Solution:

If we assume the Group Utility function equals:

$$G(x)= M\times u(qx) + I \times v(rx)$$

where:

$$u(x) = Ax+\frac bx$$

$$v(x) = Cx+\frac dx$$

and:

$$q+r=1$$
$$q,r,A,b,C,D \in\mathbb{R}(0,\infty)$$
$$x,M,I\in\mathbb{Z} [1,\infty)$$

We can state our goal as defining the boundaries of a situation where for some unfair allocation $M<I$ and $q>r$, $G(x)$ is greater than it would be if $r>q$ (a more "fair" allocation).

We can even simplify the question to finding a set of constraints on the variables $M,I,q,r,A,b,C,$ and $d$ such that would satisfy the inequality (which assumes $q$ and $r$ are equal and dropped):

$$M(Ax + \frac {b}{x}) > I(Cx+\frac{d}{x})$$

$$x\in \mathbb{Z}[1,\infty)$$

so taking the $\lim_{x\to\infty}$ of both sides, we can see that if
$\frac{A}{C}>\frac{I}{M}$ at large $x$ the inequality is satisfied.

Conversely, at $x=1$, we get $M(A+b)$ and $I(C+d)$.

This would demonstrate that a non-Pareto optimal solution exists that still satisfies the idea of declining marginal utility and maximizes overall group utility for all distributions of goods for any class of goods where there are two preference functions $u(x)$ and $v(x)$ as long as they are distributed such that[1]:

$$\infty>AM>CI+d>0$$

Or in general a "monster class" can exist as long as the coefficients of its utility function are greater than that of an innocent class times it's fraction of the population times the total number of goods the class under consideration in the economy:

$$\{c_M|c_M>c_I\times\frac{I}{M}\times x\}$$

[1] Again, assuming that $MU \geq0$ is always true


Does deflation mean drop in asset prices?

One would need access to stock price index data to make definitive statements. There are probably studies doing such analysis. However, I will make some rather safe generalised comments.

Firstly, we should expect differences in behaviour between the gold standard era and the present, as well as between developed and developing markets.

My expertise is with floating currency, developed countries. And the following statements appear safe to say.

Other than Japan, there has been no extended period of falling CPI (“deflation”) outside of recessions/global crises in the floating currency era. Pre-2020, that deflation was mainly due to falling oil prices, core CPI remained positive.
Stock markets tend to fall in recessions, for many reasons. Most modern recessions have been associated with financial crises, and leveraged investors needed to cover positions. This means that stock prices can be lower than fundamentals.
The deflation in Japan was extremely mild, and I would guess that there were some periods of rising stock prices (although the Japanese stock market was in a secular bear market after the bubble pop in the early 1990s) coinciding with deflation. Someone with access to Japanese financial data would have to verify
what happaned.

As such, we have no real guidance for what sort of “deflation” can happen in a floating currency sovereign. However, it is very hard to see what significant difference a deflation of -1% has versus an inflation of 1% for equity valuations (with those rates being similar to historical post-1990s experience). Stocks could rise and fall based on the moods of stock investors. The difference in expected nominal growth rates can easily be counter-acted by lower discount rates.


Price difference of substitute goods when the supply increases?

To use a simple example, assume that consumer $i$ maximizes

$$U(x_i,y_i,I_i) = \alpha\ln x_i +(1-\alpha)\ln y_i + (\bar I_i-p_xx_i -p_yy_i)\\
s.t. \bar I_i \geq p_xx_i +p_yy_i$$

In other words, it may spend all his income on the two goods (but not more), or he may keep some for other goods, not modeled here. Due to the inequality in the constraint we need to use a Karush-Kuhn-Tucker non-negative multiplier, rather than the usual Lagrange multiplier.
Assuming a fixed income, the lagrangean function is

$$\Lambda = \alpha\ln x_i +(1-\alpha)\ln y_i + (\bar I_i-p_xx_i -p_yy_i) +\lambda_i(\bar I_i-p_xx_i -p_yy_i)$$

Assuming income is fixed, the first order conditions are

$$\frac {\alpha}{x_i} - (1+\lambda_i)p_x \leq 0,\;\; \frac {1-\alpha}{y_i} -  (1+\lambda_i)p_y  \leq  0$$

$$\lambda(\bar I_i-p_xx_i -p_yy_i) = 0 ,\; x_i\cdot \left(\frac {\alpha}{x_i} -  (1+\lambda_i)p_x\right)=0, \; y_i\cdot \left(\frac {1-\alpha}{y_i} -  (1+\lambda_i)p_y\right) =0$$

it follows that at the optimum both goods will be demanded in positive quantities, which in turn requires that the first-derivatives be set equal to zero. This gives us

$$\frac {\alpha}{p_xx_i} = \frac {1-\alpha}{p_yy_i} \implies x_i^D=\frac {\alpha}{1-\alpha}\frac {p_y}{p_x}y_i^D \tag{1}$$

from which we can obtain the optimal relation for the Expenditure $E_i$ on the two goods

$$E_i^* = \frac {1}{1+\lambda_i^*} , \;\; E_i^* \leq \bar I_i$$

Since $\lambda_i \geq 0 \implies  \max E_i^* \leq 1$ so if it so happens that $\bar I_i >1$ then not all income will be spent on the two goods, the constraint won't be binding and so $\lambda_i^* =0 $. (Is $\max E_i^* \leq 1$ a strange result?) On the other hand if $\bar I_i \leq 1$ then we will necessarily have $E_i^* = \bar I_i$, and $\lambda_i^* = (1-\bar I_i)/\bar I_i$.

In either case, optimal relation $(1)$ remains valid. So, assuming identical consumers with respect to preferences (not necessarily with respect to Income), aggregating $(1)$ we obtain the market-level relation

$$ X^D=\frac {\alpha}{1-\alpha}\frac {p_y}{p_x}Y^D \tag {2}$$

At equilibrium, we have $X^D = X^S,\; Y^D = Y^S \tag{3}$.

Equations $(2)$ and $(3)$ hold for any supply.
We now change the supply of $X$ but leave the supply of $Y$ unchanged. Indexing  the initial situation by $0$ and the second situation by $1$, they are described by

$$ X^D_0=\frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^D_0,\;\;X^D_0 = X^S_0,\;\; Y^D_0 = Y^S_0 \tag {4}$$

$$ X^D_1=\frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^D_1,\;\;X^D_1 = X^S_1>X^S_0,\;\; Y^D_1 = Y^S_1= Y^S_0\tag {5}$$

Then we have

$$X^D_1 - X^D_0 = \frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^D_1 - \frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^D_0   $$

and using the various relations

$$\implies X^S_1 - X^S_0 = \frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^S_1 - \frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^S_0 >0   $$

$$\implies \frac {\alpha}{1-\alpha}Y^S_0\cdot \left[\frac {p_{y1}}{p_{x1}}-\frac {p_{y0}}{p_{x0}}\right] >0$$

$$\implies \frac {p_{y1}}{p_{x1}}-\frac {p_{y0}}{p_{x0}} >0 \implies \frac {p_{y1}}{p_{y0}}>\frac {p_{x1}}{p_{x0}} \tag{6}$$

$(6)$ tells us that the price of $Y$, if it falls, it will fall certainly less than the price of $X$, but in proportional terms.

It appears that in this benchmark example, we cannot say anything about the price changes in terms of levels, as the OP asks about.

If the levels of the two prices are very different we can easily have a smaller proportional fall for $p_y$, while at the same time a higher fall in monetary units. Set for example $p_{y0} = 100, p_{x0} = 1$, and assume that the price of $Y$ falls only by $10$% while the price of $X$ falls by $20$%.

In turn the levels of prices depends on the magnitude of the supply of the two goods also, which are treated as exogenous here.


A colleague of mine pointed me to Steven M. Sheffrin's Rational Expectations, where I found the explanation (on p.122):

  [L]et $P_t^*$ be the variable to be forecast. Then the optimal forecast is given by
  \begin{equation}P_t=E[P_t^*|I_{t-1}].\end{equation}
  Let $u_t$ denote the forecast error, which, by the orthogonality principle, must be uncorrelated with the forecast $P_t$. Thus $P_t^*=P_t+u_t$ and, taking the variance of both sides,
  \begin{equation}
\mathrm{var}(P_t^*)=\mathrm{var}(P_t)+\mathrm{var}(u_t)
\end{equation}
  or
  \begin{equation}
\mathrm{var}(P_t^*)\ge \mathrm{var}(P_t),\quad\text{since $\mathrm{var}(u_t)\ge0$}.
\end{equation}


Effect of minimum wage on higher levels of pay

This phenomenon is sometimes called "wage compression" because the range of wages is compressed by the minimum wage laws.
One paper on this subject is The Impact of the Minimum Wage on Other Wages


Effect of a good's inputs on elasticity of supply

Consider the extreme case: all available quantity of the no-substitute input is already employed.

To trace the supply function, let's assume that we are already at some price level, and let's increase the price of the final good: will quantity supplied increase? No, because it cannot do so -there is no available additional input to use for this increase. In this extreme case the price elasticity of supply of the final good is zero, due to the situation in the inputs market.

Assume now that a little quantity of the no-substitute input is still unemployed. Being unique, we expect that it will command a high price if it is demanded for employment. So increasing quantity supplied of the final good, bears a high cost in terms of steeply increasing production costs. In this case, the price elasticity of supply of the final good is not zero, but "small"...

...compared to the case where the inputs needed for production are generally available and so the cost to employ them is smaller than in the previous case. Here we expect that under the same as before final-good price increase, final-good suppliers will find it profitable to increase their quantity supplied more than in the former case -so here the price elasticity of supply of the final good will be "large"(er).


There is a huge discussion about structural Macroeconomics models - especially on their theoretical and practical relevance - with IS-LM being on the forefront of the battle some 40+ years ago. Many people simply reject LM existence. Advanced level textbooks ignore this approach for two decades. However it ironically dominates introductory courses.
 Here is a nice discussion from Bloomberg that claims that, perhaps, the most instructive paper on the matter was published in QJE back in 1992 - link. (Full text can be downloaded elsewhere.)
From personal experience I could add that IS-LM is a great tool to build initial schematic intuition about the issue in question even though its microfoundations are not clear at all (put it another way, we have no idea why does the model work, but it does).


Are financial markets "unique" for each "currency pair", or are they simply "translated"?

It's not clear what level of answer you're looking for, so here is a much more basic answer.
There are indeed many exchanges with many different prices. However, if you have noticed that you could make money by exchanging your BTC for USD, exchanging the USD for SEK, and then exchanging the SEK for slightly more BTC than you started with - then someone else probably has noticed that too, and they'll make free money with the manoeuvre just as you will.
As people do this, the prices gradually equalise: you're making BTC more scarce relative to SEK so are driving its price up in SEK (making the last hop less profitable), and you're making BTC more common relative to USD so are driving its price down in USD (making the first hop less profitable).
So it is theoretically possible for such an opportunity for arbitrage to exist. In practice, however, in such a simple case, there are powerful forces acting constantly to equalise the price. As long as it's easy and low-fee to make all three transactions in the chain above (that is, as long as the respective markets are sufficiently liquid), people will make those transactions and the opportunity will naturally disappear.
If you identified a really unobvious opportunity for arbitrage (so unobvious that nobody else saw it before you) and the markets were liquid enough, then you could make a lot of money off it by exactly this manoeuvre!
In fact, very early on, the Bitcoin markets were absolutely full of arbitrage opportunities. The big players were not keen to trade in such risky places (it was routine for an exchange to just close down and run away with large amounts of other people's bitcoin - see the middle point on Forbes's article about this), so there was not such a strong force equalising the prices. As it's become more mainstream, though, more people started taking the free money, and the opportunities have dried up somewhat. So you may very well find that earlier in your dataset, the markets are much more obviously distinct.


The principle supply-side policies that are currently advocated are reductions in tax rates on labor and capital income.  Its claimed that lower tax rates on wages, interest, dividends and corporate income will increase output by increasing the incentives to work, increasing in the supply of labor.

https://www.philadelphiafed.org/-/media/research-and-data/publications/business-review/1981/br81mjap.pdf?la=en

While supply-side economists expect a little government regulation of the free market, demand-side economists expect a more active government.

Supply-side works by giving incentives to businesses to expand. Deregulation removes restrictions on their growth.

A corporate tax cut gives businesses more money to hire workers, invest in capital equipment, and produce more goods and services.

An income tax cut increases the dollars per hour worked. It boosts workers' incentive to remain employed and creates more labor.

the supply-side premise is that an increase in supply (i.e. production of goods and services) will increase output and lower prices.


Why it is important to observe if the 10Y treasury yield moves nowadays?

Theoretically there could be many reasons why interest rates increase. I will present you one possible interpretation (this is the one that is currently discussed in the news, see for example NBC):
Democrats have won the congress and accordingly they have the majority to enact their political agenda. Investors believe now this will increase deficit spending for pandemic relieve plans (like the stimulus payments to low income households) or for infrastructure investments. This directly increases the supply of treasuries and prices will fall accordingly. Prices going down means nothing else than an increase in the interest rate.
Why is this potentially problematic? Well in general it isn't, prices simply adjust to the shift in demand and supply. But in this concrete situation it means states have to pay higher interest rates for their debt and this may restrict possibilities to enact investments plans. During the current global pandemic Central banks try to keep interest rates low in order to give governments greater financial leeway.


Pareto set with Cobb-Douglas and Leontief preferences

The bottom right origin is actually not in the Pareto set.  At that point, $(x_A,y_A)=(8,0)$, so $U_A(x_A,y_A)=0$.  Similarly, $(x_B,y_B)=(0,4)$, so $U_B(x_B,y_B)=0$.

As an example, $B$ could give one unit of $y$ to $A$ and thereby raise $A$'s utility to 8 without hurting $B$'s own utility (which would remain zero).  As a matter of fact, any allocation other than the top left origin is a Pareto improvement over the bottom right origin.


Arrow-Debreu Theorem of Existence: Non satiation

Converting my comments into an answer:
At the bottom of p.268, the authors say:

The set of consumption vectors $X_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is bounded from below.

[Emphasis added.]
Since the Heine-Borel Theorem establishes that $S\subset R^n$ is compact if and only if $S$ is both closed and bounded (from below and above), one cannot conclude necessarily that $X_i$ is compact. (A counterexample is that $[0,\infty)$ is closed but not compact.)


What are some good advanced level textbooks on public economics?

The best book in my opinion for your case is definitely Intermediate Public Economics by Jean Hindriks and Gareth D. Myles. It is often used as a standard text for public finance at your level and covers most of the material you need in an interesting and accessible way.


Why is the velocity of money in an economy affected by the frequency in which the worker is paid (like weekly, bi-weekly, monthly, etc.)?

This is because by definition velocity of money is basically how quickly people spend money they get. Hence any factor that causes people to hold money holder decreases velocity of money and vice versa.

The frequency people get pay might affect how quickly they spend money. For example, if you are paid quarterly basis you will probably be more conservative with your spending to make sure your money lasts for whole quarter (extreme example just to show the point) then if you are paid on monthly basis and you will be even less conservative with your spending if you get paid every week or every day.


Can intermediate good producer's profit ever be zero in new keynesian model?

Yes. In the core New Keynesian model in Galí, it is possible for the aggregate profits of intermediate good producers to be zero or negative; and it is even more likely that the profits of individual intermediate good producers, who may be stuck at prices far from the optimum, will be negative.

Since it's easier to characterize the aggregate, I'll discuss the conditions there. Note that the aggregate profits of intermediate good producers are just aggregate revenues minus the aggregate wage bill, or
$$\Pi =PY-WN$$
where $P$ is the aggregate price index, $Y$ is aggregate output, $W$ is the nominal wage (common to all producers thanks to the competitive labor market) and $N$ is aggregate labor input.

Now, consider the special case where intermediate producers have constant returns to scale and productivity is normalized to 1. There we have, to first order, $Y\approx N$ (there is a second order difference due to price dispersion). Hence, again to first order, we have $$\Pi \leq 0 \Longleftrightarrow P\leq W$$
When is $P\leq W$? Well, using lowercase variables to denote logs, we know from household optimization that $w - p = \sigma c + \varphi n$ (see Gali p.43 equation (2), for instance), where $\sigma$ is the inverse EIS and $\varphi$ is the inverse Frisch elasticity. And since market clearing and our simple production function imply that $y=c=n$ to first order, this becomes $$w-p = (\sigma + \varphi)y$$
Finally, we observe that at the "natural level" of output $y^n$, markups should equal their desired level, which is $\mathcal{M}=\varepsilon/(\varepsilon-1)$, where $\varepsilon$ is the elasticity of firm demand. Writing $\mu =\log \mathcal{M}$ and substituting into the equation for $w-p$, this implies $-\mu=(\sigma+\varphi)y^n$. Finally, subtracting from the equation for $w-p$, we have
$$w-p = (\sigma+\varphi)(y-y^n) - \mu$$
We'll have negative aggregate profits $P\leq W \Longleftrightarrow w-p \geq 0$ when $(\sigma+\varphi)(y-y^n) \geq \mu$. Note that this happens when output is high, since profits in the basic NK model are countercyclical.

Interestingly, under some not-totally-crazy calibrations this can happen. For instance, if we assume that both the EIS and Frisch are 1/2, so that $\sigma=\varphi=2$, and also assume $\varepsilon = 10$, so that $\mu = 0.105$, then exceeding the natural level of output by even 3% is enough to deliver negative aggregate profits, since $4\times 0.03 > 0.105$. And again, note that it is much more likely that we'll have individual firms with negative profits; in the case where aggregate profits are even close to negative, any firm unlucky enough to be stuck at a much lower-than-average price will have negative profits.

Finally, I should clarify that all of this is unrelated to the concept of zero long-term economic profit that you mention. This concept is often embedded in models of monopolistic competition in the form of a free-entry condition (where firms pay some fixed cost ex ante to exist), but this is very rare in New Keynesian models - presumably because NK models focus on business-cycle frequencies where the entry margin is limited.


The comments have the answer. Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of iid random variables with $E(X_1)=\mu$ and $Var(X_1)=\sigma^2$ with $\mu,\sigma^2<\infty$. Define $Y_n=\frac{1}{n}\sum_{i=1}^n X_i.$ By WLLN, $\text{plim}Y_n=\mu.$ Since $f(x)=x^3$ is a continuous function, by the continuous mapping theorem, $\text{plim}f(Y_n)=\text{plim}Y_n^3=f(\mu)=\mu^3.$


Recover cost function from profit function

I suggest that you ask yourself the following questions:

If $\tilde{z}(y;w)$ is the input demand from cost minimization, can there be any difference between $\tilde{z}\big(y(p);w\big) \equiv \hat{z} (p;w)$ and $z(p;w)$ from Hotelling's lemma and profit maximization?
Even if $\tilde{z}(y;w)$ is a correspondance rather than a function, is there any difference between $ p \hat{z} (p;w) = p \tilde{z}\big(y(p);w\big) = c(y;w) $ and $p z(p;w)$ (where the later are dot products).

Hope this helps,


When a country collects data for calculating the Gini index, do they usually neglect single household?

I am not sure this is currently correct as http://www.oecd.org/els/soc/OECDIncomeDistributionDataReview-Korea.pdf says:

  Income distribution and poverty indicators for Korea are also available from:

  The  same  combination  of  Household  Income  &  Expenditure  survey  combined  with  Farm household economy survey from previous years, using different coverage of population

  Since 2006: all households (same series as OECD reference series)

  Since 2003: only for 2 and more non-farm household

  Since 1990: only for urban salary and wage worker, 2 and more household

which I intepret as saying that all households have been counted since 2006, while only households with 2+ people were counted from 1990 to 2005


Will the goods market be in equilbrium along the AD curve?

Definition: The Aggregate Demand curve shows the combinations of the price level and level of output at which the goods market and assets (money) markets are simultaneously in equilibrium. This definition proves option a) incorrect. The correct option is d)


Price when both supply and demand have market power

I suppose one way to solve this - but I'm still looking for a proper way without - is to fall back to matching.

If there is matching between the intermediate and the final stage, at zero search cost and symmetric arrival rates, the surplus will be split given the Nashbargaining weights - which then determines the price.


Yes with reserve requirement of $0.03\%$ the maximum amount of money that can be created will be given by $\frac{100}{0.03} = {\\\$}3333.33$, with ${\\\$} 100$ reserves and ${\\\$}3233.33$ new money.
Yes if everyone repays the ${\\\$}3233.33$  and there is still demand for loans bank can lend it again. To be clear the ${\\\$}3233.33$ will not become new reserve, so the reserve ratio would still have to be maintained. If people start paying the loans back with no demand for new loans the money supply will start contracting.
However, with presently reserve requirements being set to zero by Fed the only constraint is the demand for loans. Furthermore, while in past banks tended to hold almost no excess reserves (see this graph from Fred), after 2008 they started holding excess reserves and in such situation the multiplier idea becomes less useful abstraction as banks can always borrow extra reserves from other banks that have extra reserves if they need to. For example, if system holds excess reserves when the ${\\\$}3233.33$ would be repaid (assuming all this is done through lending and borrowing with single bank) bank could just get extra reserves from other banks holding excess reserves, even when  banking system as whole would not be allowed to create new reserve (without central bank agreeing to that).  McLeay, Radia, & Thomas(2014), money creation in the modern economy is good source for explaining how fractional reserve system works in such situation.


What are some examples of goods that entrepreneurship-intense?

The term "entrepreneur" is usually taken to mean something more specific than talented and skilled business people.  Wikipedia for example gives several definitions, including characteristics such as (with my emphases):

creation of value;
launching and running a new business;
organize the capital, talent and other resources that turn an invention into
a commercially viable innovation;
willingness to take risks in the name of an idea, spending time and capital on an uncertain venture.

Many who are talented and skilled business people lack some or all of these characteristics.
Entrepreneurship can be applied to many different types of goods.  Taking your countries A, B and C, for example, consider:

Gordon Moore, co-founder of what became Intel;
Parthasaradhi Nara, Indian agricultural entrepreneur;
Nora Bannerman, Ghanaian clothing entrepreneur.

What type of good your country D would export will depend to a large extent on what other factors it possesses in addition to entrepreneurship. As 1muflon1 says there may be opportunities for it to export consulting services, but even that would depend on good transport and/or communications in order to make contact with clients abroad.  But as the above examples illustrate, entrepreneurship is typically applied in order to start and grow businesses using other available factors to develop new products, technologies or markets.


Competitive equilibrium in an exchange economy with lexicographic preferences

The preferences of agent $A$ cannot be represented by any utility function and the prefeences of $B$ not by a differentiable utility function, so forget calculus approaches.

Since $A$ has strictly monotone preferences, we must have $p_1>0$ and $p_2>0$ for every equilibrium. Also, $A$ is always willing to give up any amount of good $2$ to get more of good $1$. So he will spend his wealth on good $1$ with consumption of good 1 equal to $10 p_2/p_1$. Also, we know that $B$ must consume both goods in equal amounts, so you just have to plug the equality $x_1^B=x_2^B$ in the budget constraint (justification here). Then you have the demand for both goods and can calculate their excess demands. Since only relative prices matter, you can normalize by, say, setting $p_1=1$. Since both agents will spend their whole wealth, Walras' law applies, and in order to find an equilibrium, you just have to find prices under which the excess demand for one of the goods (your choice) is zero. If you find a unique solution, you have found a unique equilibrium (note that demand is unique given prices).


How do I calculate the impact of an independent variable on a dependent var. when the independent var. changes from 0 to some positive value?

Just thinking out loud here, not saying this is correct others may have a better idea. I'm going to roll with this assumption that you are using best econometric practices so that you trust your estimates and want to take action based on them.

(1) The partial effect of going from $I=0$ to $I=1$ would be
$E(Y_1|I=1,N=\overline{N})-E(Y_1|I=0,N=\overline{N})$

$=\hat{\alpha_3}+\hat{\alpha_4}+\hat{\alpha_5}\overline{N}$

When irrigation is already positive, the marginal effect of irrigation on yields just be

$\frac{\partial E(Y_1|N,I)}{\partial I}=\hat{\alpha_3}+2\hat{\alpha_4}I+\hat{\alpha_5}N$

(2) So you are curious what level of Nitrogen makes it so that the marginal effect of nitrogen is the same as the partial effect of irrigation? I think you could get at this by finding the $I^*$ and $N^*$ such that

$\frac{\partial E(Y_1|N,I)}{\partial I}=\frac{\partial E(Y_1|N,I)}{\partial N}$?

We've already found $\frac{\partial E(Y_1|N,I)}{\partial I}$ above, so we need to calculate
 $\frac{\partial E(Y_1|N,I)}{\partial N}$

$=\hat{\alpha_1}+2\hat{\alpha_2}N+\hat{\alpha_5}I$

From here $\frac{\partial E(Y_1|N,I)}{\partial I}|_{N=N^*,I=I^*}=\frac{\partial E(Y_1|N,I)}{\partial N}|_{N=N^*, I=I^*}$

$\hat{\alpha_3}+2\hat{\alpha_4}I^*+\hat{\alpha_5}N^*=\hat{\alpha_1}+2\hat{\alpha_2}N^*+\hat{\alpha_5}I^*$

$\rightarrow (\hat{\alpha_5}-2\hat{\alpha_2})N^*=(\hat{\alpha_1}-\hat{\alpha_3})+(\hat{\alpha_5}-+2\hat{\alpha_4})I^*$

$\rightarrow N^*=\frac{(\hat{\alpha_1}-\hat{\alpha_3})+(\hat{\alpha_5}-+2\hat{\alpha_4})I^*}{(\hat{\alpha_5}-2\hat{\alpha_2})}$

I bet the optimal irrigation and fertilizer decision would equate the marginal product of fertilizer per dollar spent on fertilizer with the marginal product of irrigation per dollar spent on irrigation which would be

$\frac{\hat{\alpha_3}+\hat{\alpha_4}+\hat{\alpha_5}N}{P_{irrigation}}=\frac{\hat{\alpha_1}+2\hat{\alpha_2}N+\hat{\alpha_5}I}{P_{fertilizer}}$


That's a big one. I'm trying without claiming to be near exhaustive.

The fundamental challenge for social sciences, as you rightly point out, is the absence of carrying out experimental designs on a large scale. Introducing a minimum wage in one half a country and not so in the other half would fail because of e.g. equity concerns (why would you advantage certain workers?) or external effects (firms moving from one region to the other, differential effects on trade).

On a small scale though, you might certainly be able to run experiments which might inform you on the large scale. An example would be random assignment in school to different pedagogical approaches. Tracking students over longer period of time might yield insights on how changing things might turn out on a larger scale.

Having said that, Economics, as widely understood by the public, deals with question on the large scale. The traditional approach to those question is to set up mathematical models which describe the behavior of economic actors. These models are not fundamentally different to what natural sciences refer to as laws (...of motion, of energy conservation).  The difference is that social models do not (and cannot) claim their general validity and their reliability about the mechanism proposed. These models have been (and still are) derived from assumptions on individual (firms, households) or collective behavior (price mechanism). It is these assumptions which can be tested. They often have been tested with macro-scale regression analysis, such as explaining GDP growth by development aid to test effectiveness of those programs. While this is still done, at the research frontier such findings are by and large ignored.

Instead, Economics has undergone a so-called Credibility Revolution. In a nutshell, this entails the development and application of quasi-experimental methods which try to imitate the experimental ideal from natural sciences. If your question at hand is not feasible for a field experiment, you may look for settings where there is exogenous variation in a treatment variable that allows you identify its effect on a outcome variable of interest. There are numerous examples for this, also cited in the linked paper. One of the most famous ones is the comparison of increasing the minimum wage in one of two similar U.S. states. Exogenous variation may be geographic (compare two similar cities across a country border) or over time (a sudden reform that was not anticipated), to name just two. Nowadays, Empirical Economics tries to exploit these kind of variations with large-scale data on the micro level (the firm, the household, the individual) to come up with causal estimates on specific settings. If their findings can be confirmed by other settings, consensus on certain mechanism emerges, which might then be used to inform large scale theories.

To answer your question rightaway: Economists are not nearly as good as natural sciences regarding to the empirical basis of their theories, but they have greatly improved. On a more fundamental note, economic "laws" can never match the universality of natural science laws, as it's human behavior underneath.


What is a good proxy for government quality?

Using corruption is part of it but a bit restrictive way to measure government "quality". You may use aggregate indicators as the one developed by the Worldwide Governance Indicators (WGI) project from the World Bank. They reports aggregate and individual governance indicators for over 200 countries and territories over the period 1996–, for six dimensions of governance:

Voice and Accountability
Political Stability and Absence of Violence
Government Effectiveness
Regulatory Quality
Rule of Law
Control of Corruption

What is Governance ?

According to the WGI project Governance consists of the traditions and institutions by which authority in a country is exercised.  This includes the process by which governments are selected, monitored and replaced; the capacity of the government to effectively formulate and implement sound policies; and the respect of citizens and the state for the institutions that govern economic and social interactions among them.


Is there a reproducibility crisis in Economics?

There is a general reproducibility crisis in most fields of science including economics. First economics is actually doing quite well compared to other social sciences for example this study shows 54% of studies that authors tried to replicate in psychology could not be replicated but this study for economics shows only about 40% failed to replicate. This being said 40% is quite a lot.

So I would say yes there is a reproducibility crisis in economics (although note what is and isn’t reproducibility crisis is to certain degree opinion based).

However, I think that examples which you gave are not the main reasons for this crisis.

The first example of that undergraduate students is basically example of conscious $p$-hacking. This certainly occurs but usually it is done by people who are not really good at research to begin with and are under extreme pressure to show some output, so they resort to this "cheating" in a similar way bad students resort to cheating during exam. I would not say that at good universities vast majority of researchers would consciously $p$-hack. However, note I on purpouse separate conscious $p$-hacking from sub-conscious $p$-hacking that can occur due to subconscious biases that any of us have which can also occur when you are simply too invested in some topic so you even without realizing become little less rigorous when creating your model. Unconscious $p$-hacking is much bigger problem because it sneaks upon you without even realizing that your doing something wrong.

In the second example, I would not even say that is an example of doing some incompetent statistical analysis necessary. In case of IV you always can in the end rely only on some logical story based on which you justify the exclusion criterion and validity of your instrument. Statistically, you can measure strength of the instrument using $F$-statistics from the first stage, and relevance of instrument from looking at the output and some auxiliary stats there, but there is no statistical test to date that would allow you to check whether instrument is really exogenous and only affects dependent variable through the endogenous independent variable. Furthermore, sometimes you cant find any perfect instrument or model specification etc. so you will just do your best and as long as you are explicit in the research findings that the results rely on some oversimplification or possibly incorrect assumption, the findings might be still useful. In that case even if the results would turn out to be non-reproducible I personally would not consider it a problem.

What is really at the heart of the reproducibility crisis is actually precisely what Ionnadis finds, that is most of the studies are underpowered. However, this is not due to some conscious attempt to make it so but because power of most statistical tests depends on the number of observations and in economics its often very expensive and hard to get more data.

Next problem is the publication bias. Even if you manage to get solid power ($\geq80$%) as this video from veritasium shows, using 5% significance and with having 80% power, and assuming that out of 1000 relationships only 10% are correct, even if you do everything completely by the book without any bias (from the perspective of scientist) or messing with the data, if there is a publication bias towards positive results, you will find that almost third of the published studies will not be reproducible at all (so now the 60% of reproducibility from that study about experiments in econ suddenly does not even look that bad). This problem is not easy to solve because if we would move to lets say 3/6/9 sigma like in Physics then just due to sheer lack of data almost no result in social sciences would ever become significant. Moreover all journals care about readership (since that usually correlated with getting more citations and thus higher ranking of the journal), so they have strong incentive to publish novel interesting research rather than replications.

So to sum up. Yes there is a reproducibility problem in economics (and most branches of science). However, its not necessary due to conscious p-hacking or incompetence. Sure that can play a role, especially at worse institutions, but generally the problem is low power, publication bias and subconscious biases. This is why the problem is so hard to solve. Low power is due to data limitations, its hard to do much about that if collecting data is expensive. Publication bias, can be addressed but every journal wants to publish `sexi' new findings, so its very hard to find proper incentives to do that (although things are getting better), and finally its very hard to deal with potential subconscious biases - you might get so much convinced about truth of some model/statement that you unconsciously p-hack by lets say trying too many different proxies for the same thing and focus just on the best results - this is probably the toughest problem to get around as it requires large amount of self-awareness and discipline.


How to interpret Whited Wu index (WW-index)

This is equation $(13)$ of Whited, T. M., & Wu, G. (2006). Financial constraints risk. Review of Financial Studies, 19(2), 531-559.

It is empirically estimated as regards the specific coefficient values. The important question is,

  What is the left-hand-side?

Looking at eq. $(12)$ of the paper the left-hand-side of $(13)$ is $\lambda_{i,t+1}$ which in turn is part of

$$\Lambda_{i,t+1} = \frac {1+\lambda_{i,t+1}}{1+\lambda_{i,t}}$$

$\Lambda_{i,t+1}$ is discussed in the middle of p. 536 (between eq. 5 and eq. 6). As the authors write,

  If the outside equity constraint is binding, the effects of external
  finance constraints show up in the term $\Lambda_{i,t+1} =
 (1+\lambda_{i,t+1})/(1+\lambda_{i,t})$, which is the relative shadow
  cost of external finance. In the absence of finance constraints,
  $\Lambda_{i,t+1}=1$. On the other hand, if the equity constraint
  binds, then generally $\Lambda_{i,t+1}\neq1$, unless $\lambda_{i,t+1}
 = \lambda_{i,t}$. As also noted in Gomes, Yaron, and Zhang (2004), this last observation implies that finance constraints can only affect
  investment if they are time varying. It is the shadow value of the
  constraint today, relative to tomorrow, that is important.

So $\lambda_{i,t+1}$ is the shadow cost of external finance at period $t+1$. Near the end of p. 538 the authors write

  The higher $\lambda_{i,t+1}$, the greater is the effect of finance
  constraints.

This is translated as "the higher $\lambda_{i,t+1}$, the more difficult (or costly) is for a firm to obtain external financing". It does not appear to relate to whether it needs more external financing or not.

Moreover, as the authors write in p. 540, it being a shadow value, it must be non-negative. So what do you make of the fact that you obtained negative values?


Why are mortgage rates in USA so high compared to Europe?

The European Central Bank has held interest rates to an extremely low level, in large part in an attempt to boost the economies of member countries in the wake of the most recent worldwide financial crisis.  The United States economy has had a stronger recovery, and so rates in the US have risen in recent years.

This article, "How Europeans' household net worth is 'now exclusively driven' by negative interest rates" provides some detail and explanation as to why rates in the EU continue to be held to such a low level.


What are the economic consequences of discounting deferred tax liability?

The answer is in the text of IAS 12, and basically amounts to "it's unreasonable to require it, and if we allow firms discretion in doing it, they will only do it when it's in their favor":

  The reliable determination of deferred tax assets and liabilities on a discounted basis requires detailed scheduling of the timing of the reversal of each temporary difference. In many cases such scheduling is impracticable or highly complex. Therefore, it is inappropriate to require discounting of deferred tax assets and liabilities. To permit, but not to require, discounting would result in deferred tax assets and liabilities which would not be comparable between entities. Therefore, this Standard does not require or permit the discounting of deferred tax assets and liabilities.


Best-responding to a stochastically higher distribution of bids

Here is a sketch of an answer.

As noted in the question, the optimal $b$ must satisfy

$$b = v - \frac{F(b)}{f(b)}$$

Let us define the function $h(b) = v - F(b)/f(b)$. Our problem is to find the the fixed points of the function $h(b)$, i.e. the $b$ such that

$$b = h(b)$$

What can we say about $h(b)$? Assuming there is a unique optimal bid, it crosses the 45 degree line exactly once. Moreover, assuming that bids cannot be negative, i.e. $F(b) = 0$, it follows that $h(0) = v$. Thus, assuming that $v > 0$, the function $h(b)$ starts off above the 45 degree line.

Now, when the opponent bids are 'stochastically higher', the function $h(b)$ moves upwards in the sense that the 'new' function lies everywhere above $h(b)$. It thus seems reasonably clear that the unique fixed point must increase, though admittedly I lack a conclusive proof of this fact.

I am also unsure about the bonus question (whether my new bids seem stochastically higher in the sense of the reverse hazard rate).


Expenditure function. Prove that this set is bounded

The set need not be bounded. To see this, just take $U$ to be constant. Then the set will either be empty or equal to $\mathbb{R}_+^n$.
It is also possible that no minimum exists. Let $n=2$, $p=(0,1)$, $v=1$, and $U$ be given by $U(x)=U(x_1,x_2)=x_1\cdot x_2$. Clearly, $px=0$ is only possible if $x_1=0$, which would lead to a utility of $0$. But for every $\epsilon>0$, the bundle $(1/\epsilon,\epsilon)$ has utility $1$ and a price of $\epsilon$. So no expenditure minimizing bundle exists. Of course, there is also no expenditure minimzing bundle if there exists no $x$ at all such that $U(x)\geq v$. This can happen if $U$ is bounded above.
However, an expenditure minimizing bundle exists if it holds that prices are strictly positive and there is at least one bundle $x^*$ such that $U(x^*)\geq v$. To see this, note that minimizing expenditure on the set
$$\{x\in\mathbb{R}_+^n\mid U(x)\geq v\}$$
is equal to minimizing expenditure on the set
$$\{x\in\mathbb{R}_+^n\mid U(x)\geq v, px\leq px^*\}.$$
The latter set is closed and bounded.


Are short-let platforms really contributing to rising rent prices?

Yes an example of such study is this Sheppard, S., & Udell, A. (2016). Do Airbnb properties affect house prices. Williams College Department of Economics Working Papers, 3.

This is their conclusion:

  We find that in New York City, the impacts appear to be that an increase in localized Airbnb availability is associated with an increase in property values. In our hedonic model estimates, a doubling of Airbnb listings is associated with increases of 6% to 11% in house values, ceteris paribus. Using a difference-in- difference approach produces an even larger estimated impact, suggesting that properties that are subject to the Airbnb treatment increase in value by about 31%. Rough calculations based on average property values, average Airbnb rentals, and an assumption that potential income streams will be fully capitalized produces an intermediate estimate of about 17.7%.

  While our results might be taken as supporting critics of Airbnb who complain that the firm’s services act to increase house prices and diminish housing affordability, we want to stress that this conclusion may be unwarranted. A service that increases house prices (such as improved police protection, making better local schools available to residents, or providing more and better public parks) need not diminish community well-being.
  Public policies that reduce house prices in pursuit of housing affordability by diminishing the efficiency with which an owner can make use of his or her property are unlikely to be welfare-improving, in the same way as a city that creates “affordable” housing by encouraging more crime hardly seems desirable.

  Evaluating the welfare consequences of Airbnb, and hence the appropriateness of any regulatory action to limit use of Airbnb services, requires deeper analysis than we have provided here and much deeper analysis than appears to have been undertaken to date.


Python vs R (vs Stata): the old battle revisited

I use all three programs.

Python can do everything that R can do and R can do everything that Python does, but I must say R is superior to Python when it comes to the packages. For that reason for most econometric analysis I usually default to R. I find also producing nice standard statistics graphics with R easier (but for maps I prefer Python).

However, Python is far superior for web-scraping, numerical analysis and sentiment text analysis (although R has some good packages for that as well). Also, I prefer to use Python when I need to set up my own program as programming in Python is more natural (if that makes any sense) than in R, unless the program can be build easily from ready made functions from various packages.

I always recommend to people around me to learn both Python and R - the difference between them is not that big and with R you don’t really need to invest heavily into programming skills but just basics and then use packages.

Also, Jupyter Notebooks that can accommodate both R and Python make using both of them easier.

When it comes to Stata I use it only for a educational purposes (I teach econometric tutorials at university). To be honest I don’t like stata for several reasons:

Stata is not a freeware and i don’t think that the price tag is justified given that it’s inferior product compared to free ware programs like R and Python. If you can get it for free from uni then you probably don’t care about that but still it’s something to keep in mind.
Stata is a program not a language so if you want to create a new complex function you need to separately get and learn Mata (statas programming language).
Stata has some serious limitations on matrix sixes even in the most expensive edition the max mat size is 11000 which is serious limit when you work with panel data and have to run some iterative model with large number of variables. You will be routinely forced to run for example panel LR heteroskedasticity test on random subsamples even with the most expensive edition.
Creating beautiful graphics in Stata is nearly impossible - now don’t get me wrong you can make some decent graphics in Stata but it pales in contrast to what you can do with Python or R.
Stata is clunky with time series analysis. If you look for easy to use program for time series analysis it would be EViews. For example Stata can’t handle if you have quarters that are expressed in date format it will think you have gaps in your time series and won’t let you run basic time series commands until you create new time series variable. Also the offer of time series models is quite low and programming your own is painful.
While technically possible webscraping or numerical analysis in Stata is hell - if you need that for your work don’t use it.

However, Stata has also few advantages:

It’s more user friendly than R or Python, and can be even used without coding through interface (that’s why we use it for tutorials for students as a first program they see so they don’t get overwhelmed).
Doing some adjustments to datasets creating dummy variables etc is easier compared to Python or R but that’s mostly because in both Python and R you can have various types of data, lists, data frames etc.
With purchase to Stata you get access to Stata forums - it’s something like stack exchange but they actually pay professionals to give you answers there and often beside support you can get there very good advice even on econometrics, and you will usually get advice really quick.

I personally would not use Stata for my own scientific work. Python and R are superior to every aspect of Stata - but it’s very good starting program for students.


Government (meaning here executive branch of government since in most countries central bank is already government institution just independent like courts) could do that, in fact in many dictatorships/monarchies central banks are not properly independent so governments are de facto doing it if not de jure. In many such places central bank as an institution exists only nominally the same way as many dictatorships hold elections or have parliaments.

This being said historical experience shows governments are very bad at managing money supply because there ultimately complicated monetary decisions are being made made by people who don’t understand them at all. Historically, politically independent central banks perform much better at keeping prices stable and also actually better job at stimulating economy exactly when it’s needed. Hence to the extend that dictators care about having good monetary policy they would opt to have independent central bank. Although let me again reemphasize that in most dictatorships/monarchies central banks are not really independent and they are de facto controlled by dictator/monarch.


Can i check monotonicity with the marginal utility?

Yes. But you should review the definition of monotonicity (and perhaps of marginal utility) so that you understand why.


The world economy is a closed system as regards economic exchanges, but it is not a closed system as regards productive resources: earth as a natural system still has resources outside the human economic system that are continuously transformed by willful human activity and enter the economic system. Also, human time (enhanced with productive know-how) is something that continuously flows into the system as a productive resource.

The issue with the quantity of money therefore, is not its absolute level and magnitude, but its quantity relatively to produced output. A small village with a monetized economy, needs a certain amount of money to function economically. A large country needs a much higher quantity of money for the same purpose.

When a person borrows an amount of money, he essentially borrows an amount of productive resources, raw (investment loan) or in finished/ready-to-consume form (consumption loan). And indeed, he undertakes the obligation to return a higher amount that includes the interest.

If, when the time comes to repay loan+interest, he liquidates some already existing asset of his, then we do not really have additional money creation to cover the interest. If it manages to pay loan+interest by the output he produced in the meantime through his activity, then the additional money that needs to be created in order for the monetary transaction to be completed, "makes no harm" since it matches and reflects underlying real value that has already been created.

Certainly, "value" is not objective, and we have the price system that fluctuates, creating defaults, bankruptcies, hyperinflations, sudden re-distributions of wealth, etc. But in the very large picture you are looking at, these are fluctuations and deviations around the structure delineated above.


Yes there are. Two big ones are:

Why are some nations rich and why are some nations poor?
What are the causes of large scale involuntary unemployment of resources and individuals (I.e. a depression).


Money Metric Utility Function: Budget constraints become Utility Functions?

$m(\mathbf p,\mathbf x)$ specifies the minimum amount of money required for a consumer to attain the same utility as consuming bundle $\mathbf x$, taking prices $\mathbf p$ as given. In other words, since all bundles on the same indifference curve as $\mathbf x$ have the same utility value, and to achieve this utility, one needs at least an income of $m(\mathbf p,\mathbf x)$, we can thus establish a one-to-one mapping between utility levels and income levels. As utility is ordinal, we could as well use income as our utility measure.

This does not, however, suggest that the consumer's budget line is converted to her indifference curve; the two remain distinct objects. To see this, consider a two-goods case. With fixed prices, the slope of the budget line is $p_2/p_1$, which is constant for all $\mathbf x$'s. In contrast, the slope of the indifference curves is generally a non-constant function over $\mathbf x$:
\begin{equation}
\frac{\partial x_2}{\partial x_1}=\frac{\partial e(\mathbf p,u(\mathbf x))/\partial x_1}{\partial e(\mathbf p,u(\mathbf x))/\partial x_2}=\frac{MU_1(\mathbf x)}{MU_2(\mathbf x)}\,.
\end{equation}

The money metric utility function simply changes the unit of utility measurement from utils to dollars, which is legitimate thanks to the ordinality of utility. It does not change the underlying mathematical objects in any substantive way, though.


Are real wages a good measure of economic progress?

Thee are several issues including what happens about incomes that are not wages, but the main one is

  Not everybody earns a wage

This is an issue for the self-employed (especially in agricultural subsistence economies), but it is also an issue in economies with large numbers of dependants, especially the very young or the very old

It also raises the issue of the employment of women: increasing the proportion of women working will tend to boost total GDP and GDP per capita even though it can in some circumstances reduce average wages if the women entering the job market earn less than men already there

There are also cyclical issues: there have been recessions where unemployment has increased substantially but wages of those still working have been less immediately affected; there have been other recessions where the reverse pattern has predominated.  In both cases GDP per capita has fallen and the economy is obviously weaker than it was before the recession


What is the opportunity cost of doing something in your spare time?

There is a crucial detail missing in the scenario you describe.  Is the worker working a five day week (and no more) because that is what their terms of employment require?  Or do they have the option of doing additional paid work (at the same rate, and for as many hours as they wish) at weekends, but on this occasion have opted to work only five days in the week?  If the latter, then it is clear that the opportunity cost of mowing the lawn is the value of the time taken at the wage rate.

If however, the worker's paid work is limited to five days in the week, then it is not paid work that is being given up in order to mow the lawn at the weekend. Although the concept of opportunity cost seems clear enough, its application in this situation can be quite difficult, because it may be hard to identify what exactly has been given up. Suppose (as suggested in the comment by 1muflon1) that an alternative use of the time was to watch a movie. Fleshing out this scenario, suppose that late on Sunday afternoon, the worker was just about to watch the movie, but suddenly noticed that the grass was getting long, and decided that it couldn't be left for another week, so instead went to mow it. So far, it seems that what has been given up is watching the movie. But suppose that, having mowed the grass, they decide to watch the movie in the evening, displacing some other activity.  Rather than a clearly identifiable alternative that has been given up to mow the lawn, there may be a whole series of activities, each of which has merely been slightly postponed to give priority to another.

A context in which economists make use of the opportunity cost of leisure time is the travel cost method for estimating the value of non-market environmental goods such as parks. In this context the value of leisure time has often been taken to be a fraction of the wage, an influential source explaining the basis for such an assumption being Cesario (1976).  Use of a fraction of the (average) wage can be a pragmatic way to avoid the complexity of identifying what an individual has given up, and for aggregation over individuals in different circumstances.

Reference

Cesario, F J (1976) Value of Time in Recreation Benefit Studies  Land Economics 52(1) Feb 1976 pp 32-41


How do oil prices rise when crude oil inventories fall?

1) Supply and Demand. When supply falls, what supply remains becomes more valuable

2) There are massive storage tanks across the US (Cushing, Oklahoma comes to mind) where oil companies can purchase/rent storage space. Refineries cannot process crude as fast as it is extracted, so the backlog must be stored somewhere.

3) Any number of reasons. Increased consumption from industry or consumers. "unexpectedly fell" could actually mean that inventories grew slower than expected because drillers have reduced output. Perhaps a refinery came online after being shut down and helped draw down the crude stockpile.


What are the economic implications of regressing the same set of independent variables with different outcome variables

Using matrix notation, denote profits by $\Pi$, revenues by $R$, and the explanatory variables by $\mathbf X$.

Assuming a linear setup, we look at

$$\mathbf \Pi = \mathbf X\mathbf a + \mathbf u \tag{1}$$

and
$$\mathbf R = \mathbf X \mathbf b + \mathbf v \tag{2}$$

Are profits correlated with Revenues? Experience say they are. So there is also a relationship

$$\mathbf \Pi = \gamma \mathbf R  + \mathbf \varepsilon \tag{3}$$

Inserting $(2)$ into $(3)$ we get

$$\mathbf \Pi = \mathbf X(\gamma\mathbf b) + \gamma \mathbf v + \mathbf \varepsilon \tag{4}$$

Looking at $(1)$ and $(4)$, we have the equivalence

$$\mathbf a = \gamma \mathbf b \tag{5}$$

Will we obtain it through estimation? Not that easily.

Least -squares estimation will give us, for $(1)$

$$ \mathbf  {\hat a} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \Pi \tag{6}$$

and for $(2)$

$$\mathbf  {\hat b} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf R \tag{7}$$

Insert $(3)$ into $(6)$ to get

$$\mathbf  {\hat a} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\left(\gamma \mathbf R  + \mathbf \varepsilon\right) $$

$$\implies \mathbf  {\hat a} = \gamma \mathbf  {\hat b} + \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \varepsilon \tag{8}$$

The interesting realization comes at this point: given the postulated relation $(1)$ it is unlikely that the variables in $\mathbf X$ will be uncorrelated with $\mathbf \varepsilon$, the error term in $(3)$.

This means that even with a large sample, calculating the ratios of the individual coefficients in the vectors $\mathbf  {\hat a},\;\mathbf  {\hat b}$, we will not recover $\gamma$, since $\left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \varepsilon$ won't converge to zero.


What does it mean to identify structural shocks in macro models?

A great up to date treatment that I keep coming back to is Ramey (2016). It discusses identification in both articles you refer to.
The short answer to your question is that obtaining structural shocks is always based on theoretical reasoning and assumption imposition including in both of your examples (e.g. Christiano Eichenbaum Evans (1999), Gali (1999)).
On page 5 Ramey writes:

I  view  the  shocks  we  seek  to  estimate  as  the  empirical  counterparts  to  the  shocks  we  discuss  in  our  theories,  such  as  shocks  to  technology,  monetary  policy,  fiscal  policy,  etc.  Therefore,  the  shocks  should  have  the  following  characteristics:

(1)  they  should  be  exogenous  with respect to the other current and lagged endogenous variables in the model;

(2) they should be  uncorrelated  with  other  exogenous  shocks;  otherwise,  we  cannot  identify  the  unique  causal  effects  of  one  exogenous  shock  relative  to  another;  and

(3)  they  should  represent  either  unanticipated movements in exogenous variables or news about future movements in exogenous variables.

Any shock satisfying (1) - (3) can be used to make statements about the causal effect of an unexpected change to a variable on contemporary and future values of all variables in the model, which is the main goal of contemporary econometrics.
It seems that in earlier times researchers were more cavalier with identification, which is reflected in for example the oirf procedure in STATA, which will happily estimate "causal" effects based on orthogonalizing the residuals from the reduced form model, but this reasoning appears to be outdated. I think its contemporary version are  sign restrictions a la Uhlig (2005), which also stem from theory.


How do central banks create money to pay interest on reserves

The payment of interest is straightforward. The central bank is a bank, and pays interests on deposits in exactly the same way as any other bank.

If the central bank wants to pay $1 in interest to a depositor, it:

increases the amount on deposit by \$1 and
reduces the central bank's equity by $1.

That is, it just reshuffles the right hand side of the central bank's balance sheet; the asset side is unchanged. (On the income statement, there is a correpsonding $1 interest expense.)

All the central bank needs to do is ensure that the aggregate interest paid on reserves is less than the aggregate interest received from its lending and bond/bill holdings to prevent equity from going negative.

Since 100% of a central bank's assets pay interest, and its currency (dollar bills) do not, keeping the net interest flow positive is not too difficult (in a positive interest rate environment).

You asked what backs the currency. The answer is the taxing power of government. There's a few theoretical justifications for that statement, One is provided by Fiscal Theory of the Price Level (edge of mainstream), and another is referred to as "neo-Chartalism" (post-Keynesian).

As for the relationship between inflation and the rate of interest, the answer depends upon which school of economics you follow. Increasing interest rates increases the interest income of the private sector (which should increase effective demand, as you note), which is traded off by the effect on behaviour of the higher rate of interest. The conventional answer is that the behavioural effects dominate; that is, higher interest rates slow the economy and inflation.


Is worsening wealth inequality intrinsic to Capitalism?

While the owners of a successful business will generally experience an exponential growth compared to their employees, there are some factors to consider:

I find it unrealistic for any business to maintain a growth rate of 100% over many years. (As per the example) You can only sell twice as much if you can find twice as many buyers.
As a business grows, it is likely to start hiring more people and paying them better as they need better skills.
The business also takes the risk. The worker is generally guaranteed his X, but stockholders may have to expect the occasional decline in profit, or even a loss. The company may even go out of business in which case they cannot recover much, if any, of their investment. Those who take the risks, get the reward. Businesses failing also mean that bad investors can often lose fortunes, it's not always a good investment.
There is nothing stopping the employee from owning part of the company by investing in some shares. Then he too may experience this exponential growth. He must also keep in mind, however, that he must now also accept the risks of such an investment.


Where to find information to establish causation between economic policies and economic data?

You might be inspired by Joshua Angrist (MIT) who talks in this podcast about the craft of econometrics--how to use economic thinking and statistical methods to make sense of data and uncover causation.
Using natural experiments is a good way to establish causation. A natural experiment is an empirical setting in which individuals are exposed to the experimental and control conditions that are determined by nature or by other factors outside the control of the investigators. A growing literature relies on natural experiments to establish causal effects in macroeconomics (see Nicola Fuchs-Schuendeln and Hassan, 2016).
For those trying to find such natural experiments as a basis for their own research, Angrist says:

One thing I learned is that empiricists should work on stuff that's
nearby. Then you can have some visibility into what's unique and try
to get on to projects that other people can't do. This is particularly
true for empiricists who are working outside the United States.
There's a temptation to just mimic whatever the Americans and British
are doing. I think a better strategy is to say, "Well, what's special
and interesting about where I am?"

Angrist has also written accessible and relevant papers on data and causation:

Angrist, Joshua D., and Jörn-Steffen Pischke. 2010. "The Credibility
Revolution in Empirical Economics: How Better Research Design Is
Taking the Con out of Econometrics." Journal of Economic
Perspectives, 24 (2): 3-30.
Angrist, Joshua, D., and Alan B. Krueger. 2001. "Instrumental
Variables and the Search for Identification: From Supply and Demand
to Natural Experiments." Journal of Economic Perspectives, 15 (4):
69-85.


If the marginal cost is equal to 1, how does that imply marginal cost is equal to marginal benefit?

You have to look at this from the derivation of the profit equation.

From the equation $\mathbb{E}(\pi_n) = e - c(e)$, you can see that the marginal benefit of increasing $e$ is equal to 1. That is, for each extra $e$ you put in, you get that exact amount back in terms of expected profits. The $e$ term is the benefits, and $\frac{de}{de} = 1$ is then the marginal benefits. The same goes for marginal cost with the $c(e)$ term.

Follow up

In any objective function, labeling things "benefits" or "cost" could be tricky. Here, one could assume, from the functional form, that the function looks like $\pi = TR - TC$. It's then quite natural to label $TR = e$ and $TC = c(e)$. The benefit is the revenue you get, and cost is, well, cost.

It could very well be that, actually, $TR = e + f(e)$ while $TC = c(e) + f(e)$, where $f(e)$ could be any well-behaved function. The "correct" marginal benefits would then be $1 + f'(e)$, while marginal cost would be $c'(e) + f'(e)$. At the end of the day, though, you'd still have

$$1 + f'(e) = c'(e) + f'(e) \qquad\Leftrightarrow\qquad 1 = c'(e)$$


Why does $\varepsilon_{x,p_x}^H =-s_y \sigma $?

First we need to define the elasticity of substitution $\sigma$. This can be a difficult and confusing concept. (If you want to have your mind blown, look at Table 2 in this survey by Stern, which classifies no fewer than 10 notions of the elasticity of substitution!).

That said, the formula that your professor mentions is true only when there are two goods, so I'm going to assume we're restricting ourselves to the two-good case - and there, defining the elasticity of substitution is (for the most part) unambiguous. I'm first going to define it as (minus) the inverse elasticity of the ratio of marginal utilities of $x$ and $y$ to the ratio of their quantities, holding the overall level of utility constant at some $u$:
$$\sigma\equiv - \left.\left(\frac{\partial \log(U_x/U_y)}{\log(x/y)}\right)^{-1}\,\right|_{U=u}$$
This measures the curvature of the indifference curve between $x$ and $y$: the higher $\sigma$ (i.e. the more substitutable $x$ and $y$ are), the closer the indifference curve is to a straight line.

Now, Hicksian demand minimizes expenditure subject to being on an indifference curve. This involves setting the ratio of marginal utilities equal to the ratio of prices: $$U_x(x^H,y^H)/U_y(x^H,y^H)=p_x/p_y$$
Hence we can reinterpret $\sigma$ as (minus) the elasticity of relative Hicksian demands with respect to relative prices:
$$\sigma=-\frac{\partial \log(x^H(p,u)/y^H(p,u))}{\log(p_x/p_y)}\tag{1}$$
Since the denominator has only the relative price $p_x/p_y$, it doesn't depend on how we change this relative price (say, by raising $p_x$ versus lowering $p_y$). Let's assuming that we're just raising $p_x$, and then use your $\varepsilon$ notation for elasticities for simplicity:
$$\sigma=-\frac{\partial \log(x^H(p,u)/y^H(p,u))}{\log(p_x)}=\varepsilon^H_{y,p_x}-\varepsilon^H_{x,p_x}\tag{2}$$
At this point, we bring in a simple identity for Hicksian demand, namely that the sum of elasticities of goods with respect to some price, weighted by their shares, is zero (proof included below):
$$s_x\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=0\tag{3}$$
Since $s_x=1-s_y$, we can rewrite (3) as
$$(1-s_y)\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=s_y\cdot(\varepsilon^H_{y,p_x}-\varepsilon^H_{x,p_x})+\varepsilon^H_{x,p_x}=0\tag{4}$$
But the term in parentheses in (4) is just the elasticity of substitution $\sigma$, as we showed in (2). Rearranging, we've proven your professor's statement
$$\varepsilon^H_{x,p_x}=-s_y\sigma$$
as desired.

Proof of (3). We know that holding income constant, Marshallian expenditure must remain constant after any change in prices. The elasticity of spending on $x$ with respect to $p_x$ is $1+\varepsilon^M_{x,p_x}$ (combining price and quantity changes), and the elasticity of spending on $y$ with respect to $p_x$ is $\varepsilon^M_{y,p_x}$. Weighted by initial shares, these must sum to 0:
$$s_x(1+\varepsilon^M_{x,p_x})+s_y\varepsilon^M_{y,p_x}=0$$
Substituting the elasticity Slutsky equation, we get
$$s_x + s_x(\varepsilon^H_{x,p_x}-\eta_xs_x)+s_y(\varepsilon^H_{x,p_x}-\eta_ys_x)=0$$
which, once we use the fact that $\eta_xs_x+\eta_ys_y=1$ (as income expands, total spending expands proportionately), has a $-s_x$ canceling out the $s_x$ at the front, and reduces to just (3):
$$s_x\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=0\tag{3}$$
as desired. This is a very important identity governing Hicksian demand: it says that as we change prices, to first order the cost of Hicksian demand at the old prices stays constant (this is related to it being "compensated" demand: locally, it would cost the same at the old prices).


Optimal strategy in a single-agent choice problem under uncertainty

Your problem can be simply expressed as
$$\arg\max_{y\in\mathcal{Y}}\sum_{v\in\mathcal{V}}u_i(y,v)P_V(v)$$
Note that the subscript in $u_i$ is un-necessary. Furthermore, this is not a game so "mixed strategies" are only a solution if the maximum is not unique, but you should not worry about them  (more on that later).

First, note that the maximization problem is equivalent to the inequality you present. Presenting it as a maximization problem is more akin to how you solve a single agent problem, and circumvents the need to specify the quantifiers. However, strictly speaking,  it only characterizes "pure strategies", (a better term would be deterministic actions), but this is, really, without loss of generality. A probabilistic action is justified only when the maximizer is not unique, in which case the $\arg\max$ operator will already be giving you a set, and any mixture between the maximizers in that set would be optimal.

Besides being more clear, another benefit from presenting it as a maximization problem is that you can use standard theorems to guarantee existence and uniqueness, for example, if $\mathcal{Y}$ is finite then $\sum u(y,v)P_V(v)$ is continuous in $y$ (using the discrete metric) and Weierstrass ensures the existence of a solution. In general  you need a $\mathcal{Y}$ to be compact and $\sum u(y,v)P_V(v)$ to be  continuous in $y$. Uniqueness is a bit more tricky, but if $\sum u(y,v)P_V(v)$ is strictly concave with respect to $y$, then uniqueness is guarantied.

In your particular example given the finiteness of the action space, existence is trivial, but not uniqueness. Knowing that $P_V$ is normal is not enough since we don't know anything about the shape of $u(\cdot)$.


How game theoretic models and analysis is different from ABM?

One of the key (and often criticized) assumptions of Game Theory is the rationality of the players/agents and common knowledge thereof. That is the agents will make the best decision for themselves given the information available.

This assumption basically says cognitive costs are negligable. E.g. I can easily tell wether it is optimal for me to buy luggage insurance. In reality I would probably need to look up statistics and make some basic calculations.

Agent-based modelling frequently uses heuristic behavioral rules instead of optimization. In some circumstances this rules are not optimal, but very close. A nice example is the gaze heuristic:

  The gaze heuristic is used by humans and animals for catching flying objects. It entails the fixation of one's gaze to the object and adjustment of the running speed so that the angle of the gaze remains constant while approaching the object (see the three decision rules in the table above). Empirical evidence shows that experienced ball-catchers use the gaze heuristic and similar heuristics, as do dogs when trying to catch Frisbees.

Similarly in the Solow model one assumes that consumers save a constant portion of their income, without any utility calculations.

By simulating models with different behavioral rules one may find a model which is a good approximation of reality or one may find a simple rule which would make near optimal decisions.

While ABM is more flexible, that flexibility is also a problem: There is a myriad of heuristics to chose from, and different heuristics frequently lead to wildly different outcomes.


Price setting firms with cost reduction technology

Your reasoning for the 1st part is correct.

The 2nd part is a 2-period game. You should try to solve it by backward induction. First you go through all the 2nd period subgames. Then you can use the payoffs of the subgames and write up the investment decision subgame as a bimatrix game. However, the question is poorly written and as stated  this game has no Nash equilibria in pure continous strategies. Let's go through the reasoning as to why.

First, let's find the NE for all possible cases in the 2nd period.

Both firms have adopted the technology and have marginal cost 3. They will engage in a Bertrand game, set their prices to 3 and make 0 profit.
No firm has invested, both charge a price of 5 and make zero profit.
Now on to tricky case. What if one firm 1 has invested and firm 2 hasn't? It is crucial to know how firms split the market if they both charge the same price in this case. Since this is not given in the question, it is not really possible give a definite answer. I will go through two cases.

First, let's assume there's a 50:50 split with identical prices. Any equilibrium price above 5 can be ruled out by the usual Bertrand game argeuments. Firm 2 will never charge a price below 5, since it will make a loss. Firm 1 will never charge a price below 5, because given $p_2 \geq 5$ it could always increase it's profits by raising the price. So really the only candidate for an equilibrium is that both firms charge 5. However, if we assume a 50:50 split in this case, then firm 1 could increase it's profit by slightly decreasing the price and gaining the whole market. As a result, there is no equilibrium in this subgame, and hence no equilibrium for the overall game.

Second, let's assume arbitrarily that if both firms charge the same price, then all customers will buy from firm 1. In this case $p_1 = p_2 = 5$ is a Nash equilibrium.

You could then go on to find the NE of the whole game for different assumptions on the payoff split.


Long run trade off between inflation and output

Note that if $\pi_t = E_t\pi_{t+1}$ (current and expected future inflation are equal) then

$$\pi_t = E_t\pi_{t+1} + h(y_t-y_t^*) \implies 0 = h(y_t - y_t^*) \implies y_t = y_t^*$$

So as the output gap tends to zero in the long run, inflation will perfectly determine next period's inflation, so monetary policy that affects inflation in the short run cannot affect output in the long run. Thus there is no tradeoff between inflation and output in the long run.


A good balance of intuition and rigour is given by a text called Chicago Price Theory and an accompanying YouTube series from Chicago Economics Dept.
https://home.uchicago.edu/cbm4/cpt/index.html
My only criticism of the YouTube series would be that the ordering of the videos is very poor. It would therefore require simultaneously consulting the text.


Why is wage-induced inflation a bad thing?

Wage induced inflation can create inflation spirals, which can lead to hyperinflation. I think everyone agrees hyperinflation is a bad thing for everyone and not just the rich. Furthermore, inflation can be a significant problem for the poor as well.

Such a case of inflation spirals was even threatening post WW2 Germany.

The spiral happens because, if workers demand higher wages that firms cannot pay, then firms may increase prices in order to make up for those losses. This causes inflation, which reduces workers' real wages, so they ask for another wage increase. Firms not in a position to deny or to afford these wages will have to comply and increase prices again. This can go on and on and easily spiral out of control.

In principle it is possible to have higher inflation and higher wage increases. This was the case in Italy before the introduction of the Euro.

However this wage induced inflation also affects non-wage earners, such as those living on welfare benefits, the unemployed living on savings and retirees.

As for whether fiscal stimulus is better put in the hands of the poor, this is a different topic related to fiscal policy and unrelated to inflation (i.e. monetary policy).


If you simply "copy" the trades of a professional trader, but with less money, isn't that *guaranteed* profits?

Within finance, there is interest in attempting to replicate the positions of strongly performing funds. So your basic premise has some merit. However, most funds attempt to obscure what they are doing to foil copycats.
However, what you describe has risks.

The investors you are copying may lose money themselves. In particular, there is no reason that people making videos on spreading them online are telling the full truth about their performance. Securities laws generally prohibit giving unsolicited investment advice, and professional investors are generally aware of those laws.
If you get information with a lag, you are not going to replicate their results. In the worst case, they buy before the price rise, and then recommend others buy at the higher price (allowing them to sell). This is a common feature of fraudulent stock trading schemes in penny stocks.
Even if you can avoid the previous problems, a smaller investor may not be able to trade all the instruments a larger investor can, and face higher transaction costs. If positions are actively traded, higher transaction costs (as a percentage of the portfolio size) can mean the difference between a profit and a loss.


What would happen if layoffs were made illegal?

We can infer from what happens in countries where layoffs are difficult, e.g. France. There employees are (or at least were) harrassed to quit instead.

There's a practice in France called being sent to the closet — "le placard."
It's when your employer makes your work life so miserable that you are forced to quit. [...]
Being sent to the closet is a pretty awful practice, but it's something that a number of French employers resort to because it's so hard to fire people legally there.
Many workers in France have contracts called CDIs — contrat à durée indéterminée — or contracts of indeterminate length. Once you have one, you can lose your job only in very limited circumstances.

I think there is/was a similar practice in Japan with the so-called "boredom room" instituted for a similar reason.
As for broader economic effects, the IMF generally has working papers favoring labor market flexibility (which among other things entail easy layoffs) because it seems to lower overall unemployment, e.g.

The aim of this paper is to analyze the relationship between labor market flexibility and
unemployment outcomes. Using a panel of 97 countries from 1985 to 2008, the results
of the paper suggest that improvements in labor market flexibility have a statistically and
significant negative impact on unemployment outcomes (over unemployment, youth
unemployment and long-term unemployment). Among the different labor market flexibility
indicators analyzed, hiring and firing regulations and hiring costs are found to have the
strongest effect.


If Eurodollar never leaves U.S., then what do we mean by bring eurodollar on shore

Eurodollars can and do leave the US. They can be anywhere in the world: any bank outside the reach of the US Federal Reserve (henceforth "the Fed"). That's what makes them Eurodollars: they're US dollars in deposit accounts beyond the reach of the Fed.

That money is said to be "brought onshore" when it is transferred from a deposit account outside the reach of the Fed, into an account that is within reach of the Fed and its regulation / protection. This changes it from being a eurodollar to a dollar.

A US-based bank can have offshore banking subsidiaries and activities, and that's how it can hold eurodollars.

The whole point of a eurodollar is that it takes it outside the reach of the Fed.


Indifference curves represent preferences. Preferences are usually assumed to be stable, i.e. they do not change. So no, indifference curves don't shift.


How do we estimate production functions?

The following is the basic idea if we are to estimate the parameters by linear regression.

Take the natural log of the production function $F(L,K)=L^aK^b$, you will then get $$\ln(F)=a\ln(L)+b\ln(K).$$
For each entity (e.g., firm) $i$, collect data on the production level $F_i$, the amount of labour $L_i$, and the amount of capital $K_i$. Note that measurement issues may arise and that it is important to be explicit about how one is to interpret 'production', 'labour' and 'capital', and how to measure them.
Apply the transformation $X_i\mapsto \ln(X_i)$ for each variable $X_i$ for each entity $i$ in your dataset. (If there is any $X_i$ such that $X_i\leq 0$ then this will not work. You may delete those observations $X_i$ for which $X_i\leq 0$. If it is real data, think about the following question. "Why is $X_i\leq 0$? Has the dataset been wrongly registered?") Economists also apply the transformation $X_i\mapsto\ln(X_i+\epsilon)$ as an approximation when $X_i\geq0$ and $X_i=0$ for some $i$, where $\epsilon>0$ is some small number.
Now, regress $\ln(F)$ on $\ln(L)$ and $\ln(K)$ using your converted dataset (cf. 3) under the assumption that the intercept term is equal to $0$. This gives you estimated values $\hat{a}$ and $\hat{b}$ of $a$ and $b$. Hence, your estimated production function, which is supposed to capture some general tendency regarding production, is $$\hat{F}(L,K)=L^\hat{a}K^\hat{b}.$$
(If the production function is written as $F(K,L)=AL^aK^b$, where $A$ captures technology for example, you will get an intercept term $\ln(A)$.)

If we are using nonlinear regression, then we take the nonlinear least squares (NLLS) estimate of $a$ and $b$ by solving $$\min_{a,\, b}\sum_i\big(F_i-L_i^aK_i^b)^2.$$ Numerical computing softwares (which uses algorithms such as the Gauss-Newton algorithm) may assist you when finding the NLLS estimate.


What was the Rogoff & Reinhart spreadsheet mistake?

Reinhart and Rogoff (2010, RR) claimed that GDP growth is much lower when debt/GDP > 90%.
Herndon, Ash, and Pollin (2014, PDF, HAP) placed their errors into four categories:

Spreadsheet error: RR omitted Australia, Austria, Belgium, Canada and Denmark from the analysis (apparently a coding error somehow dropped the first five countries when sorted alphabetically).
One small transcription error: RR incorrectly transcribed a figure of "-7.6%" in one spreadsheet to "-7.9%" to another.
Inappropriate weighting: Example: New Zealand had only one year (1951) with debt/GDP > 90% and in 1951, its GDP growth was -7.6%. The UK had 19 years with debt/GDP > 90% and in those 19 years, its average GDP growth was 2.4%. The usual practice would be to count New Zealand 1951 as one country-year observation and those 19 UK years as 19 country-year observations. But instead, RR averaged the 19 UK years as one single observation that carried the same weight as that single New Zealand 1951 observation.
Selective exclusion of available data: Example. RR exclude Australia (1946–50), New Zealand (1946–49), and Canada (1946–50) without giving any explicit reason. But these country-year observations are precisely instances of high GDP growth and debt/GDP ratios.

Errors #1 and #2 are clear cut (and may be fairly characterized as "Excel spreadsheet errors").
Errors #3 and #4 are a little more subjective.
(Neither RR nor HAP state the program used. But as reported in e.g. NYMag, following repeated requests from Herndon, Reinhart emailed to Herndon an Excel file and it was in this file that Herndon discovered errors.)


How to calculate CRRA bounds from Holt and Laury (2002) type lottery?

Let me rewrite your Option B as follows: £10 with probability $p$ and £1 with probability $1-p$ where $p\in[0,1]$ varies across the items in the list of options.

Let $\bar p$ be the point where the subject's choice switches. For example, subject chooses Option A for all $p<\bar p$ and Option B for all $p>\bar p$. Then we can use the indifference condition
\begin{equation}
U(5)=\bar pU(10)+(1-\bar p)U(1)
\end{equation}
to infer the risk aversion parameter $r$.

In experiments where $p$ is discretized, we can take $p_0,p_1$ as the values that are immediately before and after the switching point in choice list. For example, subject chooses Option A for all $p\le p_0$ and Option B for all $p>p_1$. Then the bounds on the risk aversion parameter can be inferred from the following inequalities:
\begin{equation}
U(5)\ge p_0U(10)+(1-p_0)U(1)
\quad\text{and}\quad
U(5)\le p_1U(10)+(1-p_1)U(1).
\end{equation}


Yuan to be a reserve currency, IMF is considering

Yuan will certainly appear in the reserves of oil exporting countries. The question is how much of it?

China still has many problems with its currency and a long way to go until it becomes as appealing as dollar or even Japanese yen. Whether countries include a currency in their reserve basket ultimately depends on how predictable and stable it is compare to other currencies. It is also preferred that the currency appreciates in the long term. Living in China myself and holding my savings in yuan, I try to keep an eye on factors and risks that may move it up or down in medium to long term. Here are the most important in my opinion:

Upside:

China is still a huge and rapidly growing economy, expanding at least 4% a year (~7% being the official figure). As long as it produces so much goods and services that can be bought with its national currency, yuan will be appealing to investors.
China's population is growing and with recent cancellation of a one-child policy, it will continue to grow even faster. Growing population generally leads to economic expansion.
There are still market reforms to be made that, if implemented, will make the currency even more appealing. For example allowing banks to set lending and deposit rates freely would send yuan sky high.
China's state owned enterprises (SOEs) have trillions of dollar-denominated debt on their balance sheets. If yuan were to depreciate considerably, they would go bust being unable to service debt. It limits the freedom of Chinese government to devalue yuan.

Downside:

China still has a lot of work to do before yuan becomes a full-fledged reserve currency. It is still not freely convertible, there are capital controls in place for Chinese citizens, the People's Bank of China (China's central bank) is not independent of China's government, etc.
China's economy is slowing and the reforms that are being done are not enough. To reach growth targets and maintain social stability, the Chinese government may continue the policy of devaluing yuan to boost exports. Chinese corporate sector is certainly betting on it: http://www.marketwatch.com/story/chinese-companies-are-betting-on-further-yuan-weakness-2015-11-18
China packs a lot of political risk. It is not a democratic state with thick censorship of every media outlet. If it starts the process of democratization, it may become politically unstable with disastrous consequences to the economy and its currency. If it continues the path of one-party rule with a tight grip on the economy, it may lead to prolonged stagnation and slow devaluation of yuan.


Why does a company want the value of its stocks to be as high as possible?

The board of directors is supposed to represent (or work for the interests of) share owners. Share owners prefer higher prices if and when they want to sell their shares. The board chooses the CEO and will retain the CEO if he helps share owners realize their preferences.


How to find the profit-maximizing output level?

Actually 4 in this case would be completely valid answer since the net profit at both quantity 4 and quantity 5 is equal  $\Pi(q=4)=\Pi(q=5)=40$. I would even argue 4 is potentially better answer.

I think that the reasoning of whoever made that textbook was that given the profits are equal whichever quantity has marginal costs closer to price should be preferred.

In this case the marginal costs at $q=4$ are 17 and $q=5$ are 30 and 30 is just more closer to 28 than 17. Again, I would not say that necessarily the best answer to discrete problem like this, but my guess is they want to teach to always look at point where price is the closest to MC since in optimum it should be equal.

I would not read too much into this exercise. At least this chapter of that online resource is at high-school or at best very simple undergraduate level. At such low level often rigor is sacrificed for the sake of making the material not too difficult in order to not discourage young students. In your future courses/self-study you will get more precision and nuance.


What is the advantage and disadvantage to have a high value of USD for USA?

When the USD is highly valued, it allows people from the home country, the United States in this case, to purchase goods relatively cheaply from abroad and put pressure on domestic firms to have low prices. From a consumption standpoint, this may be advantageous.

A high value USD may also make goods from the United States more expensive relative to goods from other countries, so the quantity demanded of U.S. goods abroad decreases.

Thus, the high value USD may increase the quantity demanded of foreign goods domestically and reduce the quantity demanded of U.S. produced goods abroad. Together, this can, but not necessarily, have long term consequences on the current account if imports are greater than exports over the long term.

The weak Chinese yuan renminbi allows China to export many goods cheaply and, thus, accumulate a positive current account and become richer relative to the rest of the world. In fact, China's monetary policy has been subject to criticism in the past for actively managing the renminbi's value.


Show that conditional variance of error in linear probability model is heteroskedastic?

In relation to the problem of heteroskedasticity in linear probability models, the following Ben Lambert video is a useful link:
youtube.com/watch?v=pgPhbVEbYqw


Why does an increase in money cause a proportionate increase in price level according to classical/neoclassical economists?

It's a bit hard to answer without a precise model structure, but normally an increase in money will propagate into an increase in nominal demand. For the sake of simplicity, suppose money is equivalent to nominal gdp $M$, whereas real gdp by definition is given by $Y=M/P$. In classical economics, real gdp is pinned down completely by fundamentals of the economy, so that accordingly $P\propto M$ (namely by a factor $1/Y$).


Calculating Weighted Average Cost of Capital (WACC) in this example

You can't. Beta is observed from the value of the company respective to the market (beta represents volatility).
Also the market risk premium is the expected return for the market portfolio and cannot be calculated with what you have been given.


Are all market demands residual demands to some extent?

I agree, in the real world all markets are residual. unlike the theory laboratory where a product is clearly distinct from all other products, in reality there are always substitutes, and there are fine lines between one product and another.


Difference between the minimum selling price and the actual selling price

Sale price minus cost is profit per unit or marginal producer surplus. It's not quite clear what this "planned profit" means. If you're including opportunity costs in this amount (I could spend my time working another job making X, so if I instead spend my time making Product A, I want to make at least X doing so), those are part of the costs. If you literally mean "How much I think I'll profit from this", that has no economic meaning.


If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem,
$$ x c'(x) = k c(x).$$
That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is homogeneous of degree 1, then $x c'(x) = ax = c(x)$).
Without homogeneity, I am not sure there is any insightful interpretation you can give to $x c'(x)$, but others may have better ideas. The reason I am pessimistic is the "most natural" way for $x c'(x)$ to appear is in
$$[xc(x)]' = xc'(x) + c(x),$$
which gives
$$\frac{[xc(x)]'}{c(x)} = xc'(x).$$
However, it's not clear to me how to interpret $\frac{[xc(x)]'}{c(x)}$ either. Why should anyone care about $xc(x)$ in the first place? Provided $c(x)$ is the total cost (and not the per unit cost), why would you want multiply your total cost by the number of units you are interested in (what does "the total cost of producing 5 units times 5" means?).
If you let $c(x)$ be the per unit cost, maybe you can make some interpretative sense of  $\frac{[xc(x)]'}{c(x)}$ as $xc(x)$ then becomes the total cost?


Economics of clubs(sport gym, language course, etc)

No doubt there are other topics that have network effects, but one place to look is spatial common pool or pollution literature, where actions in location A have an effect in location B, and some coordination is required.

Papers include renewable resources e.g.:

  Kaffine, Daniel T., and Christopher Costello. "Unitization of
  spatially connected renewable resources." The BE Journal of Economic
  Analysis & Policy 11.1 (2011).
  https://www.degruyter.com/view/j/bejeap.2011.11.issue-1/bejeap.2011.11.1.2714/bejeap.2011.11.1.2714.xml

The acid rain problem (admittedly quite old):

  Mäler, Karl-Göran, and Aart De Zeeuw. "The acid rain differential
  game." Environmental and Resource Economics 12.2 (1998): 167-184.
  https://link.springer.com/article/10.1023/A:1008273509255

And things like pollen dispersal (self-promotion, but it is definitely clubs):

  Punt, Maarten J., and Justus Wesseler. "The Formation of GM‐free and
  GM Coasean Clubs: Will They Form and If So How Much Can They
  Achieve?." Journal of Agricultural Economics 69.2 (2018): 413-438.
  https://onlinelibrary.wiley.com/doi/full/10.1111/1477-9552.12235


Calculus and Indifference Curves in an Urban Economics Example

The utility function under consideration is $v(c,q)$ and then
$$MRS(c,q) = \frac{\partial v/\partial q}{\partial v/\partial c} = v_2/v_1$$
make the functional denpendency of on $u$ explicit then you have
$$\frac{\partial}{\partial u}MRS(c(u),q(u)) = \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\partial q} \frac{\partial q(u)}{\partial u} = \frac{\partial p(u)}{\partial u},$$
where the last identity follows because you know that $p = MRS$. Now simply rearrange the identity
$$\frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\partial q} \frac{\partial q(u)}{\partial u} = \frac{\partial p(u)}{\partial u},$$
to get
$$ \frac{\partial q(u)}{\partial u} = \frac{\left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u}\right]}{\frac{\partial MRS(c(u),q(u))}{\partial q} },$$
then use that Brueckner has defined $\eta := \left[\frac{\partial MRS(c(u),q(u))}{\partial q}\right]^{-1}$ in footnote(3) to get
$$ \frac{\partial q(u)}{\partial u} = \left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u}\right] \eta ,$$
and finally apply the rule that $\frac{\partial c(u)}{\partial u} = \frac{1}{\partial v/\partial c} = 1/v_1$ to get
$$ \frac{\partial q(u)}{\partial u} = \left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{1}{v_1}\right] \eta.$$