Lucas Critique and RE: Did one arrive before the other?

While this probably doesn't fully answer your question, at least it might be able to serve as a start.

I think the biggest conceptual difficulty here is regarding what exactly the heart of the Lucas critique is, and why it was so influential. Before he published his seminal 1976 work a large body of macroeconomists were primarily engaged in attempting to investigate policy analysis by estimating fundamental parameters of a model (at the time, since Keynsian analysis was in vogue, it generally focused on things like the marginal propensity to consume or save, or estimating consumption functions for use in policy counterfactuals, etc.). How this was done was by using aggregated historical economic data to try and back out the parameters of interest.

The Lucas Critique points out that this effort cannot be guaranteed to be internally consistent. That is, all of the observed, aggregate historical data being used was actually the output of a complex system. When considering how a change in policy might impact welfare, the economist cannot assume that these parameters will remain unchanged.

Critically, this is true independent of whether the model is using rational expectations. No matter the macro modeling assumptions, the internal optimization that the agents are "doing" inside the model must be grounded in actual fundamentals of economic theory. That is to say, there's no theoretical reason why one consumption function should be used by an individual over any other (for example). Historical data, then, can only give insights into what the optimal decision making behavior was at the time it was observed.

There's actually a pretty good example on the Wikipedia page for the Lucas Critique in the last section (behavior surrounding Fort Knox). Rational expectations would suggest that individuals are maximizing some, known objective function for whether they should try and rob the vault or not. Given that we don't see many attempts, we can then say that, given individuals are following rational expectations and are maximizing some consistent utility function, in general, the perceived costs outweigh the benefits. The Lucas Critique, on the other hand, says that we cannot use the fact that we don't observe many attempts at robbery now to suggest that Fort Knox spend less on security. While RE theory says people aren't robbing because the costs outweigh the benefits, the Lucas Critique says we cannot use the fact that we don't observe many arobbery attempts at this level of security to say anything about how robberies will change if we change security levels. All we know is that, in this particular set of circumstances, we observe some equilibrium behavior (few robberies). To make policy statements, we have to base our models on some fundamental theory- one of the most prominent being a form of RE. But ultimately, whether we believe RE holds or not, the Lucas Critique still points out that we must base our predictions in some theoretical foundation, and not only use aggregated outcome data.

Sorry that was kind of long winded and sloppy- I'll try to tidy it up later if it would help!

EDIT:

  Is it fair to say that the LC is a criticism of keynesian models because they didn't include expectations of any kind rather than just not rational ones?

Broadly, that's exactly it. Keynseian models at the time needed lots of aggregated parameters (again, like marginal propensities, etc.) to come up with conclusions on how different policy changes, or economic shocks, might impact the economy (or industry, or group of people, etc.). The problem is, Keynsian analysis (again, at the time) requires these parameters without any theoretical context. Why should the marginal propensity to consume be X? What would cause that to change? These questions require the microfoundations that we've seen spring up in the macro field- models that attempt to examine what determines important macro parameters, and how changes to those underlying causes will change the parameters of interest. Not only did Keynsian models not include expectations (specifically forward looking variables), they tended not to include any theoretical basis for the parameters they estimated.

  And, as long as the included expectations can be connected to either A) the policy rule or B) the underlying stochastic process ( if there is no specific policy rule sometimes they seem to do this), the model is then internally consistent.

While these are certainly two good microfoundations that an economist can use to build a macro model, model consistency (again, using the broadest possible definition) can rely on an even weaker basis. For example, you could have a model of the macro labor market where labor supply is completely divorced from any sort of rational or forward looking behavior at all. You could base your model on something like:

Every period, all agents wake up and flip a fair coin. If the coin is heads, they go to work. If it's tails, they go back to sleep until the next period.

Of course, this premise is completely divorced from reality- coin flipping has nothing to do with the decision of whether to go to work. But it would at least provide a clear theoretical basis (even if absurd) for why the labor market behaves the way it does. Instead of just observing "last period, the labor market participation rate was 60%, thus, we should take that as some underlying parameter" (like the classical Keynsian methodology might), you'd say "I see last period there must have been an inordinate number of coins that landed 'heads.'" Your theory would give you a specific reason for believing the participation rate was 60% (the proportion of heads), a prediction for what it would be next period (50%, since that's the expected number of heads you'd expect to see next period), and a reason ** why ** it should be 50% (again, because everyone's flipping coins).

Moreover, using the coin-flip model of labor markets, you could consider how a policy change should affect behaviour. How would a doubling of wages change turnout? In this case, we'd predict it wouldn't have any impact (still doesn't impact the coin probabilities). And while that might seem absurd, at least you can very clearly explain why there will be no change, using some economic reasoning. The old Keynsian models couldn't provide these types of explanations, other than "because the last time wages doubled, X happened to participation." While that might be nice to know, it cannot be considered a useful answer, since there might have been any number of other things causing the wage change to happen the last time, that might not be at play now (and which might have been even more important in driving the participation changes).

Rational expectations, then, is just one useful, generalizable, easily applicable mechanism for how to bring microfoundations into these macro models.

Again, hope that helps!! Let me know if I can clear anything up!!


$
\newcommand{\Cov}{\text{Cov}}
$

  Why can the $\beta_{asset}$ be seen as if the firm had been financed only with equity?

The general idea with regard to this question stems from a discussion about how equity is a residual claim. To go through this step-by-step, recall that the value of a firm is given by the discounted present value of its cash flows. Equity is a claim to whatever is left over after other financial claim-holders have been paid (e.g., debt).  In this way, equity can be viewed as a call option and debt can be viewed as a put option. (See here and here.)

So, if the firm was financed only by equity, then equity would be the claim to 100% of the cash flow. Thus, $\beta_{\text{asset}}$ in that case can be seen as the if the firm was only financed with equity.

  We can measure the $\beta_{equity}$ by CAPM, but how can we measure $\beta_{debt}$?

You say that you can measure $\beta_{\text{equity}}$ by CAPM. However, you can measure any cash-flow by CAPM---even debt. Since $V = D + E$, keep in mind that
$$
\Cov(X + Y, Z) = \Cov(X,Z) + \Cov(Y,Z).
$$
The problem, I guess, is that you need to know the distribution of debts cash flows. This depends on the face value of total debt and the distribution of the underlying cash flows. This can get especially complicated if you're thinking about considering a dynamic setting.

This question, I'm guessing, came from a discussion of the Modigliani-Miller theorem. Remember the idea behind Proposition 2. It's illustrated in the following graph:


What are some real-world examples of the different types of trading blocs?

There are actually five different types of trading blocs: preferential trading areas, free trade areas, customs unions, common markets and economic (and monetary) unions [1].

A preferential trade area (PTA) is a trading bloc that reduces tariffs on particular goods between two countries but does not abolish them completely.  There are no particularly well-known examples, but a list can be found here [2].

A free-trade area is the agreement among all member countries to eliminate tariffs, import quotas, and preferences on goods and services traded between each other (the free-trade as defined in the question) [3].  Probably the best known free-trade area is NAFTA; but others are listed here and here.

Again, there are no well-known customs unions, but a list can be found here.

A common market has free trade and free movement of the factors of production but other economic policies of the countries may be different (one of the differences between a common market and an economic union).  You may also want to look at single market and unified market.  There are no well-known common markets, but the previous link also contains a list.

The best known economic union is the EU though more can be found here.  An economic and monetary union has the same qualities of an economic union but each member nation also uses the same currency.  One example would be the nations in the EU who also utilize the Euro.


Why is price plotted as a dependent variable?

In supply and demand model both price depends on quantity and quantity depends on price. These two variables are determined together. They are both endogenous variables. Consequently, here the rule that independent variable goes on $x$-axis and dependent variable goes on $y$-axis does not apply because neither of the two variables is exogenous.
By convention in economics price is always plotted on $y$-axis. This was not historically always the case. As you can see in this answer in past there were several authors that decided to plot price on $x$-axis as well. However, the convention of price going on $y$-axis was eventually established by Marshall who considered price to be the depend variable (although really they are both endogenous), and popularized this convention through his textbook (Principles of Economics) that was widely used.


Doesnt fractional reserve systems limit collapse of the money supply?

Without a historical reference, I have doubts about the Black Death reference. Scholarship around the Black Death has changed.
The premise behind this question is that the velocity of money is constant. Since we can see that it is not - link to FRED M2 velocity - we need to ask what else drives the price level.
The obvious concern is that there is no explanation of how output falls 90%. Is it because of out-of-control bankruptcies causing business failures, while the government does nothing? That’s a deflationary environment. Was there a war or pandemic that destroyed industrial capacity? Probably inflationary.
If we step away from the text of the question, and just look at the title - does fractional reserve banking limit the shrinkage of the money supply - the answer is technically no. For the few remaining countries with reserve requirements, if deposits fall, so do required reserves. If the central bank keeps excess reserves at zero, then reserves will shrink along with deposits. As for currency in circulation, it is determined by the desire of the private sector to hold it. If holdings depend on incomes, holdings will fall in line with incomes.
What might keep the monetary base from shrinking is a deliberate policy choice of the central bank.


How does keeping the value of the dollar by foreign US bondholders makes their exports more affordable?

Suppose a camera is made in Japan and sold in the US. If the maker of the camera wants 10k Yen for it, and the ultimate buyer is paying dollars, then somewhere along the line, dollars have to be exchanged for dollars. For instance, the company that imports the camera might go out and buy 10k Yen, then give the Yen to the company that made the camera. They then pass on the cost of buying the Yen to the camera store, who passes it on to the customer. The dollar being strong with respect to the Yen means that one dollar can buy a lot of Yen. So that means that it doesn't cost very much dollars to buy the 10k Yen, which means that the camera store in the US can sell the camera for fewer dollars, which means that people are more willing to buy the camera, which means that the camera manufacturer gets more business.

You're also asking how the strength of the dollar is affected. It's a matter of supply and demand: every time Americans buy stuff from other countries, they are supplying dollars, and every time they sell things to other countries, they are demanding dollars. The strength of the dollar depends heavily then on how much the US exports (which strengthens the dollar) versus imports (which weakens the dollar). Since transactions between the US and other countries involve trading goods and dollars, the effect on the dollar is opposite of how the goods are moving from the US perspective: when someone in the US sells a good, that can be viewed as "buying" dollars, and the US buying goods can be viewed as "selling" dollars. A strong (or "expensive") dollar means that US goods are also expensive, but foreign goods are cheap, and a weak dollar means that US goods are cheap, but foreign goods are expensive.

Companies that facilitate exports generally benefit from a weaker dollar (the goods are cheap, so it's easier to sell them). This includes export companies, and people who work in industries whose markets are largely overseas.  Companies that facilitate imports generally benefit from a stronger dollar. This includes import companies, and companies that that sell a lot of foreign-sourced goods, which in the camera example would include the camera store.

Bonds aren't generally included when calculating exports, but they do have largely the same effect: US treasury bonds are denominated in dollars, so if people in other countries want to buy them, they have to buy dollars, which drives up the value of the dollar.   You can think of it in terms of analogy with a household: exports are like people going out and working for a paycheck, while imports are like buying stuff. A bond is like getting a loan, such as a mortgage or a car loan. The more money someone makes at their job, and the more loans they get, the more the cash they have to spend. A dealership wants to help someone get a car loan, because that makes it easier to but a car. Similarly, China and Japan want the US to be able to sell their bonds, because that makes it easier to buy stuff from China and Japan. Whenever China or Japan buy a treasury bond, they give more money to the US, and the US then uses that money to buy stuff. If China were to sell off their bonds, that would make it harder for the US to sell new bonds, which would make it harder for Americans to buy stuff from China.


What is the purpose of the Treasury issuing bonds?

There are some semantic issues here. A more careful statement of the MMT position is that there is no economic reason for the Treasury to issue bonds. However, there is a self-imposed institutional need to do so: the Treasury has a balance at the Federal Reserve, and the rules it follows appears to imply that the balance cannot go negative. Issuing a Treasury increases that balance, as that is where the borrowed money goes. Note that I believe that some MMT academics have argued that there is no legal need for the balance to remain non-negative - and thus this argument does not apply - but I do not have the legal expertise to judge those arguments. In any event, the Treasury acts as if this requirement exists.
However, the economic purpose of the bonds is that they offer an alternative to holding deposits at the Fed (“reserves”), and bear interest. They create a risk-free yield curve that acts as a benchmark for private sector borrowing. The Fed can set the overnight rate, and influence the rest of the yield curve. This allows it to conduct interest rate policy.
Note that although the phrasing might be disputed, the previous paragraph describes very well all standard models that include a bond market. As such, there’s no debate about the facts, rather the interpretation of the facts.

If that is the real purpose, then what then does the Treasury do to raise money to pay off US debt?

When a Treasury bill/bond expires, it is paid out of the Treasury’s balance at the Fed. This balance comes from tax receipts, and bond/bill issuance. Like corporate borrowers, the stock of Treasury bonds is continuously rolled over.
This mechanism has very little to do with the Fed’s conduct of monetary policy, it’s just the mechanics of bond issuance.


You are missing patents.

I recommend reading the paper "Why Is There No Generic Insulin? Historical Origins
of a Modern Problem" by doctors Greene and Riggs.

  ...high costs of
  pharmaceutical research and development are
  often invoked to justify the high price tags of new
  medications. Although the price point of effective new drugs — from hepatitis cures to new
  agents for heart failure — may initially be out of
  reach for many patients, market laws predict
  that drugs with strong demand should become
  more accessible after market exclusivity ends
  and generic competition begins. Since 1984,
  when the Drug Price Competition and Patent
  Term Restoration Act was passed, pharmaceutical innovation and access have been balanced on
  that premise: innovative drugs are rewarded
  with high prices during their window of patent
  protection, and generic competition reduces
  prices thereafter.

  [...] incremental innovation has repeatedly precluded the formation of a generic insulin industry in North America when earlier
  patents expired. The history of insulin hasn’t followed the standard chronology of pharmaceutical innovation, in which patent monopolies predictably give way to generic competition.

  Viewed in historical perspective, insulin is not
  a single entity but a family of related products
  that has evolved through incremental improvements. Subsequent iterations of insulin represented actual innovations, each one being safer,
  more effective, or more convenient than its predecessor. And yet after generations of incremental
  innovation, insulin may be no more affordable
  than it was when the original patent holders
  sold their stake for $1 to ensure access to this
  essential medicine.
  Pharmaceutical-industry analysts have described a repatenting tactic called evergreening,
  in which a series of related patents — often on
  metabolites or optical isomers — extend the life
  of a product after initial patent expiration.


You can consider a single competitive firm model with quadratic cost function say $c(y) = \frac{y^2}{2}$. This firm is in Venezuela. Let $\hat{p}_c$ and $\hat{p}_v$ be price ceilings in Colombia and Venezuela respectively. Also, $\hat{p}_c > \hat{p}_v$ indicating the opportunity to smuggle. Also, assume demands are sufficiently high so that the ceilings are binding. Consider competitive suppliers' problem without smuggling:

\begin{eqnarray*} \displaystyle\max_{y\geq 0} && p_vy - \frac{y^2}{2}\end{eqnarray*}
Solving it, we get the supply curve as $y = p_v$.

Given that the price in Venezuela is $\hat{p}_v$, equilibrium quantity in this case will be $y^* = \hat{p}_v$.

With an option to smuggle with the additional cost say $c(z) = \frac{z^2}{2}$, the profit maximization problem is:
\begin{eqnarray*} \displaystyle\max_{y\geq 0, z \geq 0} && p_vy + p_cz - \frac{(y + z)^2}{2} - \frac{z^2}{2}\end{eqnarray*}

This gives us the new domestic supply function as:
$y = 2p_v - p_c$

and the quantity smuggled as
$z = p_c - p_v$

So, equilibrium quantity sold in Venezuela has fallen to
$y^{**} = 2\hat{p}_v - \hat{p}_c$ and the quantity smuggled is equal to $z^{**} = \hat{p}_c - \hat{p}_v$.


How to determine the price a monopolist will set when the demand curve the firm faces isn't known?

The monopolist is allowed to set the price. However, because his product has a perfect substitute, he can sell 0 products at prices $p>2$. He also has no incentive seting the price $p<2$, because the consumers will buy any number of products he wishes to sell at $p = 2$. This means that $p(q) = 2$. Its graph in the $(q,p)$ coordinate system would be a horizontal line at $p = 2$. Because of this, $p'(q) = 0$. Correcting for this you should be able to easily finish the exercise.


Macroeconomics, the circular flow of income and profits

The firm broke the circular flow of income in your example; profits went into a black hole.

Imagine that the firm instead distributes all profits to the owner, who spends all proceeds.

The firm gets \$6000 in revenue, pays \$5000 in wages, and pays \$1000 in dividends. The wages and dividends are enough to provide the revenue.

I am ignoring the issue of the initial money holdings. There is a number of ways these transactions can be supported: credit, or the \$1000 dividend was paid out revenue as it came in.

In order to respect the initial money holdings, and have sales of \$5000, what happens is that the firm has excess inventory (with a sales value of \$1000). That is, the workers could not afford to buy all of the output. The firm would be profitable, but with zero cash flow. Since the cash used to pay the workers returned to the firm, it would be able to then pay a dividend out of profits. That dividend could then be used by the owner to buy the remaining inventory.

It is not obvious how the firm can forecast its profits exactly in this fashion, but that would be covered up in an economic model by solving for all flows simultaneously. If all cash flows are settled simultaneously, we actually cannot see how the money flows work; we only see the net movement.


Why is the U.S. birth rate considered below replacement if more people are being born than are dying annually?

When you hear that "the U.S. birth rate is near or below replacement rate", what is probably meant is that the total fertility rate (TFR) is below 2.1 (replacement rate).
It is possible that the TFR is below 2.1, while births exceed deaths. One way this can happen is if there's a rising share of women entering reproductive age.
This phenomenon is sometimes called population momentum (more details: 1, 2, 3).
This is actually a fairly common phenomenon: Many countries have had TFR below 2.1 for many years but their populations have continued to grow. For example, China has had TFR below 2.1 since 1992 but births have continued to exceed deaths every year.


How to justify an instrumental variable?

To motivate a good IV in a paper, you need a good story. For example: schools build in Indonesia as an instrument for education outcomes, quarter of birth as an instrument for income as a result of compulsory education, in their respective papers.

The IV has to be correlated to the characteristic you're trying to fix and uncorrelated to any of the other observable characteristics.

That way, the IV is only affecting the Y variable through the characteristic you're instrumenting. As far as statistically strong, you can look at examples from a few papers and see what kind of statistical scrutiny they used.


Eliminating a column and a row strategy is a nice idea. If the remaining 2x2 game has no pure strategy Nash equilibria (PSNE) than Nash's theorem guarantues you exactly one MSNE.

However there are also very trivial 3x3 games with zero PSNE and exactly one MSNE, e.g. rock-paper-scissors.


The relationship between gross profit and tax

Most of what you have done was correct. Here are the steps to finish the argument.

Evaluating the profits at the optimum, you get
\begin{equation}
\pi_G=R(y^*(t))-C(y^*(t))
\quad\text{and}\quad
\pi_N=R(y^*(t))-C(y^*(t))-ty^*(t).
\end{equation}
Differentiate the net profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi_N}{\mathrm dt}&=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-t\frac{\mathrm dy^*}{\mathrm dt}-y^*\\&=\underbrace{[R'(y^*)-C'(y^*)-t]}_{=0 \text{ due to FOC}}\frac{\mathrm dy^*}{\mathrm dt}-y^*<0,
\end{align}
which is true assuming $y^*>0$.
Differentiate the gross profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi_G}{\mathrm dt}&=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}=[R'(y^*)-C'(y^*)]\frac{\mathrm dy^*}{\mathrm dt}<0.
\end{align}
This is true because from the FOC, we know that for any positive tax, $R'(y^*)-C'(y^*)>0$, and since you've correctly derived $\frac{\mathrm dy^*}{\mathrm dt}<0$, their product must be negative.


Modern economists believe that machines can create value. The reason why Marx thought that machines cannot create value is that he subscribed to classical notion of labor theory of value (LTV). Labor theory of value is an objective theory of value that postulates that all value is determined by labor inputs (see the Durlauf & Blume 2008 entry in the Palgrave dictionary of economics for more on LTV).
However, current mainstream economic thinking rejects LTV in favor of the subjective theory of value precisely because LTV is empirically inconsistent with valuations that we observe. Subjective theory of value postulates that value is subjective and in the eye of the beholder as in subjective theory of value value depends on marginal utility of particular person. Under subjective theory of value even object with no labor content can have value (see the Durlauf and Blume entry (2008) on utility, and on utility and decision theory and sources cited therein  - this concept is also discussed in virtually any contemporary textbook so you can learn more also from there).
Marx also thought that profit comes from exploitation of labor but modern economists do not accept that view so, yes economists do realize that profit can be made from capital alone. Profit by definition total revenue minus total cost $\Pi = TR-TC$ and there is no reason to assume that fully automated firm could not have higher revenue than cost.
Lastly, capitalism in economics is poorly defined. In most economic textbooks you wont even find the word capitalism printed. Since the term lacks proper definition there is not much to discuss about it without first defining it.


Calculating present value using Euler's number

$e^{-rt}$ is the continuous discounting factor while $(1 + r)^{-t}$ is its discrete counterpart.

Identically/equivalently, there is the continuous manner of computing factors of variation, e.g. $1 + \ln \frac{x_{0+t}}{x_0}$, and its discrete counterpart $\frac{x_{0+t}}{x_0}$.

It all depends on whether you think you are dealing with a continuous quantity or not.

In asset pricing (or finance in general) the underlying notion is that of continuous (interest) compounding.


Analytically tractable Ramsey model: how to solve ODE for optimal trajectories

The differential equation

$$\dot k =  \frac{1}{\sigma} k^\alpha - \delta k$$

has the structure of a Bernoulli equation. We solve it by the following transformation steps:

1) Mulitply throughout by $k^{-\alpha}$:

$$k^{-\alpha}\dot k =  \frac{1}{\sigma}  - \delta k^{1-\alpha} \tag{1}$$

2) Define the variable
$$z \equiv k^{1-\alpha} \implies \dot z = (1-\alpha)k^{-a}\dot k \tag{2}$$

3) Combine to get

$$(1),(2) \implies \frac {1}{1-\alpha}\dot z = \frac{1}{\sigma} - \delta z $$

$$\implies \dot z + (1-\alpha)\delta z = \frac {1-\alpha}{\sigma}$$

This is a standard first-order differential equation with constant coefficients. Solve it and then reverse the change of variable to get the result.

PS: The above requires that $k \neq 0$ everywhere, which is the economically meaningful case.


Difference between Opportunity cost and marginal cost

Opportunity cost is the price of doing something in terms of something else. For example, cost of taking trip to Prague may be giving up new bike. In this broad sense marginal cost of producing one unit of q would be also it’s opportunity cost because you could use the same resources to produce something else. Opportunity cost of producing 1 widget at 5€ might be giving up possibility to produce 2 pins at 2,5€ each.

What the quote says is not that only time has opportunity cost but that you should not forget to add the opportunity cost of the persons time to the costs.

Also opportunity cost is not producer surplus. You don’t need to give up producer surplus to produce the particular good or service you are producing. You could include that in opportunity costs if you would be looking at it from a point of view of person deciding to leave the market to go to do some other business or retiring


Why are New Zealand houses so badly insulated?

Insulation is more of a luxury than a necessity in the oceanic climate. If you don't have insulation in a continental climate, you'll die. In NZ, it will be uncomfortable, but you'll be ok.

Dunedin may provide arelevant counter argument. It gets below freezing often there. I don't know about insulation rates.

My second answer is: We're just tough mate. Harden up.


A Cake Eating Problem in Continuous Time: Hamiltonian or HJB?

The comment by user @MaartenPunt is accurate. I don't think that in general one can identify situations where one should have a clear preference over one formulation over the other. It is more of a case-specific issue (and maybe for some twisted problems where one of the two may fail for usually technical reasons). See this post for some related discussion, https://economics.stackexchange.com/a/14289/61.
...Or sometimes one may get a bit confused, for example, in the specific problem, one could momentarily stop and wonder "what is the derivative of the Hamiltonian with respect to the state variable?"
Well, it is what it appears to be: zero. Because
$$\frac{\partial \mathcal H}{\partial k}= \frac{\partial \lambda \dot k}{\partial k} = -\frac{\partial \lambda c}{\partial k} = 0,$$
because we do not differentiate the decision variable, or the multiplier, with respect to the state variable. Now, optimally, we have
$$\frac{\partial \mathcal H}{\partial k} = -\dot \lambda,$$
and so it follows that the multiplier is constant along the time axis, $\dot \lambda = 0$. Then for the other first-order condition, we have
$$\frac{\partial \mathcal H}{\partial c} = 0 \implies e^{-rt} \frac 1 c = \lambda.$$
Differentiating this with respect to time we get
$$-re^{-rt} \frac 1 c - e^{-rt} \frac{\dot c}{c^2} = 0 \implies \dot c = -rc,$$
which is what we get from HJB as "policy" function.
As for whether this is a maximum, it is, because the Hamiltonian is jointly concave in $c$ and $k$, see, https://economics.stackexchange.com/a/6063/61.


How to find daily/weekly data on the state of the economy

That frequency is very hard to come by for GDP and employment, especially in many countries.
The only thing I know that allows that level of measurement is prices: check https://www.pricestats.com/ for daily measures of inflation, purchasing power parity and real exchange rate. It's available for many countries, both in developed and emerging economies.


What is the tipping point for hyper-inflation?

There is no set tipping point. In simple terms the relationship between money supply and inflation is given by the standard monetary model which shows that:

$$MV=PY$$

Where M is the money supply, V velocity of money P the price level (and change in the price level gives you the inflation) and Y is the real output.

Hence to get any inflation the changes of M must be such that they are not offset by changes in other variables.

If money supply changes by 10%, velocity of money stays constant, but real output also increases by 10% then there will be no change in prices and hence inflation will be zero. However, in the same example if the real output would not change and all other things remained as above the inflation would be 10%.

In normal times it is valid to say that inflation is a monetary phenomenon and depends mainly on quantity of money. The economy usually grows at steady rate of about 1-2% per year (I am talking about advanced economies at the technological frontier) and velocity of money does not change that much so in that case it is valid to say that relatively large increase in money supply causes inflation.

However, in situations like when interest rates are at 0 lower bound the velocity can actually easily offset any increases in money supply and relationship may break. Moreover, more complex models show that its expectations what matters. If money supply increases today but people expect this increase will be reversed in the future it might not affect inflation (you might want to look at this paper from Paul Krugman if you want to know technical details about that).

To go back to your main question:

  At what point does introducing money to a system cause hyper-inflation?

Following Cagan hyperinflation can be defined as an increase in price level exceeds 50% per month.

To get to that rate money supply would have to growth at the rate of 50% + the rate at which economy grows per month + at any rate that would offset changes to velocity in of money. Special conditions like no zero lower bound interest rates and assumption that people credibly believe the money supply increase is permanent (or at least long lasting) must be imposed as well. In US there would also be special issues since \$ is the world’s reserve currency which helps to reduce inflation pressure as portion of the US money supply is absorbed by other countries.

So to sum up, there is no magic rate of increase in money supply. Even if money supply remains the same but output collapses you can get inflation. Alternatively even large increases in money supply might not change inflation levels at all as recent Great Recession experience shows. You have to do some complex econometric analysis to see at which rate in a given year money supply increases start to affect prices. Since that requires some historic data it’s not possible to say for sure what the level is at a given point of time.

You can make forecasts of course and central banks do that routinely. This is the FED inflation forecast for 2020 which is also based on money supply changes (aside from other things). Although it does not mention money supply explicitly because FED does most of the operations through the funds rate they factor it in their models (the money supply also expands and contracts based on the funds rate). They do not expect hyperinflation so to the extent their forecasts are accurate the recent surge in US spending as response to the recent crisis as well as the increases in money supply seem to not be large enough to cause it yet.


Non-unique utilities in a competitive equilibrium

This answer is in response to the edited (2015.10.16 ca. 12:30 CET) question.

I think the concept of equilibrium means that given the circumstances you have no incentive to alter your behavior. (This is pretty much based on the Nash-equilibrium.)
 In this situation equilibrium consists of a price and an allocation of goods. Given the price everyone makes a trade decision. The decision has to be feasible both with respect to the budget and the physical availability of the goods, i.e. you cannot buy goods that do not exist. We are in equilibrium if everyone's decision was optimal (for themselves).
In your situation for any $p_x \in [0,2]$ there exists an equilibrium allocation, where consumer 2 bought the good $x$ from consumer 1 for $p_x$ units of good $y$. There is no better feasible decision for either of them.

If you were to adhere to the strict/dogmatic interpretation there would be no equilibrium at all, because for any $p_x <3$ consumer 2 wants to buy an infinite amount of good $x$ and for any $p_x > 2$ consumer 1 wants to sell an infinite amount of good $x$. Since there is no $p_x$ such that $p_x \leq 2, \ p_x \geq 3$ you could not have an equilibrium with finite amounts.


Equivalence of inada conditions and non-negativity constraints?

In a standard growth model, people don't put non-negativity constraints as you have mentionned. Theoretically, concavity of the Hamiltonian (or Lagrangian in discrete time) is the sufficient condition for the optimality of the program.
Just for some additional details, the non-negativity constraints are so usefulin some subfields of economics, like in environmental economics.
For example, if you want to put a ceiling for stock of pollution in atmosphere as $\overline{P}$. You can use it if you want that economy do not pass beyond this critical threshold. So, basically you can make as

$$max \int_{0}^{\infty} u(c,P)  e^{-\rho t} dt $$

s.t

$$\dot{K}=F(K)-c\\
\dot{P}=\epsilon K - \delta P$$

where $\epsilon$ is the emission rate from the use of capital and $\delta$ is the decay rate of pollution.

$$\mathcal{H} = u(c,P) + \lambda (F(K)-c) - \mu (\epsilon K - \delta P) + \alpha (\overline{P} - P) $$

The constraint $\alpha$ is a non-negativity constraint which will ensure that pollution level will not pass beyond the threshold $\overline{P}$. As long as the pollution level is below the threshold, $\alpha$ will be non-binding (and will be equal to zero).

Hint : You can look at the Kuhn-Tucker conditions in order to understand better the role of non-negativity constraint.


What is the monotone hazard rate assumption used for?

The motivation for this assumption is to have a unique solution. In this article, it is necessary to prove the 1st lemma, i.e. the quasi-concavity of the profit function.  You find a similar motivation in Tirole’s book (Industrial Organization, p. 156).


Interior Solution for profit maximisation problem

Your assumption 3 is compatible with corner solutions of the kind $y_i=0$ for some $i$, and is not sufficient to avoid corner solutions for some prices small enough.
With production functions it is common to assume Inada's conditions to avoid corner solutions. With cost functions, such conditions are quite naturally expressed as
$$ \lim_{y_i\rightarrow 0} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=0 $$
$$ \lim_{y_i\rightarrow +\infty} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=+\infty. $$
In your discussion, you proposed the cost function $$c(y_1,y_2)=y_1^2+y_2^2+2y_1$$ to illustrate a corner solution. This function does not satisfy the first Inada conditions wrt $y_1$, hence the corner solution.
A general comment to conclude: it is not fully natural to forbid firm's exit from a given output market. In IO there is a quite dynamic literature on entry and exit. And no firm is producing all products available in the economy.


Proper definition of extensive form game

Let the set of players be $N=\{1,\dots,n\}$.

According to my understand your description of the game, I take the following statements to be true:

Each player chooses a number of players to integrate, not players with specific identities. As a result, the player making the choice may or may not be included in the coalition.
Each player has to choose at least 1 player, if still available.
Each player moves exactly once.
The game is with perfect information.

Under the above interpretation, player $i$'s ($i\ge2$) action space would be $A_i=\{0,1,\dots,n+1-i\}$, as, in principle, $i$ could choose at most $n+1-i$ other players (when all players $j<i$ chose $1$) and at least $0$ (when all previous players have exhausted the player list). An exception applies to the first player, where the option $0$ is not feasible. Thus $A_1=\{1,\dots,n\}$.

Player 1's strategy space would be $S_1=\{1,\dots,n\}$. Player 2's would be $S_2=\{1,\dots,n-s_1\}$, since after $s_1$ players are picked by player 1, there are only $n-s_1$ players left to be picked by player 2. Carrying this argument forward, player $i$'s strategy space would be $S_i=\{1,\dots,n-s_1-\cdots-s_{i-1}\}$.

Generalizing from the above reasoning and taking into account boundary cases, let history at stage $i$ be $h^i=(s_0,s_1,\dots,s_{i-1})$, where we set $s_0=0$. (In general, though, stages should be indexed by a parameter different from the player index. But in your game, since each player moves exactly once, we can use the same index for stages as well as players to economize notation.) The history dependent strategy space for player $i$ would thus be
\begin{equation}
S_i(h^i)=
\begin{cases}
\{1,\dots,n-\sum_{j=0}^{i-1}s_j\} & \text{if } n-\sum_{j=0}^{i-1}s_j\ge 1 \\
\{0\} & \text{otherwise}.
\end{cases}
\end{equation}


Utility Functions: Implying endless consumption?

When the utility function is strictly increasing, it follows that more consumption provides more utility. So it seems evident to say "more income more consumption". But discussing infinity outside mathematics makes no sense. So we want to examine

$$u'(c) > 0\;\;\; \forall c,\;\;\; I\to \infty \;\;\;?\implies? \;\;\;c\to \infty$$

And the answer is "the question is indeterminate" or at least "it depends on the particulars", since what will matter is the rate at which Income and Consumption "go to infinity". If consumption goes faster, it will eventually deplete the infinite income, and if the consumer has infinite horizon, it will end up with no income. Etc.

If we want to entertain concepts from Non-Standard Analysis, and treat "infinity" not as a limiting concept but as something more concrete, then the answer will be the same, because this will be indeterminate also in Non-Standard Analysis, absent the particulars of the situation.


How do increasing tax rate helps control inflation?

If the government taxes people but does not spend the money, accumulating surplus, it will decrease aggregate demand. Aggregate demand depends on consumption, which depends on disposable income and taxes reduce disposable income. A decrease in aggregate demand will decrease the price level, reducing inflation or causing deflation, if nothing else happens (see this article by Pettinger).


Transforming a matrix of explanatory variable in regression

$\boldsymbol X^*_1$ is the residual of the projection of $\boldsymbol X_1 $ on $\boldsymbol X_2 $. The 'residual maker' matrix is $\boldsymbol I - \boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T$. Therefore, $$ \boldsymbol X_1^* = \left(\boldsymbol I -\boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T\right)\boldsymbol X_1.$$


Fix the production function to a constant value $f_0$ so that

$$
f_0 = 600k^2l^2 - k^3 l^3 \tag{1}
$$

This is a figure for different values of $f_0$ (isoquants)

Note that with increasing $k$ the rate of change of $l$ becomes smaller for a given $f$ and a fixed change in $k$. We can show this more formally by taking the derivative w.r.t $l$ at both sides

$$
0 = 600\left(2kl^2 + 2k^2 l \frac{\partial l}{\partial k}\right) - \left(3k^2l^3 + 3k^3l^2\frac{\partial l}{\partial k}\right) \tag{2}
$$

You can further simplify this to

$$
0 = -3k l (-400 + k l)\left(l + k \frac{\partial l}{\partial k}\right) \tag{3}
$$

The solutions are $k = 0$, $l = 0$, $kl = 400$ or

$$
\frac{\partial l}{\partial k} = -\frac{l}{k} \tag{4}
$$

Which confirms that that increasing $k$ decreases the MRTS, in other words, the range in which the marginal diminishes is $k > 0$


Good paper/article on the mechanisms in RBC vs New Keynesian models

If possible, clarifying what exactly you're most interested in might help answers be more on point and useful. Are you interested in the mechanisms that cause a one-period shock to last (and not just immediately dissipate)? Or are you interested in understanding some of the reasons that a shock to one variable (say technological progress) leads to changes in other variables (output, employment, wage, interest rates, etc.)?

In either case, the answer depends somewhat on what flavor of the model you're considering. The reasons that shocks tend to have persistent effects in the models comes from mechanically inserting persistence (by using an AR process for shocks) and/or the interplay between stocks and flows (the most common being capital accumulation and investment).

As for sources, I personally found that lecture notes from graduate macro classes were often some of the most useful resources for understanding internal mechanisms and reasoning (Eric Simms, for example, has a wonderful series of lecture notes that include some outlines of BC facts that models try to explain (including persistence), basic RBC model notes, notes examining extensions to the RBC framework, notes on investment in RBC models, in addition to NK notes and some explaining optimal monetary policy in NK models). In addition to Simms' resources, Ordonez has created a nice, simple set that explain the derivation, which might provide some insight. Additionally, slide 15 in this set describes one example of how propogation can work, though again, that's quite a limited example.

That said, there are several classic papers (such as Resuscitating Real Business Cycles especially section 4) which are useful to keep in mind. As for how others generate persistence, you can look at habit formation in consumption (here, though there are many others, of course), variable capital utilization, or any of the other listed methods on slides 40-46.

Though again, more info on what exactly you're looking for could help me refocus a bit. I hope these at least provide a start to what you're interested in!

EDIT: Given your interest within a single period, I think the key is to keep in mind what the "engine" is behind the RBC model- ultimately, it's all about households maximizing utility. They do this by producing (through the firm side of the RBC model) and by enjoying leisure. The specific relationship between these objects leads households to respond to changes in any single value.

For example, take a technology shock (say a positive one, so now more output can be generated from the same levels of input). There are two competing forces:

The substitution effect, which leads households to "realize" that, by increasing the number of labor hours they commit to, they can get even more output (some of which can be used for consumption, since "output" in this model isn't differentiated, and can therefore be used either for consumption or investment).

The income effect, which dictates that, since leisure is a normal good (as defined by the utility function exogenously plugged in to the model), higher consumption should lead to higher leisure as well.

Generally, models are "calibrated" such that hours worked and output both rise. So this suggests that households value the additional consumption enough so they choose to work more.

In RBC models with investment, positive technology shocks also mean that capital is more productive, which means interest rates rise.

I can write a more thorough overview from my grad macro notes later in the day, but until then, I'd review this set of Simms' notes, which do a decent job explaining where everything comes from, and why variables move the way they do. Alternatively, as this source notes: "Proposed by Kydland and Prescott, the RBC theory rests on the neoclassical concept of rational expectations and constructs on the basis of expected utility maximization. Therefore, it is important to understand the central assumption in the RBC theory: individuals and firms respond to economic events optimally all
the time." Additionally pages 10 onwards might be useful in understanding where this comes from, and this set of notes also takes more time detailing where things come from, and what they mean.

Let me know if those additional resources are helpful at all!


How can the Swiss exchange rate be very biased in a free market?

Swiss people do shop outside of their border, exactly as you think they should:

  "Even the Swiss are staying away from their local stores and products—they would rather buy from neighbouring countries and take advantage of the situation. Switzerland is bound by Germany, France and Italy, and most of its big cities are located just at the border. It is relatively easy for Swiss residents to do their shopping elsewhere and get high value for their money, but it also means that the Swiss supermarkets and downtown stores have lost huge shares of the consumer market." (Source: http://www.forbes.com/sites/realspin/2015/03/30/currency-wars-the-swiss-case/)

Currency is like any other commodity in the sense that its price (i.e. the exchange rate) is higher when demand is high relative to supply. For the exchange rate to remain high even when large sections of the population are selling Francs to get the Euros they need to shop in Germany, it must be that somebody else is buying up a lot of Francs and thus providing a large source of demand for the currency.

That someone else appears to be foreign investors, who are desperate to put their money into a safe and stable environment. In 2013, investors put 101bn Francs more into Switzerland than they took out (an amount equal to 22% of Switzerland's GDP!). In 2014, this figure fell dramatically, but there was still a net inflow equating to 7.5% of GDP.

On top of investors putting money into the country, Swiss companies are also net exporters. In 2013, Switzerland exported 65bn CHF more of goods and services than they imported–15% of GDP. Every time a Swiss company sells a product to someone in the US, dollars need to be changed into Francs (either because the Swiss company sets prices in Francs, or because it gets paid in dollars and then wants to repatriate the money in domestic currency).

These two sources of money flowing into Switzerland (respectively known as the capital account and the current account, and collectively known as the balance of payments) provide a robust source of demand for Swiss Francs and ensure that the value of the currency remains high.

The source for the numbers quoted above is here: http://snbchf.com/euro-zone-swiss-macro/swiss-balance-payments-2014/


need help from theorists: proof in Cole, Mailath, and Postlewaite (2001)

This can be achieved with L'Hopital's rule, which says
$$\lim_{x\rightarrow0}\frac{f(x)}{g(x)}=\lim_{x\rightarrow0}\frac{f'(x)}{g'(x)}.$$

The result from Cole, Mailath, and Postlewaite holds fairly generally and is not special to the particular functions in their integrand.

Let
$$f(\epsilon)=\int_{l-\epsilon}^{l+\epsilon}h(x,\epsilon)dx \iff f'(\epsilon)=h(l-\epsilon,\epsilon)+h(l+\epsilon,\epsilon)+\int_{l-\epsilon}^{l+\epsilon}\frac{\partial h(x,\epsilon)}{\partial \epsilon}dx$$

and
$$g(\epsilon)=2\epsilon\iff g'(\epsilon)=2.$$

Then, using L'Hopital's rule:
$$\lim_{\epsilon\rightarrow0}\frac{f(\epsilon)}{g(\epsilon)}=\lim_{\epsilon\rightarrow0}\frac{f'(\epsilon)}{g'(\epsilon)}=\frac{h(l,0)+h(l,0)}{2}=h(l,0).$$

A final word on 'intuition'. You reason, correctly, that the integral goes to zero with $\epsilon$. But the problem is that the denominator, $2\epsilon$ also goes to zero. So, to know what the overall expression does as $\epsilon$ approaches zero, we need to know whether the numerator or the denominator approaches zero faster. In other words, for $\epsilon\rightarrow0$ are we dividing a small number by a very very small number, or are we dividing a very very small number by a small number? This, roughly speaking, is the logic behind L'Hopital's rule.


How to calculate the risk-free rate of cryptocurrencies?

Instead of looking at bitcoin like a currency, I think you should look at it like a regular stock. The "return" is the return of your fiat money (USD) that's investing in Bitcoin... so the risk free rate should be the risk free rate of your money, i.e. T-bill, etc.


Microeconomics - Expected Utility Theory - Piecewise utility index, certainty equivalence, etc.

Suppose random variable $X\sim F$.

Certainty equivalent of the lottery $X$ is the constant $c$ that solves:

  $u(c) = \mathbb{E}(u(X))$

Because $u(x) = \min(\sqrt{x}, \sqrt{M})$ is concave, by Jensen's inequality:

  $u(c) = \mathbb{E}(u(X)) \leq u(\mathbb{E}(X)) $

Agent is risk averse because $u$ is concave.
If agent prefers lottery $X\sim F$ to lottery $Y\sim G$, it is not necessary that $F$ FOSD $G$. Consider the following two lotteries:

  $X = \frac{M}{2}$ with probability 1.

  $Y = M$ with probability $\frac{1}{2}$ and $Y = 0$ with probability $\frac{1}{2}$.

Since $u$ is concave, lottery $X$ is preferred over lottery $Y$. This because $X$ gives the expected value of the lottery $Y$ with probability 1. Also, $F$ does not FOSD $G$ because $F\left(\frac{M}{2}\right) = 1 > \frac{1}{2} = G\left(\frac{M}{2}\right) $.


Comparative Methods of Econometric Analysis

In econometrics and statistics, you sometimes will truncate outliers. Otherwise, maybe it's worth mentioning that in the business of predicting, analysts are punished for either being wrong, but their punished even more if they were boldly wrong. This is a form of herding behavior. Deviating from the consensus can be disproportionately risky.

Here's a paper that talks about it: "Security analysts’ career concerns and
herding of earnings forecasts" in the RAND Journal of Economics.


To sum up the 2008 crisis in a few bullets:

Loose lending policies by government and private institution caused regular (and poor) folk to become real estate speculators.
Housing prices declined rapidly, causing foreclosures.  This caused banks and investors to lose money and put the housing/construction industry on hold, causing unemployment.
Many mortgage-backed securities were based on the value of real estate and their risks had not been taken into account by investors, who were unprepared for losses.
Financial institutions were highly levered and tightly connected with each other by contracts and derivatives that lacked transparency, causing a crisis of faith in these institutions.
Government bailouts were reasonably successful in keeping the problem contained but at great cost and setting a questionable precedent.
Since then the financial system has recovered and done very well.

The low real interest rates you mention isn't a problem with the banking system, it's a simple problem with too many parties (globally) wanting to save and not enough wanting or being able to borrow.  The latter effect is partly because of government and bank reluctance repeat the same mistakes with loose lending and partly because we don't have a lot of good real investments available to us.

  Some say "housing crash" but I suspect the existing global banking system imploded. Am I off-base, or ...?

Yes, you are off base.  The banking system, and financial system in general, temporarily lost a lot of value and suffered from a credit crunch but have since recovered and are doing very well.

  As I calculated back then, TARP and TARPII literally provided enough capital to purchase the entire US stock market via futures contracts.

Posting margin to take a large position in futures contracts is very different from "purchasing" the stock market. I don't understand how this is related to your question, though.  Using bailout funds to speculate on the market seems like an odd idea.  Note that TARP funds were loans, which have been repaid and made the government a tidy profit.

  Did the global banking system collapse in 2008, or was it simply a housing/mortgage issue?

Again, it didn't collapse, but it did suffer.  The housing situation was the precipitating event but there were a number of problems that came to fruition at the same time.

  Lastly, if it did, my understanding is that like 99% of $100 bills notes are held outside of the USA. Why didn't the Fed simply declare bankruptcy, void all those notes (and all those complex derivative contracts), and start over with new money? Why the fake? The sun's going to come up tomorrow regardless...?

Why would the fed want to declare bankruptcy? If it tried to void our currency there would be negative global economic and political repercussions the likes of which we have never seen.  I cannot imagine this happening nor why you would think it would be a good thing.

Is there a reason you think voiding derivative contracts would be a good thing?  Those are contracts parties voluntarily engaged in, many of which are necessary to provide stability to both or at least one participant.  During the crisis the fact that big banks were involved in so many derivatives undermined their credit-worthiness because people couldn't be certain how much risk they had.  But that is a very, very far cry from saying that it would be a good idea to void derivatives, or even certain classes of derivatives.


How will the UK's economy be affected, if it will be a hard Brexit?

The UK government published "Implications for Business and Trade of a No Deal Exit on 29 March 2019" on Tuesday.


Log-Linearizing a Dixit-Stiglitz function

Thanks for the hint and the link! I think I now managed to find the solution.

Putting the exponent on the LHS and replacing

$C_t^{1-\gamma}$ with $C+(1-\gamma) C^{1-\gamma}C \tilde{c}$

and

$C_t(i)^{1-\gamma}$ with $C(i)+(1-\gamma)C(i)^{1-\gamma}\tilde{c}_t(i)$

i get (subtracting Steady State values):

$(1-\gamma) C^{1-\gamma}C \tilde{c}=\int_0^1 (1-\gamma)C(i)^{1-\gamma}\tilde{c}_t(i) \ di$

Assuming steady state values of $C(i)$ are constant across $i$, i can take all values out of the integral and simplify to get:

$\tilde{c}=\int_0^1 \tilde{c}_t(i) \ di$


How to find the present value of a perpetuity with a non-constant growth rate?

Basically you want to calculate
$$
\sum_{t=0}^{\infty} a_t \cdot d^t
$$
where $d$ is the discount factor and where the value paid yearly increases linearly, that is
$$
a_t = a_0 + b \cdot t.
$$
With some rearrangements
$$
\sum_{t=0}^{\infty} a_t \cdot d^t = \sum_{t=0}^{\infty} a_0 \cdot d^t + \sum_{t=0}^{\infty} b \cdot t \cdot d^t.
$$
The first half of this is a simple geometric sequence like the present value of a perpetuity, so
$$
\sum_{t=0}^{\infty} a_0 \cdot d^t = \frac{a_0}{1-d}.
$$
To calculate the other sum we'll use a trick.
$$
\sum_{t=0}^{\infty} b \cdot t \cdot d^t = b \cdot \sum_{t=1}^{\infty} t \cdot d^t.
$$
Then write
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & d + 2\cdot d^2 + 3\cdot d^3 + 4\cdot d^4 + ...
\end{eqnarray*}
You can rearrange this (because of absolute convergence) to
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & d + 1\cdot d^2 + 1\cdot d^3 + 1\cdot d^4 + ... \\
\\
& & \hskip 11pt + 1\cdot d^2 + 1\cdot d^3 + 1\cdot d^4 + ...\\
\\
& & \hskip 45pt + 1\cdot d^3 + 1\cdot d^4 + ... \\
\\
& & ...
\end{eqnarray*}
Every line in this is a geometric sequence, so
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & \hskip 7pt \frac{d}{1-d} \\
\\
& & + \frac{d^2}{1-d} \\
\\
& & + \frac{d^3}{1-d} \\
\\
& & ...
\end{eqnarray*}
Which is again a geometric sequence, thus
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & \frac{d}{(1-d)^2}.
\end{eqnarray*}
So
$$
\sum_{t=0}^{\infty} a_t \cdot d^t =
\sum_{t=0}^{\infty} a_0 \cdot d^t + \sum_{t=0}^{\infty} b \cdot t \cdot d^t =
\frac{a_0}{1-d} + \frac{b \cdot d}{(1-d)^2}.
$$


Dynamic optimization with assets as state variable: interpreting capital gains and losses

Firstly, the $\lambda$ is by definition the shadow price of the wealth/capital. It is the price when you increase one unit of additional capital. $\lambda$ measures the rate of increase of the maximum value of the utility function as the constraint is relaxed. So, the dynamic equation of $\lambda$ is the change on this variable, which intuitively is the capital gain.

What do you mean by $(a_{t},H_{t})$ diagram ? Do you mean a phase diagram on $(a_{t},c_{t})$ ?

For your last question, you can realize on your constraint that the consumption increases at the expense of capital gain (c.f $\dot{a}$.) So, this means that every unit of consumption instead of accumulation is a somewhat loss in capital accumulation.


It is. And it is not.

Electricity markets are generally not set up for the demand-side to do much active participation at all. So the short-run demand curve as seen in, for example, Nordpool Spot, is almost perfectly inelastic. Not quite, because there are some large industrial demands that exhibit some elasticity, and are exposed to the spot market. But a very large chunk of demand isn't exposed to the spot market at all; and a lot of the demand that is exposed, is highly inelastic - particularly to half-hour-by-half-hour changes.

That tells us a lot about the structure of the markets, and only a little about the actual underlying demand curve.

Until there's a lot of demand-side response (DSR), we've only got stated-preference studies and DSR pilots to go on, to estimate what the demand curve would look like in a well-functioning market: the evidence we have (e.g. from the Olympic Peninsula study), suggests there's quite a lot of potential demand elasticity out there, just waiting to be harnessed. Note that when short-run demand elasticity has been observed, it manifests in two ways: some demand will move between half-hours that aren't too far apart, when the price differentials incentivise it. Some other demand will just vanish altogether when prices rise.

If you want to try to distinguish the elasticity signal from all the noise in the existing market, then you need to take demand and prices for a particular half-hour slot on Tuesdays-Thursdays in one particular season across several years; and look at how weather-adjusted demand varies with price, while adjusting for any long-term trends.


Yield curve flattening- abstract question

If an investor has a fixed amount to invest for a given time horizon, then a longer-term bond locks in the current long-term bond interest rate, but a short-term bond only provides this rate for a short time. After that, the short-term interest rate may be more or less favourable. Apparently the investors expect that the short-term rate will become less favourable for them after the current short-term bonds mature.

Investment is mostly about expectations, which may or may not be correct. Where the expectations come from is a more complicated question.


Is a small business owner without employees a capitalist?

(Bowing to peer pressure) According to original Marxist theory, no, he is not. Here, the defining characteristic of a capitalist is that he appropriates (as "profits") the surplus value that is created during the production process by other people's labor (capital does not produce surplus value).

Your freelancer or small business owner without employees appropriates only their own surplus value, created by their own labor. And it doesn't matter how large is their income, as long as they make it without employing other people as their employees.

According to some other theory? Well, somebody name the theory, and then we can clarify who is a capitalist and who is not, according to that other theory. Otherwise, it will be too much "personal opinion-based".


The sequence of Money Supply - Interest Rate

It’s just supply and demand. The interest rate is the price at which you can borrow money. For a fixed demand curve, if there’s less money available (the supply curve shifts left), the market will clear at a higher price (higher interest rate). Similarly, if there’s more money available for lending, the equilibrium interest rate will be lower.


What happens to the billions in Euros that Greece owes if it leaves the Euro

If you owe euros you have to pay in euros. If for example the British government owes euros, then they get some funds (perhaps take another loan) and in case this is in a currency different from euros they convert it to euros on the market. If the sum is large enough it is wise not to convert it all at once but in smaller installments. This way the exchange rate stays stable and you can also get a better deal.

So nothing will happen to Greece's debt because they leave the Eurozone. But as the whole issue is political, there could be debt relief/partial bankruptcy because of the same reasons that might cause them to leave the Eurozone.

About printing drachmas:
There is a technical issue to be addressed: It is not the government that prints money but the central bank and their goal is maintaining price stability, not providing the government with revenue. (In time of crisis this may change.)

Printing money causes inflation which in turn will weaken the drachma relative to the euro. Usually when the state prints drachmas all drachmas would be worth less, but states wealth may increase because their share of drachmas increases.
However, this is a special case because there are no drachmas at all. So initially all drachmas would be owned by the state. Hence I think if they print more the worsening exchange rate would balance it out.
What they could do would be to print some drachmas, exchange it with whomever was willing for euros, and then print more. This would only work if printing more is not expected.


What's the difference between equity capital and market capitalization?

Equity capital = Assets − Liabilities.

Market capitalization = Number of shares outstanding × Current share price.


What's the difference between Quantitative Easing and regular central bank operations?

In usual open market operations, the central bank will create money, use this money to buy short term Treasury securities from maybe banks, individuals, institutions in the open market. This creates more demand for these securities. Hence their price goes up and yield goes down. This newly generated money goes into the banking system. With more money, banks have more cash and have more reserves. With more reserves banks may be more willing to lend to each other and to individuals also. This lowers the short term inter banking interest rate. This way, there is more money in the system and there is a downward pressure on rates. With short term Treasuries, central bank is less exposed to the interest rate risk.

In quantitative easing (QE), both the purpose and the mechanism are different. First of all, QE usually will be performed in a situation which is not 'typical' money supply shortage or economic activity slowdown. This will either happen in a crisis where aggressive boost is required or when central bank has run out of its primary weapons. Maybe it already lowered interest rates to zero.   The central bank in these situations might buy securities of different kinds which can be long term treasuries, private securities, securities in a particular area of market which, the central bank thinks, needs to be revived. This certainly will not just increase money into the system, but further lower the yield and ease out specific markets such as mortgage based securities by reducing the risk spread.

One other criterion is the size of the increase in balance sheet of central bank. In QE there is usually a much larger change in the balance sheet of central bank. Also, as there are different kinds of securities (including long term securities) in the assets of central bank which are not necessarily treasuries or highly rated securities, the central bank itself is exposed to more risk.

Ben Bernanke said in 2009 that, the QE they exercised was not pure QE but credit easing, because in pure QE, the central bank will not discriminate between what securities it is buying. However, according to him, Fed did target few specific markets. Targeting few specific markets is more like credit easing and not pure QE.


With inflation so low, why is it so hard to stimulate the economy?

It's not theoretically hard to stimulate the economy. It's just the means to do so are terribly unfashionable. And that means it's politically hard.

We've had two similar depressions in the last couple of centuries: the Long Depression in the second half of the nineteenth century, and the Great Depression in the first half of the twentieth. Each saw significant monetary contraction as a policy response. Which was astonishingly daft.

But after the Great Depression, we also learnt that monetary expansion was insufficient when there's a large debt hangover and demand is suppressed: hence we got the famous analogy of monetary expansion being like pushing on a piece of string.

For many centuries, a once-in-a-human-lifetime jubilee - a cancellation of all personal debts - was seen a way to cure these huge debt hangovers. And they do, broadly speaking, provide a means of cutting the closed circle of monetary expansion which takes place through savings and loans. These jubilees used to happen once every fifty years - and that was, roughly speaking, a human lifetime. A jubilee removes all of the demand-suppression effect that personal debt has: it removes wealth from those with the lowest propensity to spend (thus has a small negative effect on demand), and increases the wealth of those with the highest propensity to spend (which has a much larger positive effect on demand). The net result is a return to increasing demand. In some ways, the bailing out of the banks in 2008 was the very opposite of a jubilee: commercial borrowers (bank bond-holders) were bailed out, and private borrowers were left hanging. Expensive and ineffective. A jubilee cancelled all private debt, but did not touch commercial debt.

The other very unfashionable intervention is fiscal expansion. Keynesian expansions were a successful response to a demand-led depresssion, when the multiplier was higher (as it is now: even Lagarde of the IMF admitted this); but they also got used in subsequent decades in other circumstances too, when the multiplier was lower, and showed less success. And that's when they dropped out of favour, and remain so in many countries now, despite them being part of the right thing to do in the current particular circumstances.


Can tariffs be said to subsidise domestic industry?

This is how I read your question:

If a state places a tariff on a good,             <- added a comma here
domestic manufacturers of that good become
more price-competitive in the domestic market
at the expense of overseas suppliers.

Could this be said to be a policy choice that subsidies that industry?

Assuming the country is a small and could not affect world price, then the answer is yes. To be more precise, an import tariff is equivalent to product-specific subsidy PLUS a consumption tax of that product.

Subsidy + consumption tax: Domestic producers gain the subsidized amount for each unit sold. Buyers pay for that extra amount the producers gain.

Tariff: The price of import goods (world price) rises, so domestic producers can sell at a higher price. The buyers are the ones who pay for that price difference.

"Credible" sources:

http://internationalecon.com/Trade/Tch95/T95-3.php
https://www.cairn.info/revue-reflets-et-perspectives-de-la-vie-economique-2004-1-page-25.htm, Section 4


If Trickle Down Economics Works, Then Shouldn't "Trickle Up" Economics Work as Well?

As you learn more about economics you should gradually come to distinguish the vocabulary of economics from that of politics (which is also not the same as political science). When politicians or political activists speak, they use a variety of terms that are very vaguely defined, and sometimes outright derogatory. They are not actually meant to convey exact economic concepts or ideas, but more like feelings. "Trickle down economics" is an example. "Neo-liberism" is another example. You won't find any formally trained economists discussing "trickle down economics" or "neo-liberist" policies simply because these are not properly defined concepts, i.e. they don't really mean anything specific. In order to discuss productively with economists and generally any scholar from a discipline that is based on the scientific method (biology, physics, statistics, etc.), it is important to acquire the right vocabulary. The dictionary or an encyclopedia is usually a good place to start.

P.S. it is not a coincidence that controversy tends to stir around topics that are not discussed in scientific terms - it is difficult for people to agree on ideas if they can't articulate them :)


When does the law of diminishing returns apply?

Law of diminishing returns refers to profits or benefit per unit of output. This will inevitably decline, profits are defined as total revenue minus costs $\pi=TR-TC$ are thought to be increasing with production but at declining rate, this can be both due to the revenue and cost part.

Costs can in principle be increasing, constant or decreasing with scale. In the first case one source of diminishing returns would be already increasing costs. In the latter case we need to examine the revenue part, but note that most economists think that even if some businesses have constant or decreasing costs along a large part of cost curve these should eventually start increasing as eventually the demand of firm for inputs would be so great to lead to higher and higher increases in the price of those inputs. If you already hired all snow collectors you need to start hiring doctors, programmers etc. to pick snow. So at some point every cost curve should be increasing due to scarcity of inputs.

Now when it comes to the total revenue part that will also be increasing in quantity but at declining rate. The reason for this is that the market price of your good is not set in stone. Increase supply and the price will drop. All people have declining marginal utility of consumption so at some point if you supply too many units to the market the price will drop. Even if you would have constant costs or decreasing costs over the whole cost curve eventually the negative change of price due to larger quantity will be larger than any cost reduction due to decreasing costs.

So in short the law of diminishing returns for firms works both because the more you consume of something the less you desire it so demand is downward sloping and because costs should increase with production at some point along the cost curve.


What are the main drivers of insolvency at a macro level?

Regarding the Index

I want to further understand how they compute this index. Unfortunately, they give very few details on this, probably because this index is a proprietary tool and they want to protect their IP

Their index is actually open access, you can access the data here and the index is based on evolution of insolvencies with base year being 2000.
They also provide data on the evolution of insolvencies but I can't see anywhere any methodology of how the evolution of insolvencies was itself created (which is a red flag if you ask me). Based on the way how data look it seems it is the percentage change in in how many insolvencies occurred. So that would imply they have access to data on number of insolvencies which is also corroborated by the their references to number of filled insolvencies within the report.
However, the report includes author will full contact information, so I would hope that if you would reach out they would share their methodology with you.
Regarding Factors affecting Insolvency:
I am not completely sure what you mean by explaining insolvency at a macro level. Any macroeconomic aggregate of insolvencies will be based on individual insolvencies driven primarily by microeconomic factors - but they would still affect any aggregate of insolvency, so I suppose they would affect insolvency on 'macro level', but I assume you wanted only macroeconomic variables.
According to Sang et al. 2013 examining impact of macroeconomic variables affecting insolvencies in construction industry in South Korea the authors show that insolvencies can be affected by:

changes in GDP (especially negative changes; recessions)
interest rates
consumer price index
corporate bond yields
exchange rate (this is most likely due to S. Korea being small open economy)

In addition other studies (e.g. Bonfim, 2009; Bhattacharjee, et al. 2009 and sources cited therein) generally find that what is driving firm insolvency from macroeconomic is general macroeconomic instability (in fact many of the above listed variables can be considered proxies for macroeconomic stability e.g. change in GDP, inflation etc.) and also to non-trivial degree country specific institutional framework (which is not surprising since true economic insolvency is difficult to observe so often firm is considered insolvent when it is declared insolvent by law).  However, the above answer is quite general the drivers can actually differ by industry as studies also show competition is important factor. Furthermore, the insolvencies in financial sector and real economy are not necessary affected by the same factors, but to go further in depth within a single answer on these issues is beyond the scope of SE.


Showing marginal product of capital is independent of the scale of production

There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).

To compute the MPK, we must differentiate the production function with respect to the current level of capital: $\partial F/\partial K$.

But in your final line, you are not differentiating with respect to the current level of capital (which is $\tilde{K}=\lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/\lambda$ of the current amount of capital.

If we compute the derivative with respect to $\tilde{K}\equiv \lambda K$ instead of just $K$ then everything works as it should:

$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$

This does not depend on $\lambda$ so MPK is indeed independent of the scale of the economy. QED


Shares, and many other financial instruments as well, are financial assets  assets, through which people save not consumption of goods and services. Consequently, neither GST (goods and services tax) or value added tax (which is in economics a type of consumption tax) can apply to stocks or many other financial instruments as they do not represent consumption of either goods nor services. Hence prima facie there is no reason why consumption taxes should apply to things that are neither consumption or goods or services in the first place. That is like asking why houses are exempt from labor taxes... well its because they are capital not labor.
For example, if you pay someone to make trades for you (and hence someone to provide you financial service), you will generally be required to pay VAT or GST on that (unless particular country decided to exempt it for whatever political or legal reason - which is beyond the professional competency of an economist to comment on).

Response to edit:
It is feasible to tax financial assets but this would be considered wealth taxation (completely different type of taxation than consumption taxes). Taxing wealth is feasible in a sense government can pass legislation to actually do it, but from economic perspective it is often considered undesirable for its negative impact on the economy.
The literature shows that optimal top marginal wealth taxes, even in case of Rawlsian social utility function (the most redistributive social function there is), are somewhere in range 0-10% (see Kocherlakota 2005; Fama 2019; Zucman and Saez 2019 or this article by Summers) or discussion in the famous Mirrlees Review: Dimensions of Tax Design). Also, note the consensus seems to still be closer to the zero result and only very few scholars would go as high as 10%.
The above being said governments typically do not conduct science based policies. Often politicians will ignore even recommendation of their own analysts when it comes to the design of tax policy. Consequently, it is beyond scope of economic profession to tell you why politicians do not base all their laws on best published science/state of the art models or why they do tax one thing but not another (for example, in the Netherlands where I live for long time food for hamsters and bunnies was taxed at different rate even though they are perfect substitutes - which is completely absurd from any economic perspective, yet that is what the government legislated).


Isn't the parameter A denoting technological progress from Cobb-Douglas already included in alpha and beta?

The word technology might have slightly different meaning depending on a field. For example, Varian in his Microeconomic analysis (3ed) calls the whole production function technology. He literally writes on a first page of his book:

The simplest and most way to describe technology of a firm is the production function

However, in macroeconomics it is often equated to the $A$ parameter (see Romer Advanced Macroeconomics chapter 1 where technology growth and growth of $A$ for Cobb-Douglass function is used interchangeably throughout the chapter and also elsewhere in the book).
In the broader (Varian Meaning) you can consider alpha and beta parameters to be part of technology or determined by available  technology (since whole production function is by definition in the Varian meaning determined by technology) but they are not technologies themselves per se (just their component). Rather they are output elasticities and in addition their sum tells you if there are increasing, decreasing or constant returns to scale, but technological progress does not necessarily mean that firms production will have different returns to scale or that output elasticity increases or changes in any way. Also, note these are virtually always considered to be constants.
In the macroeconomic sense they are not technology and the technology would be solely captured by $A$. In macro sense the $L^{\alpha}K^{\beta}$ would be some sort of ‘vanilla’ production which is then augmented by technology parameter $A$ (which could by the way also enter in a way that would augment only labor ($(AL)^{\alpha}K^{\beta}$) or only capital ($L^{\alpha}(AK)^{\beta}$) or also both ($AL^{\alpha}K^{\beta}$)).
So depending on which subfield you look at the terminology will differ, but I can’t recall ever seeing in the literature someone claiming that alpha and beta are the technology, at best you can argue they are determined by technology. Also depending on which field you specialize in you will want to follow the preferred terminology in the literature.


Intuitive explanation might help.

(i) $E[u]=0$ vs $E[u|x]=0$: Imagine partitioning the population by the value of $x$ so that each slice of population has the same value of $x$ in it. You can then get the mean of $u$ for each slice. "$E[u|x]=0$", which is a shortcut notation of "$E[u|x=c]=0$ for (almost) all $c$", means that the average of $u$ in every partition is zero. You can also get the mean of $u$ for the whole population. "$E[u]=0$" means that the average of $u$ for the whole population is zero.

For example, let $x$ be years of education. $E[u|x]=0$ means that the average of $u$ for those with 9 years of education is zero, the average of $u$ for those with 10 years of education is zero, etc., while $E[u]=0$ means that the average of $u$ for the whole population is zero.

You see that "$E[u|x]=0$" implies some sort of "$u$ being not related with $x$", and it's called mean independence. On the other hand, $E[u]=0$ does not say anything about the relationship between $u$ and $x$. $E[u]=0$ makes sense without $x$ ever being involved, while $E[u|x]=0$ makes sense only in relation to $x$.

(ii) $var(u)=\sigma^2$ vs $var(u|x)=\sigma^2$: Imagine partitioning the population by the value of $x$. You can get the variance of $u$ for every slice. '$var(u|x)=\sigma^2$' means the variance of $u$ for every partition is $\sigma^2$. Here, the key point is that $\sigma^2$ is a constant and is not involved with $x$. It means that all the slices have the same variance. $var(u|x)=\sigma^2$ is very informative. Next, you can also get the variance of $u$ for the whole population, and $var(u)=\sigma^2$ means that that variance (of $u$ for the whole population) is denoted $\sigma^2$, where $\sigma^2$ is just a notation.

$var(u|x)$ can depend on $x$ (heteroskedasticity), but it is totally nonsense to question whether $var(u)$ depends on $x$ because $var(u)$ has nothing to do with $x$ from the beginning (unless you mean $var(u|x)$ by $var(u)$).

(iii) $cov(x,u)=0$: This just means that $cov(x,u)=0$, where $cov(a,b)$ is defined as $E[(a-Ea)(b-Eb)]$. In your case, $cov(x,u)=E[xu]$ because $E[u]=0$. If you need intuition about the meaning of $cov(x,u)=0$, imagine you have $(x,u)$ values plotted on the XY plane for the whole population, with $x$ on the horizontal axis and $u$ on the vertical axis. You draw a nice straight line (where being straight is important). $cov(x,u)=0$ means that the straight line is horizontal. It says something about $u$ and $x$ being not related, and when it happens, we say that "$x$ and $u$ are uncorrelated."

$E[u|x]=0$ implies $cov(x,u)=0$ but not vice versa, which is the reason why $cov(x,u)$ is weaker than $E[u|x]=0$. You can prove it using the law of iterated expectations: $E[xu] = E[xE(u|x)] = E[x\cdot 0]=E[0]=0$. For a counter example for the other way around, suppose that $x\sim N(0,1)$ and $u=x^2-1$. Then $E[u|x]=x^2-1$, which is not zero except $x=\pm 1$ but $cov(x,u) = E[x(x^2-1)] = E[x^3]-E[x]=0-0=0$.

(iv) What is the difference between $cov(y,x)$ and $cov(u,x)$? If $y=\beta_0 + \beta_1 x +u$, $cov(y,x) = cov(\beta_0+\beta_1 x+u,x) = \beta_1 cov(x,x) + cov(u,x)$, where $cov(x,x) = var(x)$ by definition. That's the difference.

ADD:

I've just noticed the OP's confusion about $cov(x_i,u_i)$ and $cov(x,u)$. First, $cov(x,u)$ can be intuitively understood as the population property explained above in (iii). $cov(x,u)$ is about the population, and does not say anything about the sample. The notation $cov(x_i,u_i)$ is on the other hand about the sample, and should in fact mean something like "$cov(x_i,u_i)$, $i=1,\ldots,n$", i.e., $cov(x_1,u_1)$, $cov(x_2,u_2)$, ..., and $cov(x_n,u_n)$. They are all the same if the first person, the second person, ..., the $n$th person are independent random draws from the same population (which is the meaning of $iid$). If they are drawn from different populations, $cov(x_i, u_i)$ may be different for different $i$, but I am pretty sure you are assuming $iid$.

Now, what is $cov(x_1,u_1)$ then? This is the hard part. To understand its meaning, you should understand that $(x_1,u_1)$ is a random vector, whose value can change when you repeat sampling in your thought experiments. (This will keep confusing you until you understand this point.) The "first person" in the sample will keep changing when you repeat drawing the sample over and over again in your mind, and thus the $(x_1,u_1)$ value will keep changing over repeated samples. So, as you repeat sampling indefinitely, the values of $x_1$ and $u_1$ will change and make a (joint) distribution. $cov(x_1,u_1)$ is the covariance of that joint distribution. Likewise, you can understand $cov(x_2,u_2)$ as the covariance of $x_2$ and $u_2$ over the repeated samples. If you keep drawing the first observation independently from the same population as you talk about $cov(x,u)$, then $cov(x_1,u_1) = cov(x,u)$.

So when we talk about the population property, we use the notation $cov(x,u)$. When we talk about the covariance of $x_1$ and $u_1$ (over repeated samples), we use the notation $cov(x_1,u_1)$. The notation $cov(x_i,u_i)$ is a (sloppy) shortcut notation of $cov(x_1,u_1), \ldots, cov(x_n,u_n)$. "$cov(x_i,u_i)$" itself does not make much sense until you say what $i$ is, like in "$cov(x_i,u_i)$ is zero for every $i=1,\ldots,n$", or "$cov(x_i,u_i)$ is nonzero for every $i=1,\ldots,n$ but they are the same for all $i$." The statement "$cov(x_i,u_i)=0$" is usually a lazy (or clumsy) writing of "$cov(x_i,u_i)=0$ for all $i=1,\ldots,n$", which means $cov(x_1,u_1)=0$, $cov(x_2,u_2)=0$, ..., $cov(x_n,u_n)=0$.


separable utility function and cross price effect

The following argument assumes that we are dealing only with interior solutions. Let $(x_1,x_2,x_3)$ be an optimal demand bundle at prices $(p_1,p_2,p_3)$ and income $m$ and assume that $(x_1,x_2,x_3)$ is an optimal demand bundle at prices $(p_1',p_2,p_3)$ and income $m$. We want to show that one of the following three cases holds:

$x_2=x_2'$ and $x_3=x_3'$,
$x_2<x_2'$ and $x_3<x_3'$, or
$x_2>x_2'$ and $x_3>x_3'$.

What is not allowed to happen is that $x_2\leq x_2'$ and $x_3>x_3'$ or $x_2>x_2'$ and $x_3\leq x_3'$. Assume one of these cases happens, without loss of generality, we can assume it is the first.

By the strict concavity of $f_2$ and $f_3$, we have
$$MU_2(x_1,x_2,x_3)=\frac{\partial f_2(x_2)}{\partial x_2}\geq \frac{\partial f_2(x_2')}{\partial x_2}=MU_2(x_1',x_2',x_3')$$
and
$$MU_3(x_1,x_2,x_3)=\frac{\partial f_3(x_3)}{\partial x_3}< \frac{\partial f_3(x_3')}{\partial x_3}=MU_3(x_1',x_2',x_3').$$
Therefore,
$$\frac{MU_2(x_1',x_2',x_3')}{p_2}\leq\frac{MU_2(x_1,x_2,x_3)}{p_2}=\frac{MU_3(x_1,x_2,x_3)}{p_3}<\frac{MU_3(x_1',x_2',x_3')}{p_3},$$
in contradiction to the equimarginal principle.


Will a Guaranteed minimum income not eventually just be crowded out by inflation?

I will not discuss fairness (for example employers gains versus employees gains and bargaining power) principles as you did not ask about those.
Is "crowding out" the right phrase here?
Even if inflation lessens the effect of a minimal income there may still be an effect. Imagine that there are two people, $A$ with an income of 0 $B$ with an income of 60, so total income is 60. The government issues a minimal income of 30 so $A$'s income is changed to 30, total income is changed to 90. If this has no real effect on the economy then all prices rise accordingly, that is they rise by 50% because that is by how much nominal incomes haved changed. However if you adjust for this 150% price level, $A$ still can buy goods in value of $\frac{30}{1.5} = 20$ while $B$ can buy goods in value of $\frac{60}{1.5} = 40$. So there could at least be a redistribution effect as previously $A$ could not buy anything and $B$ could buy more things.
One concern is that inflation may not be even. Only the income of poor people will be raised hence demands for goods poor people buy will rise but demand for goods consumed by rich folks will not change. As a result the poor may experience larger inflation than the well to do, making the redistribution of incomes one between only poor people. The lines between poor and well to do here are murky. If (due to inflation) a pop-tart costs \$10 then people will buy quality food unless it is even more expensive. If a pop-tart's price rises to \$1,000 people will buy caviar unless its price is also raised.
A hope of governments implementing such policies is that (aside from fairness and solidarity principles) a minimal wage will also help minimum wage earners be more financially secure. Some studies have shown that in extreme poverty you make worse decisions and are less productive simply as a result of stress. If this is indeed a main effect then a minimum wage (or a higher minimum wage) can increase overall productivity. Thus inflation will not be the only effect of the policy but also an increased number of goods will be produced.


When is an OLS parameter unchanged on a subsample?

I think I figured out 1). I am still interested in 2).

OLS sets $b_0$ and $b_1$ in such a way that

$$
(b_0 \ \  b_1) = \arg\min_{a_0, a_1} \sum_j \left(Y_j - a_0 - a_1 X_j\right)^2.
$$
The first order conditions are
$$
(-2)\sum_j \left(Y_j - b_0 - b_1 X_j\right) = 0
$$
and
$$
(-2)\sum_j X_j \left(Y_j - b_0 - b_1 X_j\right) = 0.
$$
If observation $i$ is on the regression line then
$$
\begin{align*}
\left(Y_i - b_0 - b_1 X_i\right) & = 0 \\
\\
X_i\left(Y_i - b_0 - b_1 X_i\right) & = 0.
\end{align*}
$$
It follows from this and the original first order conditions that
$$
\begin{align*}
(-2)\sum_{j\neq i} \left(Y_j - b_0 - b_1 X_j\right) & = 0 \\
\\
(-2)\sum_{j\neq i} X_j\left(Y_j - b_0 - b_1 X_j\right) & = 0,
\end{align*}
$$
which are the first order conditions of the subsample's OLS problem.


How would you calculate the Price Elasticity of Demand if the Prices are constant, given that there is change in quantity demanded?

it does not make sense to use the formula for the elasticity like this.The PED is used for theoretical purposes in static equilibrium analisys. If you want to measure the elasticity empirically, then you must use the correct econometric technique, given your dataset.

One simple way is to run a linear regression (OLS) after a log-transform in your data. The parameter estimated captures de per cent sensibility of you endougeneos variable (demand) to your explicative varible (price).

Don´t forget to check if the parameter statistical significance, you may need more data.


Why would investment banks not care about their risky exposure to subprime mortgages?

The movie over-simplifies the large banks' attitudes somewhat. In reality, they differed:

some banks limited their exposure to junk debt (I will use this term because there was junk other than subprime mortgages involved). They'd buy up a little junk, repackage it into new instruments, and re-sell their instruments to clients. So their own exposure to the junk was limited, which is commendable. Their losses came a little later, when the clients who bought the repackaged junk sued them for billions of dollars and eventually won.

some banks were buying up more junk than they could use right away for repackaging - holding an inventory, intending to re-sell it eventually, and hoping to earn more money by holding the high-yielding junk meanwhile. This bet did not go well. (And they got sued too.)

a couple of banks thought they did not have junk exposure, but they were mistaken. One very prominent bank CEO went on TV to tell the public that his bank had no junk exposure. He might have believed it, but a few days later it turned out that all the repackaged junk that his bank sold to clients had a "par put" clause, meaning that the client could require ("put") the bank to buy back the repackaged junk at face value ("par"). When the repackaged junk was trading at about 20 cents on a dollar, the clients massively exercised the option, causing the bank to lose tens of billions of dollars. (And they got sued too.)


Money supply control vs. interest rate control

I think that with money supply control the central bank actively influences The money supply by adding or withdrawing money. By setting the interestrates the central bank influences the lendingcapacity of the banks and the people, lowering the interestrate creates more lendingcapacity and thus 'adds' to the moneysupply... By setting interest rates higher it diminishes lending capacity and thus 'tightens' the money supply..


There is data available on debt service for Morocco. You do not need a proxy. Google is your friend. These are some hits:

Trading Economics 1
Trading Economics 2
World Bank
Open Data for Africa
Central Bank of Morocco (open 2015 pdf, and search "debt service", and you will get the nominal value).


difference between demand for money and credit

I'm not sure what's the nature of the question but:

Credit isn't necessarily linked to money. It's linked to the fact that people desire to consume today rather than tomorrow. You can have credit, borrowing and an interest rate in a cashless economy.
Demand for money could be abstracted as an extra term in the utility function and/or a cash-in-advance restriction, meaning that the individual seeks to hold money because it makes them happy and allows him to pay for real goods. This, in turn, leads to demand for money.


An interesting question that leads to a debate among econometricians. Some consider that

Econometrics is just statistics applied to economic problems

  Econometrics is just statistics applied to economic problems—nothing
  more and nothing less. We should probably call it “statistical
  economics,” but I guess people feel that the term “econometrics” has a
  better ring to it. The only cost of using the term “econometrics” is
  that we are sometimes fooled into thinking that we work on a distinct
  discipline, separate from statistics. This is not true.

From John Stachurski at ANU (see his notes p.108 in Econometric Theory freely downloadable)

Others consider that

Eonometrics by no means the same as economic statistics

This quote is from Ragnar Frisch (one of the founders of the Econometric Society) and his Editor's Note on why the Society has decided to establish its own journal, Econometrica.

The key idea is that economic theory is crucial to understand measurement. Neither theory nor measurement on its own is sufficient to further our understanding of economic phenomena. We need both and measurement without theory is unlikely to provide a satisfactory explanation of the way economic forces interact with each other.

Another difference is, as economists, we are biased towards establishing causal relationships. So, this explains our focus on endogeneity issues and strategies of identification.