicting unemployment:
https://www.degruyter.com/downloadpdf/j/ngoe.2015.61.issue-3/ngoe-2015-0009/ngoe-2015-0009.pdf

TLDR: VAR does better than does KF. This is for Romania's economy, FYI.

Here is another paper detailing the use of VAR to forecast unemployment:
https://www.brookings.edu/wp-content/uploads/2012/09/2012b_barnichon.pdf

Here is the seminal paper that introduced FAVAR:
http://faculty.wcas.northwestern.edu/~lchrist/finc520/QJE.pdf

Wiki for VAR:
https://en.wikipedia.org/wiki/Vector_autoregression

within the wiki, you will find a link to the seminal paper by Sims that introduces the VAR methodology to economics. Fair warning: this paper is dense:
http://www.ekonometria.wne.uw.edu.pl/uploads/Main/macroeconomics_and_reality.pdf


Bayes-Nash equilibrium and correctness of beliefs

I think your definition is incorrect, or at least incomplete.** Usually, in a Bayesian game, there is assumed to be a prior distribution on $T$ (where $T = \times_i T_i$). This distribution is called the "common prior" and it is assumed to be common knowledge that types are drawn according to this distribution. In this case, each player $i$'s belief $p_i$ is given by Bayesian updating on $T_i$ and this prior; and in BNE player $i$ must be best-responding to this belief.

The assumption of a common prior and Bayesian update rule (which is what gives this solution concept its name, after all) mean that players cannot be wrong, merely underinformed. In other words, a player with posterior belief $p_i$ is correct about the distribution of types of other players conditioned on her own, even though she does not know which realizations they have.

** Edit. Osborne and Rubenstein's text mentions that it is possible to define a more general game in which each player has a different prior distribution (there is no common prior). So your definition does match their most general definition. I suppose that in such a case two players may hold incompatible views, hence you could say that someone must be incorrect. That all being said, the vast majority of Bayesian games assume a common prior.


Joint Dynamic Programming: Group Activity

If I understand correctly the temporal dimension is merely a distraction as agents face a separate optimization problem every round with no intertemporal budget constraint. Hence
\begin{align}
& \max_{h_i} h^{\alpha_i}l_i^{1-\alpha_i} \\
& \text{s.t.} \quad h_i + l_i = \bar{T} \\
& h = \min\left\{h_i, h_{-i}\right\}
\end{align}
A continuous best response function exists for each agent:
$$
BR_i(h_{-i}) = \min\left(\alpha_i \cdot \bar{T}; h_{-i} \right)
$$
The equilibrium is
$$
h_1 = h_2 = \min_i \alpha_i \cdot \overline{T}.
$$


Do local taxes help in paying back foreign debts?

If you actually raise local tax it would help you can just convert the money raised with local tax into the currency in which the debt is. For example, if UK has some debt in Euros they can raise taxes in pounds and just exchange them at an exchange rate for euros and pay down their euro denominated debt.

However, it would not be possible to pay the foreign debt by just printing the local currency as that would lead to depreciation of local currency and hence its exchange rate.


Show that the dividend price ratio is a ARMA(p, q) process

Hi: I don't know where you got the expression for $d_{t-1} - p_{t-1}$ but assuming that's true, then I think you are basically there. Take your final expression and  consider the two lagged white noise terms first. They can be added together to make one lagged white noise term because I'm pretty certain that the sum of two white noise processes is still white noise. Similarly, the two non lagged white noise terms can be added to make one non-lagged white noise term. So, you end up with an MA(1) so the whole expression is ARMA(2,1). Granted the MA(1) part will have some complicated constant coefficients on both the lagged and non-lagged term but they are not terribly important. Also, you can always normalize so that the non-lagged coefficient of the MA(1) piece is 1.0.  If you or anyone thinks this reasoning is flawed, let me know because I don't see the flaw in it and am all ears. Thanks.

$\boldsymbol{EDIT}$

Elbarto: I've dealt with ARIMA models quite a bit over the last 5-6 years so this was bothering me so I kept at it. It turns out ( after many readings that may have been in my sub-conscious initially ) that the 4 last terms of your original expression do represent an MA(1) process. The mistake I made is that my argument was el-wrongo. The following is the argument that's used in the literature-texts for showing what process any ARIMA type "process" reduces to.

1) Construct the autocorrelations at each lag of the process.

2) If that  calculation results in a functional form that behaves in the same way (in terms of its shape) as some well known ARIMA model, then the original unknown process is equivalent to that well known process.

Unfortunately, we were unable to derive the process was using your methodology which is more direct and, to me, more intuitive.  The 1) and 2) argument above stems from the fact that any stationary and invertible ARIMA model has a 1 to 1 mapping between the model parameters and the model  autocorrelations. Therefore, if you know the autocorrelation shape, you know the process. I'm not absolutely sure if the stationary part of the theorem is even needed and, in fact, I don't even know if it's an actual theorem. I just think of it as a theorem but it is true.  The best analogy to this argument would be knowing the impulse response in the DSP framework. Still, for the intuition,  I like your approach better.

So, to make a long story short,  using this approach,  take those 4 terms as some unknown  ARIMA process and calculate its autocorrelation. Since that process clearly has an non-zero autocorrelation at lag 1  and zero after that, it is an MA(1) process.  This is a specific case of the generic argument used in the literature  but it's slightly unsatisfying given how your method seems quite reasonable also. My thinking is that the  one to one mapping  theorem must be equivalent to what you (and I later) were trying to do.

Finally, if you want to be bothered ( I didn't want to be ), you could calculate the autocorrelation of that 4 term process. It will be some function of the three parameters. Then set that function equal to $\frac{\theta}{(1+\theta^2)}$ and  hopefully ( it looked ugly to me  and I'd probably make a mistake somewhere even if I could be bothered) solve for $\theta$. That implies that the four term  process is  an MA(1) with parameter $\theta$. Anyway, the usefulness of calculating it is that you would then have $\theta$ as some function of of the other parameters: $b_{x}$, $b_{y}$ and $\rho$.

With that in hand, it may become more clear as to what the algebraic argument is that would show the same thing ??? We must be thinking about it in some incorrect way that I can't see (a scaling-normalization issue perhaps ?).

I'm reasonably satisfied now so, if you can't be bothered, I understand. All the best and thanks for neat question. I learned something from you and also during my googling journey.

$\boldsymbol{Last~~ Comment}$

Elbarto: Just one last thing. Keep in mind the same autocorrleation argument could have been used for the full expression that includes the AR(2) term. I have no idea how difficult that is but it's not nearly as easy as the MA(1) because now the ACF dies out slowly or oscillates depending on the roots. In fact, I bet it would be a lot more work than you did to show the AR(2) directly so not really worth pursuing.


Under random prices, the producer solves

$$\max_x E[U(G(x,p)]$$

Compacting $U(G(x,p) \equiv U$, the f.o.c is

$$E[U'\cdot (p_x-2cx)]= 0 \implies x^* = \frac 1 {2c}\cdot \frac {E[U'\cdot p_x]}{E[U']} \tag{1}$$

or

$$x^* = \frac 1 {2c}\cdot \left[E(p_x) + \frac {\text{Cov}(U', p_x]}{E[U']}\right] \tag{2}$$

Now,
$$U' = \frac 1{2(p_xx - cx^2)^{1/2}}$$

and $U'$ is decreasing in $p_x$. Then (see for example Egozcue, Cogent Mathematics (2015), 2: 991082), Theorem 2(2) p. 6)

$$\text{Cov}(U', p_x] <0$$

So we conclude that

$$\implies x^* < \frac {E(p_x)} {2c}$$

the RHS being the risk-neutral solution.

Plugging the distributional assumption for $p_x$ in $(1)$ gives us an implicit function for $x^*$. If one is patient enough to go through some boring algebra after that, one will obtain a very simple nice looking closed-form solution.

As for the last question, it is poorly worded and it does not make much sense, or it is a trick question, since the set up implies that the producer must commit to an output level before the price is actually realized. If this was not the case, then the producer would have no reason to solve an expected utility maximization problem, but he would wait to see the actual price and optimize accordingly.

If the last question alludes to a dynamic maximization problem, then this is a whole new situation, and additional assumptions and modelling are needed in order to proceed.


Why does a loose Fed policy reduce the downward pressure on the currencies of emerging markets?

It’s because broadly speaking the exchange rate between two countries, using monetary model of exchange rates, follows the following relationship:

$$S=\ln(m)-\ln(m_f) -(\ln(y)-\ln(y_f))+\lambda(i-i_f) $$

Where $m$ is monetary base $y$ is output and $i$ nominal interest rate. Subscript f indicates foreign country. Now from perspective of developing country (so f=USA) loosen monetary policy means that $m_f$ would expand and also $i_f$ get smaller, which would lead to appreciation of the exchange rate for developing country or if there is due to $y$ or other factors already pressure for depreciation the pressure will be lower.

The intuition is that the exchange rate must be such to bring the price levels in two countries given their respective currencies to balance and the price levels in each country depend on money supply, output and interest rates. Since all currencies can be valued only relative to each other even if the home country (here developing country) does not do anything when the foreign country (here US) increases their price level and makes their currency less valuable automatically the home currency must be strengthened as its valued only relative to the other currency.

Of course in real word non of the factors are held constant and generally developing countries face strong downward pressure on their currencies because their output is so low and many developing countries also follow quite loose monetary policy often. However, again since exchange rates are relative the looser monetary policy (that is monetary expansion) in US decreases the downward pressure on the developing currencies since now there are factors that actually strengthens their currencies (namely the larger $m_f$ and lower $i_f$.

Also generally the extra riskiness of the investments in developing countries is already priced in in the interest rates companies there have to pay which already includes risk premium. If it would not be fully included there then the difference would make the exchange rate move according to above model, so that could happen as well but that would not be dependent only on whether Fed follows loose or tight monetary policy.


How does a reduction in consumer spending in favour of consumer saving, affect economic activity?

Such a "planned" and sought-after re-allocation of given income from consumption to saving, is justified only if the savings in an economy are sub-optimal (or we think so), in the sense of hurting the investment rate, which in turns hurts the (human and physical) capital infrastructure.

Think about the extremes: consume all that you produce, save nothing (as a society). Your capital base will erode, leaving in the end only the other factor of production, labor, an orphan, without the necessary capital to produce, leaving perhaps only land for the society to return to a non-industrial agricultural state.

So it is a matter of intertemporal (re)allocation of resources in order to guarantee that the economy survives, and then that it does a bit better than just surviving.

The conundrum: if businesses see consumption declining, why would they rush to make new investments even though now the "price" of these investments is lower, due to increased savings and so to increased availability of funds?

I would invoke here "Say's Law": Supply finds its own demand. (see the controversies over it).
For me, this law is better interpreted in the context of heterogeneity: no matter what overall consumer spending does, there is always some fields where consumer desire goes unfulfilled. Entrepreneurs try to identify these fields, and invest there (this has a Schumpeterian flavor).

The above discussion does not mean that we will not observe phenomena like increased unemployment. After all, investing in new fields requires re-allocation of productive resources, and this is not easily nor quickly done. And there is always the case that the "savings drive" may overshoot, and the economy will be led to a lengthy recession, Japan some 20 years ago being the classic modern example (in fact in Japan, the government tried actively to persuade the citizens to raise their consumption, but it failed).


What's the effect of the used car market on car price?

OP,

imagine that there are two kinds of people, rich and poor, and no market for used books. Rich people are willing to pay more for new textbooks. Poor people cannot afford to buy new textbooks. In the absence of a market for used books, poor people will not buy textbooks.

Imagine now that there is market for used books. Poor people are now able and willing to pay second hand textbooks. And rich people now have someone to sell the books too once they are done using these books.

A market for second hand textbooks raises the price of textbooks because some people are not able and willing to buy new textbooks but are willing to buy second hand textbooks. In the absence of such a market, they'll spend their income on other goods that they deem more important.

Think of it otherwise this way. In the absence of a market for used textbooks the full value of a textbook is not exploited because some people who would wish to trade with one another cannot trade with one another.

Hope that this helps.


Recent economics theories that involve differential topology?

The main reason differential topology had some success in economics is that supplies powerful methods to show that something holds generically, mainly Sard's theorem and the transversality theorem. Some of these methods have been generalized to contexts without differentiability, see for example the paper "A Prevalent Transversality Theorem for Lipschitz Functions" by Shannon. Differential topology is still used in general equilibrium theory, you might take a look at recent work of Yves Balasko.

In positive political theory, there are results to the effect that generically no majority winner exists in multidimensional spatial voting models.  A good introduction to this topic is the book "Positive Political Theory I" by Austen-Smith and Banks, the definitive treatment is the paper "The generic existence of a core for q-rules" by Saari.

Inspired by the work on regular equilibria in general equilibrium theory, regular Nash equilibria were also studied in game theory, starting with Harsanyi. An elegant approach can be found in the paper "The theory of normal form games from the differentiable viewpoint" by Ritzberger (preprint here). Many genericity results in general equilibrium have corresponding versions in game theory, for example, "most" normal form games have a finite and odd number of Nash equilibria.

Another area where differential topology is used is in the study of stability with respect to some dynamics. Indeed, one can define regular equilibria via the replicator dynamic from evolutionary game theory (that is how Ritzberger does it). For a recent example of using some differential topology (index theory, treat also in more generality), see the working paper "The Index +1 Principle" by McLennan.


The value of money in a model economy. Has this been addressed?

What you are asking about is "General Equilibrium with Money", and there is literature on the matter.

To get you started, check out

Duffie, D. (1990). Money in general equilibrium theory. in Friedman and Hahn (eds), Handbook of monetary economics, vol. 1, 81-100.

The chapter starts with the sentence

  This chapter provides conditions for the existence of general monetary equilibrium in a simple setting, and discusses the role of
  money in general equilibrium theory.


reference request: models of how racism or sexism affects decision making

There is a large literature on this, particularly in the context of labour markets. Some classic references:

Becker (1957) analyses employers who care about making profits
but also seek to avoid interacting with certain groups. One famous
result is employer prejudice need not lead to racial wage gaps as
long as there are enough non-prejudiced employers to hire the
minority group.
Phelps (1972) models employers who use demographic information
(e.g. gender or race) as proxies for unobserved but productivity
relevant variables. Unsurprisingly, it turns out that differences in
means or variances across demographic groups can induce employers to
discriminate in this setting.
Similar models are developed by Arrow (1971) and Aigner and
Cain (1977). I would particularly recommend reading the latter.
Here are some lecture notes which provide a useful overview of
some of the main issues and ideas.


Demand functions homogeneous of degree zero in prices and income - how this relates to budget exhaustion (solving a consumer's problem)

I would have thought the budget constraint would not have been satisfied only if one of the demand functions was not homogeneous of degree zero in prices and income.

You should look into this claim. It is true that demand functions are homogeneous of degree zero in prices and income, but not all functions that are homogeneous of degree zero in prices and income are demand functions.

Some examples with only one good, so where $p_x x = I$ should hold:

$
x= \frac{I}{7p_x}
$
is homogeneous of degree zero, but with this the consumer does not spend all their income.

$
x= 0
$
is homogeneous of degree zero, but with this the consumer does not spend all their income.

$
x= - \frac{p_x}{I}
$
is homogeneous of degree zero, but there is definitely something weird going on here.


Help finding the right term - Behavioural Finance / Pricing jargon

A related concept you might be looking for could be "mental accounting", first introduced by Richard Thaler.
According to mental accounting, people treat money as less fungible than it really is. Fungibility means that, all money is the same, regardless of any other factors, including intended use. However, people often budget money into mental accounts for expenses (e.g., saving for a home) or "spare change".
In your example, the average person in NZ may have a 10$ mental account for "spare change".
How optimal pricing works in behavioral theory is another discussion altogether though.


I thick you should return to some book about history of economic theories usch as "Modern Macroeconomics: Its Origins, Development and Current State" . Center of focus in Keynesianism/Monetarism debates was about the power of monetary/fiscal policy for combating recession.

Monetarists won and showed that monetary policy is effective at least when there is no ground for liquidity trap. Monetarists also believe that "transmission mechanisms" of monetary policy are more much broader than what Keynesian interest rate-investment mechanism ascertains. thus Monetarists as well as Keynesians accept that printing money will increase aggregate demand but through much more broader channels.

I should say that distinction between demand-pull inflation and cost-push inflation is a little tricky, because in real world (rather than in our DSGE models which are designed based on equilibrium paradigm) when central bank prints money excessively,  demand increment in  different markets will take place in a gradual manner and inflation transmits from one market to other markets step by step. According to this fact, one who just observe the input price increment in some markets, may think that there should be a cost-push inflation.

Pure cost-push inflations are mainly rooted in real shocks (namely supply shocks) rather than nominal shocks (namely demand shocks). For example droughts, wars and oil price shocks  can make pure cost-push inflation.

Finally I should say that monetary policy may be able to dampen nominal (demand) shocks, but there are some economists who argue that monetary policy should not harshly respond to supply shocks . Unfortunately in real world central banks hardly can find out whether a price increment is due to supply shocks or demand shocks.

Monetary policy is much more complicated than what Taylor Rule suggests.


Is there an economic term that means the same thing as "personal inflation"

What you describe is called "decreasing marginal utility of income". Marginal utility is just an economist's way of saying the benefit you got out of the last unit of something, and decreasing because we think this might be lower at higher levels of income.


Proving the asymptotic normality of th Non-linear Least Squares estimator

It appears that what you do is not a Taylor expansion, but an application of the mean-value theorem (as one should). If it was a Taylor expansion, apart from the remainder, the gradient would have to be evaluated at $\beta_0$. With the mean-value theorem, there is no remainder, and you evaluate the gradient at some $\bar \beta$ that always lies between $\beta$ and $\hat \beta$.

As regards the issue of second derivatives/Hessian, officially speaking they/it only "temporarily" appear in the derivation of asymptotic normality of the non-linear least -squares estimator, but they vanish asymptotically (while in the Maximum likelihood estimator the Hessian stays there).

Apart from that, we want to minimize the sum of squared residuals so, indexing the observations by $i$, and using much simpler notation (you will have to adjust it to vector-matrix notation), you set out to minimize $\sum_i[u_i(\beta)]^2 = \sum_i[y_i-h(\mathbf x_i,\beta)]^2$ w.r. to the vector $\beta$, in order to obtain a $\hat \beta$. So your FOC is (suppressing the regressors and passing the $i$ index to the function $h$)

$$\hat \beta : \sum_i\frac {\partial }{\partial \beta}[y_i-h_i(\beta)]^2 = 0 \implies \sum_i2[y_i-h_i(\hat \beta)] \frac {\partial h_i (\hat \beta)}{\partial \beta} =0, $$

Ignore "$2$" and apply the mean value theorem to the whole expression  to get

$$\sum_i[y_i-h_i(\beta)] \frac {\partial h_i (\hat \beta)}{\partial \beta} =  \sum_i[y_i-h_i(\beta_0)] \frac {\partial h_i (\beta_0)}{\partial \beta} \\
+ (\hat \beta -\beta) \sum_i\left [-\frac {\partial h_i (\bar \beta)}{\partial \beta}\frac {\partial h_i (\bar \beta)}{\partial \beta}+ [y_i-h_i(\bar \beta)]\frac {\partial^2 h_i (\bar \beta)}{\partial \beta^2}\right] =0$$

Note that $y_i-h_i(\beta_0) = u_i$, the true error, divide by $1/n$ and then multiply by $\sqrt n$ as you are allowed to and re-arrange to get

$$\sqrt n (\hat \beta -\beta) = -\left (\frac 1n\sum_i\left [-\frac {\partial h_i (\bar \beta)}{\partial \beta}\frac {\partial h_i (\bar \beta)}{\partial \beta}\right] + \frac 1n\sum_i\left [[y_i-h_i(\bar \beta)]\frac {\partial^2 h_i (\bar \beta)}{\partial \beta^2}\right]\right)^{-1} \cdot \left(\frac 1{\sqrt n} \sum_iu_i \frac {\partial h_i (\beta_0)}{\partial \beta} \right) $$

Now, we consider asymptotic normality, given that consistency holds,  $\hat \beta \xrightarrow{p} \beta_0$. Since $\bar \beta$ is sandwiched between $\hat \beta$ and $\beta_0$, it follows that $\bar \beta \xrightarrow{p} \beta_0$ also. This means that

$$\frac 1n\sum_i\left [[y_i-h_i(\bar \beta)]\frac {\partial^2 h_i (\bar \beta)}{\partial \beta^2}\right] \xrightarrow{p} \frac 1n\sum_i\left [[y_i-h_i(\beta_0)]\frac {\partial^2 h_i (\beta_0)}{\partial \beta^2}\right] \\= \frac 1n\sum_i\left [E(u_i) E\frac {\partial^2 h_i (\beta_0)}{\partial \beta^2}\right] =0$$

because $E(u_i) =0$.

So this term vanishes asymptotically and we are left with (cancelling also the negative sings)

$$\sqrt n (\hat \beta -\beta) \xrightarrow{d} \left (\text {plim}\frac 1n\sum_i\left [\frac {\partial h_i (\beta_0)}{\partial \beta}\frac {\partial h_i (\beta_0)}{\partial \beta}\right] \right)^{-1} \cdot \left(\frac 1{\sqrt n} \sum_iu_i \frac {\partial h_i (\beta_0)}{\partial \beta} \right) $$

You have to assume that the first sum converges to something positive definite, and the second converges in distribution to a normal random variable, and you do indeed make these assumptions (or deeper ones that lead to them). I don't have the specific book you mention, but you can compare the above to Davidson & McKinnon "Econometric Theory and Methods"(2004) ch. 6, around eq. $(6.30)$.


Yes. What you are doing is a solution.

The key idea here is that binary relations are necessarily ordinal relations, and do not contain cardial information. Taking (strictly positive) monotone transformations of representations preserves the preference structure. So, yes, you have shown $U(\cdot)$ (from the example) is related via a monotone transformation to an linear function with the index $ln(\alpha)$ and therefore, by the von Neumann Morgenstern theorem, satisfies the EU axioms.

Now, there is often a point of confusing here, because of the emphasis on uniqueness with regards to EU representation. But the EU representation is not the unique representation, it is the unique linear representation. While this is clearly delineated in the theorem, I think it is conflated in many people's minds with generic uniqueness.

To summarize, by choosing a particular functional form (or normalization) it is possible to act as if the preferences have a cardinal relation. But, this is just an interpretation, and not something inherent about a binary relation, even one that satisfies the independence axiom.


You will need a One sample bilateral t-test.

Let say your cum abnormal returns are:

-0,15   -0,2    -0,02   -0,19   0,17    0,06    0,05    -0,1    0,03    0,12
-0,14    0,02   -0,2     0     -0,33   -0,1    -0,14    -0,28   -0,19   -0,29
 0,09   -0,33   -0,03   -0,4    -0,24  -0,26    0,12    0,22    -0,33   -0,02
-0,15   -0,03   -0,06   -0,1    0,14    -0,14   -0,43   -0,18   0,09    0,08

Then you have:

Mean              -0,096
Standard error     0,02620139
Standard deviation 0,165712138
Sum               -3,84
Count              40

In your case the null hypothesis will be $\mu=0$.

The main statistic to be computed is the observed $t$:

$t_{obs}= \frac{\bar{x}-\mu}{s/\sqrt{n}}=\frac{-0,096-0}{0,02620139}=-3,6639278$.

In Excel you can compute the theoretical $t$ with $\alpha=5\%$. You can use the following function $t_{crit}=T.INV(0,025;40-1)=-2,02269092$. *1.

Or you can compute the p-value with the following function $pval=2*T.DIST(-3,6639278;40-1;1)=0,000737103$. *1.

In any case you reject the Null (H0)($|t_{obs}|>|t_{crit}|$, or $pval<\alpha$).

*1. It is considered the half of $\alpha$ in $T.INV$ or double the $T.DIST$ result because of the bilateral test.


Are US-branded products manufactured in China considered Chinese?

Short answer, yes, they are considered Chinese imports. Your excellent question speaks to the flaw in using Gross Domestic Product as a measurement of the health of a nation's economy.

The flaw is that the simplistic assignment of "import" and "export" completely misses the reality of modern manufacture and trade, where parts come from multiple nations.

If trade numbers more accurately accounted for how products are made, it is possible that the United States would not have any trade deficit at all with China. The problem, in short, is that trade figures are currently calculated based on the assumption that each product has a single country of origin and that the declared value of that product goes to that country. Thus, every time an iPhone or an iPad rolls off the factory floors of Foxconn (Apple’s main contractor in China) and travels to the port of Long Beach, California, it is counted as an import from China.


Inflation leads to loss of gold reserves

Inflation won't necessarily change the price of goods from country B in real terms, but it will change the relative costs of buying goods from country B now vs saving now and buying goods from country B later.

Under the gold standard, there are no central banks to force the interest rate up when inflation rises. If inflation goes up but interest rates don't go up, then consumers in country A will save less and spend more, leading to a temporary increase in imports from country B.


Here is how I would approach the first question:

We are conditioning on the fact that you have won. Given the symmetry, this implies that your valuation $x$ is greater than your opponent's valuation $y$.
Given that you have won, you either pay the 'auction price' or $B$. If you don't pay $B$, you pay somewhere between $0$ and $p(x)$. The expected auction price is the expected value of the second highest draw, so actually $p(x)/3$ - not $p(x)/2$ as you seem to be saying (intuitively, the expected valuation of the lowest bidder must be below the unconditional expected valuation).
The probability that you pay the auction price is the probability that your opponent's valuation $y$ is below your cutoff $p(x)$. This is because your opponent will drop out of the bidding as soon as the price exceeds her valuation. $Pr(y < p(x)) = p(x)$ since $y$ is uniform on $[0, 1]$.
Therefore, assuming you win, the probability that you pay $B$ is $1 - p(x)$.
Putting these claims together, I conclude that your expected payment conditional on winning equals: $(p(x)/3)p(x) + B(1 - p(x))$.
To check the plausibility of this, set $p(x) = 1$ so effectively the buy it now option disappears, and yields an expected payment of $1/3$. This is the standard result for $2$ buyers and a uniform distribution.


Characterising utilitarianism and altruism in economics

Descriptive Framework: Consider a society composed of $n$ individuals, and consider a person in this society who is faced with a decision problem where he has to choose an action $a \in \mathscr{A}$.  For simplicity, we will presume that the marginal utility from this action for each individual is known to the decision-maker, and is given by the function $\Delta U_i : \mathscr{A} \rightarrow \mathbb{R}$ for person $i$.  A positive value of a marginal utility function means that there is a gain from the chosen action to that individual, and a negative value means there is a loss.  Note that utility is interpreted here in the cardinal sense, so it is not merely an ordinal measure (i.e., it can be aggregated by summation).
The vector function of all the marginal utility functions is $\boldsymbol{\Delta U} = (\Delta U_1, ..., \Delta U_n): \mathscr{A} \rightarrow \mathbb{R}^n$, and this gives the total change in utility from the action, for all individuals in society.  We can represent an ethical system dependent on these utilities as an ordinal ranking $\succeq$ over $\mathscr{A}$ that depends on $\boldsymbol{\Delta U}$.  This ranking describes which actions are more or less good under the ethical systems, based on the vector of marginal utilities in the society.

How to describe utilitarianism and altruism: Both of these ethical systems involve judging the goodness of an action by the benefit to oneself and others, which we will take to be a judgement in relation to some aggregation of the marginal utilities of the action.  We characterise these two ethical systems with the following rankings.

Utilitarianism: This ethical system holds that a person should act to maximise the aggregate benefit of their action to the whole of society.  This is represented by the ranking $\succeq_{\text{U}}$ given by:
$$a \succeq_{\text{U}} a' \quad \quad \iff \quad \quad \sum_{i} \Delta U_i (a) \geqslant \sum_{i} \Delta U_i (a').$$

Altruism: This ethical system holds that a person should act to sacrifice his own welfare to others, to the greatest degree.  For each person $k$ in the society, this is represented by the ranking $\succeq_{\text{A},k}$ (each individual has a different ethical ranking) given by:
$$a \succeq_{\text{A},k} a' \quad \quad \iff \quad \quad \sum_{i \neq k} \Delta U_i (a) - \Delta U_k (a) \geqslant \sum_{i \neq k} \Delta U_i (a') - \Delta U_k (a').$$

Both of these rankings involve an inequality on sums of marginal utilities, but they differ in terms of the signs applied to the parts of the aggregation.  Both can be written in the general form:
$$a \succeq_{\mathbf{w}} a' \quad \quad \iff \quad \quad \sum_{i} w_i \Delta U_i (a) \geqslant \sum_{i} w_i \Delta U_i (a'),$$
for some specified vector of weights $\mathbf{w} = (w_1, ..., w_n)$.  In the case of utilitarianism we have weights $w_i = 1$, so that each individual contributes their utility equally to the aggregation.  In the case of altruism we have $w_i = \mathbb{I}(i \neq k) - \mathbb{I}(i=k)$ so that the decision-maker contributes his marginal utility negatively to the aggregation (i.e., self-sacrifice is considered good).


Unemployment and its relationship with Inflation

When agents in economy have adaptive expectations they predict future in the basis of what happened in the past. If we consider an inflation adjustment scenario, the expected inflation by the agents is directly dependent on the inflation that prevailed in the immediately previous period and the output gap.

$$Y = Y^0 + b(P - P^e)$$

where $P^e$ is expected price level.

When agents have adaptive expectations there is an inflation inertia.
To reduce unemployment rate the government will try to stimulate the economy and if the economy was at its natural level of unemployment the will be inflationary pressure an inflation will increase. If the agents do not correctly predict the inflation they are likely to produce more. Say for example agents have an expectation that inflation will be 4% based on past experience. And with the government expansion output increases and inflation actually increases to say, 8%. Their nominal wages will adjust according to 4% but the real wage will actually decrease. As a rule of thumb you can remember that if the expected inflation is lower than the actual inflation then output increases and producers hire more workers thereby reducing unemployment. So, there is a trade off here between unemployment rate and inflation level as suggested by the Okun's law and Philips curve.
However this is not the end of story. As soon as the workers realize that their real wages have reduced, they bargain for higher wages and eventually the whole change is reverted as firms' costs increase and they reduce output and fire workers to match the cost. This is also the reason why the long run supply curve is vertical.

With rational expectations ,however, it is a different story altogether. Rational expectations mean that the agents in economy apply all levels of reasoning to the available information to figure out the future scenario. They know the policy of central bank and government, they can correctly predict their actions and hence they know that inflation will be higher and simply ask for more wages. So no effect on output or employment and hence no tradeoff but only an increase in inflation. The policy will not be effective here. This is a concept in neo classical economics and is sometimes called as policy ineffectiveness proposition.


why subsidy is given why not govt direct discount the prices?

Supposing by "discount the price" you mean set a maximum allowed price that is lower than the current price.
At the current price supply is equal to demand, when the price drops, there will be less supply and more demand. Then there is more demand than supply and the government doesn't want this because some people don't get what they want. Subsidies solve this problem because the price will drop because a part of the profit isn't paid for by the consumer but by the government. Then supply and demand are equal and the price is lower.