Why are we taking the logarithm for risk-averse decision-makers?

Let's step back. In general, utility functions are (monotone increasing) functions $f(w)$ that map wealth into utility. That is, the input $w$ is money and the output $f(w)$ is utility. If we have a distribution $\vec{p}$ over possible wealth outcomes $w$, then my expected utility is $\mathbb{E}f(w) = \sum_i p_i f(w_i)$.

Now, what is risk aversion? Intuitively, it means I prefer a certain outcome to an uncertain one even though they have the same expectation.

Thus, a risk-averse utility function is defined to be a concave utility function $f$. This is natural because of Jensen's inequality: if $f$ is concave, then for any distribution over outcomes of $w$, we have $\mathbb{E}f(w) \leq f(\mathbb{E} w)$. In other words, I prefer to get $\mathbb{E}w$ for certain rather than some uncertain amount of money with that expectation.

To connect back to your question (1), you are comparing the expected utility from two different utility functions: $f(w) = w$ and $f(w) = \log w$. The latter is just a very nice and handy concave (hence risk-averse) utility function. But it is only one example (albeit an important one) of a risk-averse utility functions.

And to connect back to your question (2), to get risk aversion, any concave function will do. But the "more" concave, the "more" risk averse. Meanwhile, risk-neutral functions should satisfy $\mathbb{E}f(w) = f(\mathbb{E} w)$, so that's any "affine" function of the form $f(w) = a\cdot w + b$. Finally, risk-seeking utility functions are convex (and less-often studied as they seem less natural).


Annual Data and Heteroscedasticity (Engle's ARCH test)

Is there evidence for time varying second moments in annual economic data?

Yes, although not that much in finance in particular but in economics in general resounding yes. For example, the highly cited Engle (2001), GARCH 101: The use of ARCH/GARCH models in applied econometrics, besides examples with daily data refers also to some examples with quarterly data. Or more directly an example of this that actually uses low frequency data (yearly/quarterly/monthly) and uses variant of GARCH, GARCH-SK,  is Narayan and Liu (2018). Or another direct example is Kontonikas (2004) who applies GARCH-M models to quarterly and monthly inflation data. So GARCH is and can be to many economic series on quarterly or annual frequency.
Usually people go for quarterly data or yearly data when it comes to macro variables like GDP, unemployment, inflation etc and for higher frequencies when it comes to finance (e.g. stock prices, bitcoin etc). This is because typically having high frequency means more data. However, at the same time having more data is not always better. For example, having access to GDP data on daily frequency would likely just add noise so there is no point in trying to go for high frequency data there. However, the same does not necessarily hold for variables like exchange rates or stock prices. What the optimal frequency for particular research is has to be to an extent decided on case by case basis. Also GARCH models are somewhat more common with high frequency data as they are primarily used to solve issues like volatility clustering, which will be more common problem at higher frequencies, but as the references above show application to lower frequency data is not unheard of.

What lags are commonly used in this case?

Lags in GARCH models do not depend on frequency of data per se (of course it is more likely there will be more autocorrelation in high frequency data than in low frequency data). You include lags in order to appropriately model autocorrelation. This will be case dependent whether you have high frequency data or low frequency data. This being said evidence shows that often it is hard to beat GARCH(1,1) (see Hansen et al. 2001) and this often holds for various frequencies*, but if you would want to do it 'by the books' you should just test for this and in addition you can potentially estimate several models with different $p$ (and $q$ as well even if that was not part of your question) and compare their performance (it is really not that much extra work once you write code for the main model).

* Although note the paper has received some criticism and so you should not just necessarily take it at face value - see here


I found an answer to the second part of the question here https://github.com/oTree-org/otree-docs/issues/2.

It turns out that in oTree,  the variable form which is passed to the template is iterable. Thus one the following code does the job:

{% for field in form %}
        {% formfield field %}
    {% endfor %}

Regarding the first part of the question, I contacted a programmer working at oTree. He confirmed that, in its current version, oTree can only define a fixed number of field. He advised to define in model.py a large maximal number of fields and then use get_form_fields in views.py to dynamically return a list with desired subset of fields.

Notice that if you want to create a really large maximal number of fields, the trick presented in the question using loop and exec remains — I believe — the easiest way to proceed.


There is no trick in that Euler equation. In the New Keynesian model, the Euler equation for consumption is derived from the first order condition for $B_t$, the bond holding. You have to pay attention to the fact that bonds appear in the budget constraint at two moments in time $t$ and $t+1$. Thus you have to compute the foc as
\begin{gather}
B_t:\beta^t\lambda_tQ_t-\beta^{t+1}\lambda_{t+1}=0
\end{gather}
Then substituting $\lambda$ using the foc for $C_t$ and simplifying the betas, you obtain
\begin{gather}
Q_t=\beta E_t\{\frac{U_{c_{t+1}}}{U_{c_t}}\frac{P_t}{P_{t+1}}\}=0
\end{gather}

EDIT: In the NK model, household maximizes its utility function choosing $C_t$, $N_t$ and $B_t$. Thus, you have to compute the FOC for these three variables, i.e.
\begin{align}
B_t:&\beta^t\lambda_tQ_t-\beta^{t+1}E_t\{\lambda_{t+1}\}=0 \\
C_t:&\beta^tU_{c,t}-\lambda_tP_t=0 \\
N_t:&\beta^tU_{n,t}-\lambda_tW_t=0
\end{align}

In order to compute the consumption Euler equation, you need to substitute $\lambda_t$ from the $C_t$ foc into the $B_t$ foc. Then rearranging and simplifying $\beta$ as $\beta^{t+1}/\beta^t=\beta$ you get the equation you wrote above.

You get only one $\beta$ because it is constant over time, it is not index by $t$ as the other variables, it is a constant. Thus whatever time is $C$, you will have the same $\beta$. I think it is well explained in Gali too.


How to find price elasticity in the following demand system?

The price elasticity is the derivative of the demand for each good over price. Demand is lcons * (wi) for each wi.


Perfect risk sharing in Arrow-Debreu w/ the same subjective beliefs over states

In two details you seem to be mistaken:

You need strict concavity of $v(\cdot)$, not concavity.
According to the definition of quasiconcavity, the function $x^2$ is not quasiconcave, it is quasiconvex.

Your main point is correct. If both functions are linear, then $x_i^1 = x_i^2$ no longer needs to hold, as the goods $x_i^1$ and $x_i^2$
are perfect substitutes for consumer $i$. Hence all consumption decisions are correct as long as $i$ trades at a price ratio of 1. This is indeed the equilibrium price ratio. The same can be said for the other consumer.


Calculating natural rate of unemployment

The model you fit is simply inadequate to estimate the natural rate of unemployment any results from it will be completely unreliable, so I am not surprised if they make no sense. Furthermore, natural rate of unemployment is not necessary one where inflation is 0, that is Non-Accelerating Inflation Rate of Unemployment (NAIRU). NAIRU is often used as a proxy for natural rate of unemployment but the two are not the same (for some non-technical explanation of differences between natural rate of unemployment and NAIRU see Investopedia).
Estimating NAIRU:
First to estimate NAIRU you typically need multiple equations not just one (see Carnot et al. Economic Forecasting and Policy. 2nd ed. pp 104).
For example, NAIRU can be obtained by combining producer price equation with a Philips curve expressed as a wage setting equation in a wage-price loop. So we would start with a system like:
$$ p^s_t =  w_t - \theta_t +z_t + \alpha \tag{1}$$
$$\Delta w_t = \Delta p_{t-1} - \lambda U_{t-1} + \Delta z'_t + \beta \tag{2}$$
where $p^s_t$ are log of producer prices, $w_t$ are log of wages,  $\theta$ is productivity, $U_{t-1}$ is unemployment and  $z$ and $z$ are some control variables exerting inflationary pressure, alpha and beta are constants. After differencing the first equation we get:
$$ \Delta p^s_t =  \Delta w_t - \Delta \theta_t + \Delta z_t  \tag{3}$$
If we define the wedge between consumer and producer prices, which exists due to influence of imports and taxes, as $\phi_t \implies \Delta p_t = \Delta p^s_t + \Delta \phi_t$ and hence new producer price equation will be given as:
$$ \Delta p^s_t =  \Delta w_t - \Delta \theta_t + \Delta z_t \tag{4}$$
Combining equation (4) and (2) would give us the Philips curve:
$$\Delta p_t - \Delta p_{t-1} =  - \lambda(U_{t-1}-U^*) \tag{5}$$
where finally $U^*$ which is the NAIRU would be given as:
$$U^* = \frac{\beta - \Delta \theta_ t + \Delta z_t + \Delta z'_t + \Delta \phi_t}{\lambda} \tag{6}$$
Normally the whole system above would be estimated in a structural way where you would have separate error correction models (ECM) for producer price equation and wage setting equation  embedded in some structural model that will possibly include more equations (See chapter 3 in the above cited Carnot et al.).
There are also different approaches to estimating the above but even the basic reduced form Philips curve models used to estimate NAIRU will require multiple equations and usually will be estimated not with OLS but with something else like maximum-likelihood with Kalman filter (see this research  by Jacob and  Wong (2018) estimating NAIRU for New Zeeland).
Unfortunately, it is beyond scope of single SE answer to make an exhaustive review of all ways of estimating NAIRU but one thing that should be hopefully clear from the above is that it cannot be done with some simple naïve OLS where you regress inflation on unemployment.

PS:
The following statement is actually not entirely correct statement:

The natural unemployment rate I obtained is constant over a time period, which is not a realistic assumption given the fluctuations we observe in labor force, employment.

Natural rate of unemployment includes only frictional and structural unemployment not cyclical unemployment. Most variation in employment over time is due to cyclical or also seasonal variation. Structural changes or changes in frictions are usually small and relatively trivial compared to cyclical changes. In fact research by Fed shows that in the US natural rate of unemployment over last 100 years was remarkably stable ranging between $4.5$ and $5.5\%$. Consequently, even though there is some variation in natural rate of unemployment, it would not be unrealistic to use flat $5\%$ rate for US as a first approximation.


Your understanding is mostly correct, but the following sentence is not:

  in simple games, you often face with a situation where one of the pure strategies a player might have, say Column L of player 2, never gets underlined when you work out which ones for Player 2 is BR.

Consider the following counterexample:
\begin{array}{|c|c|c|c|}
\hline
 &L&C&R\\\hline
U&1,3&3,4&3,0\\\hline
M&2,3&0,1&0,4\\\hline
D&0,0&1,2&2,1\\\hline
\end{array}
In this game, $L$ is never a BR to any of player 1's pure strategies. However, if player 2 believes player 1 will play a mixed strategy that assigns $\frac12$ probability to $U$, $\frac12$ to $M$ and $0$ to $D$, then $L$ becomes a BR.


(1) You have already correctly pointed out that correlation makes it difficult to diversify away all the risk.

(2) You have also correctly alluded to the fact that most people are risk averse and hence that risky products tend to have a risk premium.

(3) What you have not mentioned -- and what I think is most important in the real world -- is the uncertainty premium.

The distinction between risk and uncertainty is usually credited to Frank Knight (1921, pp. 43–48).

Where we can quite precisely calculate probabilities and expected values, such as with lottery tickets and roulette wheels, we have risk. Where we cannot, such as with most real-world events, we have uncertainty.

Example. Asset $A$ pays $\$100$ with probability (w.p.) $1$ and so has expected value $\$100$.

Asset $B$ pays $\$300$ w.p. $0.5$ but loses $\$100$ w.p. $0.5$. So, the expected value of $B$ is: $$0.5 \times \$300 + 0.5 \times (-\$100) = \$100.$$

Each asset has the same expected value $\$100$.

(2) Nonetheless, because most people are risk averse, most people would prefer $A$ to $B$. That is, they'd be willing to pay more for $A$ than $B$. Or equivalently, $B$ would command a risk premium.

But any such risk premium should be small in the real world, since in the real world, as you've pointed out, there will probably be (i) ways to diversify away the risk; and also (ii) sufficiently many risk-neutral or risk-seeking individuals who bid away the risk premium.

(3) But the real world is rarely like the above example, which involves risk. The real world usually involves uncertainty:

Example. Asset $C$ is a US government bond that will pay out $\$100$.

Asset $D$ is a stock that -- based on your data, models, and analysis -- you expect will pay out $\$300$ w.p. $0.5$ but lose $\$100$ w.p. $0.5$.

Again, each of assets $C$ and $D$ has expected value $\$100$.

But now, even if you're risk-neutral, you'd probably prefer $C$ to $D$. That is, you'd pay more for asset $C$ than asset $D$. Equivalently, asset $D$ has an uncertainty premium.

The reason is that you do not (and should not) completely trust your data, models, and analysis. There are fundamental limits to your knowledge over and beyond mere risk -- we may call such epistemological limits uncertainty.


If Trump's economic policies are destructive, why is the US economy so healthy?

(1) Most of Trump's proposed policies have been implemented only to a limited extent or not at all.

He promised dramatic tax cuts. Examples from the June 2016 Moody's analysis you cited:

Replacing the current seven personal income tax brackets with three and reducing the top marginal rate from 39.6% to 25%.

The Tax Cuts and Jobs Act 2017 (TCJA) cut the top rate, but only from 39.6% to 37%. Also, the seven brackets have remained.

Increasing the standard deduction to \$25,000 for single filers and \$50,000 for joint filers, and indexing to inflation thereafter.

The TCJA increased the standard deductions from \$6,500/\$13,000 to \$12,000/\$24,000 — an impressive increase, but still less than half of what Trump had proposed. (Note also that these increases in the standard deductions were partly offset by the elimination/restriction of many deductions.)

Trump promised to immediately repeal and replace Obamacare. As of May 2019, this has not happened.

Trump also promised to deport over 11M illegal immigrants. This has also not happened. In fact, under Trump, there have been fewer deportations than under Obama (e.g. in FY2018, there were 256,085 deportations, well below the Obama 2013 peak of 438,421).

The above proposals would have severely damaged the US economy. But none have been fully implemented.

(2) The trade war.

Again it is the case that Trump falls far short of some of the wilder "promises" he made during his campaign (e.g. in Jan 2016, he proposed imposing 45% tariffs on China).

Nonetheless, to the extent that he has been "successful" at waging this trade war, there has been damage to the US economy.

Fajgelbaum et al. (2019) estimate that:

  the trade war lowered aggregate U.S. welfare in the short-run by \$7.8 billion, or 0.04% of GDP.

Amiti, Redding, & Weinstein (2019) estimate that:

  by the end of 2018, import tariffs were costing U.S. consumers and the firms that import foreign goods an additional \$3 billion per month in added tax costs and another \$1.4 billion dollars per month in deadweight welfare (efficiency) losses.

(3) It may be too soon to tell.

Or in economists' jargon, there is usually a policy lag of months or years.

For example, the TCJA just came into effect in the most recent tax season (whose deadline just passed a few weeks ago in April 2019). Even if the TCJA has any effects, it will be a little while before economists are able to measure them.

Also, as the Dec 2018 piece by Nouriel Roubini suggests, it was really only in 2018 and in particular the last few months of 2018 that Trump began to impose his economic policies:

  many trusted that at the end of the day, the “adults in the room” would restrain Trump and ensure that the administration’s policies didn’t jump the guardrails of orthodoxy. // These assumptions were more or less vindicated during Trump’s first year in office ... But things changed radically in 2018, and especially in the last few months.

(4) That the US economy is healthy is not evidence that Trump's policies are sensible.

It could be that absent Trump's policies, the US economy might be doing even better. We can't just look at things as they are; instead, we need some sort of a counterfactual analysis.


The total wealth (as is pointed out in the book) is given as :

$$
X(1)=\delta_{0}B(1)+\delta_{1}S_{1}(1)+...\delta_{N}S_{N}(1)
$$
The wealth at period one is just the sum of the securities (including amount invested in the bank) times their number. In other words, the total wealth is the amount of money this individual has invested in the different assets. It has nothing to do with a "changing process." Asssume that the individual does not buy any more assets during period 1 and the assets promise a return of $(1+r_{i})$ where $i$ denotes the asset in question. In period 2, the individuals wealth is then
$$
X(2)=\delta_{0}B(1)(1+r_{0})+\delta_{1}S_{1}(1)(1+r_{1})+...\delta_{N}S_{N}(1)(1+r_{N})
$$


Fixed Effects Estimation and Inconsistency

You can't identify the effect of oil price when Year FE are applied, since the world oil price is perfectly correlated with year Fixed Effects.
You can't identify the democracy indicator if you country does not change its value in your observed period.

For the other variables, it should be possible to obtain estimates. You should apply a fairly large sample to avoid collinearity issues (sth like 100 countries, 50 years). Further problems with your model are reverse causality, i.e. does growth cause inflation?


How can markets can be highly efficient, even if they make errors?

The theory of efficient markets is that no one can make rewards that are in excess of the level reflected by the risk they are taking, except by random chance.

So if markets can make mistakes, but not in a way that allows anyone to make disproportionate rewards except by random chance, then they can be highly efficient.


Does Dogecoin or Bitcoin have a more appropriate rate of monetary inflation?

Your coins do not have 0 inflation in the long run

First, we need to set straight that you have a critical error in your supposition: The rate of creation of money is not the only determinant of the inflation rate, see here.

Even if we ignore these equilibrium effects, is any of these coins superior in its rate of money creation?

Suppose hereinafter that the actual inflation rate mirrors the money creation rate.

The optimal inflation rate is something that is not clearly understood, but most Economists agree that it is somehow related to the current state of the economy.

Hence, an inflation rate that is independent of the state of the economy cannot be that great. It basically removes the central banking authority to manipulate the inflation rate to the greater good of society - but removing this power is also an explicit goal of many advocates of these coins.

I have answered few reasons for and again higher inflation here. These make a case for a positive inflation rate. However importantly, they make a case for positive unexpected inflation rate. If high inflation is expected, all these benefits fall down. What remains is the shoe-leather-cost of inflation, implying that the lower inflation rate, bit coin, would be weakly better.

Remember: This is based on the false supposition that the money creation rate is the only determinant of the  inflation rate.

For the optimal rate of inflation, see also

http://www.frbsf.org/economic-research/publications/economic-letter/1997/september/what-is-the-optimal-rate-of-inflation/
http://www.nber.org/papers/w16054


It depends on what you mean by "sustained growth" and "flourishing markets".

Clearly the Earth as a whole is a big market. If you do not look at human made borders: the global economy is currently growing and some would say it is flourishing. There is no physical reason why this could not be done without the borders.

There are areas where the current technology cannot support the current population without outside help. An example is the heavily industrialized North Korea (which seems to motivate your question) which gets large amount of aid in food from the UN.


Why can money be described as "government non-interest bearing debt"

Let’s unpack “government non-interest bearing debt,” assuming that we are talking about government-issued notes and coins (dollar bills). (This does not include other instruments in monetary aggregates, like bank deposits.)

Money is an instrument issued by the government.
It pays no interest.
This only leaves “debt.”

Are notes and coins debt of the Federal Government? If we look at national accounts, they are considered liabilities of the consolidated Federal government. Most people use “debt” and “liability” to be synonyms, but there are technical differences (that I will not attempt to cover here).

One could argue that money being a “liability” does not qualify it as being a “debt.” This just means that the quoted statement is sloppy.

Note that there is another component of the monetary base - deposits held at the Federal Reserve by banks (bank reserves). The Federal Reserve is a bank, and those deposits are a liability, which is exactly the same as deposits held at private banks.

Why can these instruments be considered liabilities? When someone pays a tax to the Federal Government, the money is transmitted to the Treasury. The money holding is cancelled out against the tax liability of the tax payer.

I have seen arguments online that government money does not qualify as being a liability (or debt) since there is allegedly no promise to pay anything. (The argument also implies that accounting standards are incorrect.) I disagree with that view, and would note that to be formally correct, one would need to change national accounting definitions.


This is a standard regression equation, in the context of an econometric approach. The authors postulate a theoretical relationship between the variable "Change in Cash Holdings", which is treated as the "dependent variable", while "Cash Flow" and the rest are the explanatory variables/regressors.

The "factors in front of them" are the regression coefficients, which are unknown, and which are to be estimated through a regression estimation approach, like Ordinary Least Squares or any other technique deemed appropriate, using a sample of observations, which could be either "cross-sectional" (one balance sheet for each company, many companies), "time series" (many consecutive balance sheets for one company), or "panel data" (many consecutive balance sheets for many companies).

Each coefficient represents the partial marginal effect of a change in the regressor on the dependent variable, holding the other regressors constant.

There is also a "stand alone" coefficient, $\alpha_0$: this is the "constant term of the regression". Attached to it there is an implied "explanatory variable" that takes the value $1$ throughout the sample. It captures the average value of the dependent variable after the average effect of the regressors has been subtracted. The $\varepsilon$ at the end is the regression error, and it is unknown -an estimation of it will be given by the series of residuals obtain as the difference between the estimated value of the dependent variable and the actual one.

In Table 2 p. 1794 "The Baseline Regression Model" in the paper contains the numerical estimates the authors obtained for the $\alpha$'s for the specific regression equation, based on a panel data set, it appears, and for a number of variants.

According to the authors, immediately below eq. $(8)$ they identify the "Cash Flow Sensitivity of Cash" with the unknown coefficient $\alpha_1$. As already mentioned in the question (and as can be seen in Table 2), they find that the estimated value of $\alpha_1$ is $0.05-0.06$.

As described immediately above eq. $(8)$, "Cash Flow" here is essentially Operating Profits plus Interest Costs minus Dividends. Then the interpretation is that "on average, for every dollar increase in thus defined "Cash Flow", the Cash Holdings of a company increase by 5-6 cents" (the authors use, perhaps misleadingly, the verb "save" instead of "increase". "Hoard" would also be accurate).

"Applying" a regression equation to a specific firm, means, use time-series data on that firm, run the regression as specified and obtain firm-specific estimates for the coefficients. You essentially "accept" the theoretical validity of the regression specification, and you attempt to find the values of the coefficient pertinent to a specific firm (remember, the obtained estimates by the authors are averages over data from thousands of firms). You can then compare these estimates with the estimates obtained by the authors and see whether the firm under examination is close to average or not.


It was Alfred Marshall's doing, but based on a premise that still prevails in many parts of Economics: if prices are assumed totally elastic, then, although it may be the case that they are taken as given by the individual buyers or sellers, at market level it is quantity that determines price. So price is indeed the dependent variable, if you care about the market and not about the individual.

And here is page 96 from Marshall's Principles of Economics (8th ed.1920), (the 1st edition was in 1890), the funny thing being that here Marshall talks about... individual demand! (but it would be totally confusing to swap axes between graphing individual demand and graphing market demand). Read the annotated footnote.


Is it true that when the overall money supply decreases, that the demand for money decreases and the demand for bonds and securities both increase?

The quantity of money supplied has nothing to do with either the demand function of money or the demand function for bonds.

If there are changes in monetary policy such that the quantity of money supplied in equilibrium changes, then the quantity of money supplied clearly increases. In most models it is assumed that demand for money is decreasing in the nominal interest rate, so one way to achieve an increase in demand for money is to decrease the nominal interest rate. As people will hold more of their wealth in money, they will hold less in bonds.


Depreciation is the loss in value of something. Take the car example. When you buy the car, it is worth 5000 dollars. Your net worth equals the value of the car: 5000 dollars. If you needed to use the money, you could just re-sell the car and you'd get \$5,000 (in theory).

But what about the second year? You are technically still worth \$5,000. But if you try to sell the car, nobody would buy it after you used it for one full year at the same price: the car lost part of its value. In a perfect world, the car would lose \$500 because since you can use it for ten years max, and you used a tenth of that. Thus, the car would sell at \$4,500.

From then on, the same keeps happening: the car losses 10% of its value each year, until the tenth year when it becomes valueless (at least in this perfect world). This periodic fall in value is called depreciation.

When calculating the GDP, the government sums all the goods and services consumed in a given year regardless of its depreciation. If you purchased the car on January 1st, then the GDP increses by \$5,000. The government will most certainly publish the country's gdp by the end of the year. Although the car lost some of its value during that period, this is not taken into account in the GDP. If you took away the depreciation, then it would be the NDP (Net Domestic Product).

The depreciation is also known as "capital consumption" for a simple reason. Imagine that you want to have the same value (in cars) all your life. To achieve this, you would have to save \$500 each year to replace your car every 10 years.  So, you could say that each year, you consume \$500 worth in cars. This is obviously not true, but it is a way of simplifying it.


Consider a basic case of linear supply and demand functions for oil.

Demand: $q_d = \alpha + \beta P + \gamma X + \epsilon$ Think of X as a demand shifter, so Z could move around if the Chinese economy goes into recession or changes growth speed, someone invents a cheap hybrid car, or Americans wake up one day and decide to drive less.

Supply: $q_s = \eta + \zeta P + \psi Z + \phi$ Think of Z as a supply shifter, so if Z could move around if we have a shock to oil production like an expansion of fracking, a return of Libyan oil production to market, or the Saudis decide to reduce production for strategic reasons.

Now in equilibrium supply equals demand ($q_d = q_s$):

$\alpha + \beta P + \gamma X + \epsilon = q_d = q_s = \eta + \zeta P + \psi Z + \phi$

Which we can solve for P:

$\Rightarrow P = \frac{(\alpha - \eta) \gamma X - \psi Z  + \epsilon - \phi}{\zeta - \beta}$

We can also solve for $\Delta P$:
$\Rightarrow \Delta P = \frac{ \gamma \Delta X - \psi \Delta Z  + \Delta (\epsilon - \phi)}{\zeta - \beta}$

When we observe that P has fallen, that may be because $\Delta Z>0$ or because $\Delta X<0$

But for the economy, we'd expect quite different results from $\Delta Z>0$ and $\Delta X <0$. A negative demand shock ($\Delta X <0$) for oil probably reflects people being poorer and the economy heading into recession. A positive supply shock ($\Delta Z>0$) probably reflects technological improvements that make oil extraction more cost effective. The former bodes poorly for the economy, the latter bodes well.

So it isn't enough to say that it is an exogenous change in oil prices, we have to know more about why oil prices are changing. Economics blogger Scott Sumner warns about this particular issue, advising his students and readers to never reason from a price change. However, if we can take a stand on where the shock is coming from we can get some place. For example, if we assume it is entirely a oil supply shock then because we have estimates of the price elasticity of demand for oil, we can ask how much the supply curve would have had to move to drop prices as observed. From a supply curve shift and a sense of the demand function we could then calculate the increase in social surplus(measured in dollars) from the movement in the supply curve. Or you could treat it as a pure positive income shock to american consumers, negative to American oil producers, and use estimates of fiscal multipliers and marginal propensity to consume to estimate the effects.  I suspect you could do something similar with demand shocks.

As a thought experiment, I tried to think of this as a global supply shock. In my scenario oil prices are lower because production is up but there are no additional knock on effects on demand for exports or the US dollar. We also have to take a stand on if this is temporary or permanent. In general you'll get much bigger effects from permenant shocks. I'm going to assume the price shock lasts a single year. I'm not positive that's a coherent scenario but it gives me something to plug into my calculator.

According to the US Energy Information Administration the US is a net importer of about 5,700,000 barrels of oil and oil products a day. If the price of oil falls from $\approx$\$102 to  $\approx$\$40 a barrel that's about \$127 billion in savings by the American public. That works out to about \$410 a person a year or 0.75 percent of GDP. What about the fiscal multiplier you ask? The marginal propensity to consume numbers are like 4% in representative agent models, which means that if everyone were the same and we have a multiplier of 2 we'd get something like 0.81 percent of GDP. Which suggests that the direct effect of oil price changes can't be very important. That is, if the distributional consequences are not important.

The aggregate net export numbers obscure that some people are huge net exporters (the people in oil business) but the vast majority of us are modest net importers. Aggregate production was something like 3.4 billion barrels produced by about 600,000 workers call it an even two million to include support staff and that's a massive \$100,000 less per worker (noting that capital will bear much of the adjustment here, but still, that's huge). With 9.1B barrels (5.7B imports  + 3.4B domestic production) in domestic consumption , that works out to more generous /$1,800 a year in savings for the 308 million or so of us who don't work in the oil business. But if everyone is the same, none of that really matters, income losses buy the oil guys are offset but income gains by consumers except to the extent that we are net importers. Do we think that's true here?

In general, we think that poorer people have a higher marginal propensity to consume than richer people, which generally means that shocks that affect poor people have stronger short-run economic effects than shocks of the same size affecting rich people. The people who work in the petroleum business are much higher paid than the average American (about 91k a year compared with say 43k for the typical worker and of course not everyone works).

Imagine America had one guy (some trillionare) who owned all domestic oil production and consumed so little of his money that oil price shocks didn't influence his consumption (MPC = 0). Everyone else was a consumer of oil. In that world the short run aggregate effects would be more like that of giving everyone the \$1,800 wealth effect from cheaper oil. So something like 3-4% of income is a non-trivial shocks, and by some accounts lower income people have MPC as high as 0.5, so that could be say 1.5 - 2 percent of GDP directly.  You'd also get a knock on effect of whatever you think the fiscal multiplier is. Some people would say about 2. So that tells you 3-4 percent of GDP stimulus effect should be the upper bound.

So in my thought experiment I'd say something between 0.75 - 4% of GDP effect from the oil price drop of \$62 a barrel from a pure supply effect.


Doubts in modelling the Arthur(1991) paper: Designing economic agents that act like human agent

Although I still don't understand what these "strengths" are all about, still, mechanically, step $4)$ is now clear.

At each point in time, there is a certain allocation of Strengths to actions, $\{S_i(t),\; i=1,2,...,A\}$. This allocation changes through time.

The allocation is turned into a relative frequency distribution by dividing each $S_i(t)$ by their sum $C(t)$.

But then

$$\sum_{i=1}^A \frac {S_i(t)}{C(t)} =1 $$

and (I presume), $S_i(t) \geq 0,\; \forall i$. So indeed this relative frequency distribution can be treated as a probability distribution, and each $S_i(t)/C(t)$ can be treated as a probability that characterizes action $i$ in period $t$.

So you have to code this probability distribution, and then have the software "randomly draw" from it, as though this distribution was part of  the distributions available in a "random number generator".

The floor is yours now.


Game theory with rational and irrational players

Yes, a whole book has been written on Behavioral Game Theory.

More specifically, standard solution concept such as Nash equilibrium requires that players best respond to a correct belief about other players' moves. The following are examples that relax one of these cognitive restrictions:

Quantal response equilibrium allows for the possibility that players do not always choose a best response. Instead, they "quantal respond" (or best respond with mistakes) to a correct belief about others' quantal response.
Level-k or Cognitive Hierarchy Theory relaxes the correctness of beliefs assumption, and allows players to best respond to a set of mutually inconsistent beliefs.
Psychological games allow players to derive utility directly from beliefs (and belief-dependent emotions). Relevant solution concept include sequential reciprocity equilibrium and Rabin's fairness equilibrium.

For a complete review of models of behavioral game theory, see Chapter 13 of Dhami (2016).

For a specific example of a game with mixed players, see this paper on robust protection of fisheries. A more comprehensive treatment of mixed games is available in Rani Spiegler's Bounded Rationality and Industrial Organization per Michael Greinecker's comment on the original question.


How can increasing debt boost potential returns?

The first statement refers to the return on investment to the owners of the shares. After they are paid the 1 B dollar dividend  what happens to the company afterwards is not going to change that fact. Of course they would prefer that the company continue to do well. If they sell the shares immediately after receiving the dividend then they don't care.  Since the company had  1 B dollars less equity after paying the dividend it is easy to understand that in or near 2005 they would want to replace that capital with new debt and your quote suggests that is what happened. Less equity and more debt, if all else is unchanged, results in higher ROE (return on equity) which is one of the company's measures. Owners might benefit from this measure improving if the market likes it. Of course all else is not unchanged because they have to pay interest on the new debt. The actual effect on ROE depends upon circumstances. So the dividend was certainly good for the owners and what happens afterwards is potentially good but uncertain for the owners.
The part of the second statement, "raising the company's interest costs and pushing down profit", refers of course to the company's profitability position. In other words this can be interpreted as the company's return on revenue (profit margin); not the owner's return on investment.


How can a good be considered both a luxury and a normal one?

In case of normal goods, income elasticity of demand is greater than 0 and in case of luxury goods, it is greater than 1.

Supposing the price of pizza here remain constant at $1, thus when your income increased by 10%, the demand of pizza increased by 25%

Income elasticity of demand = 25/10 = 2.5

2.5 > 0, hence normal good

2.5 > 1, hence luxury good

This way, pizza is both normal and luxury good here.


How are you supposed to calculate overhead expenses?

Estimating overhead expenses by applying a coefficient to salaries can make sense in some circumstances.  In an organisation which is mainly office-based, overheads may consist mainly of the costs of office space, office running costs (electricity, cleaning, etc), computer equipment and support, and (depending on the local jurisdiction) payroll-related and property-related taxes. Using a fixed coefficient will always be an approximation to actual costs, but sometimes it can be a fairly good one, especially if the organisation has a fairly "flat" structure with most staff being on similar salaries and having similar office and computing needs.

In a manufacturing organisation with much capital equipment this approach would be inappropriate.


Pareto efficiency (optimality conditions) in simple New Keynesian model

The Lagrangian is also a function of the multipliers $(\lambda_t, \mu_t, \omega_t, \alpha_t)$ so you are missing their FOC's (these will give you the constraints as FOC's so if you were already taking them into account, omit my comment, please). Also, $-u_{H}=u_l$ since $l=1-H$ (I would re-write the second FOC, in terms of $u_l$ in order to construct the first "target" equation that has the MRS=MRT. Finally, there are a couple of useful facts that should help you. First, note that in your 4th FOC, you have a product equal to zero, so at least one of the elements is equal to zero. Try to convince yourself, that the only possibility (so that you don't run into a contradiction with the other FOC's) is for $\pi_t=0$ as required by your second-to-last target equation, and in turn implies $C_t=Y_t$ (from your first constraint). Second, using the second constraint you can replace the integral in your last FOC with $Y_t$.

This should help for further simplifying the 10 FOC's you should have, into the 6 target equations you want.


Computing the core in a transferrable utility game

It depends on the set of feasible allocations for the coalitions $S$. Suppose for all $S$ a best allocation exists (the sum of the individual utilities of the members of $S$ is maximal). Then as in usual cooperative games each coalition can be assigned this best utility sum as its value $v(S)$. Let us denote the utility vector each player gets from an allocation by $x$, and the sum of utilites players get from this by $x(S)$. An allocation is in the core iff
$$
\forall S\subseteq N: x(S) \geq v(S).
$$
This is a slight simplification of what you wrote, because instead of comparing all allocations to $x$ we first find the 'best' allocations for each coalition and only compare these to $x$.

If you make further assumptions about the coalition values $v$, e.g. if you assume the game is convex (this is true for exchange economies) then the calculation may be simplified a bit, as the core of a convex game is the Weber set.


How to understand the USDA Demand/Supply report ?

M Acres is million acres, as you suggest
Bu is bushel, in theory a volume of 8 US dry gallons, but apparently for wheat at 13.5% moisture it represents a weight 60 pounds.  Bu/Acre is therefore an average yield: with some minor rounding $\frac{2309}{43.9}\approx 52.7$ in the first column
Stocks/Use ratio seems to be the stocks at the end of the period divided by the total use (domestic and exports) in the period so in the first column $\frac{1181}{2222}\approx 53.2\%$


When is self-selection a problem? Mincer equation

There is no fundamental difference between the choice of occupation and say region. In fact,  there exists a large literature in migration that builds on Roy's model, starting with Borjas (1987).  I'd argue that ideally you'd like to correct for self- selection/ sorting in all choice dimension. In practice this can be hard/ cumbersome/ impossible to do.

Why do we tend to worry about self- selection more in the context of occupations than regions? I suppose it boils down to the fact that everyone had to actively choose an occupation at some point,  whereas we are all born in some region. But,  again,  there is no fundamental difference.


How to perform unbalanced panel data regression in R?

Here the solution would depend on what you want to accomplish. Note the problem is not just that the series is unbalanced, for an ordinary unbalanced panel data-set where firms have different number of $T$ observation the command would still work. Here no adjustments are necessary, you can easily try it yourself:

 install.packages("plm")
library("plm")
data("EmplUK", package="plm")
result<-plm(formula = wage ~ emp, data = EmplUK, model = "within")
summary(result)

Which gives you this output:

  Unbalanced Panel: n = 140, T = 7-9, N = 1031

  Residuals:
       Min.   1st Qu.    Median   3rd Qu.      Max.
  -12.13926  -1.21442  -0.20655   1.02437  17.11197

  Coefficients:
       Estimate Std. Error t-value  Pr(>|t|)     emp -0.119813   0.031103 -3.8521 0.0001255 ***

So you can use unbalanced data without any additional adjustment to the plm code itself. I think authors there refer to adjustments that have to be made when programming the function to account for this (as this creates some problems with matrices - at least that is my understanding from the document you linked).

However, if I understand you correctly the problem you have is not just about panel being unbalanced but because there are duplicate observations like this:

Firm ID | Year  | event          | x  | Y |
  1      | 1999   |  1jan2000     |  2 | 4 |
  1      | 1999   |  20jan2000    |  24| 54 |

This is more serious problem than just unbalanced panel. Here the problem is not the 'unbalanceness' but the fact that you will have 'duplicate' observations if you will run any panel model with Firm ID and Year as panel identifiers because you cant have one panel identifier assigned to two different different observations as that means your observations are not unique.

This could be solved in one of few ways:

Run the model on dates instead of years if the events do not occur at exactly the same time. So now your panel identifiers would be: Firm ID Event instead of Firm ID Year.
Create new firm ID for each event. So you could create new firm ID like this:

Firm ID | Year   | event         | x  | Y |
  1_1    | 1999   |  1jan2000     |  2 | 4 |
  1_2    | 1999   |  20jan2000    |  24| 54 |

however, downside with within estimator would be that if these
    multiple events on same day occur only once these would drop, and also my intuition tells me this would led to some methodological issues if you would try to lets say cluster errors on firm level as one firm would be treated differently based on no of events.

You could also aggregate values of X and Y across events on a single day and just include extra dummy for firms with multiple events in a given year.
Consider different estimator. For example, instead of within estimator you could use pooled OLS where you treat each observation as individual event with firm fixed effects. I cant think of any reason why that could not be applied here.

Also here I would recommend for searching for panels with duplicates, I know this is not a genuine duplicate, but mostly when this issue is being discussed many people call these observations duplicates so it will probably give you better results than searching just for unbalanced panel data which is usually applied to panel data where you just have different $T$ for each panel ID.


How popular are alternative standard of living measures amongst economists?

This question is somewhat subjective but I'll attempt to provide an answer.

Real GDP is a very useful concept in economics. It provides a very good indicator on the size of an economy. It does have its limitations, some of which you have described.

Many economists are interested in measuring populations in different ways. At the moment, social mobility and inequality are getting a lot of attention. In analysing these issues, there is not as much value in observing the real GDP figure. Further treatment needs to be done.

Essentially, on a macroeconomic scale, GDP is very useful. When the analysis looks at individuals and concepts like welfare, it takes a far less prominent role in the analysis and other indicators are more commonly used.


How to show average cost is falling if we have IRTS

A production function $f: \mathbb{R}^k \rightarrow \mathbb{R}$ exhibits increasing returns to scale if for $\alpha > 1$:

$$ f(\alpha \mathbf{x}) > \alpha f(\mathbf{x})$$
This implies the directional derivative of $f$ at $\mathbf{x}$ in the direction of $\mathbf{x}$ is greater than $f(\mathbf{x})$. (Let $\alpha = (1+ \epsilon)$ then $\frac{f(\mathbf{x} + \epsilon \mathbf{x}) - f(\mathbf{x})}{\epsilon} > f(\mathbf{x})$ then take limit as $\epsilon \rightarrow 0$ to obtain $\nabla_\mathbf{x} f(\mathbf{x}) > f(\mathbf{x})$.)

Given a price vector $\mathbf{p}$ the cost function gives the minimum cost to produce a quantity $q$.

$$ c(q) = \min_\mathbf{x} \left\{\mathbf{p} \cdot \mathbf{x}  \mid f(\mathbf{x}) \geq q \right\} $$

The Lagrangian for this problem is $\mathcal{L} = \mathbf{p} \cdot \mathbf{x} + \lambda (q - f(\mathbf{x}))$. The first order condition is $\mathbf{p} = \lambda \nabla f(\mathbf{x}^*))$. Take the dot product of both sides with $\mathbf{x}^*$ to obtain $ c(q) =  \mathbf{p} \cdot \mathbf{x}^* = \lambda \nabla f(\mathbf{x}^*) \cdot \mathbf{x}^*=\lambda \nabla_{\mathbf{x}^*}f(\mathbf{x}^*)$.

Our previous result on increasing returns to scale then implies $c(q) > \lambda f(\mathbf{x}^*) = \lambda q$.  By the envelope theorem, marginal cost is $\frac{dc}{dq} = \frac{\partial \mathcal{L}}{\partial q} = \lambda$. Combining, this gives us marginal cost is less than average cost for an increasing return to scale production function:
$$ \frac{dc}{dq} < \frac{c(q)}{q}$$

That immediately implies average cost is decreasing in $q$.

Intuition

A more intuitive way to think about this is that increasing returns to scale in the production function implies that for $\alpha > 1$:

$$ c(\alpha q) \leq \alpha c(q)$$

If the production $f$ scales more than linearly in inputs, the cost of producing a quantity scales less than linearly. You can show this without any calculus at all.

Once you have that, divide by $\alpha q$ and you immediately obtain:

$$ \frac{c(\alpha q)}{\alpha q} \leq \frac{c(q)}{q} \text{ for } \alpha > 1 \text{ and IRTS}$$
That shows for an increasing returns to scale production function, a higher level of production has a lower average cost.


Imbens and Angrist (1994): the $D_i(z)$ variables

I hope this meets your idea for intuition, but equation 1 comes from using the Law of Total Expectation with the independence condition (Condition 1). There are four possible values of $D_i(z)-D_i(w)$. $$D_i(z)=D_i(w)=1$$ $$D_i(z)=D_i(w)=0$$ $$D_i(z)>D_i(w)$$ $$D_i(z)<D_i(w)$$. Consider the LHS of (1). $$E[(D_i(z)-D_i(w))*(Y_i(1)-Y_i(0))]$$ From the four possibilities above it is clear that this expectation only takes on a value other than 0 when the last two are true, where the third option gives a value of 1 and the fourth a value of -1. So then using the Law of Total Expectation with the independence condition yields the equality from the paper.


Equality of ordinal vs. cardinal utility under transformations - who made the distinction first?

It appears to be the Theory of Games and Economic Behavior (1944) by John von Neumann & Oskar Morgenstern. I have the 1953 edition which is counted as "3d", but by reading the included introductions to the 2nd and to the 3d editions, it appears that nothing of substance has been changed in chapter 3, where I locate the issue of equivalence up to linear transformations.

In sub-chapter 3.4 the authors discuss in general the issue of equivalence under transformations for any system of quantities, mainly from physics. They end this sub-chapter by writing (bold my emphasis)

  "One may take the attitude that the only "natural" datum in this domain (utility) is the relation "greater," i.e. the concept of
  preference. In this case utilities are numerical up to a monotone
  transformation. This is, indeed, the generally accepted standpoint in
  economic literature, best expressed in the technique of indifference
  curves. To narrow the system of transformations it would be
  necessary to discover further "natural" operations or relations in the
  domain of utility. Thus it was pointed out by Pareto (in V. Pareto, Manuel d'Economie Politique, Paris, 1907, p. 264) that an equality
  relation for utility differences would suffice; in our terminology it
  would reduce the transformation system to the linear
  transformations. However, since it does not seem that this relation is
  really a "natural" one—i.e. one which can be interpreted by
  reproducible observations—the suggestion does not achieve the
  purpose."

In sub-chapter 3.5 they start by writing

  "The failure of one particular device need not exclude the possibility  of achieving the same end by another device. Our
  contention is that the  domain of utility contains a "natural"
  operation which narrows the system  of transformations to precisely
  the same extent as the other device would  have done(*). This is the
  combination of two utilities with two given  alternative probabilities
  $a, 1 — a, (0 < a < 1)$ as described in 3.3.2. The  process is so
  similar to the formation of centers of gravity mentioned in
  3.4.3. that it may be advantageous to use the same terminology."

(*) Note: The "failure of one particular device" refers to the suggestion of Pareto mentioned in the previous passage. So what they are saying here, is that, apart from ordinal preference, there is another "natural operation" related to utility (but not the one suggested by Pareto), that will nevertheless achieve the same purpose as Pareto's suggestion, i.e. to narrow the admissible transformations to linear ones only.

And which is this "natural operation"?

  "...The combination of two utilities with two given  alternative probabilities $a, 1 — a, (0 < a < 1)$"

which ushers us into the world of expected utility.

The details come next, together with Appendix "The Axiomatic Treatment of Utility" at the end of the book.

PS: By rejecting Pareto's suggestion, the authors make clear that expected utility is not cardinal utility, since what they reject is the existence of "an equality relation between utility differences". This rejection preserves fundamentally the ordinal nature of expected utility.


Who gathers the production/use data that makes up GDP? And how?

In the UK, which uses what is know as "the output approach", the Office for National Statistics (ONS) sends out around 45,000 surveys to business, called the Monthly Business Survey (MBS).

The ONS claims that this covers "around 55% of the economy.".

In this survey, questions about turnover and purchases are asked which then allow the ONS to assign total outputs and intermediate expenditures to industry classifications. In the case of the ONS, they use Standard Industrial Classification (UK SIC 2007).

There is a separate survey for construction called the Construction Output Survey, and there are additional forecasts and models to fill in the gaps.

These output and expenditure estimates are then deflated by things like the Consumer Price Index (CPI) to produce an inflation-adjusted GDP figure.

More detail on this process along with many more links to data sources is available in the GDP quality and measurement information document.


Is Dollarization a good decision for a small open economy with an economic cycle that weakly correlates with the anchor country?

Like many questions in economics, optimal behavior depends on the set of alternatives. The alternatives of Panama, El Salvador, and Ecuador to dollarization are likely a heavily politicized monetary and exchange rate policy. And so for the hypothetical small open economy considered in the question, the key issue is "What's the institutional effectiveness and independence of the potential monetary authority?"

There are small, open economies with well run central banks (Australia and Switzerland) but also less well run ones (Turkey).  If Panama could have Australia's quality of central bank they'd probably prefer that to dollarization. That might explain why Chile and Costa Rica, with some of the best institutions in Central America choose not to peg their currency to the dollar. If Panama did manage its own monetary affairs without a peg they'd get likely get results more like Turkey. They likely prefer dollarization to the results they'd expect from independent monetary policy.

And indeed, when you peek into the past of many countries with pegged currencies, you often find an episode of poorly conducted monetary policy just before dollarization. Not Panama (never had its own currency since independence), nor El Salvador, but Ecuador introduced dollarization in response to a severe banking and financial crisis. Argentina also seemed to introduce dollarization in response to monetary crisis.

There are of course other considerations related to optimal currency areas. In particular, if you do a lot of trade with the USA dollarization is less of a sacrifice with respect to the best possible independent currency and monetary policy then if you do most of your trade with a neighboring country that uses a different currency. If Mexico adopts a pegged currency at some point the dollar makes a lot more sense than the Euro when 70% of their trade is with the USA and trade is a big part of their economy. But neighboring countries often have similar trading partners, and are in a similar position with respect to other optimal currency area criteria (no labor mobility to the reference
country, no fiscal transfer mechanism shared with the reference country, and probably similar business cycles) so institutions are going to play a important role in explaining cross-sectional differences in the decision to dollarize, and that is why I highlight that in this answer.


What are some examples of indirect mechanisms that are not auctions?

Voting mechanisms would seem to be a good example.  So are matching mechanisms such as those used to assign students to public schools.


Pricing of a call option in a one period binomial model

I assume 5% is the interest rate. If we knew stock price in each state, formula for call option price in binomial model should be
$$
C_0 = \left(S_0 - \frac{S_d}{1+r}\right) \frac{S_u-K}{S_u-S_d}
$$
Now just invert the formula to get $S_u$, given values of $C_0, S_0, S_d, K, r$.

Where did we get that formula for call option price in the first place? That's pretty important, so I'd recommend you to look into your notes more closely. In short, you want to form a portfolio of stocks and bonds that replicates the option payoff in the next period, i.e. find $a, b$ such that
$$
\begin{split}
a S_u + (1+r)b &= S_u - K \\
a S_d + (1+r)b &= 0
\end{split}
$$
(here, $a$ stands for how many stocks you buy, $b$ for how much you save into bonds)

If you solve this system, then the price of the option in current period must be just the value of this portfolio, $C_0 = a S_0 + b$ (otherwise one could make easy profits from the mispricing), and doing the algebra should yield the formula at the beginning.


What term/theory describes more efficient utilization of a resource?

At the abstract level, this phenomenon is covered by the concept of "Technological Change", which, in Economics, does not have a strict/narrow "Engineering" interpretation: here, "Technology" is not only "how things are done" but also "how things are organized", and includes also new business models like the one you mentioned (see also this post).

So the phenomenon is already embedded in the logic of economic models.

Stochastic Frontier Analysis (SFA) and Data Envelopment Analysis (DEA), are two sub-fields focusing on modeling and measuring the degree of efficiency (under various interpretations of the term), in the use of factors of production at the micro-level.

It is my impression though that since the examples you describe are relatively new, we still do not have enough data available to subject them to econometric analysis, so I cannot say if there are already papers that examine them (in a quantitative way).


In simple terms: what are the implications of homothetic and nonhomothetic consumer preferences?

From a mathematical point of view, if the function $f(x,y)$ is homogeneous (of any degree), and $g()$ is a function whose first derivative is everywhere non-zero, then the function

$$H(x, y) = g[f(x,y)]$$

is homothetic. In economics, we usually impose something more restrictive, namely that $g' >0$. But this makes a homothetic function a monotonic transformation of a homogeneous function. Now, homogeneous functions are a strict subset of homothetic functions: not all homothetic functions are homogeneous.

Therefore, not all monotonic transformations preserve the homogeneity property of a utility function. The simplest example is Cobb-Douglas utility. It is homogeneous of degree one. In an ordinal utility framework, we are ok with monotonic transformations, so we can consider the natural logarithm of it. Fine, but the natural logarithm will not preserve homogeneity. Nevertheless it will be homothetic.

The fundamental property of a homothetic function is that its expansion path is linear (this is a property also of homogeneous functions, and thankfully it proves to be a property of the more general class of homothetic functions).

In consumption theory, this means that, keeping the prices or the price ratio constant, if we vary the income of the consumer, in the $(x,y)$ plane the  tangency point of the income constraint with the highest feasible indifference curve will always reflect a fixed ratio $x/y$. This in turn implies that expenditures for each good grow all at the same rate as income, and so expenditure shares remain constant for the whole income range (always for a given price ratio).
While this may sound restrictive, in fact, it has been shown that homothetic preferences do not impose any special restrictions on aggregate demand (essentially due to the endowment vector being arbitrary and independent of preferences, which messes, or frees, things up).


The same factor affect how a product is supplied and demanded in different ways. If you want to model what the effect of change in a given independent variable has on equilibrium as a whole you need convert both your equations to a single reduced form equation.

If you have two equations a supply and demand equation both of the forms:
$$P_s=\alpha_0+\alpha_1X_i+\mu_i$$
$$P_d=\beta_0+\beta_1X_i+\epsilon_i$$

for estimating how $X_i$ affects equilibrium price set $P_s=P_d$

$$\alpha_0+\alpha_1X_i+\mu_i=\beta_0+\beta_1X_i+\epsilon_i$$

$$X_i(\alpha_1-\beta_1)=(\beta_0-\alpha_0)+(\epsilon_i-\mu_i)$$

letting the terms $(\alpha_1-\beta_1)$,$(\beta_0-\alpha_0)$ and $(\epsilon_i-\mu_i)$ be equal to $\gamma_1$,$\gamma_0$ and $z$ respectively we solve and find that the effect of $X_i$ on equilibrium price is:

$$\gamma_1=\frac{\gamma_0+z}{X_i}$$
based on this formula we see that the
coefficents value actually changes based on the the amount of $X_i $ provided.  Using this method you can get an accurate estimate of your factor's influence on equilibrium price and see how it changes.


Does an increase in the money supply increase or decrease interest rates?

I will frame this in the context of modern monetary policy and for the sake of clarity assume we are discussing the American economy.

1) Whenever the Fed wants to bring about some change in the economy, they do so by using one of three tools: open market operations, setting the discount rate, setting reserve requirements. I will only discuss open market operations. The Fed recently introduced two new tools: interest on excess reserves and forward guidance. I include these to be complete.

2) The Fed conducts open market operations through its NY branch's buy/sell desk. In simple terms, the Fed can buy/sell securities in the open market. This causes two things: a change in total bank reserves and security yields (by way of shift demand/supply curves, for example).

Your question is whether or not it makes sense to think about an increase in the money supply causing an increase in interest rates. Since we want to discuss increasing the money supply, let's assume the Fed is buying securities in the open market. Whenever the Fed buys securities, it buys securities from banks and 'pays' the banks by 'adding excess reserves' to the banks' balances at the Fed. Total reserves in the banking system rises. All else being equal, banks have more money to lend. This increases the available supply of loanable funds. Since we assume demand for loanable funds is unchanged, and since interest rates are the equilibrating force in this market, we should see that interest rates drop (because the supply curve shifts right/up).

You might also want to think in therms of what is happening to security yields. If banks start buying a large amount of securities in an effort increase the money supply, we should see a large increase in demand for securities. Assuming all else equal (in particular, assuming that there is no change in the supply of securities) we should see yields on securities fall (this is equivalent to the purchasing prices of securities rising).

And so we see that the logic, whenever we consider either adjustment mechanism, points toward a fall in interest rates whenever the Fed increases the money supply.

With this in mind, I think it is reasonable to conclude that the paper you've cited gets it wrong. And as a point of fact, we have witness recently (during QE) that money growth is not necessarily inflationary...as counterintuitive as that might be.


Microeconomic foundation of discrete choice model

Your first equation is an indirect utility function. The consumer chooses only one good and we assume that he can afford it. For simplification, the indirect utility discards the budget $m$ (often assumed be the same for all $i$).
Here, your demand function is perfectly inelastic. If the individual demand would not have been perfectly inelastic, the form would have been like $V=a-s(p)$, where $s'$ is strictly concave and increasing.

See, for example, Palma/Anderson/Thisse.


What does non degeneracy mean for a preference?

It means what Herr K. has guessed: there are acts $f$ and $g$ that the decision maker does not deem equivalent from the preference standpoint. In other words, the DM expresses a strong preference in at least one comparison he makes. a preference that does not satisfy it is a AA subjective expected utility with a constant index $u$ (i.e. such that the representation is $\int u dp$ for an arbitrary belief $p$). On the other hand if at least for one comparison the DM says $f$ is strictly better than $g$ then you satisfy it.


Why Is the West providing technology to India?

I may hazard a small guess to this. India has a very young workforce in comparison to its Asian counterparts (China and Japan). We especially see this in the field of software design where it has an even lower median age of labour than USA. As we know the West gains a lot from brain drain and stands to gain even more from young innovative minds that India produces, we can assume this access to new technology will let India produce even more bright minds.

As for India becoming a large vendor. We can see China will be facing extreme labour shortages soon due to it's high aged labour force and extreme brain drain. This in turn has China to be predicted to fall as a large vendor in the global market (Or at least possible). If it does India can take its place, easing the instability of the market, and if it doesn't we have a new youth force with amazing work ethic gain from.

Also from an Economic view more vendors in the market is decentralizing, which can be seen as good or bad depending.

Hope this had something to gain.


How does the Monetary base, Narrow money and Broad money affect inflation?

The relationship between money supply and price level (inflation is just change in the price level) is as follows:

$$P=\frac{MV}{Y}$$

Where  P the price level, M is the money supply, V velocity of money and Y is the real output.

So price level is  increasing in money supply and velocity of money, but decreasing in real output.

If everything else would be held constant price level would actually grow at the same rate as money supply, but in real life of course ceteris paribus condition is not satisfied. Real output continuously changes either due to economic growth or due to the boom/bust cycle, seasonal effects etc. Velocity of money is often though to be more or less constant, but it can change as well and in fact at zero lower bound, in a liquidity trap, changes in velocity can often offset increase in money supply preventing inflation.

Also, the above monetary formula is bit of an simplification because in more complex models not just the actual changes in these quantities but also expectations of these changes matter.


How does expected utility theory treat losses?

Gains and losses presuppose a reference point, which is not a feature in standard expected utility theory. In this theory, the only argument in the utility over wealth is $w$, the absolute level of wealth. A common form of utility function is the constant relative risk aversion (CRRA) form:
\begin{equation}
u(w)=\frac{w^{1-\rho}}{1-\rho},
\end{equation}
where $\rho$ is the coefficient of relative risk aversion. Note that this incorporates the commonly seen $\sqrt w$ and $\ln w$ forms when $\rho$ is $\frac12$ and $1$, respectively. As is obvious, there is no term that helps differentiate between gains and losses in this formulation. The graph for a standard EU function with risk aversion is the same as any strictly increasing and strictly concave function.

Prospect theory is motivated by empirical regularities found in psychology experiments. A typical utility form is
\begin{equation}
u(x,r)=
\begin{cases}
(x-r)^\alpha &\text{if }x\ge r\\
-\lambda(r-x)^\beta &\text{if }x<r
\end{cases}
\end{equation}
where $x$ is the payoff from some choice alternative (or "prospect" if the choice involves risks), $r$ is the reference point (which is $0$ in your example), $\alpha,\beta\in(0,1)$ are parameters that govern the degree of risk preference, and $\lambda$ is a parameter that separately captures loss aversion. Tversky and Kahneman (1992) estimate that $\alpha=\beta=0.88$ and $\lambda=2.25$.


Demand Function And Inverse Demand Function Representation in Books

Any time demand function is specified as function of price $Q(p)$ and we plot it using convention that price $p$ belongs on y-axis and quantity $Q$ on x-axis the plot shows the inverse demand function. However, ‘inverse demand function’ is quite a mouthful and hence it will often be just abbreviated as demand.
Furthermore, there would be nothing necessarily wrong with plotting demand function with quantity being put on y-axis and price being put on x-axis. The graph would be equally valid but it would go against established convention.
Generally in science the convention is to put dependent variable on y-axis and independent variable on x-axis. In the case of demand the price and quantity are actually jointly determined and hence both variables could be considered dependent. However, this convention got established by Alfred Marshall in his influential Principles of Economics which was extremely popular economic textbook, and if you actually go over the book you will discover that Marshall often thought of price being determined by quantity  and consequently in his own mind at least he was just using the standard scientific convention of putting dependent variable on y-axis and independent on x-axis. Marshall book was so influential that the way how we plot supply-demand in present day textbooks is practically the same way.
There is actually already an excellent post discussing this notation on the history of science and mathematics stack exchange by user EconCow (the post also gives greater detail of the Marshall’s own position and further resources). As the user shows Marshall was not even the first one to plot price on y-axis and quantity on x-axis, and there also were in past economists that did exactly the opposite plotting. To be more specific the user provides examples for the following economists and ways how they arranged their graphs:

Augustin Cournot (1838). Price on horizontal axis.
Karl Rau (1841). Price on vertical axis.
Jules Dupuit (1844). Price on horizontal axis.
Hans von Mangoldt (1863). Price on vertical axis.
Fleeming Jenkin (1870). Price on horizontal axis.
...
William Stanley Jevons (1871, figure). Price on vertical axis.

The post I linked even shows the pictures of these graphs so I encourage you to have a look at it.
As you can see from the above at the begging people plotted demand in different ways. However, after Marshall's principles of economics profession decided to favor his convention and we continue using it to this day.


What do people think are good economic policies to reduce informality?

Three part answer:

Causes:

Trust in the system (particularly trust in politics)
Attitudes to risk (i.e. being caught)/Enforceability of regulatory power
Education levels
Economic empowerment/unemployment/low wage level
Law of unintended consequences
Cultural considerations
Geographical considerations

Reduction:

Political corruption cleansing
Tighter regulation on informality
Greater communication and transparency
Demand and supply side policies

Evaluation:

Extent to which informality is a problem
'Trickle-down' effects
Ease in assessing magnitude of informality


What does Adam Smith mean by equality of condition?

The term "equality of condition" can also be thought of as "equality of outcome."

In this context, it refers to the idea that people in a society/state have approximately the same overall quality of life, regardless of opportunity, socioeconomic circumstances, prejudicial treatment, advantages/disadvantages, etc.

This is an oversimplification but, to put it another way, you could think of equality of opportunity (in regards to material wealth, access to luxuries, health and quality of life, etc.) as "equalizing how people start" and equality of condition as "equalizing how people finish."


Resources for mathematician trying to develop (macro)economic thinking?

You might be interested in David Romer's Advanced Macroeconomics. It's a grad-level textbook that includes grad-level math, but the focus is off the math and on the intuition. It doesn't spend much time going over certain tools, like dynamic programming, so it gets a bit hand-wave-y at some points, but it sounds like this might be the right balance for you.


To calculate the socially optimal level of $e$, you should look at $\pi_W+\pi_H$ (presumably that's also how you come up with the socially optimal $x$). At $x=20$, $e=1$.

The intuition is that the woodworker's production creates an externality of $-2a=-x$ on the hotel. Since the hotel has to right to ban noise pollution, the woodworker would have to compensate the hotel for its cost from putting up with the noise. At social optimum, the marginal compensation ($e$) should exactly cover the marginal external cost ($-1$). Hence $e=1$ at the social optimum.


Consider two standard relations that tend to hold:

$$i_t = r_t + \pi^e_{t+1} \tag{1}$$

where $i_t$ is domestic nominal interest rate, $r_t$ is domestic real interest rate, and $\pi^e_{t+1}$ is expected domestic inflation. The expression is known as the Fisher equation.

Also, we have the uncovered interest rate parity

$$ i_t  = i^*_t - [s^e_{t+1} - s_t] \tag{2}$$

where $i^*_t$ is "foreign" nominal interest rate, and $[s^e_{t+1} - s_t]$ is expected appreciation of domestic currency in percentage terms ($s$ is the natural logarithm of "foreign currency per unit of domestic currency" exchange rate).
Combining the two and re-arranging we have

$$[s^e_{t+1} - s_t] = i^*_t - r_t - \pi^e_{t+1} \tag{3}$$

Assume that inflation is monetary, and keep  $i^*_t,r_t$ constant. Then expectations for domestic inflation will tend to reduce expected appreciation of domestic currency, and may even turn it into an expected depreciation, especially if expected domestic inflation is high. Shortening the length of the time periods, this tendency becomes more of "actual" inflation and "actual" exchange rate.

The fundamental reason is of course what the OP was thinking: if we think of USD as another good (which it is since it is a store of value and can be exchanged for goods), then, if English pound loses value due to inflation and can buy less goods, why should it be still able to buy the same amount of the good "USD"?


Investing is about "new" things that did not exist. Any acquisition of a previously existing asset, such as a house, is considered as Savings rather than Investment. Moreover, Investment is not consumed at the present time and are expected to provide flows for longer periods of time.


Optimal production level for a typical firm in long-run

It appears we start at long-run equilibrium point.

The fact that all firms operate at the level where $q^*:MC = \min AC$, means that given demand $Q^d$ and the cost structure, what is endogenously determined is the number of firms $m$:

$$Q^d = mq^* \implies m^* = \frac {Q^d}{q^*}$$

Assume that aggregate demand increases. In the microeconomic setting we are examining things here, the cost/technology structure will not be affected by this change (we do not place bounds and the production inputs available). So in the new equilibrium, each individual firm will produce exactly the same as before.

What will change is the number of firms, which will increase, in order to accommodate the new higher required output.


Help understanding expression for continuous discounting

From time $0$ to time $t$ the profit flow is constant $\Delta$. The discounted value of the profit flow of any instance $s$ is $\Delta \cdot e^{-r s}$. To get the total discounted profit we will need the integral of this from $0$ to $t$:
$$
\int_0^t \Delta \cdot e^{-r s} \ ds = \left. \frac{\Delta \cdot e^{-r s}}{-r} \right]_0^t = \frac{\Delta \cdot e^{-r t}}{-r} - \frac{\Delta \cdot e^{-r \cdot 0}}{-r}
$$
And as $e^{-r \cdot 0} = e^0 = 1$, this is
$$
- \frac{\Delta \cdot e^{-r t}}{r} + \frac{\Delta \cdot 1}{r} = \Delta \cdot \frac{1 -e^{-r t}}{r}.
$$


PV is already cheaper than gas or coal in most of the world, on a like-for-like basis. So that sort of answers your main underlying question.

It's estimated that new PV will be cheaper than existing coal and gas within the next 2-5 years, depending on where in the world we're talking about.

LCoE is a hypothetical construct resting on a bunch of assumptions that often don't hold, so isn't wildly useful.

For real data, look at actual auction results for new generation that's priced per unit electricity supplied to the grid. That's pretty easy for PV, but you may struggle to find recent comparable auction data for dirty old generators such as gas and coal, because those are rarely procured as auctions priced per unit electricty.

You then need to adjust those costs for the factors not represented in the auction prices.

For PV, that's the cost of integration into the grid (balancing intermittency, reserve capacity, reactive power, and so on). Those costs are pretty tiny - well below USD 10/MWh - but will vary, depending on the particular grid. Look out for additional subsidies such as the CELs (tradeable clean energy certificates) in Mexican auctions, tax-breaks, free land, state-subsidised loans, etc.

For dirty generators, you'll need to add in the unpriced social costs, including the insurance risks of accidents, and the waste costs: greenhouse-gas marginal social costs are around USD 100 /tCO2e (see peer-reviewed work by Chris Hope & others).


What does it mean to make an identification assumption?

Identifying assumption: assumptions made about the DGP that allows you to draw causal inference. E.g. exogeneity assumption for IV, parallel trends assumption in diff-in-diff.

Identifying assumptions (lack of endogeneity in general) can never be
statistically confirmed (a non-reject is good, but it's not confirmation). So assessment of plausibility consists of empirical arguments based on what you know about the DGP.


Jordi Gali Book First Edition Page 47 (need help for derivation)

This simplification of the infinite sum is commonly made by a differencing approach. You can see here an example of this approach in this kind of models.
Regarding with the assumptions mentioned by you, for $\hat{mc}_t$ since it's a deviation from the levels variable steady state, in the equilibrium there's no expectations and therefore $mc_t=mc\implies \hat{mc}_t=0$ for all $t$ in the steady state. In other words in the steady state the value of this variable is deterministic and zero, but by no means the short term expectancy of this value $E_t[\hat{mc}_{t+k}]$ has to be zero, nor this is the way the series is simplified.
With respect to $\pi_t$, something similar happens, since by definition $\pi_t=(1-\theta)(p_t^*-p_{t-1})$ (Galí, 2008, p.57) and again, in the steady state $p_t^*=p_{t-1}\implies \pi_t=0$, but as with the marginal cost there's not a reason why log-linearized inflation is expected to be zero, apart from being in the steady state.


Can a monopoly supply negative units to one of its two markets?

Marginal cost is only guaranteed to be equal to marginal revenue in optima where $Y > 0$ (and when the profit function is concave, but in your case it is). If
$$
MC(0) > MR(0)
$$
the producing the first unit results in a loss, and a profit maximizing monopolist would not produce at all (if the profit function is concave).

An example: I am willing to buy golf ball sized diamonds from you. I am willing to pay 5 dollars for the first, 4 dollars for the second, 3 dollars for the third and so on, ending in 0 dollars for the sixth and any further units. How many golf ball sized diamonds will you sell me?


Is Myerson's Lemma valid for piecewise linear allocation functions?

Milgrom's excellent Putting Auction Theory to Work is a reference for what you're looking for. If I'm not mistaken, Myerson's lemma works for quasi-linear payoffs.


What would be the practicality of a widespread open-source approach in an economy?

So suppose there is new technology $A$ that only 1 person has access to. If there are multiple firms with production technology $y=k^\alpha$ then obviously if that new technology was available to them at low/no cost, the overall productivity of society would increase (each firm has $y=Ak^\alpha$)

The problem is that it is likely the case that $A$ requires expenditure to make. You need some way to incentivize the production of that technology (at least in a basic neoclassical framework). So if you make it proprietary then the other firms may have to pay for it, or you can give it copyright so that only 1 firm has access to it.

In Romer's endogenous growth model, firms have an R&D cost to produce their final good, but they have monopolistic power in the intermediate step to ensure that they can make a profit. If they didn't have this patent then they would not produce the technology.

There are newer/more advanced models in law & economics that discuss different kinds of IP arrangements. A great question is how are things like WIkipedia so prevalent where most of the authors see no profit for their effort but they do it anyway. Ostensibly they achieve some sort of utility for it. But then they are producing a 'public good' and in that case others will freeride (so its only sustained if their utility from it is really high).


Confusion on Strategy Sets in a simultaneous move game

There is no conflict. The strategy is a function which maps decision points to actions, that is it tells the player in each decision point (to be precise, in each information set) what she should do. In a dynamic game there are usually more decisions points, so this function nature of the strategy is more obvious.

In the simultaneous game you describe there is only one decision point (information set), players have to make a choice at the start of the game. A strategy maps an action to this one decision point (information set), so basically the strategy space is just the action space.

EDIT:
 You write " it would be natural for me to define a strategy as a function:
$$s_i: \mathbb{R} \to \mathbb{R}."$$
There is some murkiness here, because what is the first $\mathbb{R}$ supposed to be? The strategy set of the other player? But that is not observed by player $i$, so it cannot factor into her decision. As a function the strategy would look actually look like
$$s_i: \left\{I_i\right\} \to \mathbb{R},$$
where $I_i$ is the single information set of player $i$, so $\left\{I_i\right\}$ is a singleton set. There is a bijective mapping between the vector space of these strategies and the space $\mathbb{R}$ so it does no harm to define $S_i$ as equal to $\mathbb{R}$.


Effects of an expected increase in the inflation rate on the market for loanable funds

The Fisher Hypothesis is a standard example of the approach that argues that nominal magnitudes just reflect the real economy, they do not affect it. Denote $r$ the real interest rate, $i$ the nominal interest rate, and $\pi^e$ the expected inflation. Then the Fisher Hypothesis states that

$$i = r+  \pi^e$$

I write it this way to emphasize that it is the nominal interest rate that depends on the real one. In this framework, the real interest rate is determined in the real economy, say the marginal product of capital. So an increase in expected inflation will have the effect of increasing the nominal interest rate, and nothing else.

How is this rationalized?

The demand schedule for loanable funds is drawn with respect to their price. The price of loanable funds is the nominal interest rate. Magnitudes like expected inflation, if they have an effect, is to shift the whole demand schedule.

In a standard diagram, and with supply curve assumed unaffected, your answer appears strange: since when a "decrease in demand" (shift to the left of the demand schedule), leads to an increase in the equilibrium price? Intuitively, when customers start to demand less for any given price, this will lead to a lower equilibrium price, not higher. But the second part of your answer asserts that the price, i.e. the nominal interest rate, will increase due to this downward shift in the demand schedule.

Before any re-equilibrium effect, a rise in expected inflation reduces the real interest rate that prospective borrowers face. They may pay the same nominal interest rate, but the financial burden is smaller, due to the inflationary expectations: loans have just become cheaper, in real terms (while the current purchasing power they represent has not been affected).

So inflationary expectations shift the demand schedule upwards-outwards.

Now, it would be arbitrary to keep the supply schedule fixed. If expected inflation increases, prospective lenders should have a tendency to increase their current consumption, thus reducing the available funds: the supply schedule shifts upwards.

The Fisher Hypothesis says that these movements will lead to an increase in the equilibrium nominal interest rate, and the increase will be equal to the increase in expected inflation, since the real interest rate "should" remain constant.

$$\Delta i^* =  \Delta \pi^e \implies \Delta r =0$$

Moreover, the equilibrium quantity of loans will be unaffected, (and so real magnitudes like consumption will be unaffected).

If no choice reflecting the above effects was given to you, and if "none of the above" was not an option, then the question was deficient, or it postulated some additional assumption.


You want to find a relation between $tF(z)$ and $F(tz)$ for all $t>1$ (or $0$ for CRS).

So since $2t=tF(z)>F(tz)=2$ for all $t>1$, we see decreasing returns to scale.


Why isn't high inflation along with robust welfare a 'good' system?

There are several reason for that.
First, issue here is that you are talking about high inflation. Most economists are in favor of small inflation (around 2% per year for reasons you can read about in this and this old Economics.SE answers).
Second, inflation does not generally have the beneficial properties you ascribe to it. Inflation can stimulate output in the short-run but in the long-run the same level of inflation does not provide stimulus to the economy anymore (see discussion of this in any standard macro textbook such as Blanchard et. al. Macroeconomics: A European Perspective or Mankiw Macroeconomics). Consequently, in the long-run inflation does not increase opportunities for employment. Inflation, by itself, is not able to permanently raise real output.
Third, inflation has various costs. A non-exhautive list includes:

increase in uncertainity and volatility. This holds exclusively for high levels of inflation since modest levels of inflation are not associated with increase in uncertainty and volatility.  Higher uncertainity and volatility is generally bad for the economy as it leads to less investment and lower economic growth (see discussion in Mankiw Macroeconomics or Wilson (2006)).

next there are shoeleather costs of expected inflation. As explained in Mankiw Macroeconomic:

One cost is the distorting effect of the inflation tax on the amount of money
people hold. As we have already discussed, a higher inflation rate leads to a higher
nominal interest rate, which in turn leads to lower real money balances. If people
hold lower money balances on average, they must make more frequent trips to
the bank to withdraw money—for example, they might withdraw \$50 twice a week rather than \$100 once a week. The inconvenience of reducing money
holding is metaphorically called the shoeleather cost of inflation, because
walking to the bank more often causes one’s shoes to wear out more quickly

Of course, the above is written for undergraduates, it is not really the cost of new shoes that is biggest cost here, but opportunity cost of wasted time and general costs of adjusting to the inflation.

High inflation causes menu costs. These are costs firms incur to change prices (see Mankiw Macroeconomics).

High inflation tends to be more volatile and this complicates financial planning and distorts peoples intertemporal choices (again have look at Mankiw Macroeconomics).

The list above is not exclusive, the point is that high level of inflation has many costs associated with it. These costs increase disproportionally as inflation becomes higher, while at some point very high inflation won't even necessary have any short run benefits because the increase in uncertainty will depress economic activity more than the output get stimulated by the effect of short run effect of increase in prices.
Consequently, maintaining high level of inflation is considered very inefficient by most economists.  This, however, does not hold for low inflation (1, 2 or even 4%) as explained in the beginning.


Decision Theory Question: Existence and uniqueness of the certainty equivalent of p

Your notation is a bit misleading: it would be better to write $\mathbb{E}u(p)$ or $U(p)$ for the expected-utility associated with $p$ instead of $u(p)$, and $u(\mathbb{E}p)$ for the utility of the expected value of $p$. Formally $u$ is defined on $X$ and not on $\Delta(X)$.

Regarding your proof, it seems to me that: $(i)$ you don't explain how to find $s$; $(ii)$ you don't find a certainty equivalent, because $s$ is a lottery. What you want to find is a sure monetary prize, i.e. a degenerate lottery, that the decision-maker values equally as the lottery $p$.

For instance, you can notice that, by the monotonicity of $u$,
\begin{equation}
u(x_{*})\leq \int_{X}{u(x)p(x)dx} \leq u(x^{*})
\end{equation}

In addition, the function $u:x \rightarrow u(x)$ is continuous. Therefore, by the intermediate value theorem, there exists $CE_p \in [x_{*},x^{*}]$ such that $u(CE_p) = \int_{X}{u(x)p(x)dx}$.

For the uniqueness, imagine that $CE'_p$ is another certainty equivalent of $p$, i.e. that $u(CE'_p)=u(CE_p)$. Since $u$ is strictly increasing (which can be seen as a consequence of the monotonicity with respect to first-order stochastic dominance), we obtain $CE'_p=CE_p$.

Notice that you don't need risk aversion to prove the result, but further implies that $CE_p \leq \mathbb{E}p$.


If someone stays at home because they can't find the type of job they want, are they included in unemployment numbers?

This may vary depending on the country. But for the US, there are 3 criteria:

People are classified as unemployed if

they do not have a job;

have actively looked for work in the prior 4 weeks; and

are currently available for work.

Criteria #1 and #3 are fairly straightforward. Criterion #2 may be a little subjective and so we have the following clarification:

Actively looking for work may consist of any of the following activities:
Contacting:
    An employer directly or having a job interview
    A public or private employment agency
    Friends or relatives
    A school or university employment center
Submitting resumes or filling out applications
Placing or answering job advertisements
Checking union or professional registers
Some other means of active job search

So if your someone has actively looked for work in the prior 4 weeks (which includes the above activies), then she is counted as unemployed. Otherwise, she isn't.
Source: BLS, "Who is counted as unemployed?", which also gives some of the major questions (in the BLS survey) that determine employment status.


That is not correct  conclusion and there are several problems with the formulation you use.

$A$ is not a shock - it is a technology parameter of production function (see Woodford Interest and Prices pp 148). Heck, $A$ cannot even be zero. Following Woodford:

$A_t> 0$ is a time-varying exogenous technology factor

there can be shocks to $A_t$ that increase it or decrease it but $A_t$ itself is strictly positive at all times.

It is actually not $s(Y^n,Y^n,ξ)=μ^{−1}$ but $s(Y^n_t,Y^n_t;ξ)$. This might seem trivial but note $s(.)$ has only 2 arguments ($Y^n_t$ and $Y^n_t;\xi$  not 3 arguments $Y^n_t$, $Y^n_t$ and $\xi$). In addition, the solution to $s(Y^n,Y^n,ξ)=μ^{−1}$ gives you the natural rate of output not the equation itself. In this context how I read the Woodford semicolon (;)  is used to separate variables and shock parameters. Hence, it is just to indicate that the ξ is not a variable but given by the shock parameters of the economy (i.e. indicate variable conditional on vector of shocks.

Your definition of utility function is extremely non-standard, you define it as:

$$ u= \frac{\xi_t C_t^{1-\sigma}}{1-\sigma} $$
but why are the shocks $\xi_t$ even multiplied by consumption in the first place? In Woodford the equation (1.1) is given as:
$$E_0 \left[ \sum^{\infty}_{t=0} \beta^t \left( u_t(C_t, M_t/P_t; \xi_t) - \int^{1}_{0} v(h_t(i);\xi_t)di \right) \right]$$

first as you can see $C_t$ is not even conditional on shocks $\xi_t$, the real money holding is $M_t/P_t; \xi_t$ is conditional on vector of shocks.
you can't just assume $u_t(C_t, M_t/P_t; \xi_t) = \xi_t C_t^{1-\sigma}$, I mean that makes no economic sense whatsoever, why should utility be product of vector of shocks and consumption?

I don't think you will get the same result with some more reasonable utility function and take into account that the variables are conditional on the vector of shocks (e.g. $ M_t/P_t; \xi_t$ does not mean  $M_t/P_t$ has mean zero even if $\xi_t$ is given by Gaussian distribution with mean zero).


What is the difference between quarterly GDP and annual GDP?

You might not find a page explaining (in an easy manner) how quarterly GDP is calculated because it is done by several different means in different countries and the calculation/estimation methods can get confusing. If you want to know how a certain country generates a quarterly GDP figure, you should check the country's official economic statistics page - the descriptions of the statistic usually list at least something about the collection process.

Usually quarterly GDP is done with a smaller sample and has some estimations done (often based on last year's data). Annual GDP is usually the more accurate measure with a larger sample and more revision time between collection of data and publishing it.


How does imposing 100 percent cash margin requirement on imports help to stabilize foreign exchange reserves of a country?

Simply because when you want to buy goods from country A, you (generally) first have to buy A's currency, which means selling yours.

And given that selling a currency exerts a downward pressure on its value, the main idea behind a 100 percent cash margin requirement is about counter-balancing the depreciation by exerting an inversely-proportional upward pressure.

That being explained, if importators do so by mobilizing already-owned and in-excess cash, there is no reason for such a policy to work... which is unlikely, say, in the case of Turkey, since domestic liquidity have already been sold, which means buying them back.


Why was Swiss National Bank in red by keeping fix exchange rate to Euro?

From a central bank's point of view, printing money isn't free.  Each 1 CHF created costs exactly 1 CHF - that is, it appears on the bank's balance sheet as a liability.  They owe that 1 CHF to someone (the person they bought the Euros off - a bank).  And whenever you owe someone money, there is always the ultimate belief that you can pay it back.  Lose that and the bank fails, and the currency fails.

With central banks this isn't clear cut, since the money can only be "paid back" by destroying it, but the trust in fiat money relies on the solvency of the central bank.  A sort of, "so long as you could pay it back, I'll never ask you to".

The SNB had been buying so many Euros (and other currencies) that it was sitting on foreign reserves worth about 85% of Swiss GDP.  In the process it did make a lot of profit, but that can only go on so long.

In theory, if the SNB could have held out forever, then it could have beaten the ECB, and when the Euro eventually strengthened, unwound its position as a profit.  But that would have involved the SNB conducting massive quantitative easing in the mean time, in order to supply the world with sufficient CHF to meet demand.  Given that Switzerland is somewhat smaller than the Eurozone, it would have been a battle it was destined to lose.

Given that they knew they would likely lose eventually, it is better to lose now, when you only have 85% of GDP in Euros, than to lose later on when you have 850% of GDP in Euros!

Of course having a central bank that is completely screwed, would weaken the CHF, but they only want moderate weakness, no complete destruction of the currency!


General equilibrium effects of affirmative action policies

I found one survey which corresponds to what I was looking for: Theories of Statistical Discrimination and Affirmative Action: A Survey by Hamming Fang and Andrea Moro, in the Handbook of Social Economics (2011 edition).