The critical points here are the phrases "it is recommended that...", and "using four lags is a norm". "Recommended" and "norm" based on what? On a specific prevailing theoretical model, or on past experience with the specific kind of data? If it is the latter, you do not conflict with any economic theory, if you try something different.

This is also a good example of the need to arrive at a synthesis where the model is both theoretically supported and also statistically adequate (consult the works, especially the books, of prof. Aris Spanos on this important methodological matter).


Maybe I'm missing some information about this model, but the intuitive interpretation to me is that $$u_B = v(s) - p $$ and $$u_S = p - c(s)$$ where $p$ is the price paid by the buyer for the good.

Pareto optimality is obvious, and it's easy to check that the Pareto optimal quality is positive.


Real world production-possibility frontier example?

In the 1940's the RAND corporation put together a series of simulations about war between the USA and the USSR. In the end, they came out with a series of two "good" possibilities, degree of victory and number of crew lives saved.

  The US Air Force asked RAND to apply systems analysis to design a
  first strike on the Soviets. ....  Paxson and RAND were initially
  proud of their optimization model and the computing power that they
  brought to bear on the problem, which crunched the numbers for over
  400,000 configurations of bombs and bombers using hundreds of
  equations. The massive computations for each configuration involved
  simulated games at each enemy encounter, each of which had first been
  modeled in RAND’s new aerial combat research room. They also involved
  numerous variables for fighters, logistics, procurement, land bases,
  and so on. Completed in 1950, the study recommended that the United
  States fill the skies with numerous inexpensive and vulnerable
  propeller planes, many of them decoys carrying no nuclear weapons, to
  overwhelm the Soviet air defenses. Though losses would be high, the
  bombing objectives would be met. While RAND was initially proud of
  this work, pride and a haughty spirit often go before a fall. RAND’s
  patrons in the US Air Force, some of whom were always skeptical of the
  idea that pencil-necked academics could contribute to military
  strategy, were apoplectic. RAND had chosen a strategy that would
  result in high casualties, in part because the objective function had
  given zero weight to the lives of airplane crews.

Retrospectives: The Cold-War Origins of the Value of Statistical Life

  RAND quickly backpedaled on the study, and instead moved to a more
  cautious approach which spelled out a range of choices: for example,
  some choices might cost more in money but be expected to have fewer
  deaths, while other choices might cost less in money but be expected
  to have more deaths. The idea was that the think tank identified the
  range of choices, and the generals would choose among them. But of
  course, financial resources were limited by political considerations,
  and so the choices made by the military would typically need to
  involve some number of deaths that was higher than the theoretical
  minimum--if more money had been available. In that sense, spelling out
  a range of tradeoffs also spelled out the monetary value that would be
  put on lives lost.

The Origins of the Value of a Statistical Life Concept

Maybe that's too much like the two input story of Aerandal. Depends on if you consider the saved lives a "savings" or a "good". For a clearer example of the guns vs. butter tradeoff, consider another example from WW II.

  In February 1942, Fisher Body completely stopped making auto bodies
  and began assembling the famous M-4 "Sherman" tank in its No. 1 plant
  in Flint. The operation eventually moved to Grand Blanc and would turn
  out 11,358 tanks by 1945.

  Buick tackled the manufacture of ammunition, churning out 75,000
  casings per month for the duration. By the war's end, the division had
  supplied more than 12.5 million casings.

  Buick also retooled to meet the demands of making engines for the B-24
  bomber. At first, they talked of about 500 engines a month, but the
  government doubled its order by the time Buick had its tooling in
  place. By 1944, Buick's Melrose Park factory was regularly turning out
  2,000 engines a month....

  In all, more than 113,000 employees left GM to serve while the company
  churned out $12.3 billion in aircraft, tanks, vehicles and arms.

  When it was all counted up after the war, GM had produced 854,000
  trucks (including the legendary DUKW, or "Duck" amphibious vehicles),
  198,000 diesel engines, 206,000 aircraft engines, and 38,000 tanks,
  tank destroyers, and armored vehicles, not to mention vast quantities
  of guns and ammunition.

How GM's Divisions Tackled the War Effort


Continuous time optimization with two laws of motion (the Hamiltonian with two laws of motion)

In this case you have two state variables, so as you say, you have to check the FOC ($J$ is the Hamiltonian) for $\frac{\partial J}{\partial K}=-\dot\lambda_1$ and $\frac{\partial J}{\partial H}=-\dot\lambda_2$ (besides the FOC for control variables). In this sense the logic is the same as Lagrangian (in static context), there are as many lagrangians, as there are constraints (dynamics/laws of movement in this context). Hope this helps!


Econometric test for detecting change from trend stationary to stationary process

The OP is examining a deterministic trend, and whether it stops being present after a point in time. So the model could be something like

$$y_t = \alpha t + \beta y_{t-1} + u_t,\;\;\; t=1,...,T_A$$

$$y_t = \beta y_{t-1} + u_t,\;\;\;\; t=T_A+1,...,T$$

So we want to test whether $\alpha =0,\;\; t=T_A+1,...,T$

This is a test for "structural change". Interestingly, in the last 15 years there has been a lot of research around unit root testing if a stuctural break in deterministic components is present, but I could not find anything that tests directly for the disappearance of a deterministic trend.

At least "in spirit", it will be a Chow test (don't lookup the wiki page, it says really nothing), when the point of the structural break is assumed known -for example, we can "see it with our own eyes" because the effect of the trend was strong (steep slope) and "suddenly" the series levels off and starts to fluctuate around what appears to be a constant mean.

If we want to search for the point of structural change to a rather wide interval of observations (typically the middle 70%), then this is a Quandt Likelihood Ratio ("QLR") test.

...And the main question is: Do these tests need modification because a deterministic trend is assumed initially present and then absent?

Hamilton (1994), ch. 16.3 "Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend" (pp. 463-472), analyzes in detail the first equation in the above model. By providing the asymptotic distribution of the coefficient estimators (it is not the same as when the autoregressive term is absent - for this last case, see Ch. 16.1) I believe it provides all that is necessary to execute validly a Chow or QLR test.


The MSE is essentially a squared Euclidean distance between two vectors, say $\mathbf y$ and $\hat{\mathbf y}$, where $\mathbf y$ is the actual economic data over $T$ periods and $\hat{\mathbf{y}}$ the predicted values. A natural extension of this to matrices $\mathbf Y=(y_{it})$ and $\widehat{\mathbf Y}=(\hat y_{it})$ where $i=1,\dots,n$ and $t=1,\dots,T$ ($n$ variables over $T$ periods) would be
\begin{equation}
\text{MSE}^*=\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T(y_{it}-\hat y_{it})^2.
\end{equation}
But beware of the pitfall. Of course there are other matrix norms available as well.


What is the point of the indirect utility function?

Recall that if $x$ and $y$ are consumption bundles, $u(x)$ is the consumer's utility function, and $u(x)>u(y)$ means the consumer strictly prefers bundle $x$ to bundle $y$.

The indirect utility function $v(p,w)$ is the highest value of the utility function when $u$ is evaluated over all affordable bundles given $(p,w)$. In other words, $v(p,w)=max u(x)$ subject to $p\cdot x \le w$.

So the indirect utility function tells you the consumers preferences over price-wealth pairs: $v(p,w)>v(p',w')$ means the consumer gets to consume a strictly better bundle when prices are $p$ and wealth is $w$ than any other affordable bundle when prices and wealth are $p'$ and $w'$.


'One Unit Change in the Explanatory Variable'

You can use summary statistics to compare the one unit change to the standard deviation and could say that it represents a 1, 1.2 2 etc. standard deviation change. For example, if in the regular form:
$$y_i=\beta_{0}+\beta_{1}x_i+u_i$$
say $\beta_{1}=6$ and from your summary statistics, the standard deviation of $y$ is 3, you could in turn explain the coefficient as: "A one unit change in $x$ is associated with a 2 standard deviation change in $y$, ceteris paribus".

To interpret the unit change as a 1% change, you can convert the coefficient to read it in terms of percents relative to the levels of $x$ and $y$ or, to much more convenience, you can run the model in log-log form:
$$\log(y)_i=\beta_{0}+\beta_{1}\log(x)_{i}+u_i$$
In this form, $\beta_{1}$ is the percentage change in $y$ from a 1% change in $x$. Exactly what you are looking for!
$$\beta_{1}=\frac{\%\Delta y}{\%\Delta x}$$
where the denominator is usually a 1% change.


As @markleeds suggested, Judea Pearl made a significant contribution to the field of Causality.

His work (though not innovative) explains that causality cannot be represented by joint probabilities. In fact, even econometric models can't model causality.

So, we have to use Rubin's Causal model, Granger Causality, Difference-in-Differences or whatever, to explore causality.

So, the answer to your question is yes and no. We can use statistical learning techniques in combination with causal frameworks, but they alone can't provide causality rigorously.

For example, I came up with a very simple solution to use special impulse-responses (in my case, one-hot vectors like $(1,0,0), (0,1,0), (0,0,1)$) to clear out the partial effects of input variables to the outcome in Neural Nets. Link here: http://ravshansk.com/articles/irf-ann

But all of these are just hacks and don't quite have a rigor. Judea Pearl argues that we must have a new notation for causality or use graphical models whatsoever.


How are assets distributed by types of organisations?

The Flow of Funds (Financial Accounts of the United States) should have what you need.

For example, table  B.1 Net National Wealth Table Description Table (Billions of dollars; amounts outstanding end of period, not seasonally adjusted) might be very close to what you want.


Can monopolistic dead weight loss persist without government intervention? If so how?

If I understand you correctly, monopolistic deadweight loss persists without gov intervention because of the following:

Economies of Scale: if monoplist is able to produce at below the market prices due to the presence of economies of scale, then the monoplist is effectively able to retain its dominance in the market as new firms entering the market cannot undercut the monopolist. Thus, deadweight loss persists.

Patenting: Monoplist could patent its products and process innovation. A new entrant would have to come up with a new idea of a similar product to enter the market tha the incumbet dominates. This however is not usually possible due to very high initial R&D investment requirement.

Rent-seeking: It is also possible that a monoplist could lobby governments in order to influence policy related decisions. Tobacco and weapons manufacturng companies are known to spend large sums on lobbying efforts.

These effectively serve as barriers to entry.

In the presence of barriers to entry, monopoly profits effectively transfer income from households to the monopoly. The monopolist could also under-produce in order to maintain monopoly profits. Examples could be the production of "limited edition" goods which are usually priced above average market prices. Because the monopolist earns above normal profits, it may not invest in cost-saving technologies.

In order to minimise or elliminate the deadweight loss, governments tend to invoke antri-trust laws and regulate the markets and industries. Deadweight loss persists in the absence of action by governments.


Should Company B enter the market in the following cases

Knowing that demand is linear, we can deduce from the two points on the demand curve, $(q,p)\in\{(1,2),(2,3)\}$, that the demand function is $p=7-2q$, which is consistent with the Company A's profit maximizing quantity being $2$ units (assuming no fixed cost).

If B enters the market and competes in quantity, the Cournot outcome is $q_A=q_B=\frac23$ and $p=\frac{13}3$. Taking into account B's fixed cost of $1$ for entering the market, its profit is
\begin{equation}
\left(\frac{13}3-3\right)\frac23-1=-\frac19<0.
\end{equation}
Hence it's not profitable for B to enter.

If B enters the market and competes in price, the Bertrand outcome is $p_A=p_B=3$. In this case, revenue is equal to the cost of production, so B makes zero profit (absent the entry cost). However, given the positive cost of entry, B would be making a loss if it enters the market.


What are the contents of "Recursive Macroeconomic Theory"

I guess you mean some edition of Ljungqvist, L., & Sargent, T. J. Recursive macroeconomic theory. MIT press.

This is a very valuable book, but not for acquiring a concrete theoretical knowledge. Copying from its marketing-speak,

  Only experience in solving practical problems fully conveys the power
  of the recursive approach, and the book provides many applications.

Now really, it provides only applications. I do not mean empirical-econometric applications, but presentation of dozens of theoretical models from very different sub-fields and their workings under the Recursive Approach. Mind you, the models themselves are summarily presented, with the implicit understanding that the reader already is familiar with, or will go and study the original papers first. Moreover, a lot of theoretical/mathematical underlying knowledge is taken for granted.


Where can I find the statistics of import crude oil to China?

I'm not sure how helpful this is but the way the information is displayed is really nice. https://oec.world/en/profile/country/chn/


On the relationship between income distribution and GDP

You assume that higher spending causes higher GDP. This is not necessarily true.

Saving income will increase GDP through investments (unless you're in a Keynesian trap). Think about the most standard growth model, where the future (and steady state) GDP increases strictly in the savings rate.

Hence, whatever increases the savings rate (in your toy model, higher inequality), potentially increases future GDP. This was an argument brought forward in Barro (2000). Version with linear savings rate are in Bertola et al (2006)

Of course, this is the neoclassical answer. There are many reasons why high inequality decreases growth/GDP which work through political economy models or similar. I'll list some of them despite you not asking explicitly for them, perhaps it's of use for future visitors:

Fiscal Policies: Equality leads to less necessary government redistribution and hence more incentives for investments (Meltzer and Richard, 1981; Corcoran and Evans, 2010; Persson and Tabellini, 1994)
Crime: Inequality reduces opportunity costs for illegal activities that harm GDP (Alesina and Perotti, 1993; Barro 2000)
Imperfect Credit Markets: Fixed investment costs or similar may lead to higher growth in unequal societies (Barro 2000)
Saving Incentives: With "not much to lose", poor people face moral hazard in unequal societies, a mechanism that decreases growth with inequality (Banerjee and Newman, 1991)
Fertility-Education: Equality in human capital leads to smaller fertility rates, and hence lower GDP. The Argument is too long to be summed up here; see Perotti (1996) and Croix and Doepke (2003))


The basic intuition is that the companies either directly know by looking at what the central bank is saying, OR (and more importantly) the money growth means that money is now put into the hands of various agents in the economy. The second case is the important one.

When that money is put into the hands of various people they go out and spend that money. Thus there is increased competition to buy the same amount of goods. (More $$ chasing less goods). That competition to buy means that firms will profit by raising the price, and so they do that when they see the increase in demand.


The optimal price for a demand curve with a steep slope

That portion of answer about decreasing price is not generally correct as a is both in numerator and denominator and you don't know what $c$ is. You can answer the question about what happens to $p$ as $a$ becomes arbitrary high using limits. Taking your solution for optimum price and taking limit as $a$ goes to infinity gives:

$$\lim_{a \to \infty} \left(\frac{A+ac}{2a}\right)=\frac{c}{2}$$
(here we had to use L'Hôpital's rule as the limit was in indeterminate form of $\infty/\infty$). So if $a$ is arbitrarily high the price will be half of $c$

You do not mention the whole problem is so it is hard to be precise here (you say $c$ is cost but $c$ is usually marginal costs in these problems), regardless optimum price can be below costs that would just mean that firm would eventually exit the market and stop providing the service. That is because the demand is simply too much decreased by even small increases in price.
This is essentially situation where people would want some goods or service but only if price is so low that firms are not willing to provide, so the market won't exist. This can happen in reality. You probably had at least once in your life thought that it would be nice if there would be movie/game/book/some service etc that caters to some niche taste but you see that nobody is providing it. Well one of the reasons why nobody is providing it might be that there is no demand at the price where it would actually be economically viable to provide the good or service.


Why are there not more competitive nonprofits?

There is a Hungarian economist by the name János Kornai, who has renowned work on eastern block socialist systems. He has devised a system theory on coordination mechanisms which guide the way people work together and allocate resources. Based on this theory you can explain different economic and political systems as well. The four coordination mechanisms are:

market (for-profit business - capitalism)
bureaucratic (the state - socialism)
ethical (non-profit, charity organizations, the church etc.)
aggressive (organized crime etc.)

From these four, the ethical is the mechanism which guides charity and non-profit organizations, which is actually the weakest in terms of efficiency. The guiding objectives are grounded on moral and altruistic behavior, but as humans are collectively more selfish in reality, this does not work too well. There is a crowding-out process, by which market coordination takes over in that certain field. Which is in your case the FMCG industry. Even if there are some moral initiatives to simply provide consumers with a healthy and affordable morning meal, but not to make a profit, slowly the enthusiasm erodes and people start to behave more selfish, and exploit the the brand and image to make money. Look at part III. section 1. of the article titled, Neither bureaucracy, nor market? for a good explanation.


Cyclicality of Real Wages? Counter, pro or acyclical?

Are Wages Procyclical?

Here's the entry from Table 1 in King and Rebelo (2000; Resuscitating Real Business Cycles). The variable of analysis is log-wages, detrended with HP filter. It has a std, relative std, first-order autocorrelation and correlation with output of 0.68, 0.38, 0.66, 0.12.

This leads the authors to conclude that

  The real wage is much less volatile than output.

You can see that for their long US period, 0.12 is a positive number, but pretty close to zero. This is why different authors take this number as either acyclical or weakly procyclical.

Price rigidity required for procyclical wages?

Assume that some sort of fluctuation in TFP "drives" the business cycle. Furthermore, if markets are competitive, the wage rate should be set by the marginal product:

$$ w = AF_L(K, L)$$

If $L$ stays constant over the business cycle (e.g. with inelastic labor supply), this setup will generate wages $w$ that vary with $A$, and hence any change in $A$ will result in a one-for-one change in $w$ and in $Y$: $Y$ and $w$ are perfectly correlated, while $L$ is fixed.

This is pretty much the prediction of the RBC model, and it is the opposite of what we observe in the data: $L$ is volatile, while $w$ is fixed. If you'd assume that $L$ responds strongly to changes in $A$, $w$ might be pretty constant over the business cycle. This is what we observe in reality.

And most of the labor-side of RBC research is to come up with ways that force $L$ to respond so much.

King and Rebelo (2000) should be a key paper of any introductory reading list for macroeconomics and I strongly recommend reading it.

The case of less competition

If there is literally zero competition, then firms will at all times only pay the household his outside-option value, making him indifferent between working and not working. If that outside option is constant over the business cycle (for which we have to some extent theoretical and empirical arguments), wages will be acyclical.

However, to the extend that there is no perfect monopsony, firms have to compete for workers and offer higher wages whenever workers are worth more. Hence, wages will be procyclical even if, to some extent, firms are not perfectly competitive on the labor side. The more competitive they are, the more pro-cyclical wages should be.

Alternatives and extensions to the neoclassical model

The above model was just a simple one to generate a case in which pro-cyclicality of wages can be generated without price stickiness. There is much more to the subject than an answer here can provide. Here's some more references:

Barro (1977): Long-term contracting, sticky prices, and monetary policy. *Wages are long-term contracts, hence if firms are less risk-averse than workers, firms will insure workers against wage changes over the business cycle and we can observe (optimally) constant wages
Beadry and DiNardo (1991) is a follow-up on that discussion
Haefke et al (2013, JME) is also a very recent follow-up
Bewley (1999): Why wages don't fall in a recession
Shimer and Hall (two separate papers, both in the same 2005, AER) are the standard references on whether the more modern and reasonable frictional DMP model can match labor (and wage) volatility over the business cycle.


Properties of Financial Markets in Real Life

Equilibria: in the macroeconomic sense of aggregate equilibrium where all markets clear, markets are most likely never in any equilibrium but rather in constant flux between different equilibria, because the market clearing macroeconomic equilibrium always depends on real and also in short run nominal factors which constantly change. Hence it does not make sense to say whether the markets are currently in equilibrium or not. However, textbooks use the concept of such equilibrium because it is very useful way of describing states toward which the markets gravitate - you can imagine them as a screenshot of a continuously playing movie.

Completeness: this condition, in Arrow-Debreu sense, requires there to be very small or non-existent transaction cost and perfect information, this is almost definitely not satisfied in many cases and consequently you will see a lot of research that looks at what happens when there are transaction costs and information asymmetries in markets.

Arbitrage: empirically arbitrage opportunities are definitely present in financial markets but they are usually small (i.e. no million dollar bucks left laying on a side walk) and they tend to disappear really quickly for example many studies of covered interest parity in foreign exchange markets show that arbitrage opportunities usually don’t last more than few hours, sometimes even seconds and arbitrage opportunities lasting few days are already considered long (Mark, 2000). History also shows longer arbitrage windows but they usually come from times where information technology was not as sophisticated as now. So you could say that the condition of no arbitrage in real life holds most of the time approximately.


When gradient of utility function is a zero vector

This concerns the partial derivatives of the utility function with respect to goods, and not the partial derivative of the Lagrangian of the maximization problem.

So a zero derivative, and moreover at the optimum, would imply a threshold quantity after which utility diminishes.

In the real world, we all know that consuming excessively may result in utility reduction (think eating too much food too quickly).

In the theoretical world such utility functional forms have been used, especially "quadratic utility" in intertemporal representative consumer models in macroeconomics with a single good,

$$u(x) =ax - bx^2$$

In microeconomics, all abstract development of the theory usually assumes that utility is non-decreasing in each good separately. But in cases where one would want to allow for decreasing utility, then, indeed, the reason why we would not expect to find the optimum at the zero-gradient point, would be the workings of the budget constraint together with relative prices.


Robust Standard Errors for Control Function Approach?

123, the CF method is a specific case of when we have generated regressors. Check the appendix to chapter 6, pages 157-160, for the asymptotic covariance matrix of $N^{1/2}(\hat\beta_{CF}-\beta)$. The expression is truly enormous. ;)

Instead I will just cite wooldridge on page 160, 2nd paragraph:

«If $\mathbf{G}\neq \mathbf{0}$, then (...). Neither the usual 2SLS variance matrix estimator nor the heteroskedasticity-robust form is valid in this case.»The $\mathbf{G}\neq \mathbf{0}$ is the case when we have generated regressors.

Good luck ;)


What does the Fed mean by "smoothening money market functioning"?

Money markets are where financial institutions and corporates access short-term funding.
One money market where the Fed has intervened directly is the overnight repo market where overnight loans ("repurchase agreements") are made in exchange for high quality collateral such as Treasuries.
It is arguably the most important money market in the world, with volume---the demand of the financial system for overnight cash---up to 1 trillion USD.

The overnight repo rate is determined by supply and demand for overnight cash.

  ...what [potential] problems in the money market is the chairman referring to...

If the spread between overnight repo rate and the Fed's policy rate (fed funds rate)
is too wide, it could be symptomatic of:

The Fed's monetary policy is not being effectively transmitted.
Shortage of cash in the financial system.

Typically, the spread is in single-digits basis points.
This slightly higher rate of interest in the repo market is normally sufficient to  induce (big) banks to lend out their cash reserves held at the Fed.

As mentioned by @Henry, there was a September 2019 episode where the overnight repo rate shot up to 10%, or about 800 basis points over the then-policy rate.

This points to severe imbalance of supply and demand for overnight cash.

  ...why the Fed's policy actions would resolve this problem...

After the September 2019 spike, the Fed responded by simply flooding the market with cash, increasing supply to meet funding demand.
The NY Fed started making overnight loans in the repo market, later expanding into two-week loans, as well as purchasing short-dated Treasuries.

(The fact that such measures were effective indicates that the spike was a liquidity event, rather than credit event. The unwillingness of banks to lend was not due to concerns of counterparty credit-worthiness.
There are several possible reasons that bank were unwilling to lend. One is that reserves held at the Fed is allocated to satisfy capital requirements, as mentioned by @Henry. There was also a concurrent unwinding of the Fed's QE balance sheet, which takes cash away from banks.)

The Fed is currently maintaining its repo facilities, By doing so, the Fed signals to the market that it is willing to step in as, to mis-use terminology somewhat, "lender of last resort".

In another money market, the purchase of short-term commercial paper by the Fed has a similar effect, on funding for businesses instead of banks.
It is meant to ensure that demand of short-term credit for businesses is met.

  ...it seems that the most relevant ones would be the lowering of interest
  rates and the increase in asset purchases.

I would say those measures constitute a different aspect of monetary policy. (Whether purchasing short-dated securities is the same as QE is a different question.)


Solow model: Partial derivative of k* with respect to g

When you are using implicit differentiation along a level curve you will treat the variable with respect to which you are differentiating as a single variable, rather than function.  This is because the formula for implicit differentiation along level curve is already based previous derivation where you already solve for $y'$.
For example, for general function $F(x,y)=c$ where $y=f(x)$, we get by implicit differentiation:
$$v(x) = F(x,y) = c \implies v'(x) = F_x' + F_y' y' = 0 $$
Consequently we get the result that:
$$y' = - \frac{F_x'}{F_y'} \tag{1}$$
Hence, if you just apply formula given by 1 you will not use chain rule anymore, since it was already used in derivation of the formula. For example:
$$xy=5 \implies y'= -\frac{y}{x}  $$
so in your case the correct implicit differentiation along a level curve would yield:
$$\frac{\partial k^*}{\partial n}=  -\frac{F_n'}{F_k'} =\frac{k^*}{sf′(k^⋆)−(n+g+\delta)}$$
You could verify it by actually implicitly differentiating the original function which would yield:
$$\frac{\partial k^*}{\partial n}[sf(k^*)=(n+g+\delta)k^*] \implies sf'(k^*)\frac{\partial k^*}{\partial n} = k^* + (n +g + \delta )\frac{\partial k^*}{\partial n}$$
Solving for $\frac{\partial k^*}{\partial n}$ yields:
$$\frac{\partial k^*}{\partial n} =\frac{k^*}{sf′(k^⋆)−(n+g+\delta)}$$
so it clearly works.


Unbalanced panel data in prodest package in R

A common technique we use in Python through Pandas is the introduction of means, medians and modes (depending on the structure of data) in the stead of these missing values.
Another great technique is introducing fitted values by a regression model, thus, predicting the missing values. In this case you'd want a command to drop your NaNs, estimate a model, and replace empty cells with their fitted value. Depending on your data's structure, you might encounter problems with this technique.
As a final comment, I'd suggest you use Python. Here's the pertinent website.


Would healthcare costs increase (in the long term) if all smokers quit?

It is a reputable journal and the results are surely correct on a "total dollars" sense. I guess the point they are making is that because health care expenditure of old age individuals is very likely to be high, there is some 'social benefit' of dying before old age.

However, it seems somewhat meaningless:

A) First, by this logic, murders also decrease health care costs. We're not trying to keep smokers form smoking solely because they will be a health care burden. We also want these people to live long, rewarding lives.

B) People contribute to society, their families, their friends and so on. They also save and, specifically, they pay health insurance premiums while they are alive. So, while they might require more care in the long run, non-smokers might well be able to pay or it with the taxes on their income and the productivity they bring to society. Similarly, we all invest through our taxes, on public education, and it goes to waste when somebody that has received public education dies.

C) There are enormous variations in the amount of healthcare spending that old people receive across countries, so the estimates of how much health are old people will receive has got to be very imprecise.

D) Investments pay a positive rate of return, so if we saved the money now we could pay for a lot more care later. Put differently, all the resources that would not be devoted to the care of smokers now would be used to improve our lives meaningfully, potentially even helping us do research into how to take care of old people effectively, which would again reduce the future expected cost of caring for the smokers.


There are different notions of neutrality of technical progress in macroeocnomics.

You can have Hicks-neutral technical progress - that is a technical progress that increases the marginal productivity of all factors of production by the same proportion at the same capital-labor ratio. An example of Hicks-neutral production function would be one where $Y^*= F(AK, A L) = A F( K,L)$ and consequently $\frac{\partial Y / \partial K}{\partial Y / \partial K}= \frac{AF_k}{AF_L}= \frac{F_K}{F_L}$. Also note the technical progress would be neutral when its neither capital or labor saving, according to Hick's definition in his the theory of wages, labor saving technology would be one that improves the marginal product of labor relatively more and capital saving that which improves the marginal product of capital relative to labor.

I dont recall ever seeing Leontief production function that would be assumed to be Hicks-neutral in research papers but I think it should be possible to have special cases such as $ F = \min{ \{A K, A L}\}$ where $K$ and $L$ have to be consumed exactly in the same proportion you could say that the technology is Hicks-neutral as it is neither labor nor capital saving, also the $K$ and $L$ ratio would have to remain constant as they are perfect complements and here we would need one unit of capital for every one unit of labor.

Whether the production function is Hicks-neutral depends also on factor elasticity of substitution as well as product elasticity of demand but that is more in sense that the same production function can be or not be Hicks-netural depending on the elasticity of substitution it has.

Then you can also have Harrod-neutral technical progress. In this definition the technological change is neutral as long as capital-output ratio and the rate of profit stays constant. An example of such function would be a function where technology is labor augmenting like $F(K, AL)$.

Next, you can also have Solow-neutral technical progress which is defined as technological change that leaves
the relative input shares  unchanged for a given labour-output ratio. With example being capital augmenting technology $F(AK,L)$.

Moreover, you can even have Leontief-neutral technical change. This occurs when the augmentation effect of technology would be different for both factors of production so you would have $F(A_1 K, A_2L)$ where $A_1\neq A_2$.

None of the above definitions seem to exclude Leontief production function per se.


Price Elasticity of Demand for Positive Price Increases

Except of the purely descriptive aspect, "elastic demand", or more accurately, regions of the demand schedule where demand elasticity with respect to price is higher than unity, in absolute terms, is linked to the basic monopoly theory, since the monopolist maximizes profits at a point of the demand schedule where "demand is elastic".

Define the demand point-elasticity with respect to price as

$$\eta = \frac {\partial Q }{ \partial P}\cdot \frac {P}{Q} \Rightarrow \frac {\partial Q }{ \partial P} = \eta \cdot \frac {Q}{P} \tag{1}$$

Note that algebraically, the elasticity is a negative number, with the sign indicating the direction of influence, since $\partial Q / \partial P <0$.

The profit function of a monopolist is

$$\pi = P\cdot Q(P) - C(Q(P)) \tag{2}$$

The first-order condition for a maximum with respect to price is

$$\frac {\partial \pi}{\partial P} = 0 \Rightarrow Q + P\frac {\partial Q }{ \partial P} - MC\cdot \frac {\partial Q }{ \partial P} = 0 \tag{3}$$

Inserting $(1)$ into $(3)$ we have

$$Q + P\cdot \eta \cdot \frac {Q}{P} - MC\cdot \eta \cdot \frac {Q}{P} = 0$$

$$\Rightarrow 1 + \eta - \eta \cdot \frac {MC}{P} =0$$

$$\Rightarrow - \eta \cdot \frac {MC}{P} = -\eta -1 $$

$$\Rightarrow  |\eta| \cdot \frac {MC}{P} = |\eta| -1$$

$$\Rightarrow P^* =  \frac {|\eta|}{|\eta|-1} MC \tag{4}$$

$(4)$ is essentially an implicit relation since $\eta$ is a function of price also, but it provides a specific insight: Since we naturally expect that price will be positive, we see that we must have $|\eta| >1$: the price will necessarily be set at a level where "demand is elastic", i.e. at a point on the demand schedule where the point price elasticity of demand is higher than unity, in absolute terms.


Evaluation around steady state for a specific DSGE Model

I found the error in my derivation: I mistakenly supposet that the steady state of $p_t+1$ equals $\bar{\rho}$.
I did not recognize that p was different from $\rho$ because of the poor quality of my printout.
Reading the paper again on a screen I immediately spotted the mistake.


Would a car that was produced last year but bought this year be accounted for in the GDP?

GDP is actual production. Since the car was produced last year, it adds to last year's GDP.  Since it wasn't sold, it would show up in inventory investment (which seems to be included in I in your equation). This year, when the car is sold, inventory investment goes down and consumption goes up by an equal amount, so there is no effect on GDP.
Pop quiz: analyze this example using the income approach to GDP.


Solve for the bundle component that makes one as 'well off' as earlier

Converting my comments to an answer:

I think your answers to 1 and 2 are correct. For 3.1, your intuition that "as well off as" should be interpreted "indifferent" is correct, and indifferent is about comparing utility levels, not MRS. You answer to 3.2 looks correct.


Consider an investor who splits up his portfolio into stocks and bonds according to some ratio that considers all things that need to be considered. These include returns etc. Assume the portfolio is "correct" i.e. in equilibrium. Through the GDP revision, stocks become more attractive compared to bonds for the individual investor, so each individual investor has to adjust his ratios and increase the nr. of stocks, which means he has to sell bonds. The individual investor does not expect his action to actually influence government bond yields as he is small compared to the market. If all investors do this, then we get the result mentioned in the link.

Further dividends are not the only reason stocks have value. Many companies don't pay out dividends, because the shareholders then have to pay tax on them, but rather re-invest the profits which increases the stock value, which can then be sold.

Also its not clear that the government bond interest rate is the relevant interest rate for discounting for these investors (the effect you are referring to).


Markov decision processes, contractions and value iteration

The value iteration operator is a contraction with respect to the supremum norm. Your example probably provides a counterexample for the statement that it's a contraction with respect to the Manhattan norm.


Economic schools of thought VS Econometrics

I think I can explain the problem as to why the schools persist.  For simplicity we will limit ourselves to the Marxist, Austrian, Neoclassical and Neo Keynsian schools.  I am not sure where to put the Feminist school except under the broad heading of heterodox with Marxists and Austrians as I don't think of it as a school so much as an attitude or direction of thinking.  The same thing would be true for Green economics, it isn't so much a school as a domain.  There are many small groups out there.

For starters, economics is a broad umbrella that prefers dissent and maybe even vocal and extensive dissent.  Part of economics is part of the humanities and part of economics is in the sciences.

Austrian economics as thought through by von Mises argues that there can be no way to measure what they are saying and so they are part of the humanities.  The neoclassical and the neo Keynesian school say any claim that is stated should be measurable and falsifiable, so it is in the sciences.

That said, any serious review of von Mises and related writers would tend to drive one down the path well laid by Leonard Jimmie Savage and his personalistic statistics and in doing so would have a complete empirical methodology.  I believe the branch of the Austrian school that says things cannot be measured simply hasn't read enough statistics.

The two primary schools, the modern Keynesian and Classical forms, are empirical and the reason you cannot rule one out is that both are empirically supported even though they are contradictory.  This is similar to the problem between quantum physics and relativistic physics.  They make contradictory assertions.  It is impossible that both are true, except where they don't overlap.  Yet they both work everywhere else.

The best tool for model selection in that it is both disciplined and founded on well understood mathematical principles is Bayesian model selection.  Non-Bayesian methods assume that you know the null model and you assert the model is true.  This works great for binary problems but is terrible for non-binary problems.

If we define a model space as indexed on the set $I$, with $i\in{I}$ and a set of models, $M$, so that $\mathcal{M_i}\in{M}$ and with vectors of parameters $\theta_i\in\Theta_i$ in a parameter space $(\theta_i,\mathcal{M}_i)\in\Theta\times{M}$ then we can solve the problem of the best model.  Best in this case being understood as being most probably close to the data generating function.

The probability that a model is the true model, from a set of models and given a data set, $X$, is $$\Pr(\mathcal{M_i}|X)=\int_{\theta_i\in\Theta_i}\frac{\Pr(X|\theta_i;\mathcal{M}_i)\Pr(\theta_i;\mathcal{M}_i)}{\sum_{\mathcal{M}_i\in{M}}\int_{\theta_i\in\Theta_i}\Pr(X|\theta_i;\mathcal{M}_i)\Pr(\theta_i;\mathcal{M}_i)\mathrm{d}\theta_i}\mathrm{d}\theta_i$$.

While this methodology works generally and works especially well on a problem by problem basis it won't answer which school is best since it will provide contradictory solutions across specific problems.

What is really needed is a better, overarching explanation of what triggers Keynesian-like effects and what triggers Classical-like effects.  My suspicion is that it is stochastic shocks in the planetary budget constraint compared to where that constraint is expected to be at by the individual actors.


Robert Mundell originally wrote about Mundell-Tobin effect in response to the Fisher effect, which states that "the real interest rate is unaffected by monetary policy and hence unaffected by the expected inflation rate." In other words, nominal interest rates will always adjust such that real interest rates stay at whatever the market rate is. Hence, if inflation rises, then nominal interest rates must rise. Which brings us to the first point of confusion...

  Mundell-Tobin effect supposed to result in 'higher' nominal interest rates

This is the Fisher effect not Mundell-Tobin. Note the Mundell-Tobin effect explains why nominal interest rates will not rise as high as expected by the Fisher effect - it will be less than one-to-one. This is not a contradiction as much as a competing factor.

  I believe that an increase in the demand for real capital means the same thing as the increase in the demand for loanable funds

The reduced demand for money increases the supply of loanable funds, which induces more demand, which is then used to purchase real capital. This in turn reduces the expected rate of return on this capital. If expected rate of return is not as high on capital, real interest rates do not have to be as high to compete in terms of opportunity cost.


Capital Asset Pricing Modeling for Consumption

I guess the question is : which asset would yield the higher risk premium or expected return?

Risk premium depends indeed on the intrinsic characteristics of the asset, and on the pricing kernel, in particular the covariance between the pricing kernel (which is proportional to marginal utility) and the asset return. Therefor the risk premium will depend on the utility function you specify. But the general idea is that an asset that pays high when consumption is high will have a lower price and then a higher return to compensate for that risk. Under complete market, the general formula is :
\begin{equation}
E(R^{e}_{t+1}) = -R_f\cdot cov(m, R^e_{t+1})
\end{equation}

where $R^e_{t+1}$ is the excess return, $R_f$ the risk-free rate, and $m$ the stochastic discount factor. $m$ will depend on the utility you specify but will be proportional to the marginal utility. Because of the (generally assumed) concavity of utility function, $m$ will be low when consumption is high and high when consumption is low.

In your example, the expected excess return of asset B would be 0, otherwise stated the expected return of asset B would be the risk-free rate because its return is uncorrelated with consumption.

Asset A pays high when consumption is low, so when marginal utility is high, so the risk premium will be lower than the risk-free rate. The main idea is that since this asset can be use as insurance against variation in consumption, agents will want to hold it, so its price will be higher and then its return lower.

So you were almost right, asset A will have a higher price than B, but a lower expected return.


just economic interpretation: Is every steady state a golden rule steady state?

No. There are many steady state levels of per-capita consumption. The maximum of these is then defined to be the golden rule (steady state level of per-capita consumption).


Inc Linear Transformation of Bernoulli Utility

This is a subtle topic. Any strictly increasing transformation will preserve the ordering of lotteries. This is the standard result saying that utility is just an ordinal thing, we really don't care about the level of it.

In the other hand, there are representations os someone's preferences that are easier to work with, from the calculations point of view, or improve our understanding of the underlying economics of the problem. One example is when we have quasilinear preferences on good 1 (check MWG, chapter 3 for this example). When this is the case, we have that $U(x_{1},...,x_{I}) = x_{1} + \phi(x_{2},...,x_{I})$. Of course, if we take, say, the exponential of $U$, we will preserve ordering, but not this very useful representation.

When it comes to chapter 6 and lotteries come on the stage, the same happens. One's utility over a lottery (think about it as a distribution of consumption over different scenarios) might be a very complicated function. Maybe, the utility you get by holding an umbrella if it starts to rain also depends on the utility you get when your football team wins and you are not at home watching the game. The V.N.M. utility representation is useful exactly because it separates utilities of different scenarios.

So, when you start comparing lotteries, the effect of an increase in the probability of holding an umbrella when it rains in a given lottery is completely isolated from the effect of a higher probability of another scenario. If we keep consumption fixed over scenarios and let only the distribution of probabilities describing what can happen vary, the V.N.M utility form will let us isolate effects of different probabilities over scenarios and express difference of total utility by difference in probabilities over scenarios between two in-comparison lotteries.

The reason behind only linear transformations conserve this format is that the V.N.M utility form is linear, by definition, and non-linear transformations of linear functions are not linear. This is somewhat related to Jensen's inequality, from my perspective (that's at least how I intuitively interpret that).

I hope this helps!


When there is a third degree price discrimination, why is the total MR curve kinked at the top?

I believe this is just a horizontal sum. At the very top of your graph, only adult has a part of its curve above 10.25. So when your summing horizontally student =0  Adult = x, This changes once you are below 10.25, then all of a sudden you have student = x  and adult = x. This additional amount from student causes the kink. If pretend student was always =0 then the total on the right would be identical to adult.

I hope this helped. I'm sorry if it's not phrased better.


Difference between Value-Added TFP and Gross-Output TFP?

Your assumption is correct.  Value added is Gross Output-intermediate consumption(inputs).

value-added approach is a simple measure that ignores the difficulties of dealing with inter-industry and intra-industry flows of goods and services. Intermediate inputs are simply excluded here.

The value-added approach provides a simple link of industry-level MFP
and sectoral or aggregate MFP growth.

For gross output-based estimates of MFP growth they acknowledge and allow for intermediate inputs as a source of industry growth. In this method they provide a more complete picture of the production process.

If you need more information this manual can help: https://www.pc.gov.au/research/supporting/comparison-gross-output-value-added-methods/cgovam.pdf


What are the effects of a 12 month sudden stop/shutdown of a capitalistic economy like the US?

In the most optimistic case, there will be a difficult 12-18 months, followed by a return to normal. The factories are still there, but most things are just put on hold in the mean time. That return to normal will involve some changes (e.g. commercial real estate will hurt as many workers continue working remotely), but it could be that the economy loses no real productive capacity. There would be a big increase in public and private debt, but basically it would be like stepping back a few years to a time that was fairly peaceful and productive, just less wealthy.

In the most pessimistic case, all the physical stuff remains, but all the social stuff (contracts, org charts, institutional knowledge etc.) crumbles and has to be rebuilt. Your local bar goes under, and your local economy loses the knowledge currently embedded in that bar's operations/management. Although the building and bar stools remain, the business can't be spun up smoothly because running a business just isn't that easy. It would be sort of like landing on an abandoned alien planet, finding a lot of empty factories, and trying to get them operating again.

Then there are all the political-economy issues. Government debt, political instability, and all the rest. As a general rule, the unscrupulous are best positioned to take advantage of a crisis, so we have reason to expect some degradation of political institutions costing us future productivity.

These are some pretty wide bands on the range of possible outcomes: from "Dang, it's going to take me an extra couple years to pay off my mortgage!" to "Holy Hell, how did this turn into post Soviet-collapse Uzbekistan?!" Of course the truth is going to be somewhere in the middle.

What I can say pretty confidently are these very general points:

There will be shifts in what sectors get resources/attention. People will buy more rubber gloves and chest freezers and will spend less money on something else.
There will be shifts in political coalitions. People will spend more attention to public health issues (and maybe less on national parks). Consequently, lobbying groups will become more/less important.
There will be a recovery period. Once we're back to work, people will be trying to pay down their debts for a while and will delay that vacation they might otherwise have taken.

The first two points are barely predictions at all. Change is constant in markets and politics. The real question is "how dramatic will these shifts be?" My prediction: they'll be big enough to be measurable, but not so big that the world we end up with is unrecognizable compared to what we started with.


Derivation of demand for intermediate goods in DSGE model

So I've seen this sort of thing where you basically differentiate with respect to each Y_j so the integral drops out (seen this in moral hazard chapter from
MWG and Romer's endogenous growth model).

The foc becomes:

$P_t(Y_t(j)^{(1-v)})^{v/(1-v)}Y^{-v}-P_t(j)=0$

This gets the result.


As Kitsune rightly points out the usual reason for the inclusion of the time preferences for money into the discount formula (or Ramsey fomula) is that we observe this kind of behaviour in people. Given that many economists take a positive approach towards the study of economics, that is a logical stance.

That being said the last word is not spoken on this, and discounting and its use in benefit cost analyses for especially long-term policies is an active area of research. The Stern review on the economics of climate change sparked intense debate on whether we should include a pure rate of time preference and if so, what its size should be. On the one had you had economists like Stern who argued for a very small rate (0.01 % if memory serves), only to account for catastrophic risk, on the other hand there were economists like Nordhaus who argued for a rate close to 3%, based on observed choices by people in real life and the interest rates. Wikepedia covers the debate here

Martin Weitzman did a study among 2160 economists what the real discount rate should be  for climate change (that is pure rate of time preference plus discounting because of GDP growth) called Gamma discounting and published in the AER. He found a range from -3 to 27%, which may give you an idea of the disagreement among economists on the issue of discounting.

Drupp, Freeman, Groom and Nesje did a more detailed follow-up in 2015, called discounting disentangled. Their range for the full discount rate is lower, from 0 to 10%, and they also asked explicitly what the pure rate of time preference should be. The mean recommended value for the latter by their 200 surveyed experts was 1.10%, but even that rate had a range from 0 to 8%.

Interestingly, the person who first came up with the Ramsey formula, Frank Ramsey, in a mathematical theory of saving, argued for a rate of time preference for societies of 0, calling a positive rate "ethically indefensible", although he does use a positive rate later on in the paper.


Example of information structure in a one-player Bayes Correlated Equilibrium

The signal structure must specify what the DM learns in all possible states of the world, so I think your question should read:

"Suppose that the DM discovers whether the state of the world $v\in V$ is in $[a,b]\subset V$ or not. Can this be written as an information structure?"

or should read:

"Suppose that the DM discovers the state of the world $v\in V$ whenever it is in $[a,b]\subset V$ and learns the minimum otherwise. Can this be written as an information structure?".

From the comments of the question, I think you are more interested in the first one. Regardless, a couple of comments are important:

You want the  DM to assign positive probability to an interval, and there are many ways to do so. There are some restrictions that are coming from your prior, but they are relatively mild.
The most important one is that $P_V(v)>0$ for all $v\in [a,b]$. That is, the prior must assign positive probability to all numbers in the interval.
Even when you decide how much probability you want to assign to each point (assuming it is feasible given the restrictions mentioned above), there are many signal structures that can achieve this (Thus, I will be providing just a couple of examples).

Example 1:

Suppose $V=\mathbb{R}$ (the real line), and $P_V$ is the normal standard distribution over the reals, suppose that you are interested in finding a signal such that the posterior $P_{V|T}$ is the normal standard distribution truncated to $[a,b]$. Then the signal structure can simply be as follows:

Let $\mathcal{T}=\{blue,red\}$
$$P_{T|V}(blue|v)=\left\{\begin{array}{c c}
1 & ; v\in[a,b]\\
0 & ; v\notin [a,b]
\end{array}\right. \ \ \ \forall v\in V
$$

Of course $P_{T|V}(red|v)=1-P_{T|V}(blue|v)$.

Notice that if the DM gets a signal "blue", they infer that the state is in $[a,b]$,  and because the signal "blue" is equally likely to be received for any state in $[a,b]$, then the posterior has the same shape as the prior (i.e. it will be the truncated normal). Contrast this  with the following signal: (for simplicity, let's assume that $[a,b]=[1,2]$:

$$P_{T|V}(blue|v)=\left\{\begin{array}{c c}
\frac{1+v}{3} & ; v\in[a,b]\\
0 & ; v\notin [a,b]
\end{array}\right. \ \ \ \forall v\in V
$$

and $P_{T|V}(red|v)=1-P_{T|V}(blue|v)$

Now, receiving the signal "blue" also makes the DM  infer that the state is in $[a,b]$, but the  signal structure sends message "blue" more often when the state is closer to $b$, so the posterior will assign a higher probability to states closer to $b$ than what the truncated normal would do, and less probability to the states closer to $a$.

Example 2:

If instead, what you wanted from your signal was the second interpretation. Namely,  "Suppose that the DM discovers the state of the world, $v\in V$, whenever it is in $[a,b]\subset V$ and learns the minimum otherwise. Can this be written as an information structure?"

You  can simply let $T=\mathbb{R}$ and define the signal structure as follows: (for simplicity, assume again that $[a,b]=[1,2]$)

$$P_{T|V}(t|v)=\left\{\begin{array}{c l}
1 & ; \ t=v \ \& \ v\in[a,b]\\
1 & ; \ t=0 \ \& \ v\notin[a,b]\\
0 & ; \ otherwise.
\end{array}\right. \ \ \ \forall v\in V
$$
(Here the message "0" can be any other random number that is not in $[a,b]$.)

Therefore the DM receives signal "0" whenever the state is outside the interval [a,b] (so they learn the minimum possible) or they learn the state of the world (by getting message "v") when the state is in the interval.


A simple question on a relationship between output growth rate and unemployment rate

I get the same equation as you do. So you are correct. But tough usually $u_t - u_{t-1} \neq -g_{y_{t}}$, for small values of $g_{y_{t}}$
$$
u_t - u_{t-1} \approx - g_{y_{t}}
$$
does hold. In macroeconomics these kind of (in my opinion annoying) approximations are used every now and then. The author perhaps neglected to mention that he was using an approximation. For example the Fisher equation is sometimes written as $i = r + \pi$, even though it is actually $i \approx r + \pi$ and that only for small values of $r,\pi$.


Difference between objective and subjective distributions in asset pricing models?

Probability judgments are based on an investors' private information sets, i.e. subjective probabilities. In contrast, objective probabilities are the underlying real probabilities that the world will end up in some state. It is necessary to make the distinction because traders do not have perfect information. See Markowitz (1991) for further explanation.

But objective probabilities remain a remarkable concept. If the market reveals objective probabilities, then these are in fact known, and traders can update their subjective probabilities accordingly. But if all traders agree on the probability distributions, they would have very little reason to trade. Hence one implication of portfolio theory is that there is too much trading going on. See Odean (1999) for empirical work on this, or Grossman & Stiglitz (1980) for a theoretical critique.


Log linearization of the technology dynamic

The shock process is already linear. any log linearisation will result in identical expression. The underlying non linear shock process can be something like this:$

$$ Z_t = Z^{(1-\rho)} Z_{t-1} e^{\epsilon_{t}}$$

This when you linearise, you obtain your linear shock processes.


etymology of TFP (total factor productivity)

It is total because it takes into account all output per all inputs - hence it is the total productivity of all factors.
Total factor productivity as that excerpt from Wikipedia states is given as:
$$TFP=\frac{\text{GDP}}{\text{All Factors}}$$
that is where the total comes from. In practice, the factors used in denominator are capital and labor, but they are measured very broadly to include other factors (e.g. human capital will often be 'hidden' in the labor variable). However, this being said some factors are very hard to measure (e.g. some economists consider entrepreneurship a factor of production), this is why some economists prefer to call it multifactor productivity (i.e. it is productivity of multiple factors but not all), as mentioned in the OECD glossary. However, it has nothing to do with complementarity which in economics has completely different meaning than what you imply in the question so CFP would make no sense.
Also, the passage you cite as reason for your confusion is actually not about total factor productivity but about growth in total factor productivity, i.e. not TFP but $\Delta$TFP. Those are two different variables in the same way as GDP and growth rate of GDP are not the same.
The idea here is simple, if output (GDP) grows, but all factors remain same, that means that the productivity of all the factors used must have increased.  However, in real life things usually don't remain constant so you will observe that quantity of inputs economy uses increases over time, but if output increases even faster that has the same implication as the case when inputs remain constant and output grows, namely it means productivity of all those factors is increasing.


Are equity prices and stock prices the same?

Stocks:

Stock (also capital stock) is all of the shares into which ownership of a corporation is divided

Equity:

In finance, equity is ownership of assets that may have debts or other liabilities attached to them. Equity is measured for accounting purposes by subtracting liabilities from the value of the assets


Why are required reserves so trivially low?

The world changed in 2008, and commercial banks substantially increased their reserves held at the central banks in many countries including the US, mainly because they did not want to lend to each other in the unsecured interbank market

At that point, required reserves failed as a policy measure for fine-tuning money markets in many countries, since it amounted to pushing on a piece of string; the Bank of England removed them entirely in 2009.  Several central banks moved to alternative instruments, including:

quantitative easing
interest on excess reserves (in the US they were small and positive but some cases such as Euroland the equivalent could be negative)
bank capital ratios, as a solvency regulation


Has worker productivity decreased since the election of Donald Trump?

I think it's too early to identify econometrically an effect of Trump in the economy. Recall that the election was the 8th of November, and his victory came partly as a surprise. He took some time to define his cabinet and got into office the 22nd of January. Thus, any effect of a Trump's "dummy" might only partially be reflected in data for Q4 of 2016, and perhaps not fully in Q1 2017. The release of (worker's productivity) for Q2 2017 is the 9th of July (soon), where we might see more of Trump's effect on the economy and productivity.

The best we can do at the moment is simply look at how the economy has evolved since Trump (see here for a related exercise for Obama).

For example, the favourite measure of labour productivity among economists slowed down in Q4 of 2016 and did not increase in Q1 of 2017:

(Source here).

Similarly, there was an increase in workers productivity in the manufacturing sector in Q4 of 2016, which slowed down considerably in Q1 of 2017:

(Source here).

How these developments are related to Trump is not clear though. Some news reports about these data releases make no mention of a Trump effect (e.g. here (Q4 2017), here or here (Q1 2017)).

Another example comes from the stock market. Here is an analysis showing the performance of the stock market under different presidents, after their 100th days in office. I show the main table below:


Why is the consumption line and 45degree line's intersection not the equilibrium income level?

For most people consumption does not equal income. Many consume less (they save) and others consume more (they take on debt) than they earn. In an economy with financial markets and heterogeneous consumers, discounted lifetime income should equal discounted lifetime consumption, but there is no reason why consumption should equal income at every point in time.


Does constant money supply hinder economy?

The second is true. The reason why the second is true are macroeconomic friction and the fact that constant money supply in presence of growth would lead to monetary instability.

The mechanism works as follows; If you fix money supply and your economy grows you will get deflation as you can see from the standard monetary equation $MV=PY$ where $M$ is money supply, $V$ velocity of money, $P$ aggregate prices and $Y$ output. Since the M is fixed by assumption and V does not usually change much, any economic growth (increase in Y) must be accompanied by deflation (decrease in P).

Now depending on how high the deflation is this would be bad in itself. Even if there would be no macroeconomic frictions deflation similarly to inflation is form of price instability and even frictionless economy would be able to handle only little bit of it. In presence of deflation people will unnecessary hoard money instead of actually using it in the economy, it redistributes wealth from debtors to creditors and so on. Hence, even in completely frictionless world it could be proven that the optimal monetary policy should be such that change in aggregate prices is exactly 0.

However, unfortunately we are not living in frictionless economy. In real life there are sources of frictions which make deflation dangerous. There are several of them but to keep this concise I will focus only one of the biggest one sticky wages.

Sticky wages mean that companies cannot change their wage costs in short run. This is because usually contracts have fixed wage and to change them you need to renegotiate them. If wages are sticky in presence of deflation firms that can’t decrease their wages quickly enough (and workers usually oppose pay decreases - even nominal ones very hard) will see their labor costs significantly increasing forcing them to unnecessary lay off workers thereby creating  unemployment that is higher then socially optimal. Unnecessarily high unemployment has its toll on economic growth especially when it leads to some people being permanently unemployed. Due to sticky wages and other frictions economists are usually quite vary of deflation and hence predominant mainstream view is that countries should run inflation of about 2% on average, by growing their money supply in ways that achieves that target.

An example that shows that deflation can really have serious bite is the Great Depression. As famously proven by none other than Milton Friedman and Anna Schwartz in their monetary history of US, Great Depression at the beginning seemed like a relatively mild recession, however the recession lead to contraction in money supply due to bank failures and consequently to high deflation. Instead of doing anything about it Fed was completely passive, and what according to Friedman and Schwartz could be short recession became the worst US recession in recorded history (excluding maybe the current covid recession which might beat it) and this was mainly due to extreme deflationary pressure that lead to excessive unemployment and exacerbated the negative demand shock.


Does it matter which is the dependent variable in regression of time series data?

Asymptotically, you should be able to interchange the role of $Y_t$ and $X_t$ and estimate lets say $$X_t =\alpha^* + \beta^* Y_t+u^*_t$$ where $\alpha^* = -\alpha/\beta$ and $\beta^*= 1/\beta$ and still get cointegrated relationship.

However, crucial caveat here is that this holds only asymptotically and only for $R^2$ close to unity. In finite samples you can find discrepancy between the two. Moreover, if $R^2$ is not high you will also find discrepancy and in that case you should not even model cointegrated relationship with standard OLS but rather with something like Fully Modified OLS (FMLS) or Canonical OLS. Also, which variable is exogenous and which endogenous will matter if the variables are not integrated of the same order or one of them is stationary. Maybe you just got false negative in one of the previous tests.

All this aside if you believe that there could be a cointegrated relationship that could go both ways it is not really appropriate to use Engle-Granger approach to test for it by two Engle-Granger tests. Rather you should use Johansen cointegration test that tests for it by estimating simultaneous system of these all in one model.


How does buying more of a foreign currency devalue a nation's currency?

Simply because buying country A's currency priced in country B's currency means selling country B's currency.

Simplified but not simplistic example: say that you have 80¥ and want to buy 10€. When buying/demanding these 10€, you will actually simultaneoualy also sell 80¥ i.e. by doing so you become a supplier of (€-priced) yuans. And because selling an asset generally exerts a downward pressure on its price, the price of 1¥ expressed in euros will decrease. Hence the explanation you read.


I want to know if it's valid to talk about the economic growth and development of a city instead of a country

In principle there is nothing wrong with narrowing your research question down to municipal level. There are studies that look at economic growth on urban level. Here is just an example that came out of quick google scholar search:

Ding, Chengri, and Erik Lichtenberg. "Land and urban economic growth in China." Journal of Regional Science 51.2 (2011): 299-317.

However, it is not possible to give more specific answer without knowing what $x$ is. There is no guarantee that something that determines growth of an economy as a whole will be also important determinant at municipal level and vice versa.

  If yes, when I talk about growth, do I talk about the increase in AD/GDP or the increase in government revenue for city y etc?

Ideally you would look at growth of metropolitan GDP. In some countries it is possible to get Urban/Metropolitan level GDP data. Here is an example of such data set from US Bureau of Economic Analysis. However, not all countries provide metropolitan level GDP so you might have to use some proxy, like average incomes or nighttime light is often used as a proxy for economic activity (see for example this paper). The list of possible proxies is not exclusive, what will make the most sense will depend on specifics of your research.


When the ECB does QE, does it print new money?

First of all, I'm not sure you are reading that article correctly, as I cannot find any mention of the claim you make. In any case, you can find here a superb comparative analysis of QE for the Fed, the Bank of Japan, the ECB and the Bank of England.

Regarding money creation, the key difference between conventional monetary policy and QE is the type of asset being transacted (short-term government debt versus long-term government/private debt, respectively; again it depends on the central bank, see referred document). There is, to my understanding, no difference on how the monetary operations are carried out in each policy.

In fact, these operations are quite standard. The central bank buys assets by crediting the seller's bank account, in exchange for the ownership of the assets, which are then added to the central bank assets and eliminated from the sellers assets. This is akin to you receiving a deposit in your current account. You have been credited with some amount. There is no physical money creation or physical money transfered in the process itself. There is indirect money creation (all electronic) if such transactions enable the banks to expand their credit to other institutions or companies (provided demand for such credit exist and banks are willing to do so, e.g. in an economic boom), whilst still complying with the regulation.

To learn more about money creating and modern banking, I suggest this page, this page or this post.


Given a discount function $g(t)$, the discount rate is the rate at which the discount function declines over time. If time is discrete, then the (possibly time-varying) discount rate is
\begin{equation}
\rho(t)=-\frac{g(t)-g(t-1)}{g(t)}.
\end{equation}
If we divide each period into $n$ equal intervals and let $n\to\infty$, then we get the discount rate in continuous time as
\begin{equation}
\rho(t)=-\frac{g'(t)}{g(t)},
\end{equation}
which is the same as your expression.

The (possibly time-varying) discount factor is defined as
\begin{equation}
\delta(t)=\frac{g(t)}{g(t-1)}.
\end{equation}

With exponential discounting, the discount rate is constant, i.e. $\rho(t)=\rho$ for all $t$, and the (discrete time) discount function is given by
\begin{equation}
g(t)=\frac{1}{(1+\rho)^t}=\delta^t
\end{equation}
where $\delta=\frac{1}{1+\rho}$ is the constant discount factor.

The continuous time analogs are
\begin{align}
g(t)&=\mathrm e^{-\rho t}\\
\delta(t)=\delta&=\mathrm e^{-\rho}
\end{align}
Again, these are obtained by dividing each time period into $n$ equal intervals and observing that $\lim_{n\to\infty}(1+\frac{\rho}{n})^{-nt}=\mathrm e^{-\rho t}$.


Why don't governments exploit negative interest rates more than they currently do?

(The arguments you listed are questionable, so I will not discuss them.)

One thing to keep in mind is that we need to distinguish between the central bank and the fiscal authority (e.g., the U.S. Treasury). This is doubly true in the euro area. I will put aside what the central bank is doing, to match the spirit of the question.

If we are in a place that has limits on debt issuance - e.g., the US debt ceiling, or countries in the Maastricht Treaty - there are limits on debt that need to be respected. (Why these limits should be implied when it is profitable to borrow is a mystery to me, but that is a different question.)

If we put those legal limits aside, there is a very practical issue: when the Treasury issues bonds (which is a form of borrowing), it is removing money from the “private sector” (or more accurately, the “non-government sector”). Under normal circumstances, borrowing is aligned (roughly) with deficit spending, and the deficit spending sends money to the private sector

As a simple example, imagine a government runs a deficit of \$100 million. The deficit results in sending \$100 million in extra payments to whomever, and that will roughly coincide with what it needs to borrow.

If the government just issues bonds, there is no source of money available to buy them. (The central bank can operate to make it possible, but this is limited.) The government will need to do something with what it borrows, and that would most likely be buying financial assets from the private sector.

(Since this explanation was not adequate, I will explain further. If the government issues \$100 million in bonds, private sector entities need \$100 million in balances at the central bank to pay for them. However, banks need to keep balance levels near minimums to meet liquidity needs. If they are already at the minimum, there is no capacity to buy the bonds at auction. The only way the bonds are sold is via some arm of government to transfer \$100 million to banks to allow the auction to proceed. If the central bank is not able to do it, the only course of action is for the Treasury to buy something worth \$100 million from somebody in the private sector.)

It is entirely possible that the securities it buys are worthless, which poses a risk to the government. Also, it will hand windfall gains to people selling assets, which raises political issues.

They could borrow to fund investment, which is exactly what many people have argued for governments to do. However, the capacity to so is limited by useful projects to invest in, and any projects raise other political issues.


Price elasticity of demand in the point of economic equilibrium

Your formula is overly simplified.

The elasticity is supposed to be "how much does supply change if the price changes". Now, the natural way of looking at this is $S'(p)$, the derivative of $S$. However, the units of that are hard to make use of. Hence, we normalize $S'(p)$ by multiplying it with $P/S(p)$, hence getting the "percentage response of $S$ to a percent change in $P$".

Long story short, your $k$ should satisfy

$$ k = S'(p)$$

Note that this is the price elasticity of supply. For the price elasticity of demand, you would instead use $D'(p) P / D(p)$.


Selecting the best utility function for households

According to Wikipedia and this notes from HKU, the indirect utility function is part of consumer theory, and is defined as the maximum utility that can be attained given a consumers' money income, and goods prices.  It reflects both the consumer's preferences and market conditions.  It is a model which helps to understand how consumers optimize their expenditure considering the price of products.

The model considers a consumer who faces a world with only two products with fixed prices and one limited budget, and is represented by a perpendicular axis graph, with each axis representing the quantities of each good, and both axis converging on quantity 0.  A descending diagonal line is then drawn to represent all the possible quantities of good A and good B the consumer could buy by spending his entire budget.   This line is called the goods bundle line, the price of both goods is reflected in the steepness of this line.

Another descending line is then drawn called the marginal utility function, this line tends to be concave and not straight, due to the theorem of diminishing marginal benefits.  The line represents all the combinations of quantity of good A and good B that give the consumer the same level of benefit.  In fact you could draw a whole map of concave lines, all parallel to each other, and each would represent the total quantity of good A and B needed to achieve a certain fixed level of "satisfaction" by the consumer.  This map of lines would represent the consumers' absolute "tastes and needs" sort of speak.

The consumer, in consumer theory, will try to select a bundle which combines just the right amount of both products, and will spend the whole entire budget on doing so, in order to obtain the greatest possible benefit in his consumption process.   The decision of how much to spend on each product is represented in the graph by the intersection of the goods bundle line and the one marginal utility function it intersects tangentially with.  Any other products bundle in the goods bundle line will not be as beneficial to the consumer as the one where the goods bundle line tangentially intersects one of the utility functions.

This is how you can finally build the indirect utility function, by fixing the price for both goods and making the budget a variable.  Now for each level of budget you´d get one single point in the goods bundle line where the benefit is maximized for the consumer, and if you put together all the dots, one representing the optimal goods bundle for each level of budget, you'd construct the indirect utility function.  It is called "indirect" because of its' indirect correlation with the price of goods.  Budget level thus now becomes irrelevant, and what molds the consumers decision making are prices of goods in the market and the consumers' preferences.

So my answer to your first question, "In general terms, how does one go on to select a given functional form.?", would be that I don´t think the techniques for figuring out a particular utility function, for one or a group of specific individuals, belongs to the economics realm.  Because it would involve practical modeling and not just the economic theories any more. I'd say that question could find a better answer in the finance realm.

My answer to your second question, "Are there a class of utility functions for
 households that are better than others?", is yes.  Consumer theory describes one utility function for any given person and price level of goods A and B.  But if the price of any of the products in the consumer's bundle changes, the utility function reshapes given the consumers preferences.  Also if the consumers' preferences or needs towards any of this products change, that would result in the reshaping of the utility function.

My answer to your third question, "How does one even judge better, in this context?", is you can't judge better in reality.  Perception of the benefit for consuming is subjective, and this subjectivity is embedded in a consumers' map of marginal benefit functions, and through it in the resulting utility function. This makes it impossible to quantify and thus to measure.  Making it only possible to estimate.  So people use this theory to understand the phenomena in a general level, like predicting the behavior of entire economies, or a whole segment within them.

And for your last question, "Does the selection of a utility function depend on the context and analysis at hand?", my answer is yes; and I quote from this book:  -What exactly is a “good”? The answer lies in the eye of the modeler. Depending on the problem to be analyzed, goods might be very specific, like tickets to different world series games, or very aggregated like food and shelter, or consumption and leisure. -

In any case I strongly suggest you specify if your question is theoretical or practical, in order to get a more accurate answer.


What arguments are there in favor of national food self-sufficiency?

Have there been any economists who have argued in favor of national
  food self-sufficiency and what are some of their arguments?

I am not knowledgeable of literature in that direction, but some of the arguments in the aforementioned excerpts are flawed.

Simply put, self-sufficiency does not preclude free trade. The excerpt of The Economist certainly admits that "The notion that free trade precludes food security is plainly wrong-headed", but it still misses subtle differences between security and self-sufficiency: (1) the latter intrinsically entails more autonomy; and (2) the cost of reliance on international markets might ultimately outweigh the costs of pursuing self-sufficiency.

The undeveloped mention of "starving North Korea" in The Economist improperly conflates the notion of food self-sufficiency and North Korea's policy of generalized isolationism. But, more generally, a country may prioritize food self-sufficiency without necessarily imposing trade barriers.

Note that The Economist article mentions Vietnam and India as examples of economies with export restrictions. But an export restriction is consistent with the purposes of self-sufficiency: It aims at ensuring that the domestic production of the commodity is not depleted by international markets prematurely (that is, prior to satisfying the domestic demand). An increased volatility in international markets as a result of the export restriction becomes irrelevant to the domestic economy if the net effect is that its population is not deprived of the commodity it produces.

Likewise, the issue of "domestic farmers [being] discouraged from producing more" has to do with policy (sub-)optimality regarding the excedents from self-sufficiency. It is not a defect inherent to self-sufficiency.

Mohanty's position fares better in that he rather discourages "pursuing self-sufficiency at any cost". But his argument of climate change predictions is self-defeating because a concentration of supply exacerbates a bottleneck risk --and therewith its potential severity-- of a commodity trade. The global impact of extreme weather conditions is predictably worse in the context of fewer exporters than if sources of supply are geographically more diversified because more economies pursue self-sufficiency.


Why not say the buyer will *definitely* buy at price X?

Under Yatharth's approach, the set of prices at which Bob will definitely buy an apple is the closed interval $[\$0,\$1]$, which is nice. However, the set of prices at which Bob will definitely not buy an apple now becomes an open interval $(\$1,\infty]$.

In contrast, under the conventional approach, the set of prices at which Bob may buy an apple is closed: $[\$0,\$1]$. So too is the set of prices at which he may not buy an apple: $[\$1,\infty)$. So, under the conventional approach, we're dealing solely with closed intervals.

Both approaches have their merit. But I believe that all things considered, fewer inconveniences arise with the conventional approach than with your suggested approach. In general, it's "nicer" to not have to deal with any open intervals.

Example 1. Under Yatharth's approach, there is no answer to the question "What is the lowest price at which Bob may not buy an apple?"

Example 2. Under Yatharth's approach, the demand curve will be discontinuous and with jumps. In contrast, under the conventional approach, it is continuous (albeit with kinks).

Elaborating on Example 2.

The demand function $D:\mathbb{R}_{0}^{+}\rightarrow\mathcal{P}\left(\mathbb{R}_{0}^{+}\right)$ maps each price (a non-negative number) to a set of quantities demanded (a subset of the non-negative numbers). The economic interpretation is that at each price, $D$ tells us what numbers of units of the good the consumer most prefers to buy.

We assume that the good is infinitely divisible.

Then here are the graphs of the demand function under the two approaches:

The demand function is continuous under the conventional approach, but not under Yatharth's. (Nonetheless, under Yatharth's, it is lower semi-continuous.)


First order conditions in a Moral Hazard problem

Notice that MWG choose a wage function $w(\pi)$ to maximize profit, not a wage level $w$. As footnote 6 on p.481 of MWG says:

  The first-order condition for $w(\pi)$ is derived by taking the derivative with respect to the manager's wage at each level of $\pi$ separately.

In other words, you'd treat the wage function as a variable, e.g. $w(\pi)=z$, and solve
\begin{equation}
\max_{z} \int [\pi-z]f(\pi|e)\mathrm d\pi +\gamma \left[\int v(z)f(\pi|e)\mathrm d\pi-g(e)-\overline u\right],
\end{equation}
which will yield the FOC
\begin{equation}
-f(\pi|e)+\gamma v'(z)f(\pi|e)=0.
\end{equation}
Substituting $z$ with $w(\pi)$, you'd have condition (14.B.3) in MWG.


Homogeneous Utilities: Anything other than CES?

There is an infinity of such functions. You can for instance construct a linear homogeneous function $u$ from any utility function $U$ by using a linear homogeneous function $h: \mathbb{R}^J \rightarrow \mathbb{R} $ as follow:
$$ u(x) = h(x)U(x/h(x)). $$
Example:
$U(x)= \alpha x_1^2 + \beta x_1x_2 + \gamma x_2^5 $,
$h(x)=x_1+x_2$
yields
$$u(x) = (x_1+x_2) \left(  \alpha(\frac{x_1}{x_1+x_2})^2 + \beta\frac{x_1x_2}{(x_1+x_2)^2} + \gamma(\frac{x_2}{x_1+x_2})^5 \right). $$
If you also want to ensure that $u$ is a utility function (increasing in $x$) it is a bit more tricky, however.


When a stock market crashes, how does money just disappear?

Stocks are not money.

The valuation of a company - the market captialisation - is the number of shares multiplied by the share price.

The share price is the price people are willing to trade at right now. It does NOT mean all the shares have been traded at that price.

If a company issues 1 million shares, at a starting price of £10, the market cap is £10 million. They may only sell a few of these shares, so much less than £10 million actually changes hands. If the price rises to £20, only a few of the shares changed hands at that price - but the market cap is reflected in all the shares, and is now £20 million. People have commented on the flaws in this methodology.

The same happens in a crash. Only a fraction of the shares actually change hands, but the media report on the change in the market cap.

If you want an analogy, consider a house burning down. You buy the house for £200,000. Then it sadly burns down (and without insurance). It is now worth £0. However, no money has been created or destroyed; it's a loss of value of an asset.


Instrumental variables vs control function: Which approach and why to handle endogeneity?

The Control Function Approach (proposed by Heckman-1979 if I remember correctly) , is similar to an IV strategy in spirit, except that it can be used in a non-linear setting, such as Probit/Logit models etc. For instance, if you have endogeneity in a logit model, you cannot just run a 2SLS procedure as the first step is a linear projection of your x on your z, and the second step is nonlinear. There are two ways of handling endogeneity in nonlinear models: control function approach and a special regressor approach.


What utility functions are equivalent to additive functions?

Ted Bergstrom has Lecture Notes on Separable Preferences that seem to have what you are looking for. For example:

  When are preferences additively separable?

  The most useful necessary and sufficient condition for preferences to be additively
  separable is that every subset of the set of all commodities is
  separable. The proofs that I know of for this proposition are a bit
  more elaborate than seems appropriate here. A somewhat more general
  version of this theorem can be found in a paper by Gerard Debreu (Topological methods in cardinal utility).
  Debreu’s paper seems to be the first satisfactorily general solution
  to this problem. Other proofs can be found in (Foundations of Measurement) and (Utility theory for decision making).


A question regarding definition of Pareto efficiency

the process of moving from one point to another i.e. reallocation makes one person better off while making another person worse off. for instance, moving from the lower part of the contract curve to a higher part makes Octavio better off as Octavio's indifference curve goes further away from the origin. However, the process makes Abby worse off as Abby's indifference curve moves closer to the origin.

There exist multiple points of pareto efficient outcomes which are stable, i.e. no more trading occurs.


I think that the wikipedia steps are not completely correct. When you perform Johansen cointegration test you first have to pretest the data to find if they have the same order of integration. That is all variables have to be either $I(1)$ or $I(2)$ etc. (see Verbeek, 2008). There are some cointegration tests and models that relax this assumption but Johansen is not one of them. But otherwise the steps 1-3 are correct.

$\Phi D_t$ are indeed seasonal dummies. However, note they are actually not standard part of the model, they are added when you think series can be affected by seasonal trends and if you dont use some other seasonal adjustment technique or if you dont already have data provided with seasonal adjustment from the source.

$\mu$ is the constant or 'drift' term. It allows your regression to have different intercept than zero.

You are correct $\Pi$ are matrices of coefficients but dont forget about $\Phi$. Any coefficient you add there has to be estimated - I know its just nitpicking but still.

The co-integration vectors come from the $\Pi$ in the second equation. This matrix can be decomposed as $\Pi = \gamma \beta'$ where $\gamma$ is the matrix of weights with which the cointegration vectors enter the model and $\beta$ is actually the matrix of cointegrated relationships. I am not really exactly sure what you mean by choosing cointegrated vector from this matrix. Its not like you just somehow extract one of the vectors somewhere into some separate vector.

Afterwards the Johansen test basically tests the rank or the number of columns of the cointegrated matrix (With dimensions $k$x$r$ you test for # of $r$). Intuitively this works because you need one column to represent each cointegrated relationship (note there might be multiple cointegrated relationship at the same time).

The rank of the matrix is tested either by trace or eigenvalue test as you correctly mentioned. The null of trace test is $H_0: r \leq r_0$ against $H_a: r <r_0 \leq k$  and the max eigenvalue test has the same null but alternative hypothesis is slightly different $H_a: r = r_0+1$. However, this might be hard to read in plain English. In practice if you would have only two variables both tests would boil down to first testing 0 cointegrated ($r=0$) relationship vs 1 $(r=1)$, second less or equal to 1 cointegrated relationships $(r\leq 1$) vs 2 ($r=2$), third less or equal to 2 coint. rel. $r\leq 2$ vs 3 ($r=3$). You dont really pick from these vectors although the number of cointegrated vectors can affect functional form of VECM, but with just two variables you are looking for $r=1$, if your test finds there should be more you are missing some important variables in your model.

Also yes $\Delta$ is the first difference $\Delta y_t = y_t - y_{t-1}$. Also including first differences assumes that the variables are all $I(1)$ if they are $I(2)$ you need at least second differences. The reason for that is that outside the cointegrated lagged variables the 'contemporaneous' variables must be stationary otherwise your ECM will be biased.  The whole justification of including level variables in ECM is that as long as there is a cointegrated relationship between those their combination will be stationary and their coefficient will be super consistent, but this does not extend to the variables that are added to the model outside the cointegrated relationship.

References:
Verbeek, M. (2008). A guide to modern econometrics. John Wiley & Sons.


Does creating more jobs actually help the economy as a whole?

Generally, whenever people have more income, it doesn't only benefit those people with more income. Imagine you somehow received \$100 that fell from the sky (or through NASA job). Would you save the entire \$100 and never spend it? More likely, you would spend some fraction of it... let's say you spend \$60. This money you spend will go to a shopkeeper, who then keeps a fraction and spend some of that \$60 on something else, and so on. So this extra money benefits other people too, not just the person receiving the money.

If you're interested, you can read more about this here.


Was the 2008 financial crisis caused more by the US subprime mortgage lending or more by an international energy and oil pricing crisis?

It's worth differentiating between the financial crisis of 2007-2009, and the recession that began in December 2007.

The financial crisis was indeed caused by a run on wholesale funding that was precipitated by significant losses on real estate, both subprime and otherwise. As noted in the first link, stresses in wholesale funding first appeared in August 2007. At that time, oil prices (WTI) were still \$70-\$80/barrel.

Losses to financial institutions during the crisis that can reasonably be linked to oil are limited, and the timing is all wrong, so it's not credible to suggest that the increase in oil prices would have led to a financial crisis absent the actual mechanism (run on wholesale funding, significant losses on real estate assets) by which the financial crisis occurred.

That said, it's certainly likely that oil prices contributed to the recession that began in December 2007, as documented by James Hamilton, though the severity of this recession was in large part due to the fact that there was a financial crisis:

  Recessions associated with financial crises have been more severe and
  longer lasting than recessions associated with other shocks.
  Recoveries from such recessions have been typically slower, associated
  with weak domestic demand and tight credit conditions.

Thus, oil prices are likely to have contributed to pushing the economy into recession, which may have worsened credit losses, fueling the stresses in the financial sector, but it's not credible to claim that they were a primary factor in causing the financial crisis, as compared to the structural vulnerabilities relating to wholesale funding and securitized assets.


You seem to think of a game where objects are obtained at a fixed rate by farming. Fix supply of work for a certain object (bc. Only a few people know where to find things to farm or are high enough level or have the right profession) and increase demand (new players coming in or more players buying in game currency for real money) then prices for the object will soar over time, people producing the object will find it easier to sell it, so much so that they will increase the price until people stop buying it. This will inflate the price of the object. Whether this will last over longer period of time then depends on many factors : ability of supply to adjust (e.g. How much are player willing to switch to farming that object?), rate of new player characters coming in, their propensity to buy the object etc.

There are  many other ways. One example I could think of: as a player on a WoW private server with moderate population I would routinely buy all objects from the auction house that I thought of as undervalued and put them back in auction at higher prices. This allowed me to gain arbitrage profit from less careful players and also to craft and sell the same object at higher prices.


Essential goods: How does one restrict the utility function?

In order to have $x^*_j>0$, it is possible to impose either
(i) a condition on the marginal utility: $\lim_{x_j\rightarrow0} \partial{U}/\partial{x_j}(x)=+\infty$
(ii) or an inequality $x_j \geq a_j>0$ where $a_j$ is interpreted as a subsistence level of $x_j$
Often the utility function is reparameterized and written $U(x-a)$ with the constraint $X:=x-a\geq 0$, which ensures that $x^*\geq a$ at the inner optimum.
Example: in the Cobb-Douglas case with subsistence levels, we have
$$ U(X_1,X_2)=U(x_1-a_1,x_2-a_2)=(x_1-a_1)^\alpha(x_2-a_2)^\beta $$
Provided that the income $m$ is high enough, the Marshallian demands are given by:
$$x^*_1 = a_1 + \frac{\alpha}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_1}\\
x^*_2 = a_2 + \frac{\beta}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_2}. $$