This is unlikely to be a published result in its own right, simply since it is publicly available here  (click on the row Firm Age).


In which market structures does the producer employ price discrimination?

Market power required for price-discrimination?

In my understanding, no. I'm pretty sure that one can show this mathematically, but I'll just give you the intuition. Price-discrimination (PD) increases profits, because it incorporates different price-elasticities by having different prices. Therefore, a firm that incorporates PD should make, c.p. more profits than a firm that does not.

Perfect competition means that less profitable firms are forced out of the market. In this case, perfect competition will lead to firms that act with price-discrimination whenever they can.

Think about restaurants, where there is a lot of price-discrimination going on, either via lunch offers or menu pricing. Restaurants really shouldn't have market power, should they?


World Inequality Database: Top 10% Income Share U.S. vs. Top 10% Average Income

I have actually found an answer to my own question. In case anyone's interested, I found the data on the top 10%'s income share going back to 1913 on dr. Mark Frank's website - he collaborated on the construction WID. I suppose the series have gone missing / were excluded for some reason on the WID's platform?

https://www.shsu.edu/~eco_mwf/inequality.html


The equation-wise OLS estimators are consistent under the further normalization that $E(u_{i1})=0$ and $E(u_{i2})=0$, which are naturally assumed to be satisfied. For GLS, I don't know what you mean by GLS, but if you mean the SUR estimator, then I see no reasons why inconsistent. The fact that $E(x_i' u_{i1}) \ne 0$ is irrelevant because $x_i$ is not a regressor in the first equation.


Is there really an "efficient equilibrium *price*" with externalities?

The blue lines are labelled as being costs, not prices. The line labelled "Private Cost" represents quantity supplied for a particular cost to the consumer. The line labelled "Social Cost" represents the quantity supplied for a particular cost to society. While, for the former, the cost is equal to the price received by the supplier, for the latter the cost is equal to the price received by the supplier plus the externality. $P_S$ therefore represents the price paid by the consumer, not the price received by the supplier.

In the case of steel smelting causing air pollution, the free market price represents the equilibrium if steel consumers are required to compensate steel suppliers for the cost that suppliers incur in smelting steel. The ideal price represents the equilibrium if steel consumers are required to compensate steel suppliers for the cost the suppliers incur, and also compensate the general population for the cost that they incur through pollution.


International Capital Asset Pricing Model (CAPM)

When using DCF you would discount using the weighted average cost of capital; and yes you can then find the expected return on equity using CAPM - if you have beta. But how do you know what beta is in your case?

That being said it seems like you are just using DCF for the 6 years where you know the cash flow? Will the company seize to exist after that? If not, you need to discount all future cash flows as well making assumptions on how income/profit will continue in the future (usually a constant growth rate from e.g. Year 10-15 and into all eternity).


Proof that the diff-in-diff (wrt sample size) of the expectation of a first-order statistic is positive (Stigler 1961)

Well, you cannot take the derivative of $E(n)$ with respect to $n$, because $n$ is an integer variable.

More generally, you want to prove a property with respect to $n$. The problem you have is that the corresponding domain is not a convex set: say, for $n$ and $n+1$, $0<\lambda <1$, the value $\lambda n + (1-\lambda) (n+1) = n+1-\lambda$ does not belong to the domain since it is not an integer. But if the domain of the function, with respect to the variable you are interested in, is not a convex set, the concept of convexity/concavity is not even defined (perhaps in higher mathematics they have some more nuanced concept to accommodate that).

To determine whether the inequality you want holds, a brute-force algebraic approach is to examine its sign directly: we have

$$[E(n+2) - E(n+1)] - [E(n+1) - E(n)] =\\= (n+2) \int_0^\infty p (1-F)^{n+2-1} F' \mathrm d p \\- 2 (n+1) \int_0^\infty p (1-F)^{n+1-1} F' \mathrm d p\\ +n \int_0^\infty p (1-F)^{n-1} F' \mathrm d p$$

$$= \int_0^\infty p (1-F)^{n-1} F'\cdot\Big[(n+2)(1-F)^2 - 2(n+1)(1-F)+n\Big] \mathrm d p $$

Doing the algebra

$$(n+2)(1-F)^2 - 2(n+1)(1-F)+n  \\= n(1-F)^2 + 2(1-F)^2 - 2n(1-F) - 2(1-F) +n \\
= n\Big[(1-F)^2 - 2(1-F) +1\Big]  - 2(1-F)F \\
= n [(1-F) -1]^2 - 2(1-F)F = nF^2 -  2(1-F)F = F\Big[nF-2+2F\Big]\\
=F\cdot \Big[(n+2)F-2\big]$$

So we have arrived at

$$[E(n+2) - E(n+1)] - [E(n+1) - E(n)] = \int_0^\infty p (1-F)^n F'\cdot\Big[(n+2)F-2\Big] \mathrm d p$$

Breaking up and manipulating,

$$[E(n+2) - E(n+1)] - [E(n+1) - E(n)] = \\=(n+2)\int_0^\infty p (1-F)^{n+1} F' \mathrm d p \\- \left(\frac{2}{n+1}\right)(n+1)\int_0^\infty p (1-F)^n F'\mathrm d p$$

$$\implies [E(n+2) - E(n+1)] - [E(n+1) - E(n)] = E(n+2)-\frac{2}{n+1}E(n+1)$$

Simplifying $E(n+2)$ from both sides we get

$$-2E(n+1) + E(n) = -\frac{2}{n+1}E(n+1)$$

$$\implies E(n) = \frac {2n}{n+1}E(n+1)$$

and by induction

$$\implies E(n+1) = \frac {2(n+1)}{n+2}E(n+2) \implies E(n+2) = \frac {n+2}{2(n+1)}E(n+1)$$

Therefore

$$[E(n+2) - E(n+1)] - [E(n+1) - E(n)] = $$
$$= \left(\frac {n+2}{2(n+1)} -1\right)E(n+1) - \left( 1- \frac {2n}{n+1}\right)E(n+1) $$
$$= \left(\frac {n+2}{2(n+1)} -1 -  1+ \frac {2n}{n+1}\right)E(n+1)$$

$$=\frac {n+2 - 4(n+1) + 4n}{2(n+1)}E(n+1) = \frac {n-2}{2(n+1)}E(n+1)$$

So $n=1$ is not a search, for $n=2$ we have equality, and for $n\geq 3$ the inequality holds.


Can saddle path not go through the origin in Ramsey model?

I guess you already went trough the algebra below, but just for context, the problem you're trying to solve is

$$
\max_{c}\sum_{t=0}^{+\infty}\beta^t u(c_t) \\
\text{s.t.}~~ f(k_t) + (1- \delta)k_t = c_t + k_{t+1} \tag{1}
$$

where $f(k_t) = k_t^\alpha$ and

$$
u(c_t) = \frac{c_t^{1-\gamma}}{1-\gamma} - 1 \tag{2}
$$

The problem in (1) can be cast into the two coupled equations

\begin{eqnarray}
u'(c_t) &=& \beta[1 + f'(k_{t+1}) - \delta]u'(c_{t+1}) \\
k_{t+1} &=& f(k_t) + (1-\delta)k_t - c_t \tag{3}
\end{eqnarray}

where $u'(x) = x^{-\gamma}$, and $f'(x) = \alpha x^{\alpha-1}$. These first of Eqns. (3) can be inverted to obtain an expression for $c_{t+1}$ in terms of $(k_t,c_t)$, leading to

\begin{eqnarray}
c_{t+1} &=& \beta^{1/\gamma}c_t [1 + \alpha[k_t^\alpha + (1-\delta)k_t - c_t]^{\alpha-1} - \delta]^{1/\gamma} \\
k_{t+1} &=& f(k_t) + (1-\delta)k_t - c_t \tag{4}
\end{eqnarray}

Which can be expressed as

$$
{\bf x}_{t+1} = {\bf F}({\bf x}_{t})~~~\mbox{with}~~~ {\bf x}_t = \left(\begin{array}{c}c_{t}\\k_{t}\end{array}\right) \tag{5}
$$

A fixed point ${\bf x}^*$ of the map ${\bf F}$ is such that

$$
{\bf x}^* = {\bf F}({\bf x}^*) \tag{6}
$$

that is, a point for which the system does not evolve. If you use $\gamma=2$, $\beta=0.9964$, $\alpha=0.36$, $\delta=0.025$ this point is (found by solving Eq. (6)),

$$
{\bf x}^* = \left(\begin{array}{c}c^*\\k^*\end{array}\right) = \left(\begin{array}{c}2.84829\\52.2808\end{array}\right)  \tag{7}
$$

which is clearly different from zero! You can linearize ${\bf F}$ around ${\bf x}^*$ and write the result as

$$
{\bf y}_{t+1} = {\bf J}{\bf y}_t ~~~\mbox{where}~~~ {\bf y}_t = {\bf x}_{t} - {\bf x}^*, ~~~ {\bf J} = \left.\frac{\partial{\bf J}}{\partial {\bf x}}\right|_{{\bf x} = {\bf x}^*} \tag{8}
$$

Is this last system the one that has a saddle point at ${\bf y} = 0$


A general formulation to capture price stickiness

I will answer the first question, I believe the second one can be found in the book's appendix.

"Price stickiness" is defined with respect to the optimal price level for the period (here denoted by a star) or equivalently in inflation terms. We do not have price stickiness if the current inflation equals the optimal inflation:

$$\pi_t = \pi^*_t$$

Looking at eq. $(9.26)$, written

$$\pi_t = \alpha\pi^*_t + \beta \pi^e_{t+1}$$

to allow for any expectations formation hypothesis (we will need this flexibility), here are the only two scenarios under which we do not have price stickiness, while also Wickens' "general formulation" holds:

A. Assume $\beta = 1-\alpha$ and $\pi^e_{t+1} = \pi^*_t$. Then we get $\pi_t = \pi^*_t$. Now, $\pi^e_{t+1} = \pi^*_t$ can be observed, it may be the case under Rational Expectations to expect next period inflation to equal the current optimal inflation. But assuming $\beta = 1-\alpha$ is a very special and uninteresting case.

B. Impose immediate and full adjustment, $\pi_t = \pi^*_t$, while respecting equation $(9.26)$: we get a very specific rule for expectations formation:

$$\pi^*_t = \alpha\pi^*_t + \beta \pi^e_{t+1} \implies \pi^e_{t+1} = \frac {1-\alpha}{\beta}\pi^*_t$$

This is clearly an ad hoc expectations formation assumption.

It follows that in any other case, eq. $(9.26)$ implies that $\pi_t \neq \pi^*_t$, i.e. we have partial adjustment. It is perhaps more illuminating to perform the above "what ifs" using eq. $(9.27)$ of the book that is written in terms of the price level (starting in long-run equilibrium, i.e. assuming $p_{t+1} = p^*_{t+1}$).


How did the authors of the (Credit Suisse) Global Wealth Databook calculate the Gini coefficient?

The GINI has fairly standardized calculation so you can just follow the standard formulas. The GINI is the ratio of the grey area on the plot below (taken from wikipedia - which also mentions the formulas below) to the total area. Hence if you already have data where these areas are calculated you can simply compute GINI as:
$$GINI= \frac{A}{A+B}$$
If you have raw data then you can use formula:
$$GINI= \frac{\sum^n_{i=1}\sum^n_{j=1} |x_i-x_j| }{2n^2 \bar{x}}$$
where $x_i$ is the income of person $i$ (or if you are calculating wealth inequality wealth of person $i$). The formula above can be also rearranged in a different ways, so some textbooks/sources can express it in different way but the results will match.
Next the formula could be also expressed in terms of income distribution:
$$GINI= \frac{1}{2\mu} \int \int p(x_i) p(x_j) | x_i - x_j | dx_i dx_j $$
where $\mu$ is the mean of distribution of $x_i$.
You can find more on these in Sen, Amartya (1977), On Economic Inequality (2nd ed.). If you want something solely focused on GINI then Yitzhaki and Schechtman. The Gini methodology: A primer on a statistical methodology. Vol. 272. has an exhausting literature review on GINI.


statistical discrimination VS taste based discrimination

You will find below two good and recent surveys on the literature on economics of discrimination and they all point out a clear difference between taste-based and statistical discrimination. We can imagine a situation where the employer's information about workers' productivity is perfect, such that there is no statistical discrimination, and still observe a racial wage discrimination for example because the employer has a taste for discrimination. Knowing the source of discrimination seems crucial to fight against discrimination and its consequences.

Charles, K. K., and J. Guryan (2011): “Studying Discrimination: Fundamental
Challenges and Recent Progress,” Annual Review of Economics,
3(1), 479–511.

Lang, K., and J.-Y. K. Lehmann (2012): “Racial Discrimination in the
Labor Market: Theory and Empirics,” Journal of Economic Literature,
50(4), 959–1006.


Ordinal axiomatization of proportional division

The answer hinges on the level of 'atomicity' dictated by the problem. If the decision making unit is non atomic- i.e. can make infinitesmal decisions- then there is always an algorithm corresponding to the competitive allocation such that it is known also to be a 'cardinal' value allocation under certain conditions- vide Aumann & Shapley. However this would not be robust to even a small perturbation towards atomicity. My understanding is that this field is interesting because of how small changes in the lower limit to what is within the agent's decision making scope can lead to big changes in behavior

D.G Saari's work may be of interest if only as an awful warning- vide http://socioproctology.blogspot.co.uk/search?q=dark+matter


One reason why there exists no ultimate overview is that these issues are still under debate. A great entry point would be the survey "Foundations of Strategic Equilibrium" by John Hillas and Elon Kohlberg in Volume 3 of the Handbook of Game Theory with Economic Applications. A preprint version of the survey can be found here.


How much inflation will changes to official wage benchmarks cause?

.

From Investopedia

  Inflation is the rate at which the general level of prices for goods and services is rising and, consequently, the purchasing power of currency is falling. Central banks attempt to limit inflation, and avoid deflation, in order to keep the economy running smoothly.

The Consumer Price Index (CPI) is your best measure of inflation which is (put simply) the change in cost of a representative basket of goods. Each country measures this in a slightly different way.

If you are looking for a numerical increase in inflation, you'll be disappointed. A Basic Income has quite a few effects making this difficult to model. There are anticipated significant impacts on labour/leisure as well as consumption bundles. Also, inflation itself is difficult to model. To make things worse, many reserve banks target inflation with their monetary policy. So you're unlikely to ever get any empirical evidence as well.

Anyway:

Why would increasing income of lower income earners have an inflationary impact at all?

An increase in welfare should result in higher incomes. Higher income leads to higher consumption (or quantity demanded) which should lead to an increase in prices. But, this is only a one-off increase, not an ongoing source of inflation. I would be hesitant to call this inflation and would prefer to call it a permanent demand shock.

Unfortunately, I am not aware of any estimates of this impact but I would expect it the long term impact quite small. The higher the basic income, the greater the impact.

Also worth noting:

I can't think why this would be any more inflationary than any other form of welfare.


Examples of risk-neutral firms or people in business

Its very hard to give an actual real world example of risk neutral person (firms are just run by people only people can have attitudes to risk) not because they do not exist but because research on micro-level risk neutrality does not publishes name of its participants and all data are anonymized. You also can't say someone is risk neutral just by pointing toward some non-experimental decision they made in real world as in such situation there might be a lot of different effects in play that you cant control for. That being said, there are some studies that try to empirically measure risk attidudes for industries just not really giving particular examples.
For example, Pennings & Philip Garcia (2001) examine the risk attitudes in Dutch farming industry using the certainty equivalence technique. The authors report that in their sample $6\%$ of farmers were shown to be risk neutral, $34\%$ risk averse and $60\%$ risk seeking (see their table 3), but of course they will not provide names of those individuals. So based on this we can conclude that $6\%$ of Dutch farmers out of 373 included in the study were risk neutral.
When it comes to investors, for example Dorn and Huberman (2005) use survey data on self reported attitudes to risk of German investors. The data show that actually most investors are neither too risk averse neither too risk loving. Unfortunately, this does not necessary imply they are exactly risk neutral as they could be just slightly risk averse or risk loving. It is hard to measure what proportion of people are exactly risk neutral outside some experimental framework, but this would seem to suggest some risk-neutral investors probably exist.
I could not find any example of an industry where empirically risk neutral people would dominate. That being said, absence of evidence is not necessary evidence of absence, as it might be due to lack of available data. However, unless there is strong competitive advantage of being risk neutral in some industry I would not expect most people there being risk neutral. The study of Dohmen et al. (2005) conducted on a general population in Germany (they had a nice representative sample of 22,000 individuals) showed that $78\%$ of people are risk averse, $13\%$ risk neutral and $9\%$ to be risk loving. Since the study is quite representative I would expect that it would be generally true for most people. Unless there is some competitive pressure for certain type of individual in the particular industry attitudes of people there should follow the distribution in general population.
Risk attitudes definitely evolve over time (see for example this and this paper), it can even be molded by researchers priming the subjects (see here), but there seem to be no research suggesting that some periods favor risk neutrality as opposed to others. In fact, risk attitudes can be affected by your emotions as well (see Loewenstein et al., 2001) so they are definitely not stable.


Utility function of habit formation model

I believe $n_{t-1}$ is a quantity from the past, so you should be able to treat it as an exogenous variable


What prevents a government from issuing debt to finance riskier investments?

Here are some hand-wavy arguments to suggest that the "no-arbitrage" rule applies here:

Bond-buying at the scale that nation-states would do it would drive up prices.  So you'd see yields on Brazilian bonds fall even if coupon rates don't move.
Germany would have to finance such a play by issuing yet more bonds, which would drive German bond prices down and therefore German bond yields up even if coupon rates don't move.
Credit markets would likely punish Germany for taking that kind of risk, pushing German bond yields up further because coupon rates would rise.
Ironically, the piling-in of high-quality foreign money into Brazil could serve to stabilize it, which would again lower Brazilian yields because coupon rates would fall.
and 6. Movement in foreign exchange markets that would further close the gap. (credit to @Gio in a comment above).

The net effect: the arbitrage opportunity you've spotted would quickly vanish.  In essence, your plan would amount to Germany volunteering to prop up the Brazilian economy in exchange for being made worse-off.


How can I motivate a dynamic model using utility and consumer preference where only one good is affected by past consumption?

I just wanted to clarify on Starfall's comment a bit. $\phi$ can be a bit confusing.

Let me appeal to Chris Carroll who uses the CRRA utility function $v(\frac{c}{x^{\gamma}})$ where $c$ is consumption, $x$ is the habit, and $\gamma$ is the "importance" of habits. In this framework, $\phi$ can be seen as the speed at which habits catch up to consumption. If $\phi = 0$, then the consumer is constantly maximizing their utility with a constant multiplicative factor in the denominator of the function's argument. If $\phi=1$, then habits collapse to last period's consumption.

There are a number of interesting features from this model. For instance, if persistence is low enough then the intertemporal elasticity of substitution tends to zero. It is used famously in the asset pricing literature to explain the excess return premium puzzle. The development of habit formation theory in microeconomics (both theory and econometrics) is slow but finally arriving.

Resources:
http://www.econ2.jhu.edu/people/ccarroll/HabitsEconLett.pdf
http://www.isaacbaley.com/uploads/6/7/3/5/6735245/lecture_7_baley.pdf


Why did foreigners keep buying Swiss francs at -41% (not a typo) interest rate in the 1970s?

I am not sure if 41% is correct I wanted to double check it but could not find a reference to primary source for 41%, but the overall narrative seems to be true. For example, Reinhart, C. M., & Smith, R. T. (2002), in their paper that you can read here argue that

  Switzerland  in  the  late  1970s  in  which  substantial  capital  inflows  –  partly because  of  German  residents’  desire  to  escape  a  new  withholding  tax  in  Germany  –  prompted  the Swiss  National  Bank  to  impose  a  100%  reserve  requirement  on  non-residents’  bank  deposits  in Switzerland.  The  result  was  a  negative  nominal  interest  rate  on  foreign  deposits  in  Swiss  banks,  as these  banks  demanded  a  fee  to  accept  foreign  deposits.

The above quote already gives you a partial answer (it was German savers), but actually the second part of the original quote of paragraph you gave gives you hint at the rest of the full answer. CHF was appreciating really fast at the time. Standard monetary model for exchange rate tells you that exchange rate depends on fundamentals, output, money supply, price levels, interest rate and also based on future expectations of these.

Now consider western world in the 70s. Half of Europe was under uncertain totalitarian regimes, eastern Europe under iron curtain, Spain was still under Franco, even democratic regimes across the Europe seemed very uncertain. You had a lot of right or left wing radical terrorism across many democratic countries in Europe. Major developed countries like USA or Britain were hit hard with oil shocks. France and west Germany were not doing very good as well. Many countries were experiencing stagflation. Plus in 70s it was popular to have punitively high marginal tax rates and Switzerland always was good tax haven.

In such crazy world there were not that many options to invest your capital in so safe havens like Switzerland experienced unprecedented capital inflow. If you are faced with bad options a rational person picks the least bad option. It is like asking why did you got your arm cut. Well if other options involved you dying then suddenly loosing a limb is not so bad.

Investors expectations of fundamentals in all other countries were probably so bad that simply Switzerland came on the top even if you face negative nominal interest rates. They rightly expected large appreciation of CHF and also expected that their savings will be safe even in near-apocalyptic scenarios - after all if you saved your money in Switzerland during I or II world war you did not lost your money, but if you did so in many other European countries your savings could be often confiscated or wiped out by inflation. It is also the real not nominal interest which drives decision making and I am sure that must have been less negative or even positive. Some irrational exuberance could play role too. Once everyone is saving in Switzerland it sort of becomes self fulfilling prophecy, but in this case Swiss fundamentals seemed to be really strong so this was at the best just part of the story.


Set of rationalizable strategies for this 4 x 4 matrix

The set of rationalizable strategies is the set of strategies that survive the iterated elimination of strictly dominated strategies, i.e., strategies that are never a best response. It is a weaker concept than Nash equilibrium.
For player 1, you can eliminate strategy M, which is strictly dominated by T. You cannot eliminate any strategy for player 2 as there is only a weak dominance.
In a second step, you can eliminate A which is dominated by C.
As far as I can see, you cannot eliminate any other strategy in the next step and, hence, all remaining strategies are rationalizable.


It's possible that the stories about drastic drops in Japan refer to working-age population, not the population itself.

This is a link to the U.S. Census methodology.

I have only casually looked at these data over the years, but I would summarise the general idea as follows.

We already have the current population distribution. We work with the data based on the number of people in each age cohort.
We have estimates of life expectancy based on age, and actuaries spend a lot of time working on developing projections. (The defined benefit pension and life insurance industry depend upon such actuarial projections.)
We usually have a good idea of the birth rate, based on the characteristics of under-30 population. Even if we miss, it takes a long time for the errors to really add up. (People have to reach puberty before the prediction error compounds.)
This covers the existing population; what remains to do is estimate the effect of net migration.

For the developed countries, the big unknown is typically net immigration. In the euro area, the collapse of economies has often caused a considerable amount of emigration.

Outside of net immigration, the only thing that could cause the projections to off is a massive change in the death rate (or birth mortality). Birth rates tend to be slow-moving; the baby boom after World War II was a response to highly unusual earlier conditions (Great Depression, WWII). Life expectancy is another wild card, but it is unclear how much it will change.


Is nominal/real income and nominal/real wage the same? If so, is it possible for your nominal income to rise as your real income falls?

Wages are a subset of income.  Your income includes your wages but it also includes other sources of income such as dividends and interest.  If your only source of income is a job, then income and wages are the same thing.

It is certainly possible that nominal wages would rise while real wages fall.  Imagine that you earn 50,000 at a job in 2019.  Inflation runs at 5% a year.  And say that at the end of 2019 your boss says that you're getting a 2,000 raise so your salary in 2020 will be 52,000.  In nominal terms, your wages increased from 50,000 to 52,000, a 4% increase.  But since inflation was 5%, your 52,000 salary in 2020 will buy less than your 50,000 salary bought in 2019-- you would have needed a 5% raise (2,500) in order to keep pace with inflation.  Your nominal wage increased but your real wage decreased.


Representing these lexicographic preferences by non-continuous utility function?

As @HRSE pointed out, your terminology is a bit misleading, and these preferences are not exactly what we refer to when we use the word 'lexicographic'. You might be tempted to consider every $R>0$ as a two-dimensional vector $R=(\mathbb{1}_{R \leq Q},R)$ and to apply the lexicographic order on these vectors, but I would say it is a bit unnatural since the two dimensions of the vector are not independent (you cannot vary the second freely without affecting the first).

That said, the usual observation that lexicographic preferences can be represented by a utility function (but not by a continuous utility function) applies to your case as well. The common feature of both situations is that preferences satisfy standard rationality requirements (weak order) but are discontinuous. You can refer to this question for a formal discussion of these issues:

Lexicographic preference relation cannot be represented by a utility function


How does an increase in USD money supply affect inflation?

Yes there are other factors at play. Inflation is change in a price level. The price level, according to classic textbook monetary equation, is determined as follows:
$$P = \frac{MV}{Y}$$
Where $M$ is the money supply, $V$ velocity of money (how much is one dollar used in the economy) and $Y$ is the real output.
So beside money supply and real output inflation depends also on velocity of money. Furthermore, the inflation also depends on peoples expectations of these quantities. That is on what people expect the money supply, real output and velocity will be (although this is not shown in the simplified textbook model above).

Would this average American family's \$50,000/yr income in February equivalent to a \$42,000/yr salary in May?

Not likely. According to the International Monetary Fund the the annual inflation forecast for USA for 2020 and 2021 is $0.6\%$ and $2.2\%$ respectively. These are mean forecasts so of course the actual realized values will not be exactly the same, but I would be surprised if even $99\%$ confidence interval would even include double digit inflation. The assumption that output does not change is currently also not realistic. USA lost $6\%$ of GDP just in 2020Q1 according to the Bureau of Economic Analysis and the fall for Q2 is projected to be even worse.


Growth rate of variables on a balanced growth path (BGP)

The concept of "balanced growth path" in economics incorporates three characteristics at the same time (related to the main macroeconomic aggregates):

1) Growth rates are constant  (reflecting a notion of equilibrium/stability)

2) Growth rates are strictly positive  (otherwise the economy would eventually vanish)

3) Growth rates are equal to each other (hence there is "balance" and no macroeconomic aggregate becomes negligible relative to another)

To obtain the first characteristic you have to impose that the derivative of the growth rate with respect to time is zero. Nothing weird about that, just a valid mathematical operation.


Why is the 10-Year Treasury Constant Maturity Minus 3-Month Treasury Constant Maturity Curve significant, but not the 5-year minus 1-year?

As a first approximation, we can think of the 3-month bill as as being the “market forecast” of the (geometric) average of the overnight rate for the next three months, while the 1-year is the average for the next year.

In a rate-hike cycle, the 1-year rate ends up much higher than the 3-month rate. This means that it will get close the the 5-year rate much faster.

Furhermore, there is presumably a greater risk premium - term premium - in the 1-year versus the 3-month maturity. This will also tend to compress the 1-/5-year slope faster.

Once the yields on the 1-year and 5-year are close, small “technical factors” in yields are much more likely to lead to an inversion. Therefore, it would be more likely to generate false positives.

The reality is that market commentators and economic researchers have searched through all the various slopes to find the “best” ones for forecasting. So it is probably not safe to say that the 1/5 slope is “irrelevant.”

In any event, yield curve slopes are hardly perfect. I discussed some of the issues in this question. Since slopes are not infallible guides to recession forecasting, one can argue that the choice of maturities (technically, tenors is a better term) is somewhat arbitrary.


Can we stabilise the price of a cryptocurrency?

First it might be useful to list the reasons behind cryptos' volatility.

One of the reasons is that cryptos have no underlying value. In contrast to other assets like oil, gold, company shares, etc. that you could make use of them, cryptos are similar to fiat currencies in that they're not fully backed... but in the case of fiat currencies, at least (in my understanding) there are laws that make them valuable ("these notes can be used to settle debts" or something similar.)

Value of most cryptos are determined, therefore, by "investors"' beliefs.

To "fix" this, you could, as you suggested, peg it to fiat currency (or basket of currencies), or you could back it up with other assets/commodities.

Other reasons (you could Google why cryptos are volatile) include having many retail investors holding a significant portion of the asset, not having that many institutional players in the market, etc. could be analyzed accordingly to make cryptos more stable.


Expected Utility with expected value and variance

It would suffice to show that $U$ is linear. But is $U$ necessarily linear if it satisfies the vNM axioms?

Hint: No.


Are there any instances where short run demand/supply is more elastic than long run cases?

It is possible.  We normally think demand and supply are more elastic in the long run because consumers/firms have more options in the long run.  For example if gas becomes more expensive consumers can buy more more fuel efficient cars.

So what if consumers or firms are more constrained in the long run?  This could be the case when goods are storable.

Suppose I need to drink coffee every day or I get a headache.  At home I have a canister of coffee that I use every day.  If coffee doubles in price I can hold off buying and hope the price comes down, so I am fairly elastic.  In the longer term though I'll run out of coffee and I'll need to buy more regardless of the price.

Of course in the even longer term maybe I can stop my caffeine addiction so I become more elastic again.

This maybe isn't the best example, but the key thing to think about is how constrained the agent is at each point in time.  When people are more constrained in their options they are less elastic.


Value vs. Price and its relation to tariffs

You asked "why is cheaper always being preferred". Well, it's not. People routinely pay more for quality. Cheaper is almost always preferred for identical products. But cheaper is not always preferred for goods that are only partial substitutes. For example, a well-made pair of shoes that will last ten years, will cost more than an equally-comfortable pair that will last one year. But will cost less than ten times as much, and many people who can afford the well-made pair, will buy them.

"I've read numerous publications citing American Steel & Aluminum to be of far greater strength and quality than overseas Steel"

There are many different grades of steel. These guarantee certain proportions of metals, that in turn guarantee certain physical properties. And they come at different costs, and thus different prices. It doesn't make sense to say "this country's steel is better than that country's steel" - that's jingoism, not economics, not materials science. What matters is the grade of steel being produced, and its price.

"these Tariffs are so minuscule no one will notice them": if that was the case, there'd be no point in flouting international agreements to implement them. The big issue is that this signals an administration that is willing to cause itself and other countries economic harm, in the interest of pursuing an ignorant, silly ideology.


Looking across countries, there is not much of a relationship.

The following chart is taken from the latest IMF World Economic Outlook database, comparing General Government Gross Debt as a percentage of GDP (you could look at net debt) and General Government Revenue as a percentage of GDP (which includes more than taxation) both in 2014.

The extreme debts are for Japan and Greece, while the extreme revenues are for Tuvalu, Kiribati, and Micronesia (foreign aid related), and Kuwait (oil revenues/taxes).  Hong Kong, Brunei, and Saudi Arabia have small gross debts while Nigeria and Bangladesh have the lowest revenues.  But even without these special cases the relationship is weak.

You could instead look at individual countries over time, but some of the relationship may be cyclical (either political or economic) as lower debt encourages lower future tax rates, but lower tax revenues can lead to higher future debt rates.

Of course the true relationship also involves government expenditure rates and GDP growth rates among other significant factors.


Can a spending freeze solve the problem of the Italian enormous government debt?

This question is somewhat political, and runs into the area of “opinion-based,” which this site has to avoid. I am only going to attempt to discuss part of it. There’s some long-running economic debates involved, so I will try to describe the debates. I obviously have my own opinions.

One side issue: it is unclear in what sense is an “enormous” debt a “problem”, other than the technical issue of the European treaties? But since those treaties exist, I will put that aside.

As background information, we normally think about the size of the government as a percentage of GDP. So if nominal GDP is growing in steady state, so would government spending. Therefore, a “freeze” in euro terms, although that might be interpreted as unchanged, would be thought of as shrinking government spending relative to trend. In commonly used terms, it would be viewed by many as a mild austerity policy.

This leads us to the debate about expansionary austerity. Alberto Alesina is the economist most associated with the revival of this policy. I am not familiar with the more recent literature, but this paper by the IMF was critical of idea of expansionary austerity - IMF paper link. I would summarise the conclusions that there was a mild depressing effect on growth (“contractionary” versus “expansionary”), but one may note that this is just reducing the growth rate of an already growing economy.

I am not going to claim to give the “correct” answer, but the debate is this: will the cut in spending reduce growth, so that the effect on the denominator of the debt/GDP ratio of reduced spending is cancelled out? Realistically, it probably depends on the state of the cycle. Cutting spending right after a crisis when confidence is low may have a more dampending effect than when the economy is starting to grow more rapidly. In Italy’s case, one would need to do a detailed forecasting exercise to see how large the margin of safety is.

However, freezing spending would only reduce the deficit if the tax take rises along with nominal GDP. The suggested policy is to cut tax rates. That enters into another bitter debate in economics, the supply side effect of tax cuts.

(As an immdiate disclaimer, it is possible to redesign the tax system so that a lower tax rate will imply a higher tax take; for example, eliminate a popular deduction. It does not sound like that is the case here.)

The original supply side debate was very strong in the era of Ronald Reagan, the “Laffer Curve” (an idea of economist Art Laffer) was that very high tax rates were counter-productive. Cutting taxes would raise incomes through growth, and so revenues would increase. In 2018, I do not know of anyone who would argue that this would happen right now in the short term (one or two years, say). I believe the consensus is that the improved growth will appear on a longer horizon. In the United States, there is a debate about “dynamic scoring” that discusses this. There are plenty of resources for dynamic scoring on the internet, this is a recent presentation by the CBO - link to CBO presentation.

So one could expect that such a policy might result in a higher deficit in the short term (1-2 years), but will the tax cut lead stimulate enouh new activity on a longer horizon - that leads to a falling debt-to-GDP ratio? That question would either require analysis by experts on the Italian economy, or it would be largely opinion-based.


Why doesn't government work more like a company when it comes to taxes?

Getting the facts straight
First before addressing the question, lets get the facts straight as arguing based on incorrect premises is not good and in this case this is also relevant to the answer.

In government, the revenue is fixed (most of the time)

This is completely incorrect. In fact government revenue varies quite significantly most of the time.
According to OECD data on government revenue the tax revenue varies a lot as you can see in the graph I made in R below. Even the countries for whom the lines look flat they look that only because of the scale but if you would 'zoom'in you would see quite a large variation in government revenue.

Second, it is also not true that government has no incentive to operate efficiently. There is already an incentive for government to be cost-efficient as in the long run government must balance its budget. That is in the long-run all taxes $T$ must be equal to all government expenditure $G$.
A less cost effective government will have to ultimately impose higher taxes than more efficient government. Since people dislike paying taxes government officials already have some incentive to be cost effective. Arguably this incentive is not strong due to all political and public choice considerations. See for example Public Choice III by Dennis Mueller which provides wide overview of literature on public choice which also discusses why inefficiencies arise in government provision of public goods and redistribution.

Addressing the question
There would be several downsides to such a proposal. You state that:

The government & parliment (as usual) outline what they what they will fund in the next year, and outlines the overall cost. Then, the percentage of taxes (income, wealth, capital gain) are determined ... according to that cost.

This is terrible idea as it does not allow government government to have counter-cyclical tax policy which is undesirable and destabilizing for economy.
The standard textbook policy advice (see for example Blanchard et al. Macroeconomics: an European Perspective)  for a government when it comes to spending ($G$) and taxes ($T$), over business cycle is that government should run deficits during recessions $G>T$ and surpluses during expansions $G<T$ in order to help smooth the business cycle. Yet at a core of your proposal is that in short run $G=T$ which would prevent government to fiscally respond to recession. Now, as discussed before in long-run $G=T$ as government has to ultimately respect its intertemporal budget constraint, but this should be done by balancing deficits and surpluses over business cycles not every year.
Moreover, you make your tax proposal too vague to make any historical comparisons. On the balanced budget part there were historically countries that run or tried to run balanced budget. A recent example would be Germany, but such proposals are usually always abandoned during the recessions for the aforementioned reasons.


Depends on WHY you have the Euros.

If you are saving up with a BMW or a holiday in France, it is very different to you holding them just hoping the Euros will go up.

I expect that if Greece leaves OR if Greece remains after a REAL agreement is made and KEPT TO, then the Euro will not be effected much.   But if Greece remains, but nothing changes in Greece, then the Euro will be dragged down by the next set of counties that think they can getaway with it.


How have central banks been injecting currency after crises since 2007?

The question touches several points, but I can try to explain the "where did the money go in the crisis" with an example from Portugal:

Through automatic stabilizers, and
due to acquisition of bankrupt banks and companies, absorbing the losses in the public budget.

With the crisis several companies and financial institutions went bankrupt, meaning that the government increased spending with the automatic stabilizers such as unemployment subsidies => more public spending due to more subsidy and loss of tax collected from company and salaries

Allegedly due to international toxic assets, as well as some media speculation, some banks couldn't face their responsibilities anymore, declaring bankruptcy - see BES bank for example. People were in panic because the money they had in the bank could disappear suddenly, so the government nationalized the bank, guaranteeing everyone's deposits, assuming the losses of the bank. This means that the government increased its costs in millions of euros to save this and other bank, in the interest of national financial stability.

If you are more looking at how central banks inject money in the economy, take a look at money supply tag.


Consumer Utility Maximization equivalent of economies of scope

Nothing explicit, but I think that there is no real need for a special consumer theory...

If we see diversity as similar yet not identical products, for example red and yellow shirts, then for the theory to hold, you should have specific demands for yellow shirts and red shirts

We touch marketing basics, but this is the core element of differentiation:
Markets split into sub-markets until an imperfectly imitable product is being made, then a premium can be collected on
A premium labeled red shirt for example


Calculate Elasticity of demand in this example

Yes, the answer is -0.8.

Price Elasticity of Demand is defined as:
$$e_{\langle p \rangle} = \frac{\mathrm{d} Q/Q}{\mathrm{d} P/P}$$

You're given :
$$\mathrm{d} Q/Q = -0.08$$
and
$$\mathrm{d} P/P = 0.1$$

ergo:
$$-0.08/0.1 = -0.8$$

I'm reasonably certain that by "Interpret your answer," your professor means that s/he wants you to explain what that number means, in layman's terms.

You can always check the Wikipedia entry.


Labor time and the exchange value (Marx). What about the quality of the raw materials?

The example you've picked is slightly complicated, because I think there are two possible reasons for the difference in price.  Both could contribute at the same time, depending on your view of how wood production works in your example.

1. The labour value invested in growing, harvesting and processing better quality wood (prior to furniture manufacture).  It is natural to suspect that higher quality wood might be harder or more time-consuming to grow, otherwise nobody would bother growing low-quality wood.

2. Rent.  Marx wrote at length about how rent is extracted in capitalist economies.  If growing the best wood is a matter of owning the most fertile land, then the landowner would naturally extract that surplus value from the wood production process.  This rent then shows up in higher prices for higher quality wood.


What's the econ word for "Savings and efficiencies gained via consolidation"?

If you are looking for term for the reason why the efficiencies occur it would be the increasing returns to scale.


A feasible rational payoff that is not an equilibrium payoff in the repeated game

It is true because of the discounting. If the discount parameter were $\delta = 1$ then players 1 and 2 could alternate playing TL and MR in equilibrium and then reach the average payoff vector $1,1,5$. But if $\delta < 1$ someone will not play along. Suppose the players are supposed to start with TL. Then MR would follow in the next round then, then TL again, and so on. In this case
$$
U_1 = 0 + 2 \cdot \delta + 0 \cdot \delta^2 + 2 \cdot \delta^3 + ...
$$
Yet by always playing B player 1 would achieve the payoff
$$
U_1' = 1 + 1 \cdot \delta + 1 \cdot \delta^2 + 1 \cdot \delta^3 + ...
$$
which is larger then $U_1$. Similarly player 2 would not want to start with MR. So these are not equilibrium playoffs. (Not even Aumann's correlated-equilibrium would do the trick.)


Per capita income growth rate is difference between income growth rate and population growth rate?

Knowing that:
$$\frac{\Delta PCI}{PCI}=
\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) $$

If we rearrange the terms on the right hand side:

$$\frac{\Delta PCI}{PCI}=\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

In order to get your last equation we would have to log the variables, for logged growth rates. This makes things easier because we can think of the variables in terms of percentages.

Logging the variables we get
$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\Delta Y}{Y}\right)-\ln\left(\frac{\Delta P}{P}\right)$$

in more formal notation used in macroeconomics:

$$\hat{PCI}=\hat{y}-\hat{p}$$

hope this helps.


What does it mean when R&D is "3% of GDP"?

Expenditure, which can be thought of as a form of investment that does not involve the production of physical capital.

  First, the satellite account provides estimates of expenditures on R&D that are designed to be used in conjunction with the national income and product accounts measures. Second, it treats R&D expenditures as a form of investment, recognizing the role R&D plays in adding to knowledge and in developing new and improved processes and products that lead to increases in productivity and growth. Third, it provides estimates of the stock of knowledge capital.

[...]

  Expenditures on R&D can be viewed as generating future income and product. With this view, a case is made for treating them as investment, paralleling the treatment of business expenditures on structures and durable equipment, and for recognizing a stock of intangible capital, just as there is a stock of tangible capital.

Source: BEA’s R&D satellite accounts.


Nash equilibrium two players in a joint effortful project

You are actually almost correct. Your first approach shows the normal form of the game, but in an incomplete way, which is why you find "extra" Nash equilibria. The possible strategies of the players include all positive effort levels not just the ones you show.

The NE (0.25;0.25) is not actually an equilibrium because either player can reduce their effort a tiny amount (but less than the amount in your game) increasing their payoff. For example the optimum amount of effort for player 1 if player 2 uses effort level 0.25 is 0.1875 which is not in your game. If player 1 opts for an effort level of 0.1875 player 2's optimal effort is 3/4*0.1875=0.140625.
Continuing this reasoning you'll find eventually there is only one NE: (0;0) which is what you had correctly identified using your second approach.


Which elements to forecast/estimate Oil/Pretroleum derivatives, would you consider?

Often people will first find a spread between the derivative products and the base one. This spread, is likely to be a reflection of the marginal cost of refining that last unit of the derivative product. Therefore it is likely to depend on demand relative to supply of refining capacity: If there's a new refinery for this product coming online, then you'd expect the spread to fall. If the demand for construction goods that use this refined product is increasing, then you'd expect and increase in the spread.

Often several plastics and refined produces are substitutes of each other, like natural gas and gasoline are somewhat substitutable for each other. I would use the recent price of the other refined products to predict the price of polyethylene or polyurethane or its spread w/respect to crude oil.

This wikipedia article has more on the spread and how people think it is determined.


What would happen in Venezuela if the currency exchange control is removed overnight?

You could probably think about the policy trilemma. It's the idea that a country can only have up to two of the following: free capital flow, a fixed exchange rate, and independent monetary policy. Right now it seems that Venezuela wants a stable currency exchange (so they control it) and they want reserve currency available to stabilize the economy if need be and have independent monetary policy.

So if you got rid of currency exchange controls so people could buy foreign currency more readily, in the short run, you'd lose control of the exchange rate without gaining anything. Long term, since the money market is clearing, and assuming Walras' Law holds, then capital will be able to flow more freely and contribute to economic growth. It might force Venezuela to be serious about not causing hyperinflation for that to happen that.


Can printing money with simultaneous government intervention to increase supply prevent inflation?

Yes it can. There is nothing wrong in the above. The money market equilibrium is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing we get:
$$\ln P = \ln M + \ln V - \ln Y $$
Consequently, ceteris paribus, if $M$ increases by $1\%$ and $Y$ increases also by $1\%$ then the net effect on price level will be zero. The crucial assumption is of course that $V$ remains constant otherwise there could still by inflation/deflation.
However, this is not because this would balance supply and demand - or at least not as understood in econ jargon. When people talk about balancing supply and demand they just talk about them being in the equilibrium. Aggregate supply and demand can be in equilibrium even if inflation increases. Rather here it would be more appropriate to say that both supply and demand shift to the right by amount that makes leaves the price unchanged.


Why are German bonds used as the reference for the calculation of Eurozone bond spreads?

German bonds placed as a reference in the eurozone is a form of tacit understanding and i don't think you will find official conventions about this. This practice comes from market Finance pricing technics, as the CAPM, which use a risk (-free-rate) referential to price different kind of assets. Given the higly positive trade balance of Germany, the stability of its revenue base, etc... the country is seen by financing markets as higly creditworthy, i.e. viewed as the "most risk-free" of the eurozone.


What is the purpose of the local non-satiation assumption in the first welfare theorem?

The assumptions are different. First one states that if a bundle is better than the optimal one the consumer cannot afford it. The second one states that if a bundle is as good (not necessarily better) than the optimal one it has to cost as least as much, not less.

Consider a space with just one type of good, $x$, and a utility function $U(x) = 0$. Let the endowment of the consumer be $w = 1$. While
$$
\text{if } x_i \succ x_i^* \text{ then } p_ix_i > p_i w_i
$$
is still true,
$$
\text{if } x_i \succeq x_i^* \text{ then } p_ix_i \geq p_i w_i
$$
isn't, because $x^* = 0$ is both optimal and feasible, thus
$$
x^* \succeq x^* \text{ AND } p x^* < p w.
$$
More complicated examples (multiple goods, global non-satiation fulfilled) may also be constructed.


Adjusting GDP for Environmental and Resource Impacts

A good place to start is a report by the U.S. National Academy of Science called Nature's Numbers. The upshot: There is a consensus among economists that it is possible to refine standard measures of GDP rigorously to incorporate adjustments for impacts on environment and natural resources. Implementation is in practice difficult due to data problems.

Most alternative indicies, e.g. GPI, are not consistent with national income accounting principles. But since there is a demand for such indicies, there is a supply response.


What is the logic behind this sentence in Keynes's General Theory?

This sentence says something very simple: If labor supply is a function of many arguments, not just the wage, $L^s = f(w,X,Y,...)$, then the supply curve, which is a line in a two-dimensional space of $(L^s,w)$ keeping everything else constant, will shift as a whole("bodily") if these $X's$ and $Y's$ change.

What could these "$X$'s and $Y$'s" be? First, if the author defined labor supply as a function of nominal wages, then certainly, a change in prices of consumption goods would shift the labor supply curve.

But even if the labor supply is defined as a function of real wages already, then there are other "$X$'s and $Y$'s" that may shift the whole curve, even if the real wage has not changed: for example, direct transfers from the Government (not as a windfall, but as a systematic policy). Since we assume that working is a "bad", people will want to work less for every real wage level, if they get an increased supplementary income through the Government.


The use of preference relations in choice theory and the $\succsim$ symbol

About your second question, Gérard Debreu, in his book Theory of Value, says in the notes of chapter 1 that the $\succsim$ notation is due to Herstein and Milnor. So, probably, this is the origin of symbol for preferences.


Clarification on amortization of constant payments

Let $L$ be the loan amount, then $L=P\frac{1-(1+i)^{-n}}{i}$

The balance at time $t$ is defined as the present value of the remaining payments,

$B_t=P\frac{1-(1+i)^{-(n-t)}}{i}$

By using the following equations

$I_k+C_k=P$ and $I_k=iB_{k-1}$ where $I_k$ is the interest paid at time $k$ we derive the following equation

$C_k=\frac{P}{(1+i)^{n-k+1}}$ and after a simple rearrangement we arrive at the formula you presented.


Why does price elasticity of demand formula work?

Why are dividing the change in quantity and price by the averages of the quantity and price change? Is that a way of calculating the percentage change?

Yes, elasticity is a measure of proportional response.  The math you've shown is the "midpoint" method.  It's basically a linear approximation of elasticity, and you'll abandon it once calculus starts showing up in your learning.  I would suggest not spending a huge amount of time trying to understand it, but only for this reason.  As long as you understand that elasticity of demand says something about how consumers respond to price changes at a certain price level, that's the key message.

  And why is elasticity a fraction and not a ratio?

It's really neither - it's a quotient of quotients, which yields a pure (that is, unitless) number. You can check this for yourself if you explicitly write out the units for the things you're looking at.  This is called "dimensional analysis" and can be extremely useful for interpreting complicated expressions.  The way you do it is to replace each quantity with the thing it "counts" in [square brackets]:

$\frac{2(Q_2 - Q_1)}{Q_1 + Q_2} \longrightarrow \frac{[purenumber]([apples]-[apples])}{[apples]+[apples]} \longrightarrow \frac{[purenumber][apples]}{[apples]} \longrightarrow [purenumber]$

  Would it be better to write: as price changes by 5 percent, quantity decreases by 10 percent, and write that as 5:10?

In general I would caution against using ratio notation except when playing with statistics (and even then, I personally find them tricky to work with). Here, you may risk confusing a ratio of absolute changes (not what you want to express) for a ratio of percentage changes (this is what elasticity actually is).


Preference relation value representation without uncertainity

First, consider the preference relation $\succeq$ defined by
\begin{equation*}
x \succeq y \text{ if } x_1 + \cdots + x_n \geq y_1+\cdots+y_n
\end{equation*}

This preference relation satisfies your assumptions. Indeed, suppose that $x \succ y \succ z$. You can check that $y \sim \lambda x + (1-\lambda) z$ for a unique $\lambda$ defined by
\begin{equation*}
\lambda = \dfrac{y_1+\cdots+y_n-(z_1+\cdots+z_n)}{x_1+\cdots+x_n-(z_1+\cdots+z_n)}
\end{equation*}

And these preferences obviously admit the utility representation $u(x)=x_1+\cdots+x_n$.

Now, consider the lexicographic preference relation $\succeq$ defined (I take $n=2$ for simplicity) by
\begin{equation*}
x \succ y \text{ if } (x_1>y_1) \text{ or } (x_1=y_1,x_2>y_2)
\end{equation*}
Lexicographic preferences have two important properties:

$x \sim y$ if and only if $x=y$
they have no utility representation

If $x=(1,1)$, $y=(0,1)$ and $z=(0,0)$, we have $x \succ y \succ z$. In addition, any $\lambda>0$ is such that $\lambda x + (1-\lambda) z \succ y$. Thus assumption 2. is not valid, and the conclusion of the theorem (existence of a utility representation) is not valid either.


When recession strikes, it's prone to the currency equivalent of "bank runs" where everyone attempts to trade in their paper money for gold at once, causing a drastic reduction in the money supply, rising of interest rates, and the dreaded deflationary spiral. In the face of a recession, you want the opposite to happen.

During the Great Depression, every major currency abandoned the gold standard, and the time of abandonment is seen as being correlated with their subsequent recovery. The U.S. abandoned it for good in 1971 while the economy was facing stagflation and foreign countries were demanding to exchange their U.S. dollars for gold, of which we didn't have enough gold reserves to fully satisfy.

A fiat currency gives policymakers many more tools to stimulate the economy to meet their objectives. Tying your currency to gold limits the extent to which you can expand your money supply. And don't forget that under fractional reserve banking, it's the banks that create most of the money supply, not the Fed.

On top of that you're prone to sudden random money supply expansions due to new mining windfalls, wasting a commodity that otherwise could be put to use in the real economy, and have to deal with the increased transaction costs of shipping and storing physical gold.

http://www.bloomberg.com/news/articles/2011-08-04/the-nixon-shock


Has anyone studied how people's indifference curves can change?

One important thing to remember is that people don't necessarily "have indifference curves". More correctly, indifference curves are a tool used in economic literature to model human rationale and decision making.

Transitivity is one very common assumption regarding utility functions, and it lays on a  fairly reasonable explanation, which basically assumes people are "smart enough to do the math" and figure out that if they prefer A to B and B to C, then they should also prefer A to C.

So really what you're saying is that in your experiment, you found evidence that this doesn't model human behavior all that well. It's a perfectly reasonable claim, and if you think of it, such things make sense - since economic models are, almost by definition, a huge simplification of reality. Trying to adjust the models to better fit observed human behavior  is one of the things that drives the field forward.

It turns out that the issue you brought up - whether people actually display this "transitivity" quality -  is a pretty fundamental issue and has been studied quite intensively do date, in different scenarios.

To read more about it  and get to know the work of people who previously challenged this assumption, try to look up "Transitivity of Preferences". This is one example of such paper which should get you into the subject:

  Any claim of empirical violations
  of transitivity by individual decision makers requires evidence beyond a reasonable doubt. We discuss
  why unambiguous evidence is currently lacking and how to clarify the issue

Good Luck!


Is there a financial hypertrophy ? Could you make a case for finance?

Financial markets can look unimportant to layperson but empirical research shows that well functioning financial markets have direct impact on economic growth (have a look at Levine,
1997 and Levine 2005), which in turn benefits society greatly. For example, well functioning financial sector is crucial for economic development which helps to rise living standards of the poor (see Beck et al., 2007, Honohan, 2004). In addition, medium and small firms get large
share of positive benefits brought about by financial development (see Beck and
Demirguc-Kunt, 2006).
As a matter of fact non-trivial portion of development economics is dedicated to figuring out how to bring well functioning capital/financial markets you can see in the west to the poor countries (see Banerjee and Duflo Poor Economics: A Radical Rethinking of the Way to Fight Global Poverty).
The studies above look at financial sector very broadly and consequently we can conclude that actually most of the financial sector is useful and beneficial for the society. This does not mean that every single activity that occurs on financial markets is but on the net empirical literature shows large benefits of financial markets for the economy.
For example, derivatives you mentioned in your question help price discovery and are often used for hedging (which you actually put as an example of beneficial activity).
Unfortunately, it is not easy to just list useful and harmful activities. The same activity like mortgage backed security can be excellent tool for financial development depending on the characteristics of the market and appropriate regulation, or it can be harmful to the economy if market has some serious imperfections or is improperly regulated.
In fact the same holds even about less exotic types of financial activity, such as simple insurance. For example, adverse selection (an example of market failure) can in principle lead to undesirable outcomes in the insurance market (see some examples in Freixas Microeconomics of Banking) if it is not corrected by some regulation or private mechanism.
Consequently, one can't really provide any simple list of socially 'bad' and 'good' assets. It would be possible to discuss examples when different financial activities cause problems and when they are beneficial, but providing literature review on all financial instruments would go well beyond the scope of one SE answer (heck it would likely take more than one volume as a book). If you are interested in this sort of discussion you might refer to books such as:

Faia & Ester, Financial Regulation: A Transatlantic Perspective
Armour et al. Principles of Financial Regulation

However, as noted previously if you just want big picture the literature shows that on the net financial markets are quite beneficial for the economy.


Necessary conditions for the existence of a competitive equilibirum

If the competitive equilibrium exists, it must lie on the contract curve where two consumers' marginal rate of substitution are the same. Also, this equilibrium lies inside the zone of mutual advantages (area where the indifference curves intersect with the endowment) because trade is voluntary. If consumers’ marginal rates of substitution vary continuously with their consumption levels then a competitive equilibrium exists


Is it possible to have a preference relation that is complete but not transitive?

I'm quite surprised nobody has picked the obvious one:

I prefer rock over scissors.
I prefer scissors over paper.
I prefer paper over rock.

Complete, definitely not transitive.


Is a hyperlink a kind of informational public good?

In the Samuelson's definition of public good, the good must be non-excludable and non-rivalrous.

Hyperlinks (if public and pointing to public pages) are non-excludable, and --to the condition of not overloading the hosted content-- non-rivalrous.

The frequency of publication (and other criterions) of hyperlinks pointing to specific pages/domain, is used by ranking algorithms as a proxy of relevance for the users.

Hyperlinks through ranking algorithms create a positive externality (i.e. access to more relevant web-pages).
To the extent that the access to the result of those ranking algorithms is not excludable (you don't need to log in the use google, duckduckgo, etc), you can say that the result of this positive externality is a public good.


inversely proportional utility functions

Converting my comment to an answer...

I suppose the expression you wrote is almost correct, except perhaps for a notational issue. I would write $u(x,y)=[w(x,y)]^{-1}$, because $w^{-1}(x,y)$ is usually used to denote the pre-image of $w(x,y)$.

More generally, inverse proportionality between $u$ and $w$ is defined up to a multiplicative constant $K$; that is, $u(x,y)\cdot w(x,y)=K$ for some $K\in\mathbb R$. In the above case, you implicitly assumed $K=1$. So generally, you'd have
$$
u(x,y)=K[w(x,y)]^{-1}=\frac{K}{w(x,y)}.
$$


What is the upper limit of circulating money on earth?

What you are asking, dressed up in economics lingo, is as follows:

Is there a limit to real gross world product growth?

Short answer

A qualified no.

Longer answer

First of all, developed countries do indeed heavily benefit from the work of developing countries. Globalization has found a way to incorporate cheap labor from overseas for economic growth. However, incorporation of more labor is not the only way the economy grows. Innovation such as new ways to manufacture things more efficiently is another. Just search for "ways in which GDP grows" and most results should apply. But the bottom line is one could imagine a situation were autonomous robots produced expensive cars that were freely available to everyone without having to exploit human labor. Of course, whether the environment could sustain such intense resource usage or whether humans can bring automation/AI to such a point may put practical limits.


When she chooses the bundle (6,6) when the prices are (6,7), she spends
\begin{equation*}
6*6+6*7=78
\end{equation*}
The total price of the bundle (10,0) given this price system is
\begin{equation*}
10*6+0*7=60<78
\end{equation*}
Therefore she could afford the bundle (10,0) if she wanted to. The fact that she chooses (6,6) reveals that she prefers (6,6) to (10,0).

Given the price system (5,5), when she chooses the bundle (10,0) she spends
\begin{equation*}
10*5+0*5=50
\end{equation*}
And the price of the bundle (6,6) would be
\begin{equation*}
6*5+6*5=60
\end{equation*}
Therefore (6,6) is not necessarily affordable if her budget is comprised between 50 and 59. This is why (10,0) is not revealed preferred to (6,6) in that case, and thus there is no evidence that she violates WARP.


Is there any overlap between evolutionary and behavioral game theory

I don't think behavioral game theory is inspired much by biology. Rather it's motivated by the discrepancies between theoretical predictions and the choices observed in lab experiments. It seeks to improve the predictive accuracy of game theoretic models by introducing behavioral assumptions that allow players to behave in ways different from the traditional Bayesian rational benchmark.

Many behavioral game theories such as level-k and cognitive hierarchy are non-equilibrium theories. This is in contrast to evolutionary game theory which studies the stability of equilibria with respect to selected evolutionary dynamics.


Are there any studies that show that "Affirmative Action" hurts the economy?

Very interesting question. Those at cafehayek.com seem to dislike it a lot but don't really back it up which is disappointing. At first glance, it is very easy to see how affirmative action could hurt the economy, and construction a model to show this would also be pretty straight forward.

See here(paywalll) (Moro & Norman, 2003). There is pretty strong intuition both ways so I'd think that any valuable study would be empirical.  This article looks pretty light empirically but does support the view you are looking for and provides a useful model.

  We consider a model of endogenous human capital formation with competitively determined wages. Discrimination is sustainable in equilibrium in the presence of two distinguishable, but ex ante identical groups of workers. An affirmative action policy consisting of a quota may ‘fail’ in the sense that there still may be equilibria where groups are treated differently. However, the incentives to invest for agents in the discriminated group are improved by affirmative action if the initial equilibrium is the most discriminatory equilibrium in the model without the policy. The welfare effects are ambiguous. It is possible that the policy makes the intended beneficiaries worse off: even if the starting point is the most discriminatory equilibrium the expected payoff may decrease for all agents in the target group.

Most work in this area suggests that the effects are welfare ambiguous/slightly positive.

This Paper (Leonard, 1990) discusses the idea that affirmative action does not work.

To borrow from the conclusion:

  Direct tests of the impact of affirmative action on productivity find no significant evidence of a productivity decline. However, since the productivty estimates are not measured with great precision, strong policy conclusions based on this particular result should be resisted.

This paper (Long, 2004) looks at the impact of the removal of affirmative action on college applications.

  This paper evaluates the behavioral response of college-bound high school seniors to changes in college affirmative action admissions policies during the late 1990s. After the elimination of affirmative action in California and Texas, the gap between the numbers of SAT score reports sent by non-minority and minority students to in-state, public colleges significantly widened...  predicts a large decrease in the number of reports sent by minority students and a large increase in non-minority score reports sent to top-tier public colleges as a result of changes in the students’ probabilities of acceptance.


Measuring income effect before knowing substitution effect

You can definitely do so! There are two versions of the income effect. The one that you are referring to is called the Equivalent variation. The two ways of looking at the problem were introduced by Sir John Richard Hicks.
To see the impact of a price change in monetary terms, we ask how much money should have been taken before the price change to leave a consumer at the same utility level he attains after the price change.
This is the same thing as you had asked: can we reverse the two procedures and measure the income effect first by drawing a line parallel to the original budget line and tangent to the new indifference line?
I hope this has helped but if you wish to explore further, see the 'Consumer Surplus' chapter in Hal Varian's 'Intermediate Microeconomics'.


Bayesian Correlated Equilibrium in a one player case: relation with literature

There are a few imprecisions in the way you formalize things. For example saying " is the baseline choice problem" does not make a lot of sense, because a problem should include the DM's utility, the available actions, and the belief about the state of the world. You only included the latter.

Regardless of notation, I think that you are missing the key connection between these two papers. Let me summarize what each paper does first and then draw the connection for you.

The concept of BCE characterizes the answer to the following question: If the DM were to observe some signal (informative or not) what are all the actions that would be optimal?
Of course the DM can use mixed strategies, so we talk about distributions over actions. Also, it is clear that depending on what information is contained in the signal, the actions of the receiver can vary. So the set of BCE collects all possible actions for all possible signals. Note that the concept of BCE is agnostic about where this extra information is coming from.

So yes, the distributions of actions that are in the set of BCE are obedient (i.e. optimal given some belief), and the beliefs that rationalize these actions are consistent with the prior (I.e. they are the Bayesian posteriors derived from some signal given the DM's prior).

In contrast, Kamenica and Gentzkow (KG) set up a game where a sender chooses what signal to give a receiver and then the receiver takes an action that affects both players. Kamenica and Gentskow presented the simplest case, where the sender knows the prior of the DM and there is only one sender.

Their approach has some limitations because finding the optimal signal can be complicated; especially if the state space is not binary. In their paper, it is also not clear at all how to solve the problem if the receiver has more information than $\underline S$, or if there is more than one receiver. So yes KG assume that the receiver has some prior information $\underline S$, but in equilibrium the sender usually provides more information, so your statement "Kamenica and Gentzkow (here) characterize the set of distributions over actions and states that the sender could induce through picking the null experiment" is false.

Now, let me try to connect the two papers:

Bergeman and Morris (2019) found that we can drastically simplify the analysis of the game presented by KG if instead of maximizing over signals, we maximize the sender's utility choosing the distribution of actions that can be induced using some signal. They make the connection that if a distribution of actions can be chosen by the sender, it must be because it is optimal for the receiver to choose those action after observing some signal $S\geq\underline S$, i.e. if the distribution over actions is a BCE.

This is nice because the set of BCE's is relatively easy to find and to work with. So the game that KG setup can be dramatically simplified into a game where the sender chooses its favorite BCE.

Once you realize this connection, you can overcome many of the limitations in KG's approach. The concept of BCE can be easily extended to multiple receivers, and even receivers with private information. The authors showcase in their 2019 paper, the power of making this connection, and show how to think formally of receivers with more information than $\underline S$, among other things.


Are Open Market Operations a Gift to the .1%?

Yes they are "gift" (or rather a stimulus), but not exclusively to the 0.1%. There's basically "trickle down" by inflating the M2 money supply, i.e. bank loans, which hopefully include loans to companies where the "average guys" work.

So in theory, banks have more funds to lend, at a low rate of interest, encouraging households and businesses to borrow. [...]
The Bank of England estimates that ultra-low interest rates and QE meant that the average household's income was £9,000 higher than it would have been by 2018.
It argued that younger people actually benefitted most, as they were more likely to be in work and reliant on wages for income.
By contrast, savers, who tend to be older, received negligible return on their money.

It's also true that QE and cheap credit in general have been a boon to the those well off, fueling asset prices (and not just stocks, but also real estate etc.)

But the starkest impact of QE has been on wealth, the value of assets, from housing to pensions.
The money received by financial institutions in return for bonds sometimes got diverted into other investments - bulking up the prices of the value of shares or property, benefiting those already at the top of the tree the most.
Shares in the FTSE 100 index have, on average, more than doubled in value over the past decade. [...]
property prices have risen by 43% over the past decade - far more than they would have done in the absence of the emergency injection of funds.
So they've vastly outpaced earnings growth; buying your first property and moving up has got even harder.
A 30-something today is less likely to own property than their great grandparents.

This is for the UK but the effect of QE and cheap credit have been pretty much the same everywhere.
So yeah, if you want to be a cynical Marxist, you  could say QE has created even more "wage slaves" than before, i.e. an increase in wealth inequality in some sense. The ECB hower disagrees with such a conclusion:

Quantitative easing in the euro area through the ECB’s asset purchase programme (APP) has stimulated economic activity and asset prices, affecting income and wealth inequality among households. It has decreased income inequality, mostly by reducing the unemployment rate for poorer households, but also, to a lesser extent, by increasing the wages of the employed. Quantitative easing has also helped to reduce net wealth inequality slightly through its positive impact on house prices.

Meh. There's a more rounded review of the research on the effect of monetary policy on inequality in an open access paper. Basically the effects on inequality seem to be ambiguous or modest at best.

Our review suggests that empirical research on the effects of conventional monetary policy on income and wealth inequality yields mixed findings, although there seems to be a consensus that higher inflation, at least above some threshold, increases inequality. In contrast to common wisdom, conclusions concerning the impact of unconventional monetary policies on inequality are also not clear cut.

And it also features a quote from Bernanke:

the effects of monetary policy on inequality are almost certainly modest and transient. (Bernanke, 2015).

N.B. When you say "trickle down doesn't work" (with respect to inequality), you're probably mixing the jars with the fiscal policy, i.e. tax cuts... which might be good separate question, although it might have been asked here before. Fiscal policy is generally considered the main "tweaking knob" for inequality.


Why isn't the cost of shoes affected by their size?

Most of the price of creating a shoe is the cost of labour to make it and the cost to ship it to the store. The cost of the materials needed to make the shoe is negligible. So, they can ignore the difference in material costs without losing too much money.

But why would they want to lose any money at all? Logistics. Keeping track of the different prices per shoe, printing the different labels, making sure the shoes are sold at the right price, paying people to determine how much of a price difference the company should charge are all things that cost money.

So the company is looking at creating a giant logistical nightmare for themselves, costing them who knows how much money, all for a penny difference between a size 8 and a size 13.

Trying to charge a different price would be a case of being 'penny smart; dollar stupid'.


Here is a link to a blog post aiming to answer your question.

Also, I am not sure if you mean the NFP is problematic or if general payroll estimations are problematic. I would say the later is, and the blog post highlights some reasons why.


Log-linearisation Euler Consumption Equation

To simplify matters, let's call the right-hand side of your starting equation $X_t$ Then, I start just like you did with

$1 = X_t$

The difference between your solution and Gali's is that you took the Taylor expansion around

$\log(1)=0=log(X_t)$  which implies that also the steady state equals 0, so we can simply subtract it to get log-differences,

whereas Gali used

$1 = \exp(\log(X_t))$ with steady state $1=\exp(\log(X^*))$

Let's define $x_t=\log(X_t)$.

By definition of the Taylor expansion this gives

$T(1) = \exp(x^*)+\exp(x^*)(x_t-x^*)$

Note that the exponential function is identical to its derivative. We know that $\exp(x^*)=1$ from above, so we can simplify to

$T(1)=1+(x_t-x^*)$

At this point, you are correct that the $\rho$ cancel out. Instead of defining $\hat{x}_t=x_t-x^*$ but Gali chooses to replace $\rho=\pi +\sigma \gamma-i$, which is the steady state condition, which is why $\rho$ stays on the paper.

Note, however, that the "approximation error" from taking log differences instead of Gali's approach is tiny, i.e. for $\beta=0.99$ it is only $-\log(0.99)\approx 0.01$

Have a look at appendix 2.1 of Gali's book! It is quite complicated, but together with this post, I hope you get it!


How to derive formula for marginal probability of choosing nest in nested logit model?

As it turned out, my previous logic was wrong. Here is how it should be done.

Marginal probability of choosing nest $k$ is
$$P_{nB_k} = P\left[\max_{j\in B_k} U_{njk} \geq \max_{j\in B_s} U_{njs}, \forall s\neq k \right]\\
= P\left[W_{nk}+\epsilon_{nk}+\max_{j\in B_k}(Y_{nj}+e_{nj}) \geq W_{ns}+\epsilon_{ns}+\max_{j\in B_s}(Y_{nj}+e_{nj}), \forall s\neq k \right]$$

Then as $e_{nj}$ is iid $Gumbel(0,\lambda_t)$, $\max_{j\in B_k}(Y_{nj}+e_{nj})$ is iid Gumbel with location parameter $\lambda_kI_{nk}$ and scale parameter $\lambda_k$. The Gumbel distribution is preserved over linear transformations so
$$ \max_{j\in B_k}(Y_{nj}+e_{nj})=\lambda_kI_{nk}+\xi_{nk}$$
where $\xi_{nk}$ is iid $Gumbel(0,\lambda_t)$. Substituting back to marginal probability, get
$$P_{nB_k} = P\left[(\epsilon_{nk}+\xi_{nk})-(\epsilon_{nk}+\xi_{nk}) \geq (W_{nk}+\lambda_kI_{nk})-(W_{ns}+\lambda_sI_{ns}), \forall s\neq k \right]\\
=\frac{e^{W_{nk}+\lambda_kI_{nk}}}{\sum_{s=1}^Ke^{W_{ns}+\lambda_sI_{ns}}} $$

the last equality follows from the fact that $\epsilon_{nk}+\xi_{nk}$ is iid $Gumbel(0,1)$ by assumption on $\epsilon_{nk}$.


Who really pays transaction taxes like VAT or sales tax?

The issue you are asking about is tax incidence. This Wikipedia entry provides a good introduction. In short: it depends on how easily supply and demand respond to price changes relatively to each other, the so called price elasticity of demand and supply.

According to standard theory, it doesn't matter who pays (in the legal/fiscal sense) the taxes, as the burden of the taxes depend on economic circumstances. If demand falls rapidly with higher prices while supply does not, sellers/producers will lower their price (before tax), and therefore carry a relatively large part of a tax.

Of course, there may be all kinds of administrative costs and other 'frictions', that make it more or less efficient to have one group or another pay taxes. Also, there is evidence that people do change their behavior according to how the taxes are paid; see for example this paper, or this document.


Generating profit by leveraging fluctuations

Generally, it's a combination of 2 and 3.
You can't predict that a stock will go up and down in cycles. Stock prices move randomly. Let's say you're looking at one stock, and it goes down so your algorithm buys some and waits for it to go back up. What happens if it never goes back up higher than it started as? Then you lost money.
Even worse: Even if it does go back to where it started with, you still lost money. Stocks have an overall upward trend, so you have to make more than a certain amount of money, or otherwise you would've been better off by just buying random stocks!
Why do stock prices move randomly? According to the "Efficient Market Hypothesis", stocks move randomly because all non-random information is already being exploited by other people (point 3) until it stops being profitable to do so. Thousands of people might get a couple of cents each from the fluctuations.
When something has a regular fluctuation - like ice-cream prices perhaps - there's a reason. Maybe (hypothetically, if ice-cream prices fluctuated a lot) you could make money by buying ice-cream in winter and selling it in summer. You'd have to have a big refrigerated warehouse, and that would be a legitimate way to make money, it wouldn't just be a financial gain.


"manufacturing sector" vs. "manufacturing industries"

It depends how industry and sectors are defined.  In a few settings they could be used to mean the same thing.

However, usually, the term industry refers to a much more specific group of companies or businesses, while the term sector describes a large segment of the economy.

Industry can be used to define the group of organizations or firms that are engaged in the production of related goods or services.  The industry name will depend on the type of service or product made.     Sectors describes the division of the whole economy, which can consist of business groups(multiple industries) which are engaged in related or similar services or products.

Industry are a cluster of different firms while a sector are a group of different related industries.

Since an industry involves a particular process the scope it is more narrow while for a sector it relates to business activities which makes its scope wider.

https://www.wallstreetmojo.com/industry-vs-sector/


Utility Maximisation Subject to Income and Time Constraints

Despite its funny title and although it is quite old you may find this paper interesting:

The economics of the afterlife by Scott Gordon in the JPE. His argument is basically that in the presence of an infinite amount of time there would still be a rate of time preference.

A follow-up paper disproving Gordon's claim was written by Amegashie, but I haven't checked that one in detail. You can find that  here.

Nb credit where credit is due: I would probably never have read the first paper without Yoram Bauman's presentation "Hyperinflation in hell"


With my very limited knowledge of development economics:

$\left(\frac{x_m}{x}\right)^\alpha$ represents the proportion of the population that has an income larger or equal to $x$ where $x\geq x_m>0$ and $x_m$ is the minimum income amount.

Example 1:

Suppose $\alpha\rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $50,000$?

$$\lim_{\alpha \rightarrow 1} \left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{50,000}\right)^1=1$$

This suggests that $100\%$ of the population has an income larger than or equal to $50,000$.

Example 2:

Suppose $\alpha \rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?

$$\lim_{\alpha \rightarrow 1}\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^1=0.1$$

This suggest that $10\%$ of the population has an income larger than or equal to $500,000$.

The story does not end there. What about $\alpha$? It must be the case that $\left(\frac{x_m}{x}\right)^\alpha$ is bounded between 0 and 1 (it is a proportion). As such, it must be the case that $\alpha>1$. In the literature, $\alpha$ is refered to as the Pareto Index. Let us consider another example.

Example 3:

Suppose $\alpha=2$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?

$$\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^2=0.01$$

This suggest that $1\%$ of the population has an income larger than or equal to $500,000$. From this, a high $\alpha$, or a high Pareto Index, suggests a small proportion of high-income individuals.

What is a reasonable value of $\alpha$? We can normalize $x_m = 1$. Suppose we wanted to know the share of total income that is received by those individuals with an income above, say $\tilde{x}$. We take the following integrals:

$$\int_{x_m}^{\infty} x^{-\alpha} dx = \int_{1}^{\infty} x^{-\alpha} dx = \frac{1}{\alpha-1}$$

$$\int_{\tilde{x}}^{\infty} x^{-\alpha} dx = \frac{\tilde{x}^{1-\alpha}}{\alpha-1}$$

Therefore the share of total income that is received by those individuals with an income above $\tilde{x}$, call it $q$, is the ratio of the two integrals:

$$q = \tilde{x}^{1-\alpha}$$

We also want to define the proportion of the high-income population with an income larger than $\tilde{x}$ as $p$:

$$p = \tilde{x}^{-\alpha}$$

The next logical question would be what value of $\alpha$ allows us to reconcile the empirical observation that $20\%$ of the population has $80\%$ of the total income? This implies that $p=.2$ and $q=.8$.

$$q = p^{\frac{\alpha-1}{\alpha}}$$

Taking the log of both sides:

$$\ln(q) = \frac{\alpha-1}{\alpha}\ln(p)$$

Solving for $\alpha$:

$$\alpha = \frac{\ln(p)}{\ln(p)-\ln(q)} = \frac{\ln(.2)}{\ln(.2)-\ln(.8)} = 1.16$$

Therefore, an $\alpha=1.16$ is consistent with what is observed empirically. To answer the question, $x_m$ is the minimum income amount and is usually normalized to $1$. $\alpha$ is the Pareto Index which is a measure of the proportion of high-income individuals. A high $\alpha$ suggests a low number of high-income individuals.

The transformation $L(F)$ simply allows you to compute what % of the total wealth is controlled by those with an income below $\tilde{x}$.


Very unlikely. Terrorism events are not economically significant enough to move a commodity like gold.  In fact, terrorist attacks only effect markets if they are likely to lead to a major economy entering a full-scale war like it was after 9/11, 2001. The cost of terrorism, while at historical height, is still very small to effect the markets significantly: http://fortune.com/2015/11/17/terrorism-global-economic-cost/. Even economy of a country like France that have been recently hit by a terrorist attack is not likely to be affected by it too much. Politicians may exploit the terrorist attack to push their agenda though, which may have negative effects on the economy, like this articles argues: http://www.marketwatch.com/story/frances-real-challenge-is-growing-stagnant-economy-not-fighting-in-syria-2015-11-19.

Yet, gold is a very volatile market and the price of gold can move a lot in any direction for no obvious reason. Purchases and sales of gold by central banks and institutional investors often go unannounced.

Another thing you should keep in mind if you want to speculate on the rate hike is that some big players may be waiting for that moment to BUY gold in large amounts, because that is when they can expect the most people placing sell orders. In that case the price of gold can jump contrary to expectations.


I bought a house on a mortgage. If its price changes later, how will it affect the ease of repaying the loan?

If a mortaged house decreases its market price after the mortage was made, the bank is in risk in case of default - it might be unable to recover the amount it loaned. In practice this doesn't change the amount you owe to the bank or the monthly payment -  you owe the same regardless of the value of the house after making the mortage.

I believe the author meant that the people the banks were lending to would only be able to afford to pay the mortages if the economy kept on booming, and they would remain employed.


Bellman equation for this dynamic programming problem

The "second" constraint appears redundant and it confuses matters. Re-arrange the first one to obtain

$$\tilde{a}_{t+1} = \big[\tilde a_t+(1-\delta )Y_t-\tilde{c}_t\big]R_t$$

This tells us that wealth in the beginning of next period is fully determined by current-period decisions and known states, without any uncertainty whatsoever: we start with the given wealth at the beginning of the period, our income and the interest rate becomes known, we decide and "set aside" the full amount of consumption (for the whole period, at the beginning of the period), and the rest becomes interest-bearing asset.

Why the second constraint, which looks into the future, confuses matters? Because in order to apply Dynamic Programming/Bellman equation, it has to be the case that our optimization problem can be formulated as a recursive one, meaning that the whole many-periods or even infinite-horizon problem can be broken down to a two-period problem. And we have this condition in the above constraint, we do not need the second one.

If $V(\tilde a_t)$ is the value function, the Bellman equation is then

$$V(\tilde a_t) = \max_{\tilde c_t}\big [u(\tilde c_t) + \beta V(\tilde a_{t+1})\big]$$

and we must find the policy function $h(\tilde a_t)=\tilde c_t$  that satisfies

$$V(\tilde a_t) = u[h(\tilde a_t)] + \beta V\left[\big(\tilde a_t+(1-\delta )Y_t-h(\tilde a_t)\big)\cdot R_t\right]$$


utility maximization with nested Cobb–Douglas–CES preferences

Take the following utility function:
$$
\prod_{s \in S} \left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} \right)^{\frac{\sigma_s}{\sigma_s-1}\mu_{s}}
$$
Which is a CES nested in a Cobb-Douglas.
Consider the sub-utility function:
$$
Q_{s} = \left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s - 1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s - 1}}.
$$
Then we can re-write the utility function succinctly as:
$$
U = \prod_{s \in S} Q_s^{\mu_s}
$$
To solve the utility maximisation problem. Notice that $U$ is separable in the subgroups. As such, we can solve the problem using two stage budgeting.

In the first step, we can determine the optimal allocation within each subgroup $s$ by maximize the sub-utility functions $Q_s$ given the total expenditure $E_s$ given on each subgroup $s$.
In the second step, we determine the optimal allocation of income across each subgroup $s$.

For Step 1, we solve the following problem:
$$
\max\left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s - 1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s - 1}} \text{subject to } \sum_{i \in N} p_{i,s} x_{i,s} = E_s.
$$
This is a CES utility function. The first order condition gives:
$$
Q_s \frac{x_{i,s}^{-\frac{1}{\sigma_s}}}{\sum_{w \in N} x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}} = \lambda p_{i,s}.
$$
Then:
$$
x_{i,s} = \lambda^{-\sigma_s} p_{i,s}^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-1\sigma_s}
$$
Multiplying by $p_{i,s}$ and adding over all $i \in N$ gives:
$$
\begin{align*}
E_s = \sum_{i \in N} p_{i,s} x_{i,s} &= \lambda^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-\sigma_s} \sum_{i \in N} p_{i,s}^{1 - \sigma_s}\\
&= \lambda^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-\sigma_s} P_s^{1 - \sigma_s}
\end{align*}
$$
Where $E_{s} = \sum_{i\in N} p_{i,s} x_{i,s}$ and $P_{s}^{1- \sigma_s} = \sum_{i \in N} p_{i,s}^{1- \sigma_s}$.
Then:
$$
x_{i,s} = E_{s} \frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1-\sigma_s}} \tag{1}
$$
Let's compute the value of $Q_s$:
$$
\begin{align*}
&x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} = E_{s}^{\frac{\sigma_s-1}{\sigma_s}} \left(\frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1 -\sigma_s}}\right)^{\frac{\sigma_s-1}{ \sigma_s}},\\
\to &\sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}}= E_{j,s}^{\frac{\sigma_s-1}{\sigma_s}} \left(\frac{1}{P_{s}^{1- \sigma_s}}\right)^{\frac{\sigma_s-1}{\sigma_s}} \sum_{i \in N} p_{i,s}^{(1-\sigma_s)},\\
\to& \sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} = E_{s}^{\frac{\sigma_s-1}{\sigma_s}} P_{s}^{\frac{1 - \sigma_s}{\sigma_s}}\\
\to & \left(\sum_i x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s-1}} = E_{s} P_{s}^{-1}\\
\to& P_{s} Q_s = E_{s}.
\end{align*}
$$
This shows that the price index $P_s$ and the quantity index $Q_s$ are exact price and quanty indices for this subgroup (as is expected as we were maximising a CES utility function).
Let's now go to step 2. Notice that the budget constraint can be rewritten as:
$$
E = \sum_{s} E_s = \sum_s P_s Q_s.
$$
We want to determine $E_s$ to maximize:
$$
U = \prod_{j \in S} Q_{s}^{\mu_{j,s}} \text{ s.t. } \sum_s P_{s} Q_{s} = E.
$$
This is a simple Cobb-Douglas utility function. As such, we know that expenditure shares are proportional to income:
$$
P_s Q_s = E_s = \frac{\mu_s}{\sum_w \mu_w} E
$$
Finally, substitute this into the equation $(1)$:
$$
x_{i,s} = \frac{\mu_{s}}{\sum_w \mu_w} E \frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1- \sigma_s}}
$$


Was the average person no better off in 1800 than in 100,000 BC?

In a detailed review of Clark's book, concluding that its "central theses ... are contradicted by well-known evidence" (p 969), Robert Allen addresses the claim about 1800 and 100,000 BC in a section headed "Malthus in the Very Long Run" (pp 951-5). His criticisms of Clark include:

That he has no evidence of living standards in 100,000 BC, but relies on an assumption that living standards at that time were the same as those of modern foragers (p 951).
That he draws inferences about diet (a key aspect of living standards) from male heights (from skeletons) but ignores evidence from female heights and from other skeletal features, which some have claimed to support different conclusions (p 952).
That his argument that humans of 100,000 BC had more leisure time (another aspect of living standards) because of their forager life-style is not supported by skeletal evidence comparing (more recent) foragers and agriculturalists (p 955).

Some other criticisms in this section are hard to summarise briefly. In conclusion of this section, Allen states: "Economic growth before the Industrial Revolution was not rapid, but it did generate a higher standard of living for most people than that enjoyed by ancient foragers".