To what extent are renewables like solar driving down the cost of oil?

There are several reasons from supply and demand side. From production side, for example, contrary to anouncement of cutting production (from Saudi Arabia, Venezuela and Qatar), Saudi Arabia continued to increase its production.
(There are some political reasons behind like Saudi Arabia would like to make some pressure on Russia and USA but I don't enter into this discussion as it does not concern an economic mechanism.)

Also, there are some other countries like Canada, Iraq and Iran where the oil production rises, which pushes prices to fall. (I think you can easily find some data about that, I have seen somewhere in web.)

From demand side, there is a risk of deflation in Europe and contraction of demand in most of developping countries, which also affects production side and could make a downward pressure on energy prices.

Another reason could be use of energy efficient technologies (like some cars or machines in some industries.)

What theory says about this issue ?

In fact, the existing literature on environmental economics says that price of exhaustible resources tends to increase as they become scarce. (it is the famous Hotelling rule)

There is a relationship between the use of renewable energies and exhaustible resources. Normally, an economy starts with use of exhaustible resources. Theory says that when economy grows in time, the exhaustible resource becomes scarce and its price increases. In this case, economy will use exhaustible resources until the marginal benefit from its use is equal to price of backstop (solar, wind etc...)

(A very interesting paper to see about subject is Withagen and Van der Ploeg (JEEM-2012))

In this current situation, it is sure that there are less incentives for economies to switch to clean technologies as price of exhaustible resources decreased to much.

To respond to your question, the use of renewable energy is increasing in world but I don't think at all that the huge fall in oil prices is due to "increasing" use of renewables.
This trend for renewables is always present since some decades but until now, oil prices have always increased.

As I have tried to explain, I don't really think there are robust correlations between the drop of oil prices and renewable energy use.


1929 stock market crash and unemployment rate

One caveat to the author you refer to, Thomas Sowell is a staunchly conservative economist, so most arguments you hear from him will stick to the Econ 101 wisdom that government intervention is usually bad, tariffs are always bad (to be fair, it is hard to use them to correct externalities for global "public goods"), etc.

Although tariffs probably were not necessarily the right choice for a tightening economy, the general story is that our central bank in America back then was a lot less independent from politics and less knowledgeable of how to deal with liquidity crises. It ended up tightening monetary policy instead of loosening it to ease the economic contraction, exacerbating employment problems for a long time. Keep in mind as well that this period of unemployment over 10% lasted for over a decade or so; it's not quite feasible to tell what all the factors were that caused the high unemployment, especially without official data on unemployment for a lot of that period.


Continuous Preference Relation Imply Continuous Utility Fn Existence

I think you're overthinking this proof. MWG only say that such a compact set exists because, by monotonicity, you can always find $\alpha_0$ and $\alpha_1$ such that $\alpha_0*e<\alpha(x')*e<\alpha_1*e$ (with $<$ I mean strictly preferred). Remember that, by monotonicity, if $y>>x$, then $y$ is strictly preferred to $x$. Therefore, we only need to find some $\alpha_0$ and $\alpha_1$ such that $\alpha_0<\alpha(x')<\alpha_1$ for all $x'$.

This is because the set $\{x':||x'-x||\le\epsilon\}$ is bounded and $x'∼\alpha(x')*e$.

(Note that with local non-satiation the result does not immediately follow).

You're right, $\alpha(x')*e$  is on the diagonal line Z and $x'$ is not (note that $\alpha(x')$ is mutiplied by $e$, a vector filled by 1's). However, working with $\alpha(x')$ reduces the complexity of the problem. Instead of working with two (or more) dimensions, we just work with one (the diagonal line Z or one axis). We can do that because for any element $x\in \Bbb R_+$, there is $x∼\alpha(x)*e$ which is in the diagonal line.


Self policing incentive in Goods and Services Tax

Self-policing incentive, in theory, works similarly for both Value Added Tax (VAT) and Goods and Services Tax (GST). The improvement, however, could come from the reduction in the number of instances where suppliers can circumvent the self-policing system, to evade taxes.

VAT is collected by every supplier in the supply chain and paid by every buyer in that supply chain. At every incidence of tax there is an opportunity for tax fraud. One example of VAT fraud is: the supplier provides an invalid VAT number on the invoice and collects the appropriate amount of VAT from the buyer and instead of transferring the collected tax to the government, the supplier pockets it. Of course, such fraud can be detected by the customs department and when audited the supplier may claim that there was a printing error in the VAT number on the invoice. But if suppliers are not audited (perhaps because the cost outweighs the benefit) then they can plausibly pocket all the tax they collected from the buyer. It is also plausible that such a fraud can be carried out by many suppliers, if not every supplier, in the supply chain. Hence, even though there is self-policing mechanism fraud can still be committed by one or more suppliers in the supply chain under the VAT regime. In contrast, under the GST regime there is only one supplier in the supply chain who can theoretically commit that tax fraud. Since there are less instances under GST to evade taxes it is plausible that the amount evaded will be less and that it would be easier to audit.


Why are the YTMs of TIPS of longer maturities more sensitive to current interest rates and inflation rates?

{Comparison of TIPS yields}

In order to understand the TIPS yield curve, you need to look at the breakeven inflation curve. Breakeven trades literally pay off based on the gap between realised inflation rates versus the entry breakeven rate. This is the fundamental valuation metric for these bonds; the quoted yield is the residual.
(There is the issue of embedded risk premia. This is complicated by the breakeven being a spread; the premium in the breakeven is the difference between risk premia in the conventional and the inflation-linked. There are factors that could push the risk premium in the breakeven to be positive or negative.)
The implication is that investors project breakeven inflation, and the shape of the indexed yield curve (quoted inflation-linked yields) is forced to price that inflation trajectory. The resulting shape can be very unusual, particularly at short maturities.

On the other hand, the site writes that the shorter-maturity TIPS are less sensitive to current interest and inflation rates. Why is this so?

Short-dated bonds have lower durations. As for the statement about inflation, your phrasing implies an incorrect statement.

The sensitivity to breakeven inflation depends upon the duration, with longer duration bonds being more sensitive (by definition).
The sensitivity to realised price changes is equal for all TIPS: a shock to the CPI index moves the value of all bonds by the same percentage amount. (All cash flows are indexed, and if the start point of the index is shocked, all cash flows have the same percentage shift.)


One can prove the existence of such optimal plans using the extreme value theorem of Weierstrass, but it requires some advanced math.
Here is a toy version of the model without energy and emissions. Both the instantaneous utility functions $u:\mathbb{R}_+\to\mathbb{R}$ and the production function $f:\mathbb{R}_+\to\mathbb{R}$ are assumed to be continuous and nondecreasing. Moreover, $u$ is assumed to be bounded (!). There is given initial capital stock $k_1\geq 0$. The space of feasible consumption and production plans is defined as $$F=\big\{(c_1,k_1,c_2,k_2,\ldots)\mid 0\leq k_{t+1}\leq f(k_t-c_t)\}, k_t\geq c_t\geq 0\big\}.$$
$F$ describes all feasible paths of consumption and capital.
This set is a nonempty compact subset of $\mathbb{R}^\infty$ endowed with the product topology. Here is why: By Tychonoff's theorem, the set $$\prod_{t=1}^\infty [0,f^t(k_1)]^2$$ with $f^0$ the identity function $f^{t+1}=f\circ f^t$ is compact and $F$ is a subset of this compact set.  By definition, all coordinate functions are continuous. Each of the relevant inequalities defines a closed set and $F$ is, therefore, a closed subset of a compact set and, therefore, compact itself. Also, the plan that never invests and consumes everything is in $F$, so $F$ is nonempty.
The utility function $U:F\to\mathbb{R}$ given by
$$U(c_1,k_1,c_2,k_2,\ldots)=\sum_{t=1}^\infty \beta^t u(c_t)$$
is well-defined and continuous in the product topology. Since the utility function is bounded, $U$ will always be finite and, therefore, well-defined. To see that it is continuous, note that since $u$ is bounded, there exists for each $\epsilon>0$ some $T$ such that the utility of any two paths that only differ at times later than $T$ can differ by at most $\epsilon/2$. Since all coordinate functions are continuous, if the two consumption plans are close enough in the first $T$ coordinates, they will differ at most by $\epsilon/2$. So if the paths are close enough at finitely many coordinates, the corresponding utility will be close enough. So $U$ is continuous in the product topology.
So $U$ is a continuous function on the nonempty compact set $F$. By the extreme value theorem of Weierstrass, $U$ takes on a maximum at some point in $F$ and such a point is an optimal plan.
You can find a more general proof of the existence of optimal plans along these lines in the book "Dynamic Programming in Economics" by Le Van and Dana. The assumption that $F$ is increasing is not needed, you can replace the product of intervals that includes $F$ by bounding it with the largest values that can be produced instead of the one that comes from always reinvesting capital. That $u$ is bounded was assumed to guarantee that $U$ is finite. One can replace this by an assumption that guarantees that the instantaneous utility cannot increase to
fast along a feasible path.


What Is the Economic Meaning of "Structural Reforms"

Economic variables can be "real" or "nominal". So when policymaker discus influencing the economy, they look at implications of influencing these economic variables.

An example of a nominal variable is the price level influenced by money supply. Money is directly created by the government and can take on any value per se and is not "real. E.g. if we multiply all prices and bank notes by 10 nothing in the economy changes.

Real variables are for example output, productivity etc.

However, it turns out that these nominal variables can influence the real ones. Increasing the money supply can increase output (at least temporarily). This is also easy to implement, so monetary policy is often used to fight a recession. Note that lowering the interest rate and increasing the money supply are two sides of the same coin and both part of monetary policy.

An alternative to monetary policy (based on nominal variables) would be "real" economic reforms. One form of "real" policy is a "structural reform". In this case you would try to increase output by changing something in the real side of the economy. For example lowering (wage) costs through changes to labor protection laws. Fiscal policy (government spending or tax cuts) is another form of "real" policy, however it is not a structural break.

To fully understand structural reforms we must introduce the concept of output potential. This is the output that would normally occur in absence of disturbances. It only depends on the underlying structure of the economy. These are factors such as costs (e.g. ease of doing business), labor market flexibility and other things that influence output in general in "real" terms. The economy generally comes back to that level and fluctuates around it in the medium run. (Note this level may be growing over time, e.g. by 1-2% each year).

However, sometimes it happens that our economy is operating below its "natural" potential. In this case we can employ monetary and fiscal policy to get back to that natural level. We can also push the economy beyond the natural level, but we will eventually come back, so this is not a good idea since the side effect that brings is typically more inflation.

If the crisis is because we are simply below potential, we do not need structural reforms. However if we are in a crisis because the potential or "natural" output has risen, then we need "structural reforms". An often cited recent example is the Greek crisis where Greece's wage rate increased a lot over time making its exports too expensive, meaning the economy did not have its past potential to export anymore. In this case many economists suggested "structural" reforms (law changes) that would lead to lower wages.

Monetary and Fiscal policy are often used to correct short-term problems or deviations from output potential. Structural reforms are supposed to change the (medium to long term) output potential itself.

The type of policy best suited often depends on the type of crisis at hand and its underlying causes.


Walrasian demand with a twist of Leontief function

Why don't you just plug in some values for $x_1$ and $x_2$?
Start with something like
$x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$.
Then increase $x_1$ or $x_2$ and let
$x_1=2, x_2=1$ and find utility $u(2,1)=1+5*2=11$, and
$x_1=1, x_2=2$ and find utility $u(1,2)=1+5*2=11$.
The utilities of both bundles are identical, and the goods seem to be substitutable.
In fact for any $x_1<x_2$ you will find
$u(x_1,x_2) = x_1 + 5 x_2$, and, similarly,
for any $x_1>x_2$ you will find
$u(x_1,x_2) = x_2 + 5 x_1$.
Now, just for now, assume both goods have the same price $p_1=p_2=1$ and you have a budget of $w=3$ such that you can buy both bundles giving utility 11 up there.
You can see that it would be better to put all your money in one of the goods and get
$u(3,0)=u(0,3) = 0 + 5*3=15$. Now think about different prices and a general income $w$. Does it again make sense to buy only one of the goods? Is it the cheaper one?


Empirical measurements of consumption and production elasticity of substitution?

The elasticity of substitution of production factors derived from empirical data depends on the data. In the case of simple production, you can estimate the production function (fit your data to the functional form of the production function) and from it derive the elasticity of substitution and other functional parameters.
The elasticity of substitution in the case of a production function (accounting for output) has it's roots in the classical consumption function (accounting for utility), as you mention.
Empirical studies are plenty and the numerical results vary depending on country, industry (aggregate level) or type of company (micro level). However, the robustness of the results may not be up to par, because aggregate (and otherwise), especially neoclassical, production functions are subject to numerous restrictions and assumptions.
For empirical studies you can start here Chirinko R., Fazzari S., Meyer A., 2004

In the case of utility (elasticity of substitution in consumption function), I don't know of any empirical studies. I'm not one to be asked this, but my first hunch is - additive utility is a REALLY strong assumption. Classical and austrian economists have argued that utility is subjective. So, numerical studies of elasticities of substitution in consumption would have to be verry biased.


Understanding minimums in the calculation of Human Development Index

This means that if a country would have most of its population living out of subsistence farming, which in practice could be registered as zero income, this would lead to an estimated/assumed GNIpc equivalent to \$100 per year per capita, which no country would ever go below.
Having this value makes sense when reading the starting notes of that section:

  Minimum and maximum values (goalposts) are set in order to
  transform the indicators expressed in different units into indices on a scale of 0 to 1. These goalposts act as the “natural zeros”
  and “aspirational targets” (...)

Said in other words, having this minimum means that \$100 would be the GNI per capita that would provide the lowest or minimum standard of living possible.

Answering to the "isn't \$100 too low":

  For income, it is set at \$100 per capita GNI, which is lower than the lowest value attained by any country in recent history (Zimbabwe in 2008). Should any country’s per capita GNI fall close to or below $100, the minimum will be changed accordingly.

I can recommend to read the source from UNDP, gives interesting inputs on the other dimensions too.

There are mathematical reasons to chose a low minimum

It is also convenient to choose minimums that make the dimensions non-zero, or not too close to zero even for countries which would score close to the minimum. Since the HDI is all about the combination of several aspects, and the formula is:

$HDI = (I_{health} \cdot I_{education} \cdot I_{income})^\frac13$

If any country would score a zero in any dimension this would dominate the whole HDI (make it zero), which defeats the purpose of the index. The index aims at acknowledging average human development balancing the three aspects.


Stationarity, ADF/KPSS, Autocorrelation and Heteroscedastiy

First, it is actually heteroskedasticity or heteroscedasticity and homoskedasticity or homoscedasticity (depending on whether you prefer to be faithful to the Greek root of the word or prefer the latinized English spelling).
Unit root test only tell you if the dependent variable has constant variance. That is for a dependent variable $y_t$ that is non-stationary (among other things) it will hold that:
$$\operatorname {Var}(y_{t})=\sum _{{j=1}}^{t}\sigma ^{2}=t\sigma ^{2}.$$
However, note this is variance of $y_t$. When you test for presence of presence of heteroskedasticity in subsequent model that uses $y_t$, for example:
$$y_t = a + b x_t + e_t$$
you are testing for constant variance of $e_t$ not $y_t$, since one of the Guass-Markov assumptions behind OLS is that errors should be homoskedastic (although this can be adjusted for by reestimating the errors with different method as it only affects efficiency of OLS estimator).
When it comes to autocorrelation, the same issue as above applies. In addition, unit root tests won't even tell you if $y_t$ is autocorrelated or not. In fact, most unit root tests will be biased in the presence of autocorrelation (e.g. Augmented Dickey-Fuller test) so that is something that you should test separately before you even perform your unit root test.


What would an investment bank have to gain by warning of an impending
  financial crisis?

An entity might have an incentive to deceitfully alarm about an "impending crisis" if the entity's portfolio largely consist of positions which lead to profit when the price of other asset(s) goes down. Examples of such positions are put options, short selling of stocks, and certain side in interest rate swaps. The same applies if much of the entity's liabilities is positively correlated with the price of assets which are representative of the market.

The entity could hedge against the risk of a generalized fall by getting long positions in "very" Out-of-The-Money put options. Being far OTM implies that this protection for drastic scenarios is not significantly expensive.

Likewise, the rest of an entity's portfolio could also consist of assets deemed stable enough to withstand crises.


Why is market equilibrium pareto efficient?

A situation can be Pareto efficient without also being a Pareto improvement over every other situation.

So, here for example, equilibrium is Pareto efficient, but -- as you have noted -- is not a Pareto improvement over every other situation.

Example. Say I have 100 apples to divide between A and B. The allocation (60, 40) is Pareto efficient, but it is not a Pareto improvement over the allocation (59, 41).


When do trades between two countries not result in gains for either of them?

The reason you probably find it hard to think of a case where this happens is because rarely do countries go around trying to make poor trades (read: never). If you trade real goods, there will almost always be a comparative advantage of trade to be had as long as countries have different production possibility frontiers in different ratios. Different opportunity costs of making more of one good means you might want another countries' labor to offer that service.

I can't give any specific examples of trades that were bad for each party where each party also was trying to act in there own self interest. Maybe if two countries were trading public debt with each other before a recession, they might end up with riskier assets than they bargained for, so even though the expected value of the debt was worth enough before a negative growth shock, you might get something less favorable in practice. Even then I don't think a situation like that is very likely to happen.


Is there any scientific proof that 2%-3% target inflation rate is ideal?

Science doesn't work on the basis of "proof". Proof is something that belongs only in the realms of mathematics and philosophical logic. Whereas science works on the basis of the best-available explanation that fits the available evidence.

That's as true of physics as it is of economics.

The challenge in economics is that it's much harder to do experiments to gather additional evidence, particularly in macro-economics. So decisions have to be made on the basis of quite thin evidence. Nevertheless, the principle is the same: we look for the best-available explanation that fits the available evidence.

The point of a 2-3% inflation target was that it was neither too high nor too low. And that's based on the harm we see being done during periods of deflation (such as the Great Depression) and during periods of high inflation (the oil crises in the early and late 1970s).

And because it's a target, rather than a cap or a floor, that means that it's applicable in a period of high inflation and high interest rates; and it's applicable in a period of near-zero inflation and near-zero interest rates. The evidence that supported this target pre-2007, is still the evidence that supports it now.


How do saving and investment work in macroeconomics?

Kenny LJ and FooBar already did a great job of clarifying the broader conceptual issue here: in equilibrium, prices don't just increase or decrease without a cause (a shift in supply or demand), and any deviation from the equilibrium price will result in a surplus or deficit.

But since you're talking about interest rates, I assume that you're interested in what happens when a central bank changes the interest rate. This is legitimately very confusing, because at first glance it doesn't seem like the central bank is affecting either the supply of savings or the demand for investment - and hence, by changing the interest rate, it seems like the central bank will break the equality of savings and investment.

This isn't true. But it took me years to wrap my head around this, and I suspect that the vast majority of economists would get it wrong if asked to explain it (as evinced, perhaps, that someone as smart as FooBar misfired a little on this point). Indeed, John Taylor - an extremely well-known macroeconomist - recently fell victim to a fallacy here, as Miles Kimball pointed out (see point 7 here; as further proof of the difficulty of this topic, I'm not sure that Kimball got it quite right either!).

To clarify, let's split into two cases.

Case 1: flexible prices.

Note that what really matters for savings and investment is the real interest rate, which is the nominal interest rate minus expected inflation, while the central bank sets the nominal interest rate. Suppose that the central bank increases interest rates, and at first suppose that inflation expectations remain constant. Then as you point out, desired savings will increase while desired investment falls.

This means that there will be a glut in the economy: there will be more goods supplied than demanded. How is this resolved? In a world where prices are flexible, the price will fall until inflation expectations rise exactly enough to offset the central bank's increase in the nominal interest rate, leaving the real interest rate ultimately unchanged (so that desired savings and investment balance again). For instance, if we have a certain expectation of where prices will be next year, then when prices fall today, that fall creates more expected inflation over the next year.

This is the neoclassical, flexible price case, and it resolves your paradox in the simplest possible way - by saying that expected inflation will adjust such that real interest rates, and hence the desire to save and invest, are unchanged. But as you can probably guess, this is fairly unrealistic: prices don't always adjust that quickly, and it's counterfactual to say that higher nominal interest rates result in an equal rise in expected inflation.

Case 2: sticky prices.

Let's suppose that prices are "sticky", meaning that they don't change in response to events very quickly. This is the kind of assumption made in "New Keynesian" models (and, to some extent, "Old Keynesian" and monetarist models before them).

Indeed, for our purposes, let's suppose that prices are so sticky that expected inflation is a constant, so that a change in the nominal interest rate translates exactly one-for-one to a change in the real interest rate. (This is not a bad assumption empirically, according to Nakamura and Steinsson!) This means that we can't use the dodge from before: when the central bank pushes up the nominal interest rate, the real interest rate that matters for the savings/investment market rises too, and it seems like we should get some kind of disequilibrium.

Indeed, if not all sides of the market react right away, something like this might happen. Suppose that in response to an interest rate hike, consumers quickly increase their savings, cutting back consumption, but producers are slow to respond and keep producing the same amount of goods. (Let's ignore the response of desired investment to interest rates, which complicates the story but doesn't change the basic lessons.)

Then there will be a surplus of unsold goods, which will accumulate as inventories. This inventory accumulation is a form of investment (e.g. line 14 in the US national accounts here), so $S=I$ will continue to hold - but it will hold in a weird way, where consumers' saving is matched by producers' unintended inventory accumulation.

Alternatively, suppose that producers react right away. Then when consumers try to save more and consume less, producers will produce less, and there will be less income earned by workers and capitalists. How will consumers deal with this fall in income? Partly, they'll save less - because it makes less sense to save when your earnings are low relative to their usual level. Partly, though, they'll consume even less - which means that producers make less and incomes fall again, and so on and so on.

This is just the Keynesian multiplier, and the process ends when income $Y$ has fallen by $\Delta Y = \Delta S/MPS$, where $\Delta S$ is the initial increase in desired savings and $MPS$ is the marginal propensity to save out of income. At the point, the decrease $MPS\times \Delta Y$ in desired savings from the $\Delta Y$ fall in income precisely offsets the initial increase $\Delta S$ in desired savings arising from the higher interest rate. (If there is an initial fall $\Delta I$ in investment in response to the higher interest rate too, then this becomes $\Delta Y = (\Delta S - \Delta I)/MPS$; and it becomes more complicated still if investment changes in response to income, etc..)

In short, $S=I$ continues to hold because desired savings depends on income - which you can't see from the simple, partial equilibrium diagram showing S and I relative to the interest rate. Consumption (and therefore income) will eventually fall enough to cause a decline in savings that restores equilibrium.


Divergence of a variable in forward-looking models. Blanchard-Khan conditions

QUESTION 1
By inserting $(6.28)$ into $(6.29)$, re-arranging and applying direct forward substitution , we obtain

$$y_t = \lim_{n\rightarrow \infty}\left[\left(\frac {\theta}{\theta + b}\right)^nE_t[y_{t+n}]\right] +\frac {\theta}{\theta+b -\theta \rho}u_t \tag{1}$$

Then forwarding $n$ periods and taking the conditional expectation with respect to information available at $t$ we have

$$E_t[y_{t+n}] = E_t\left(\lim_{n\rightarrow \infty}\left[\left(\frac {\theta}{\theta + b}\right)^nE_{t+n}[y_{t+2n}]\right]\right) +E_{t}\left(\frac {\theta}{\theta+b -\theta \rho}u_{t+n} \right)$$

$$\implies E_t[y_{t+n}] = E_t\left(\lim_{n\rightarrow \infty}\left[\left(\frac {\theta}{\theta + b}\right)^nE_{t+n}[y_{t+2n}]\right]\right) \tag{2}$$

The conditional expectation is an integral. Can we interchange integration and limit? To do that Dominated Convergence & Co should hold (they are only sufficient conditions, but we do not have something better). For our case, Dominated Convergence requires that the expression inside the limit is bounded("dominated"), and that the limit is finite. If it is then, using also the Law of Iterated Expectations, and the fact that under the limit $t+2n$ is equivalent to $t+n$ we have

$$E_t[y_{t+n}] = \lim_{n\rightarrow \infty}\left[\left(\frac {\theta}{\theta + b}\right)^nE_t[y_{t+n}]\right] \tag{3}$$

Note carefully that in the left-hand side, $n$ does not go to infinity. Also, remember that we obtained $(3)$ under the assumption that the limit is finite. So $(3)$ tells us that for every finite $n$, the conditional expectation must be equal to the unique limit, i.e. equal to the same number, i.e. it should be a constant.

Can this constant be anything else than zero? Assume the limit is not zero. Then one can verify that the only other finite value that the limit can take is $1$. But for the limit to be $1$ we must have
$E_t[y_{t+n}]\rightarrow \left(\frac {\theta + b}{\theta}\right)^n$, which goes to infinity with $n$. So the left hand side would go to infinity and will not be equal to the limit. Hence we conclude that the limit must be zero, for $(3)$ to hold.
This can happen if we assume that the expectation alone is a non-zero constant, or that it is growing but boundedly (say, it has an asymptote to which it converges from below). But then again, in both cases the left hand side will not be equal to the now zero limit. So we see that the only way that $(3)$ can hold is to assume that the expectation alone is zero.

So I would say that the approach "if we assume that the limit is zero, then we obtain etc" (as presented in the quoted text), is wrong: looking at our solution, which is equation $(1)$, we must first assume/impose that the expectation is zero, to make it all consistent, and finite (note: if the solution $(1)$ included also a constant term in the right-hand side, things would be more flexible).

Sticking with $(1)$, we also see that the only other consistent solution is to assume that the limit goes to infinity -and this can happen only if we assume that the expectation on its own goes to infinity.

QUESTIONS 2 & 3
As I have wrote in another answer, everything in real-world economics is finite -even prices, even expectations. The above mathematical formulations and the mathematical possibility of an infinite solution represent in a colorful way the phenomenon of "bubbles". In real life, bubbles eventually burst -but what we have managed to capture by using the "infinity" concept, is the abnormal inflation of the bubble while it hasn't burst yet. So agents can "expect infinity", in the metaphorical sense just described. But since "bubbles" is not your everyday economic phenomenon, these models usually assume them away in order to study the stable situation.

Marx was the one that argued that capitalism will explode, or rather, implode. So you have an articulated theory that advances the view that a specific economic system possesses the property of (eventual) divergence.


How are workers harmed, from firms' payroll tax revenue aimed at reducing workers' tax?

As the answer says, it depends on the elasticity. The following is a short counter example.

A Short Example

Suppose that the firm is perfectly elastically demanding labor based on

$$L^D (w) = 5 - w$$

And workers supplies labor:

$$L^S (w) = w$$

In equilibrium, we have $L = 2.5, w = 2.5$. Total income = $2.5^2$ Now install a tax:

$$L^D (w) = 5 - (1+t)*w$$

Then, we have that $w = \frac{5}{1+1+t}$ and $L = \frac{5}{1+1+t}$

Hence total labor income is: $wL = \left(\frac{5}{2+t}\right)^2$. We give the payroll taxes to the workers, who then receive $(1+t)*wL$. Let $t=1$.

Total worker income is then $2*\left(\frac{5}{3}\right)^2$ = 50/9. Previously, they got 50/8. They new total income, including payroll taxes, is smaller than before.

So what is the math indicating?

The reason behind this is ignored externalities.

The payroll tax increases the labor cost, making the firms want to hire less
The workers do not take into account that they are going to get the payroll taxes out later on, so they do not compensate for the higher labor costs by accepting smaller wages (see last paragraph)
Given higher costs, we produce less. There is a deadweight loss (DWL) caused by disturbing taxation (production changed), and the only question is: how this DWL is shared. This depends on the elasticities of supply and demand w.r.t. wages. In this scenario, some of the DWL was taken over by the workers. They got partly compensated through taxes, but not by enough

Negative externality

Why doesn't the worker take into account that the more they work, the higher the taxes they receive are going to be?

Basically, because each worker is very very small compared to the total labor force. Think about it this way: Compensating for the increase in labor cost means that the worker has to work for a smaller wage. This will hurt him a lot. The benefits of this $t\cdot wL$ are going to be shared with all other fellow workers. That means that his own share of his "sacrifice" is going to be small. His private gain is too small from doing this. On the other hand, he would receive the positive externality from all the other workers working more. To wit, he would receive some of their tax share - but since the other workers' incentives aren't set correctly either, they don't work "as much" either.


Economic models failing to predict inflation

Depends on what the author meant by:

  All their models show its indeterminate.

If you would want to take this in a literal sense then no. An easy counter example would be some bachelor level monetary model or Philipps curve.

However, there is a famous indeterminacy problem in macroeconomics which is real in many models. The problem is that in many (but not all) macroeconomic models forward looking monetary interest rates setting rules such as Taylor rule with finite horizon may lead to real indeterminacy which means that when you shock your system there might be many different paths that will make real variables return to the long run equilibrium, and consequently also to indeterminacy in the rate of inflation.

But there are also mainstream models outside MMT that solve this problem. For example, even in this paper from ECB authors show that just change in interest rate setting rule can resolve this issue. Moreover, again this is definitely not issue in “All economic models”.

However, if the author was exaggerating and by that meant that there is a serious indeterminacy issue in many models based on simple finite horizon interest rate setting rules, which is still an active research topic in mainstream macroeconomics then in that sense the statement would be accurate.


What's the difference between loans, notes, bonds and mortgages?

Loan is an umbrella term for individual(s)/organization(s)/government(s) borrowing from another individual(s)/organization(s)/government(s).

Mortgage is a type of loan for the purpose of obtaining funds for real estate which also puts lien on the property (meaning if you don’t pay the loan you will loose the property).

Note is a contract for loan with specific repayment date and interest. However, I am surprised that they are listed as long term finance as these are usually payable within 1 year (although there can be exceptions).

Bonds are very similar to notes except they have usually higher term to maturity and interest payments on bonds are usually called coupons.

These are by the way not even all possible long term debt instruments, you could for example have debentures and many more. Hence, the text probably just wanted to give you some examples of these.


Making sure the p-values of my OLS estimates are correct

The answer to your question:

I wanted to know if it is possible to have an exhaustive set of things I need to test to be reasonably confident about the p-values I get, both overall and for each specific regressor's coefficient

is "no", but for a different reason than is stated in the comments.
To start this, let me ground you in what a p-value is and what it is not.  First, it is not a probability statement.  It is a conditional frequency statement.  It is conditional on a utility function and a model of the world.  If you have a bad model of the world, then your p-values are suspect.
Frequentist tests, as opposed to Bayesian tests, attempt to show that some null hypothesis is false.  It does so by assuming that the null is true.  That is not a trivial statement for two reasons.  First, the null hypothesis acts as information.  You are conditioning all the observations on that assumptions, as if true.  If you would change your null, you would change the inference on your assumptions.  Second, there is no way to distinguish a chance effect from a false null.
A p-value is a statement that if you would repeat your experiment, possibly an infinite number of times, then the frequency of seeing the result you saw or a result that is more extreme is no more common than the stated p-value if the null model is the true model.
So, let us look at the things you need to get right for a p-value to imply its abstract meaning.
First, your utility function needs to be correct.  Least squares models assume that losses are quadratic, that is to say $$U(\hat{\theta},\theta)=-c(\hat{\theta}-\theta)^2.$$  If you change your model, then you change your estimator, which implies that your choice of p-values changes.
For example, if you would assume that $U(\hat{\theta},\theta)=-c|\hat{\theta}-\theta|$, then you would end up with median estimators and would use things such as Theil's or Quantile regression.  The results are more robust but less powerful.
Second, you need to get your specification correct.  P-values assume the null model is true, if you have a misspecified model, then your p-values will be suspect, although not automatically so far from reality as to discard them.
Third, you need to match the correct assumptions to the real null model.  This is the danger in testing for heteroskedasticity.  If you have heteroskedasticity, but it fails to reject the null, then you have a bad set of assumptions.  If you do not have heteroskedasticity, but it rejects the null, then you have a bad set of assumptions.
The actual fix for problems like that is to approach the assumptions logically.  That is why there is no single book or reference set for this problem.
Diamond prices are heteroskedastic in size but in a weird way.  Simple logic would tell you that they are probably heteroskedastic with respect to size.
Very small diamonds have only two purposes, industrial uses such as drilling, and as additions to jewelry.  As jewelry, their price is very high.  As an industrial material, the price is very low.
Most medium-sized diamonds are either close to their desired size for jewelry or need cut into very small pieces for industrial use.  However, the bulk of their use will be to make medium-sized diamonds cut to specification.  It will be easier to estimate their value as is.
Very large diamonds might remain highly valued, very large diamonds.  Think of the Hope Diamond as an example.  Others may make hundreds of small diamonds for industrial use.
The result is that you get a bowtie version of heteroskedasticity.  Small diamonds and very large diamonds have enormous price variability.  Medium diamonds are quite predictable as to the final cut and value and so are priced in a much narrower range.
You should never test for heteroskedasticity with diamonds because there is a way to think this through.  Testing should be reserved for things that you have no subject matter knowledge regarding.
Most p-values are robust because most Frequentist estimators are minimax estimators, that is to say, the estimator minimizes the maximum amount of risk that you must take under a specified utility function.
As a result, p-values tend to be conservative in their use of information and any resulting inference.
The reason that there is no one list is that if you can assert something about the population as true, such as diamond prices being logically heteroskedastic in size, then you can condition your model with that knowledge.
As a result, if you were to test diamond prices and they tested as homoskedastic, then you could ignore the test result as irrelevant.
There isn't a list.  Instead, you need to determine what researchers in a narrow domain do.
Let me give you one more example to see why that matters.
If a brick-and-mortar retailer needs to alter its revenue, it can issue coupons. It has the ability to control its revenue and its sales volume.
Electrical utilities, generally, cannot control their volume or price.  Even if they could mail coupons to use additional electricity, how would anyone monitor this sale?  Electric utilities control their costs, not their revenue.
That discussion alone tells you that the accounting information that management sees as important for the marginal decision is different.  If you give them a common specification, then your model will be misspecified.  Your p-values are off.  You omitted a variable, industry type.
Finding correct p-values are about more than overcoming technical obstacles, such as multicollinearity and heteroskedasticity, it is about understanding your domain of research.
Fortunately, p-values are usually robust.  If you set $\alpha=.05$ and $p<.01$ then you are probably safe.  On the other hand, if $p<.04999$ and your work is a bit shaky, then do not feel confident even if you followed the technical rules.
There is no one book, no simple list.  There are just domains where researchers discover unexpected things and correct for them.


A few variables expressed in units of a standard normal distribution in a regression

In general, there is no expectation that right hand side variable in a regression are in the same units. We often have combinations of variables like years of education, price, regional dummies, etc. that are all included together. So in principle, there’s no reason why you should have to convert your controls to some other scale.

From a numerical standpoint, it can be useful to normalize variables to maintain numerical stability. The eigenvalues of the covariance matrix can be a little funky if the relative scales are way different (for example, one variable ranging between 1-4 and another variable ranging between 1-4 billion). It can also be useful to normalize variables to help with interpretation of the coefficient values (easier to compare 1 and 2 than it is to compare 1 and 0.000000002).

From your example, though, if you’re just considering adjusting from something that ranges on the order of -2 to 2 to something that ranges from 0 to 1, neither of these reasons would probably hold. As such, based on the info you’ve given, I would probably just leave the RHS variables as they are.


Firm 1's objective is to maximize profits :
$$\max_{q_1\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_1 - cq_1$$

Solving the above problem, we get the best response function of firm 1 as
$$q_1 = \dfrac{a - c - bq_2}{2b}$$

Firm 2's objective is to maximize revenue subject to the constraint that its profits are non-negative :
$$\max_{q_2\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_2$$ $$\text{s.t. }\left(a - b(q_1+q_2)\right)q_2 - cq_2 \geq 0$$

Solving the above problem, we get the best response function of firm 2 as
\begin{eqnarray*} q_2 =\begin{cases}  \dfrac{a - bq_1}{2b} & \text{if } a - bq_1 \geq 2c \\ \dfrac{a - c - bq_1}{b} & \text{if } a -  bq_1 < 2c \end{cases}\end{eqnarray*}

Solving the best response functions for $q_1$ and $q_2$ yields
\begin{eqnarray*} (q_1^*, q_2^*) = \begin{cases} \left(\dfrac{a-2c}{3b}, \dfrac{a+c}{3b}\right) & \text{if } a \geq 2c\\
 \left( 0, \dfrac{a-c}{b}\right) & \text{if } a < 2c \end{cases} \end{eqnarray*}


Understanding the Zellener-Revankar Production Function

Caution: Heavy citing

"In his doctoral dissertation Revankar (1967) expounded his generalized
production functions that permit variability to returns-to-scale as well as elasticity of substitution. In contrast with the production functions that (rather unrealistically) assume the same returns to scale at all levels of output, Zellner and Revankar (1969) found a procedure to generalize any given (neoclassical) production function with specified constant or variable elasticities of substitution such that the resulting production function retains its specification as to the elasticities of substitution all along but permits returns- to-scale to vary with the scale of output. Their Generalized Production Function (GPF) is given as
\begin{equation}
\ Pe^{\theta P} = c^h f^h
\end{equation}
where f is the basic function (e.g. Cobb-Douglas, CES, etc) as the
object of generalization, c is the constant of integration and θ, h
relate to parameters
associated with the returns-to-scale function. In particular, if the Cobb-Douglas production function is generalized, we have
\begin{equation}
\ Pe ^{\theta P}= AK^{\rho\alpha} L^ {\rho (1- \alpha )}
\end{equation}. This function is interesting
from the viewpoint of estimation also. It has to be estimated so as to maximize the likelihood function since the Least Squares and Max Likelihood estimators of parameters do not coincide. The return to scale function is given by
\begin{equation}
\rho (P) = \rho /(1 + \theta P)
\end{equation} . Depending on θ the sign of θ , the returns-to-scale function monotonically increases or decreases with
increase in P. However, as we know, the returns to scale first increases with output, remains more or less constant in a domain and then begins falling. This fact is not captured by the Zellner-Revankar function since it gives us a linear returns-to-scale function."

from A Brief History of Production Functions by SK Mishra 2007

So, I would guess, the  \begin{equation}  \alpha  \end{equation} parameter
in your log-linearized function would have something to do with returns-to-scale.

Regarding estimation, I'm way too much of a newb. But this might be something at least similar to what you are looking for: https://link.springer.com/chapter/10.1007/3-540-28556-3_2
Variable Elasticity of Substitution and Economic
Growth: Theory and Evidence by
Giannis Karagiannis,
Theodore Palivos,
and Chris Papageorgiou,
2005


How prevalent is the use of machine learning among economists?

I will attempt to answer the first (How prevalent is it?) and last question (Where can I see their work?).

There was a recent post in The Economist about the emergence of machine learning into standard economic papers. They produced a nice graph, reproduced below:

Albeit machine learning is confounded with big data, it shows a rise in their use since 2014. Surely, this is a very narrow selection of the literature (NBER working papers, and only abstracts), but is a start.

Regarding where you can find papers using machine learning, you can use "keyword" search here (26,665 documents are found), or here (337 documents are found). The latter allows you to select dates, which you could use to build your own indicator of machine learning related publications per year.

Finally, related to blogs, I am not aware of a dedicated ML/DL economist's blog. However, here there is a very useful blog aggregator, which has indexed most of the "serious" blogs available. The full list of blog posts indexed is here. A search for "machine" or "deep" does not yield a result. However, there are some hits for "econometric" and "statistics" which might help you.


Dynamic programming with housing consumption and labor

$$V(b_{t-1})=\max\limits_{C_{t},H_t,N_t} \Bigg\{\ln C_t+j\ln H_t-\dfrac{(N_t)^\eta}{\eta} +\beta E_tV(b_t) \Bigg\}$$

so

$$\beta E_tV(b_{t})=\beta E_t\left[\max\limits_{C_{t+1},H_{t+1},N_{t+1}} \Bigg\{\ln C_{t+1}+j\ln H_{t+1}-\dfrac{(N_{t+1})^\eta}{\eta} +\beta E_{t+1}V(b_{t+1}) \Bigg\}\right]$$

But forwarding the budget constraint

$$C_t + b_t +q_t(H_t-H_{t-1})=\dfrac{R_{t-1} b_{t-1}}{\pi_t}+w_t N_t+F_t$$

we get after re-arranging,

$$C_{t+1} =- b_{t+1} -q_{t+1}(H_{t+1}-H_{t})+\dfrac{R_{t} b_{t}}{\pi_{t+1}}+w_{t+1} N_{t+1}+F_{t+1}$$

So $\beta E_tV(b_{t})$ is also a function of $H_t$ through $C_{t+1}$.


What would be the GDP of Algeria without Oil & Gas?

If I interpret your question correctly, you're asking what percentage of Algerian national income is derived from oil and gas extraction. The answer appears to be 30% from the website.

I think the problem is you're assuming that the 30% figure means something like '30% of private spending', and so you're trying to account for government spending as well.

The website says 30% of GDP comes from oil and gas, that means 30% of income in Algeria is derived from oil and gas. It should already include government spending, which is taxed from income. And government spending should already include the fake jobs you mention.


What are the prerequisites to study Mathematical Economics?

In the preface, Takayama writes that the book was written with the intention to keep the prerequisites to a minimum: elementary calculus and matrix algebra.

Perhaps he was exaggerating a little, but I suspect, after skimming the table of contents, that knowledge of the aforementioned subjects and experience working with (i.e. reading/understanding and independently constructing) mathematical proofs should be enough.

I'm assuming you've taken calculus and linear algebra -- in which case you might want to get more experience with working with proofs. I'm quite fond of Axler's Linear Algebra Done Right, and I've heard good things about Abbott's Understanding Analysis.

If you haven't studied those two subjects, work on learning those first. You can try the two textbooks I suggested to do so, but that might be a bit rough-going. I'm sure your math teacher might have more helpful suggestions for a first reference to learn calculus and matrix algebra from.

I think with a bit of what mathematicians like to call 'mathematical maturity', you should be able to manage Takayama just fine.


Yes it can be used. The theory applies regardless of what’s the source of comparative advantage.

For example, if country A is able to produce 200 widgets and 50 apples and country B is able to produce 5 widgets and 45 apples you get comparative advantages - A in widgets and B in apples - and it does not really matter if these are due to technology, economic development, work ethic etc.

As a matter of fact differences in technology are sometimes mentioned in undergraduate class as source for the comparative advantage.

Actually in the original example of comparative advantage used by Ricardo the example was of England that had comparative advantage in cloth production due to new industrial technologies and Portugal which was more rural and hence had comparative advantage in wine. Also at the time England was more economically developed than Portugal- at least to the extent that can be seen from historical estimates provided by Maddison.


Estimating causal effect for educated people on wages

Yes, you have to worry about the difference between correlation and causality. In these situations, it helps to try to force a case of ommited variable  bias.

In this case "talent" or "effort" is unobserved. Both this might make people more likely to pick up additional education, and also be more successful at their jobs.

Or family background: If your parents are high achievers, they might induce you to take more education, and they also are well-linked to ensure you get a well-paid job.


What exactly is on the Fed balance sheet?

See this federal Reserve webpage that provides links.

Note that these are aggregates; there are specialised reports on various components of their balance sheets.

The FRED website has these series in an interactive database.


How were unit labor costs estimated by Gali and Gertler (1999)?

In Galı́ & Gertler (1999) the authors state that marginal costs is given by:
$$MC_t = \frac{S_t}{\alpha}$$ where $\alpha$ is the labor elasticity (from the Cobb-Douglas prod. function), and $S_t$ is the labor share of income (i.e. what portion of output GDP goes to labor). Next they actually get around estimating $\alpha$ by expressing everything in percentage changes so they actually use:
$$mc_t = \frac{MC_t - MC_{t-1}}{MC_{t-1}}= \frac{\frac{S_t}{\alpha} - \frac{S_{t-1}}{\alpha}}{\frac{S_{t-1}}{\alpha}}$$
where alpha cancels out so you are left with just  change in the labor share of income $\Delta S_t = s_t$.
They even state in the paper:

The data we use is quarterly for US over the period 1960:1}1997:4. We use the
(log) labor income share in the non-farm business sector for $s_t$

So in this case they just seem to have directly taken the labor share of income, which is plausible data for labor shares of income exist without any 'assembly' required (e.g. see this Fred dataset as an example).


is it better a low or high WACC for a company valuation?

WACC is the weighted average cost of capital - the price of money for the firm. All else equal, lower is always better.


What is the significance of the Hansen-Jagannathan bound?

Fundamental theorem of asset pricing tells us that if there is no arbitrage, there must exist a positive random variable $M$ (also called stochastic discount factor) such that for any return $R$, we must have $1 = E[M R]$. Thus to understand asset prices, in cross-section as well as across time, we need to understand which factors affect $M$.

Now, the most basic Hansen-Jagannathan bound can be stated as

$$
\left| \frac{E[R^e]}{\sigma[R^e]} \right| \leq \frac{\sigma[M]}{E[M]}
$$

where $R^e$ is an excess return. The main contribution is that it allows us to say something about moments of $M$ (right-hand side), which is unobservable, in terms of moments of returns (left-hand side), which can be (in principle) observed. Specifically, given the observed Sharpe ratio (say, around 0.4), the bound tells us that the SDF must be at least just as volatile.

The problem is that standard consumption-based models for $M$ have problems matching this volatility. For example, your basic Lucas-tree economy with CRRA utility leads to $M = \beta \exp(-\gamma \Delta c)$, where $\Delta c$ is log consumption growth. Since $E[M] \approx 1$, we have $\sigma[M] \approx \gamma \sigma[\Delta c]$. If $\sigma[\Delta c]$ is let's say around 1% yearly, we need to set $\gamma > 40$, an unrealistically high value of risk aversion.


Neoclassical model with proportional taxes

In your household problem you have a typo, they maximize their utility so I think you meant $\sum_{t=0}^\infty\beta^tu(c_t, L_t)$. Now their income comes from the profits of the firm $\pi$ their wages $w_tL_t$ and what they borrow-lend (not directly the production $Y$ or government expenditures $G$) and their expenditures are consumption, paying their debts (or receiving money from what they lend yesterday) and what they pay in taxes. Obviously they only pay consumption taxes, bond taxes, and labor income taxes, since consumers don't directly invest. We usually model that investment decisions are made by firms, so investment taxes will enter the firm's problem.

In contrast, the social planner chooses everything so all taxes will be relevant for his problem.


Let's write
\begin{align}
  x&=ap+(1-a)r\\
  y&=aq+(1-a)r
\end{align}
You want to show that $x\sim y$. But
\begin{equation}
x\sim y\quad\Leftrightarrow\quad x\not\succ y\;\text{ and }\;y\not\succ x.
\end{equation}
So the given proof tries to establish that $x\not\succ y$ (and in a similar way, $y\not\succ x$) is true.

To show this, use proof by contradiction. Suppose the contrary is true: $x\succ y$, and try to derive a contradiction. This is what the given proof is doing.


No Ponzi game condition and transversality condition are the same?

Is it right to interpret the no Ponzi game condition as a finite
  horizon version of the transversality condition?

No. The "No-Ponzi-Game" or "solvency" condition is an external constraint imposed on the individual by the market/other participants. The individual would very much like to violate it.

The Transversality condition must be satisfied in order for the individual to maximize indeed its intertemporal utility. It is an optimization condition.

So they are conceptually very different aspects of the problem.

Finally the No-ponzi-game/solvency condition is not inherently of finite horizon -it extends to the infinite horizon also.


How are the price signals of stocks in the stock market centralized?

Think of the stock certificate as a physical object being traded around. It may be traded on more than one market. The centralisation you speak of depends on the source for the quote. The quote, generally (some exceptions), is a last trade price (a delayed last trade price). It is not global. I have trading software that allows me to see quotes with less delays, I can choose to see it only on certain markets or last trade from any market. I see different signals than you do.
The most direct answer to the question in the title is "they're not". There is no central, global signal. What you are likely referring to is just the (delayed) last trade price of the security on the markets that the quote service pulls from.


If I employ someone to destroy a building, what is the net GDP impact?

It certainly does affect DGP, because what you destroy is part of the capital base $K_t$, while the building activity is part of gross investment. The capital base is not part of the DGP, while investment $I_t$ is.

A related question is, what is the net effect on the capital base? That will depend on the current value of the buildings being destroyed, compared to the value of the new buildings.

The more interesting question is : how are we to classify the value of demolishing services alone? Labor and capital will be expended to bring down the old structures, so certainly production factors will get rewarded.

From an accounting point of view, these constitutes also part of the cost of the new buildings, and this has a wise economic rationale: the returns from the commercial exploitation of the new buildings, must be evaluated against the total cost incurred so as to be able to exploit them, and this includes the cost of demolishing pre-existing structures.

Do the economics discipline treats such demolishing services the same way? yes, they are also part of gross investment.

Assume now that we are only demolishing an old building, without building a new one. It seems a little weird to count the cost of demolition as "gross investment" that will increase GDP and tend to positively offset the reduction of the capital base due to the elimination of the old buildings. Investment to what? - we have not erected new structures.

Well, what increases here by the demolition is the value of the land. Why? Because prior to demolition, the land was committed to a specific building, and so the land's uses were restricted. Now that the land is land only, it can be used for many more purposes than before. This has economic value.

Usually, when a piece of land has buildings, we see it as a whole, and we don't make the above distinction. But the distinction is there and justifies treating the cost of the demolition alone as investment.


Finding the conditional input demand function

Output $z$ is given as $z = x + y$ where $x=min(a,2b)$ and $y = max(3c,4d)$.
So assume that you want $x=12$ then $a=12$ AND $b=6$. Since this part of the production delivers only the minimum of serveral production processes $a$ and $2b$ the producer must get insure that ALL subproductionprocesses $a$ and $2b$ deliver the minimum.
The producer therefore has to buy both $a$ and $b$ in the required amounts and therefore the price for $x$ is composite including both the price of $a$ and $b$.
However, when you want $y=12$ you must have $12=3c$ OR $12=4d$ implying that $c=4$ OR $d=3$. Since the productionprocess $y$ delivers the max of the subproduction processes the producer only has to make sure that AT LEAST ONE of the subproduction processes delivers the wanted amount $y$.
The producer will therefore always produce $y$ using the "cheapest" input and as dependig on input prices. The input price will therefore only depend on the factor $c$ or $d$ chosen to achieve the max.


Indirect Impacts Of The Netherlands Cutting Greenhouse Emissions?

global
The Dutch energy system is small enough that the ripples to the global supply chain will be very small. However, it is looking possible that there might be some temporary tightness in the global PV module supply chain coming up, after years of capacity expanding faster than demand, so there might be a price-spike up, there, if the Dutch go for rapid PV deployment 2015-6. Conversely, there is a bit of a dip coming in the global offshore wind pipeline, so the Dutch might help there; they do have a deployment pipeline set out to 2023, so one approach would be to accelerate that to deliver all of those windfarms by 2020, smoothing the work of the global supply chain, and reducing wastage there. But the Dutch energy system is small, and so all of these consequences would be small at the global scale.
The much larger potential impact is that of the ruling itself, on other countries. This successful application of tort in one country may well prompt similar cases in other countries, as tort is a principle found in legal codes around the world. Further successful tort cases might help push the world along the decarbonisation trajectory much faster than it is at present. Now, as there is a non-negligible chance that business-as-usual could result in the destruction of much of the global economy, the option value of reducing the probability and scale of that wipe-out is very high; so the ruling may yet have very high, very positive economic consequences beyond the Netherlands.
overview
In many ways, the Dutch are about to go through, at small scale, what the rest of the world is about to go through at much larger scale. We've known for about 30 years that almost-complete global decarbonisation was necessary, and that the later we left it, the more expensive it would be. And then there was very little action in most countries. So now, not only do we have to decarbonise in about half the time we originally had (which will thus be more expensive), but we'll also have to spend a huge amount on adaptation too, as a lot of global warming is already in the pipeline and almost completely unavoidable over the next 40 years.
costs, benefits, prices, quantities
The negative externalities of the current system outweigh the benefits; so once we've transitioned away from fossil fuels, we'll be more economically efficient that we would otherwise have been; but there are short-term costs attached to that transition, and so resources will be diverted from other activities and into an accelerated transition.
The accelerated action required in the Netherlands 2015-2020 will be to temporarily increase prices for the mitigation measures deployed; and those prices will then, post-2020, be lower than they would otherwise have been, as the supply chains will have expanded.
Regarding demand for oil, coal and gas: they are all internationally traded commodities, so a decrease in Dutch demand will add a little downward pressure on their prices across Europe; however, Dutch demand is a small proportion of total European demand , so this price effect will be very small.
The Dutch have been investigating various forms of road pricing for around 30 years; and planned out to some detail. So we may see some version of that finally implemented. Now, that would be more economically efficient in the long run: fossil-fuelled car travel has a lot of externalities in addition to greenhouse gas emissions. But in the short run, there are additional costs as land-use patterns change, and people change where they shop, where they live, where they work.


Can a monopoly INCREASE the market surplus compared with a competitive market?

Under standard assumptions (some of which you state in your question: no externalities, etc.), no. This follows from the First Welfare Theorem.

Perhaps there are departures from standard models that would support something resembling your conclusion, but my guess is that most economists would view any such departure as the absence of “perfect competition”.

It might be a good exercise for you to write this down in a model (even a simple one) to help you figure out what those departures might be.

Regarding the example you put in your edit:

First, your "monopoly" is not a monopoly in the sense that economists understand it. Monopolies have market power, and thus have no supply curve. The firm in your example has a supply curve, and has no market power.

Second, the example you give does not seem to all support your argument. The firms in the two different situations you present seem to have markedly different production technologies (reflected in the differing supply curves). Your argument requires that the firms have the same technology, only that perfect competition prevents the exploitation of "economies of scale" of their (common) technologies. In other words, this result is driven by differences in production technology, and not from the fact that one market is perfectly competitive and the other is monopolistic. (I believe this is what some comments to your question mean when they say that this is a weird comparison.)

Moreover, where exactly is economies of scale in the second situation? How am I supposed to be able to interpret what is going on as the firm being able to take advantage of economies of scale? Bear in mind that in the second situation, you would have the same outcome (with respect to surplus) if there were many firms, as long as the aggregate supply curve were the same. The fact that there is a single firm is not at all important to the result you've found.

Let's illustrate this previous argument specifically in your example. It seems the correct intervention for the state is not to allow the monopoly to continue but to regulate it. Instead, what the state should do is to make the monopolist's production technology available to other firms that want to enter the market to foster competition.

Suppose that the state does this, and another firm enters the market with the same technology. Let's also suppose now that the market is perfectly competitive. Each firm's supply curve must be

$$ q_s = -100 + 5P $$

which makes industry supply

$$ Q_s = -200 + 10P. $$

Given the demand curve $ Q_d = 1000 - P $, the equilibrium price and quantity is

$$ P^* = \frac{1200}{11}; \, Q^* = \frac{9800}{11}. $$

Total surplus is then $$ \frac{1}{2} (1000-20)\cdot \frac{9800}{11}, $$ a $7 \%$ increase in surplus. Surplus should also increase further when more firms enter.


McCloskey's example of fungibility in basketball

In economics and finance, fungibility means that we cannot distinguish between instruments. For example, we normally do not care whether we have one \$20 bill or two \$10 bills.

McCloskey's example is poor. In one sense, it is correct: when points are scored do not matter in the final tally. However, it is psychologically incorrect when applied to sports.

For example, in ice hockey, it is somewhat common to pull your goalie if you are behind at the very end of the game. You have a slightly greater chance of scoring since you have an extra attacker, but it is very easy for the other team to score if they get control of the puck. If you were leading 1-0, and then scored a second goal because the other team pulled their goalie, no sensible hockey fan would weigh the second goal as much as the first. The game was already effectively won by the earlier goal, and the second was just a statistic.

More generally, a team's effort during the game probably reflects the score. Piling up points early may be dangerous, as the team will then relax and sit on its lead. Therefore, later points are effectively more valuable. In order for McCloskey to be correct, we have to believe that the score of the game does not influence behaviour.

These problems do not crop up when using \$10 bills in your example.


Differentiating through this integral in banking loans?

In generic notation, we examine

$$\frac {d}{df} \left(\frac {f(x)}{\int_0^1 f(x) dx}\right)$$

Note that we take the derivative with respect to $f$ not with respect to $x$. Applying the standard rules we have

$$\frac {d}{df} \left(\frac {f(x)}{\int_0^1 f(x) dx}\right) = \frac{\int_0^1 f(x) dx-f\cdot \left[\frac{d}{df}\int_0^1 f(x) dx\right]}{\left(\int_0^1 f(x) dx\right)^2}$$

But

$$ \frac{d}{df}\int_0^1 f(x) dx = 0$$

because this definite integral, having specific values as integration limits, is a number, a constant, not a function. So we end up with

$$\frac {d}{df} \left(\frac {f(x)}{\int_0^1 f(x) dx}\right) = \frac {1}{\int_0^1 f(x) dx}$$

that validates the obtained f.o.c.

Leibniz rule for differentiating under the integral sign does not enter the picture here.


Publicly funded healthcare and social welfare schemes typically serve redistributive functions and therefore can be considered as promoting equity (among the well-off and the poor). At the same time, these programs must be financed through taxation such as income tax that's usually distortive, thereby weighing down on efficiency.


When everyone sells at an event of a recession, who buys?

the news of a downfall of stock prices hit investors, so they obviously
  wanted to get rid of their shares.

This is not the way you want to think about prices. Stock prices are determined via the equilibrium of demand and supply. It sounds like you are assuming that they drop from the sky and investors react, but they are determined via the actions of investors. What happens is that news of the fundamentals of the economy (Things like interest rates, employment numbers, job forecasts, etc.) hit, investors react, and then price falls.

Suppose there are two investors A and B and a single stock. Suppose that $t=0$ both investors 'subjectively value' the stock at 10, this valuation is determined via each investors subjective expectations of future earnings and future prices. So the price at $t=0$ is 10 and lets suppose that investor A owns the stock. At $t=1$ news about the fundamentals of the economy hits, each agent updates their subjective value of the stock. Suppose that the 'value' to agent A is 5 and to agent B is 6. Its clear that Agent A will trade the stock to agent B at a price between 5 and 6, hence price falls endogenously not exogenously.  As this example illustrates, trade occurs between pessimistic sellers and relatively optimistic buyers.


Doesn't the Pareto-extension rule invalidate Eliaz's (2004) unified theorem of social choice?

Acyclicality as defined by Eliaz is simply the condition that
the values of a social aggregator are transitive. It is different from the usual acyclicity condition according to which a relation is acyclic iff it has irreflexive transitive closure. The formulation of Eliaz is a bit cryptic, but he actually shows in his Observation 1 that his acyclicality implies transitivity. The converse is also easy; the simple details are below. The result of the Pareto extension rule need of course not be transitive.
Formally, acyclicality as defined by Eliaz says that for all $R\in\mathbf{R}$ and for every three alternatives $a$, $b$ and $c$ in $A$, if $aRb$ and $\neg(cRb)$, then $\neg(cRa)$.
But the condition that for every three alternatives $a$, $b$ and $c$ in $A$, if $aRb$ and $\neg(cRb)$, then $\neg(cRa)$ is simply the transitivity of $R$, formulated in a slightly weird but equivalent way.
Here is the simple proof: Assume that $R$ is transitive and $aRb$ and $\neg(cRb)$. If it were the case that $cRa$, then $cRb$ by transitivity, contradicting the assumption. For the other direction, assume that the condition holds and that $cRa$ and $cRb$ holds. We want to show that $cRb$. Indeed, if not then by the condition $\neg(cRa)$, contradicting the assumption.


Is it correct to say opportunity cost is cost of giving up quantity of what you are selling?

It seems the instructor is referring to the opp. cost of giving up the profit from selling x amount of apples. Which would be an opportunity cost. Opportunity cost isn't strictly alternative based, for example: If I'm selling 10 apples today, the opportunity cost of be only selling 9 apples tomorrow is whatever profit I stand to lose from not selling the 10th apple tomorrow.


Basic Solow Growth Model: Stability Proof

For completeness, let me illustrate this in the continuous time framework. The Solow equation, in the simplest of cases, is

$\dot{k} = s f(k) - \delta k = \phi(k)$

Then we have

$\frac{\partial \phi}{\partial k} = s f'(k) - \delta = \frac{sf'(k)k - \delta k }{k}$.

In steady state (i.e., $\dot{k} = \phi(k^{\ast}) = 0$), we have $\delta k = s f(k)$, hence

$\left. \frac{\partial \phi}{\partial k} \right|_{k=k^{\ast}} = \frac{s f' k - s f(k)}{k} = \frac{s}{k} f(k) \left[ \frac{k \cdot f'}{f(k)} - 1 \right]$

Since both $\frac{s}{k}$ and $f(k)$ are always positive, the sign of that expression hinges on the term in brackets. But remember that $\frac{k \cdot f'}{f(k)}$ is the output elasticity of capital, which is smaller than 1 if $f$ is concave in $k$, which means $\frac{\partial \phi}{\partial k}$ at $k^{\ast}$ is negative. Therefore by theorem 3.2.1 in Zhang (2005), the steady state in the Solow model is stable.


How can I calculate with the average of the consumption in an RBC framework?

Consider the risk free steady state. In that situation the Euler equation becomes:

$$ \left(\frac{C_{t+1}-b \overline{C_{t}}}{C_t-b \overline{C_{t-1}}}\right)^{-\sigma}=\frac{1}{\beta \cdot (1+r)}$$

Which implies

$$ \Rightarrow  \ln[C_{t+1}-b \overline{C_{t}}] -\ln[C_t-b \overline{C_{t-1}}] = \frac{\ln[\beta] + \ln[1+r]}{\sigma}$$

Assume that in the non-stochastic setting that growth is constant of the form:
$$ C_{t+1} = \gamma \cdot C_{t}$$

Note that by definition
$$ \overline{C_{t}} \equiv \frac{1}{t}\sum^t_{\tau=1} C_{\tau}$$
Under a constant growth rate
$$ \Rightarrow \overline{C_{t}}=\frac{1}{t}\sum^t_{\tau=1} \gamma^{\tau-t}C_{t} = \frac{C_t}{t} (1+\frac{1}{\gamma-1}) = \frac{C_t}{t} (\frac{\gamma}{\gamma-1})$$
$$ \Rightarrow \overline{C_{t+1}}= \frac{C_{t+1}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \frac{C_{t}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \overline{C_{t}}$$

$$ \Rightarrow  \frac{\ln[\beta] + \ln[1+r]}{\sigma} = \ln[C_{t+1}-b \overline{C_{t}}] -\ln[C_t-b \overline{C_{t-1}}] $$
$$ = \ln[\gamma C_{t}-b \gamma \overline{C_{t-1}}] -\ln[C_t-b \overline{C_{t-1}}] = \ln[\gamma]$$

Therefore the one period growth rate in consumption ($\gamma$) in the non-stochastic model is $$ \gamma = e^{\frac{\ln[\beta] + \ln[1+r]}{\sigma}}  = [\beta(1+r)]^\sigma$$


Interestingly—and much in contrast to recent research—our data supports the conventional wisdom that persons with a higher inclination towards risk have a significantly higher probability of becoming entrepreneurs. However, sensitivity analysis reveals that this result holds only for those individuals who were previously employed. For previously unemployed or inactive persons, we find no significant influence of risk attitudes, indicating that other variables drive their decision towards self-employment.

Caliendo et al. (2009)

The Conclusions section of the paper lists several other research papers with slightly different conclusions, you may also want to read those.


You need a clear understanding of what is exchange rate and how it is determined.

Consider two countries with free flow of goods and services that is there is no transaction, transportation, tariff costs etc. [I am assuming an ideal world here].

say,

India (Currency: Indian Rupee abbreviated as INR) and USA (Currency: Dollar abbreviated as $ ).

Exchange rate:

$ 1 = INR 50

Consider a commodity let us say pencils:

Assume that in year 2018 - one pencil costs $ 1 in USA and INR 50 in India.

Now let us suppose that India faces an inflation of 10% per annum and there is no inflation in USA.

Hence new prices of pencil in year 2019:

India - INR 55

USA - $ 1

Since, there are no transportation and transaction costs Indians will start purchasing pencil from USA since 1 US $ still equals INR 50 and Indians get pencils relatively cheaper in USA after converting their currency into dollars. This will push up Indian imports and hence Indian currency will depreciate and US dollar will appreciate because US exports have gone up.

This process will continue till 1$ = INR 55.

Now coming back to your question say home country is USA and due to relatively lower inflation rate in USA, the US dollar appreciates (NOT depreciate).


Do carbon taxes in place today apply to carbon embedded in imports?

I think the answer is generally/overwhelmingly "no" there usually not a embedded carbon (import) tax in most jurisdictions. A 2019 paper says:

  The idea of a BCA is not new. In the US, for example, lawmakers have tabled several proposals to introduce a carbon tax. So far, though, no country in the world has an operational BCA.

Supposedly the EU Green Deal was supposed to include one, but I haven't followed the developments. (There have been arguments against it already. And the US has already threatened to retaliate should the EU adopt it.)


Does good geographical location count as land in economics?

In Economics "Land" as a form of capital is those parts of the natural environment that had become a subject of economic transactions (market or otherwise).

Consider a piece of land that is public property, say, a mountain. No economic activity is taking place related to that mountain. Then, it is not consider part of the Fixed Capital base of the national economy. Certainly the mountain may have minerals. The mountain contributes to the local climate which in turns affects the weather and so the local economy. But to the degree that it is not used productively through purposeful economic behavior, it is not considered part of the Capital base (that estimations of natural resources are calculated for various economic plans has to do with future actions, not with the capital base at present).

Assume now that the Government actually sells part of this mountain to a company that wants to build a winter resort. This part of the mountain becomes now a real-estate asset of that company, and so part of its Fixed Capital and so part of the Fixed Capital base of this economy.

The features and characteristics of this piece of the mountain (that it has, say, good slopes for skiing, low temperature that is good for snow, etc) through the transaction are priced, i.e. they are (somehow) evaluated and acquire a specific monetary value, usually as a whole, and not as distinct entities from the piece of land itself.

Assume now that the company, after buying the piece of the mountain, seriously delays building the resort, say, for some years. Should we count this asset as part of the capital base of the economy for these idle years? Sure: it is now "idle capital" not "potential future capital".

At the micro-level, the performance of the company must be judged against the total investment that it has made. If it has made an investment (i.e.it has spent company resources) that it doesn't exploit economically, this worsens the company's performance overall. This in turn affects the private sector of the economy on aggregate: there is an amount of Fixed Capital that "sits idle" while it shouldn't (and it shouldn't, because by buying it the company revealed a decision to use it productively, and has sacrificed resources for this decision).


Impact on interest rates of massive government spending

You could say IS-LM is too simplistic.

The fundamental discrete time consumption-based asset pricing equation says that if $ r_t $ is the risk-free interest rate at time $ t $, we have

$$ \beta (1+r_t) \frac{u'(C_{t+1})}{u'(C_t)} = 1 $$

(I'm dropping the expectation since I will be working with a perfect foresight model later on) where $ u $ is the utility function of any consumer, $ 0 < \beta < 1 $ is a discount factor which measures "impatience" of the same consumer, and $ c_t, c_{t+1} $ is the consumption of the same consumer in time $ t $ and $ t+1 $ respectively. The most elegant form of the relation is with the assumption of power utility $ u(c) = c^{1 - \gamma}/(1 - \gamma) $ with $ \gamma > 0 $, in which case we get (after log-linearization)

$$ r_t = R + \gamma \Delta c_t $$

where $ \Delta c_t $ is log consumption growth and $ R = -\log(\beta) $ is the real interest rate consistent with flat consumption. The intuition is simple: people want to smooth consumption over time. If they expect their consumption in the future to be much higher than their consumption today, they will try to borrow against the better future and push real interest rates up, and vice versa.

Consider a baseline economy with no investment, perfect foresight, and a representative consumer first. (This is unrealistic, but IS-LM is even more unrealistic, so...) In this case, the resource constraint of the economy reads $ Y_t = C_t + G_t $ in every period, where $ G_t $ is government spending. If you assume consumers have utility that is additively separable over time (the asset pricing relation assumes this as well), then the no investment assumption turns their decision problem into a set of one-period decision problems. (This is a technical condition to ensure what I am doing is well-defined.) Define $ D_t $ as the level of output consistent at time $ t $ with $ G_t = 0 $, and let $ m_t $ be the "fiscal multiplier", defined by

$$ m_t = \frac{dY_t}{dG_t} $$

Most models won't give constant $ m $, but if necessary we may linearize the model around somewhere other than $ G_t = 0 $ and make this assumption locally. These assumptions imply $ Y_t = D_t + m_t G_t $ at any time $ t $, so we may express consumption as $ C_t = D_t + (m_t -1) G_t $. Plugging into the relation gives

$$ r_t \approx R + \gamma \Delta d_t + \gamma (m_t -1) \Delta \left( \frac{g_t}{1 - m_t g_t} \right) $$

where $ g_t = G_t/Y_t $ is the ratio of government spending to GDP and $ \Delta d_t = \Delta \log(D_t) $. (Ignore the pole of the denominator - we're working around $ g_t = 0 $.) The fraction $ \frac{g_t}{1 - m_t g_t} $ is increasing in $ g_t $ with $ m \geq 0 $, so we see that the real interest rate at time $ t $ depends on two things: the "natural" growth rate of the economy given by $ \Delta d_t $, and the consumption growth induced by changes in government spending rel to GDP $ g_t $. However, the sign of the effect of an anticipated increase in government spending depends on the value of the fiscal multiplier $ m $. If $ m < 1 $, then government spending crowds out consumption, and rising government spending is associated with low real interest rates, i.e expectations of high spending in the future would push rates down today, and vice versa if $ m > 1 $. Most everyone would agree that the fiscal multiplier in the US economy today is less than $ 1 $, so news of higher spending in the future should be associated with lower real interest rates now, if with anything at all.

It also matters, of course, if government spending affects $ \Delta d_t $ - if we're speaking of an infrastructure investment program, for instance, then we should expect that the change in government spending also leads to a change in the path of $ D_t $. The two effects for such a program act in opposite directions, so the net effect of a government investment program tomorrow on real interest rates today could go either way, even in this simple economy.

I won't work out the model with investment (allowing consumers to invest) here, but the effect of investment is to allow consumers to trade consumption now for consumption in the future in aggregate; so it dampens the impact of a change in government spending on real interest rates (since now consumers have a tool for smoothing consumption in the face of government expenditure shocks).

It's also true that the announcement in the State of the Union address probably carried less information than an announcement in this model, since market participants were already aware of such an infrastructure investment plan for months (maybe a year?), but with a fiscal multiplier less than $ 1 $ it's not easy to say anything about the direction of the real interest rate response.


There are several exchange rate concepts that need to be distinguished. There are bilateral and multilateral (aka "effective") exchange rates, nominal and real exchange rates, and market-price versus PPP (purchasing power parity) exchange rates.

Let's start with the simplest concept: A bilateral nominal exchange rate, e.g., 117 yen/(US dollar), 0.8 pounds/euro, 7.7 (Swedish krona)/(Swiss Franc), etc. These are the exchange rates that are commonly reported in newspapers and in any number of online outlets.

Aside: There are two distinct quoting conventions for exchange rates: the "E" system (with the home currency in the numerator) and the inverse, the "R" system. A nominal exchange of 1.2 Swiss francs per euro is expressed in the "R" system from the point of view of a euro area resident, and in the "E" system from the point of view of a resident of Switzerland. The "R" convention is conceptually easier to work with, since an increase of the numbers of foreign currency units per unit of home currency corresponds naturally to an appreciation of the home currency. Using the "E" system, the opposite is the case. There seems to be little or no standardization with regard to quoting conventions of bilateral exchange rates. However, multilateral exchange rates are always expressed using the "R" convention, such that a larger numerical value corresponds to an appreciation of the currency, and vice versa.

A bilateral real exchange rate is the bilateral nominal exchange rate multiplied by a ratio of price indices of the two currencies. (How the ratio is taken depends on whether the currency pair is quoted in the E or R method.) Naturally, many different price indices can be used; most common, though, is the use of headline CPIs (consumer price indices). Because CPI series are generally monthly (or even lower frequency) time series, real exchange rates too are generally monthly time series. The home currency appreciates in real terms against a foreign currency either if the home currency appreciates in nominal terms or if the home country's inflation rate is lower than that in the foreign currency.
While bilateral exchange rates are straightforward to report, interest often centers on multilateral exchange rates. The home economy has trade and financial ties not just with one other economy but (usually) with many different economies. To form an index of the multilateral exchange value of the home currency, it's necessary to weight the various bilateral exchange rates according to the importance of the foreign economies. Again, many different weighting schemes are possible; a common scheme uses merchandies trade weights based on imports and/or exports between the home economy and the most important foreign economies. Because bilateral exchange rates are ratios of relative prices, an exchange rate index (aka a multilateral or efffective exchange rate) is computed as a Tornqvist index (a geometric rather than arithmetic average).
Up to now, we've mainly dealt with market prices for both nominal and real exchange rates. While that's perfectly OK for many applications, it's not the most helpful way to proceed if one wants to make comparisons of standards of living across countries. Purchasing power parity (PPP) tries to facilitate such comparisons by figuring how much a standard "basket of goods" -- frequently, but not always, the goods in the CPI baskets -- costs in various countries.

A famous case of a bilateral PPP exchange rate is the "Big Mac Index" of The Economist magazine: Since a Big Mac is pretty much the same in all countries where it's sold, one can work out the PPP-based bilateral exchange between the U.S. dollar (say) and all other countries as the exchange rates that would make the cost of purchasing a Big Mac the same. (Obviously, The Economist makes no claim that the Big Mac features prominently in anybody's consumption basket.)

By comparing nominal and PPP-based exchange rates it is possible to make statements such as "currency A is overvalued (or undervalued) against currency B". Of course, different choices of consumption baskets will lead to different PPP exchange rates. Hence, depending on the choice of basket, conflicting answers may arise as to whether a currency is over- or undervalued.

For more about the Big Mac index, check out the Economist Magazine's own website.

For a non-technical discussion of the dollar indices used by the staff of the U.S. Federal Reserve Board see, e.g, the 2005 article Indexes of the Foreign Exchange Value of the Dollar in Federal Reserve Bulletin.


(Edited)
Here is the working paper. Enjoy.

Rothschild, M., and J. E. Stiglitz, "Equilibrium in Competitive Insurance Markets," Technical Report No. 170, IMSSS Stanford University, 1975.

https://drive.google.com/file/d/1MXb3OcOQc_lxNYC4CzsNTTax9_3Q_4UN/view?usp=sharing


As regards dynamic convergence and stability, complex roots are associated either with a stable orbit around the fixed point, or with convergence to the fixed point by damped oscillations.

Check also https://wordpress.com/post/alecospapadopoulos.wordpress.com/197


Where can I find data on market capitalization?

Market capitalization of listed domestic companies (% of GDP) across years and countries from the World Development Indicators in the World Bank data website or the United Nations one.
FRED economic data on market capitalization.
Focusing on European Countries: the ECB website.


How Are The Interest Rates Affected By Inflation Changes?

I would note this similar question on interest rates and inflation.
I would summarise the answer there as follows.

Short-term interest rates are set by the central bank (in most developed countries; things are different in places with currency pegs).
Longer-term bond yields should roughly equal the expected path of the short-term rate (plus a term premium). One can debate this point, but it is a fairly standard assumption.

The answer therefore ends up: what is the effect of inflation on how the central bank sets policy? This is the subject of monetary economics, and is an open-ended topic that cannot easily be answered here.

In practice, we often see the central bank raising rates as the expected rate of inflation rises. Since they are worried about expected inflation, they do not automatically react to every wiggle in measured inflation. For example, a central bank may not react to an oil price spike. (Reactions varied during the 2008 oil price spike, with the ECB hiking, and the Fed not.)


Print money to buy and nationalise companies

This is a transaction where productive assets change hands, it is not a transaction that reflects production.

Moreover assume that the company oparates exactly as before, producing the same output in quantity and quality and with the same prices and with the same costs.

Then nothing has changed in the economy as regards production (and so income).

The only difference is that the previous owners of the company now have in their hand 4 billion pounds. What will they do with it?

Assume they just keep it in the bank. Then under fractional reserve banking, the bank will be able to offer more loans at a total principal value a multiple of the 4 billion. Will the bank attempt to offer more loans? If it does, will it have to offer lower interest rates to attract willing (and solvent) borrowers? The questions have no ending.

Assume instead that the owners try to spend the 4 billion by purchasing goods from domestic businesses. Is a 4 billion-increased-demand large enough to push prices up (is supply capacity at its limits and stock low) or suppliers will be happy to accommodate, no problem? The questions have no ending.

Assume instead that the owners attempt to import goods from another country...

My point is that such questions can in principle be answered only after assumptions as to the state of the economy have been clearly spelled out (i.e. only in the context of a model). Moreover, for a specific case study, the relative size of the transaction compared to the economic system matters. How is a 4-billion increase in government debt perceived by economic agents?

...and I haven't even touched and the "burning the profits" strategy proposed in the question (and it wouldn't burn it in any case, it would repay its debt to the central bank, which, granted, is the economic equivalent of destroying money).


Hint: To find equilibrium price and quantity, you want to find when the system is in equilibrium. In equilibrium, quantity demanded should equal quantity supplied.

$$Q_d = Q_s \implies -300,000P + 6,000,000 = 200,000 \sum_{i=1}^n (c_i'(q_i))^2 - 200,000$$

Where $c_i(q_i)$ is the cost function for firm $i$ with respect to $q_i$, firm $i$'s production. I'm assuming when you wrote $MC$ you meant marginal cost.

$$\implies 6,200,000 - 200,000\sum_{i=1}^n (c_i'(q_i))^2 = 300,000P$$

$$\implies P = \frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c_i'(q_i))^2$$

Now that we have an inverse demand function with respect to $q$, we have to solve the firms' maximization problem. Since there's no information you've given to us directly about that, let's assume technology is homogeneous and that firms compete via Cournot competition.

$$\Pi_j = q_j(P(q)) - c(q_j)$$

$$\Pi_j =  q_j(\frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c'(q_i))^2) - c(q_j)$$

Now take the derivative w.r.t $q_j$ and set to zero to find the maximum.

$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \sum_{i=1, i \neq j}^n (c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$

Where that derivative is evaluated with the product and chain rule.

From here, you solve for $q_j$ as an expression of all $q_{i, i \neq j}$. Since the cost functions are homogeneous, it shouldn't be that bad to do. Also remember that the number of suppliers is 1000, so the problem can be further simplified:

$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \cdot 999(c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$

$$\implies \Pi_j' =  (\frac{62}{3} - 666(c'(q_i))^2) + \frac{2}{3} (2c'(q_j) \cdot c''(q_j) \cdot q_j + c'(q_j)^2) - c'(q_j) = 0$$

and then you solve for $q_j$ in terms of $q_{i, i \neq j}$ and do the rest of the Cournot derivation.


It's true that given the utility function the $y$-good is a normal good, so the question is quite odd. Ignoring this, your calculations are correct, but you could simplify to $y(p_x,p_y,b)=\frac{b}{2p_y-p_x}$.

Your curve contains a mistake, since $x(1,3,48)=19.2$. Indeed by substituting your expressions for $x$ and $y$ you can show that $2x+y=\frac{b}{p_x}$ as long as $p_y\ge p_x$, so the price-consumption curve is piecewise linear.


This depends on the person you are asking.

Suppose an individual's utility function only depends on hotel rating scores and money (which is used to purchase other things). Then it is completely consistent for two individuals to prefer A=(9.8,m-10) to B=(9.6,m-8) (and vice versa). Thus, there is no value metric for all individuals. However, if you focus on one individual, you could ask how much cheaper/costlier you would need to make each hotel such that the individual would be exactly indifferent to A. This idea is captured by the concepts of equivalent and compensating variation. However, for this you would need to know a lot about each consumer.

A simpler measure of value (but less meaningful to each individual) would be to measure the proportion $p(A \succsim B)$ of individuals which would prefer A over B. To each individual, this number would of course be meaningless. Notice also that the numbers obtained need not be consistent across hotels. Indeed, $p(A\succsim B)> .5$ and $p(B \succsim C)>.5$ does not imply $p(A \succsim C)>.5$. This follows from the cyclicality of pairwise majority voting. Arrow's impossibility theorem tells us that under some reasonable assumptions, there exists no aggregation procedure of ordinal preferences. Thus, unless you are willing to make stronger assumptions on preferences (than most economists are willing to make), it will be impossible to measure the value of hotels.


The new "equilibrium" would have been at a lower price on the assumption that the firm has full Monopoly over the market (and hence cannot instead just compete with some other firm for a larger share of market demand) and demand remains constant.

Moreover unless you know about the elasticity of demand (how sensitive people's wish to buy is with regards to commodity price) you cannot be certain if selling a significantly larger amount at a slightly smaller price might in totality fetch the producer a larger sum of money.