There are some mistakes in your presentation of the Dornbusch model.

  $(∗)$ is the domestic (foreign) price level, $m_s$ stands for nominal money supply, $\bar{y}$ is potential output,

These actually stand for the log of price level, log of money supply and log of potential output. In international macro an unwritten rule is that lowercase letters represent logs of uppercase letters. For example, money supply is $M$ and $m = \ln M$. To be consistent you cannot add to the Dorrnbush model just government spending but you should add log of government spending. So $g$ instead of $G$:

$$y^d=h(e−p)+g$$

  From what I can gather, real shocks in the model won't lead to any changes in long-run prices. Indeed, if we plug this equation into the model, only $\bar{q}$ and $\bar{e}$ change.

The permanent government spending can in principle affect the price level through $\bar{y}$, but it depends on the background assumptions of the model. In Copeland, exchange rates and international finance (6th ed), you can see on pp207 that long run equilibrium values are derived based on arguing that in long run equilibrium $y^d =\bar{y}$. So in principle it could be affected by $g$ through $\bar{y}$ unless they are offset by changes in the other variables.

However, Copeland is not really clear on all background assumptions he makes in the model that he presents (at least not in the edition I have). For example, if we assume flexible exchange rate perfect capital mobility and the shock was completely unanticipated the $\bar{y}$ wont change in response to fiscal expansion as any change in $g$ will be just offset by changes to the exchange rate (due to capital inflows) and fiscal policy will only be able to expand output of its recipients at the expense of tradable sector in the economy. If some of the assumptions above are violated then the price level can actually change in the Dornbusch model.

In case you are interested in this for your own sake and you want to understand how price level responds to fiscal policy in general I recommend the following paper Devereux & Purvis (1990). Fiscal policy and the real exchange rate. They actually show that in the Dornbusch model fiscal policy can affect the price level and also discuss in greater detail how the sign of the effect depends on model parameters and assumptions of the model. If you are studying for an exam I would recommend clarifying this with your supervisor because Copeland does not provide detailed list of assumptions (at least not in the chapter 7.3 where he gives formal exposition of the model).


The purpose of part b is probably to illustrate what happens during hyper inflation: people spend there money immediately while they get anything for them.

Note that there is probably an implicit constraint that $c_2$ cannot be lower than 0, so the answer would be $c_1 = \$6, c_0 = \$0$.


Blackwell's Sufficiency Conditions for a Contraction:

Let $X$ be a nonempty set, $B(X)$ be the space of bounded functions from $X$ to $\mathbb{R}$, and $\|\cdot\|_\infty$ the sup-norm on $B(X)$. Write $f\leq g$ for $f,g\in B(X)$ if $f(x)\leq g(x)$ for all $x\in X$. Let $\mathbf{1}\in B(x)$ be the function with constant value $1$.
Here is the result of Blackwell:

Theorem: Let $T:B(X)\to B(X)$ be a function such that for some $\beta\in(0,1)$

Whenever $f,g\in B(X)$ satisfy $f\leq g$, then $T(f)\leq T(g)$ (monotonicity).
For all $f\in B(X)$ and $\alpha\geq 0$ one has $T(f+\alpha\mathbf{1})\leq T(f)+\alpha\beta\mathbf{1}$ (discounting).

Then, $\|T(f)-T(g)\|_\infty\leq \beta\|f-g\|_\infty$ for all $f,g\in B(X)$.

The following argument should make things a bit more transparent. Take $\alpha\geq 0$ large enough that both $f\leq g+\alpha\mathbf{1}$ and $g\leq f+\alpha\mathbf{1}$ holds. Since both $f$ and $g$ are bounded, this is certainly possible.
By monotonicity,
$$T(f)\leq T\big(g+\alpha\mathbf{1}\big).$$
By discounting,
$$T(f)\leq T(g)+\beta\alpha\mathbf{1}.$$
Therefore, $$T(f)-T(g)\leq \beta\alpha\mathbf{1}.$$
By switching $f$ and $g$, we get
$$T(g)-T(f)\leq \beta\alpha\mathbf{1}.$$
It follows that
$$|T(g)(x)-T(f)(x)|\leq \beta\alpha$$
for all $x\in X$,
which implies
$$\|T(f)-T(g)\|_\infty\leq \beta\alpha.$$
To finish the proof, simply note that we can choose $\alpha=\|f-g\|_\infty$.
Here, monotonicity guarantees that $T$ preserves the pointwise ordering of functions, and discounting guarantees that if we increase every coordinate of a function by the same amount, the resulting value of $T$ increases at most $\beta$ times this amount.
We want to use discounting even for two functions in which neither equals the other plus a constant amount. Monotonicity guarantees us that if one function is bounded above by another function plus a constant value, a similar bound can be used for the transformed functions under $T$. This makes discounting much more powerful.


Effort setting game - no idea where to start

This problem is similar to first-price (sealed-bid) auctions with independent private values for the buyers. You can read about those for exaxmple here.

Your first question was already ansewered in the comments: given others' efforts the probability of getting the reward is  or 1. Let us assume that $e_1 \geq e_2 \geq \dots$ are the other players' efforts and $e$ is own effort. Then
$$ \Pr[\text{getting }R | e, e_e, e_2, \dots] = \begin{cases} 1 & \text{if } e > e_3 \\ 0 & \text{if } e < e_3 \\ \dots & \text{otherwise} \end{cases} $$
What happens in case of ties is not exactly specified in the question, but as $\theta$ has a continuous distribution, if $e(\theta)$ is strictly monotone
edge cases happen with 0 probability. Based on this, you can also calculate what happens if you do not condition on other players' strategies (you should probably look into order statistics).

Now the player want to maximize the following:
$$ EU(e) = \Pr[\text{getting }R | e]\times R - c(\theta, e) $$

You can start by assuming that players follow pure strategies, so $e_i = e(\theta_i)$.
Also assume that effort is strictly monotone decreasing in $\theta$.
And last, assume that players have symmetric strategies.

Now you have to write down the expected utility defined above in terms of integrals. As of assumption 2. (and the convexity of $c$) you can differentiate it with respect to $e$ to get a (sufficient) first order condition. Then using assumption 3. and luck will get you a solvable differential equation, and hopefully you can find a closed form for $e(\theta)$. That is going to be your equilibrium strategy.

Or to be precise, one of the equilibrium strategies, as we have only looked at a subset (pure symmetric strategies). But finding others (or proving that none exist) is much more challenging.

I hope this is enough to start. If you get stuck, I will gladly expand this answer with some concrete formulas.

Disclaimer: I have not solved this problem, but as it is remarkably similar to the previously mentioned auction problems, this method should work.

There might be another way using the revelation principle. You could come up with a (second price auction-like) game where reporting the truthful $e$ is the dominant strategy, and has the same expected payoffs. Then you could relatively simply calculate the expected payoffs, and based on that, getting the strategies might be quite easy. I am not sure this method works either, but if you are interested I can look into it.


Does the amount of inward investments affect inflation?

Yes, FDI and Inflation are correlated.

Think of it this way, if foreigners buy assets, that reduces the supply available for domestic consumption, driving up prices, and if they buy them from citizens, it increases the domestic supply of money.

That being said, the relationship can be complicated because if there's substantial trade of real assets between foreigners, the effect might not trickle down to the domestic populace.

Ex 1:

  Alice and Bob are citizens of Country X. Charles is a citizen of Country Y. At equilibrium, sans FDI, Alice and Bob price the only pineapple at \$3. If Charles buys that pineapple for \$4 from either of them (assuming symmetric preferences for A & B), he has injected \$1 into the economy of Country X and removed 1 pineapple, consequently the supply of money has increased relative to the supply of assets, and prices should increase.

Ex 2:

  If Charles had left the pineapple in Country X, and David, a citizen of Country Z then bought it from him for \$5, and Charles did not invest any of that profit into Country X, the supply of capital in X would not be increased by this transaction, nor would the supply of assets, consequently the ratio of money to goods hasn't changed, and country-wide price levels shouldn't fluctuate.

Ex 3:

  Assuming there was a second pineapple that Alice or Bob held, they could then borrow against it at a higher value (given the new equilibrium price and FMV), leaving more money in the economy and consequently increasing the price of sponges for Bob as the ratio of assets to money would have changed in situ.

Note: This is an off the cuff explanation, not a formal model


Think about a player (female) that moves first and is considering the profitability of deviating. Let's say that according to the strategy she is supposed to play action $x$ and then the second player (male) is supposed to play action $a$. It will not be correct for the first player to think that if she instead chooses $y$, the second player will necessarily still play $a$. Remember this is a sequential game, so after seeing that she chooses $y$ instead of $x$ player 2 can also deviate to some other action,  say $b$. Therefore, by specifying the actions that each player will take in each node (even in the ones that are not supposed to be reached according to the strategy) we completely specify the strategy of each player, and can correctly check that no player has profitable deviations.

In fact, in many games, the only reason why the player that moves first chooses an action $x$ is because of the response of the other player after observing $x$, and the knowledge that choosing something else will also affect the decisions of other players, etc.

In a practical example, if you are playing chess and consider moving your queen to get one step closer to eat an important piece of the other player, you must anticipate they will react to this movement and protect their piece, or attack something more valuable of yours, etc. This is why when thinking about the strategy of the other player you must consider how different they will react depending on which piece you move next and to where.

BTW, chess is really hard to fully specify as a sequential game precisely because the number of possible nodes is massive, and properly specifying a complete strategy for each player has proven extremely hard. However, hopefully it gave you some insight of why actions and strategies are not the same things in an extensive game.


How can I know whether a good is inferior or normal? I can't determine elasticity with this?

Suppose we have a function of income for a particular good, $f(x)$ for income $x$. Then its derivative is the income elasticity of the good (at a certain level of income, though for linear models it's all the same). The consumption of a normal good grows with income while the consumption of an inferior good decreases with income: we have $f'(x) > 0$ for a normal good, $f'(x) < 0$ for an inferior good and $f'(x) = 0$ when the consumption of the good does not depend on income.
That the consumer income doesn't appear in the demand function means just that, that demand isn't affected by changes in income.


The Utility function here is the average score that Polly gets i.e.

$U(x_2,x_3)=\frac{max\{x_2,x_3,60\}+max^2\{x_2,x_3,60\}}{2}\\$

Where: $max^2\{.\}$ stands for second highest number.
Now this utility function can be rewritten as:

$U(x_2,x_3)= \begin{cases}\frac{x_2+x_3}{2}, & \text{ if } x_2,x_3\geq 60,\\ \frac{60+x_3}{2}, & \text{ if } x_2<60, x_3>x_2,\\ \frac{60+x_2}{2}, & \text{ if } x_3<60,x_3<x_2,\\ \frac{60+k}{2}, & \text{ if } k<60,x_3=x_2=k.\\
\end{cases}$

Plotting its level curve i.e. the IC for utility level $c$ would be a horizontal line from point $(0,2c-60)$ to $(60,2c-60)$, a vertical line from point $(2c-60,0)$ to $(60,2c-60)$ and a line from $(60,2c-60)$ to $(2c-60,60)$.

So if $(x_2,x_3)=(50,70)$ option 2 satisfies the IC's description.


How to find the elasticity of substitution for the general CES-function?

four things :

The desired result is actually
 which is the answer you are getting if you multiply the negative into the denominator( you can cross-check on https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution)
. However, you have made the following mistakes.
NON CONSEQUENTIAL ERROR: Your first error is when you differentiate to get MP1 and MP2. In both these cases, the long term should have power : (1/p - 1 ). This term gets canceled hence doesn't affect the solution.

Further MRTS(marginal rate of technical sub) = MP1/MP2 and not - MP1/MP2. This is because MRTS is the absolute number of factor x2 that has to be given up to employ one more unit of factor x1 while keeping the production constant( moving along the same isoquant curve). MRTS is always taken as positive. This is because the negative sign of the slope simply implies the "giving up" which is implicit in the definition of opportunity cost.
However, in both of your mistakes, your error got canceled due to the symmetry of the numerator and denominator. Hence you are actually getting the right answer.  Your method is a little long. I am posting here the way I do it :


Why is investment not a cost in firms profit functions?

Both in Economics and in Accounting, there is the following fundamental principal: we have to subtract revenues generated in a given time period from costs incurred in the same time period (because this is what makes basic sense).

Now, "cost" is the value of productive resources absorbed into production, in the given time period. The total value of an "Investment" is not absorbed in the usual time period where firms record their profits (month, quarter, year).
When we buy a machine, we use it for many years.
So only a part of "Investment" is absorbed in a period (essentially what constitutes an "investment" and what an "(operating) expense" is based on this criterion exactly -whether it is fully absorbed in production in one time period or not). The total value of Investment done in say one year is an outflow in the Cash-Flow statement, but it is not a Cost in the Profit and Loss statement.

And, as you mentioned, that part of the Investment that was absorbed in production during the period, is reflected in the calculated Depreciation, which is chraged as a cost to the Profit and Loss statement.


What would be the defining characteristic of New-Keynsian thought?

Gali's Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, provides an advanced undergraduate / first-year graduate student introduction to these models, and I'd recommend it for self studies. According to chapter 1, available online, which offers an overview of the  New Keynesian Model, the key elements are:

Structural characteristics of Real Business Cycle (RBC) Dynamic Stochastic General Equilibrium (DSGE) models
Monopolistic competition
Nominal rigidities
Short run non-neutrality of monetary policy


Data: Trade imbalance at the County level in the US

The data you're asking for doesn't exist.

Most data on things like this are captured through the taxing authority, which means that there are entities that want to collect taxes and therefore care very much where you do something and where your inputs came from. That's very true when there are international borders involved, and it's true to a much lesser degree when there are state borders involved, but in the US, there's generally no administrative interest in tracking the flow of goods across county borders.

Source: I used to create state and local area GDP estimates for the Bureau of Economic Analysis.


The existence of solution for profit maximization problem

A possible approach is to find a compact set $Z$ of inputs and show that the PMP has an optimal solution if and only if the PMP has an optimal solution in $Z$.
If so, we can replace the PMP by the following problem.
$$max_{z \in Z} \,\,p f(z) - w z.$$
If $f$ is continuous and if $Z$ is compact, the existence of a solution follows from the Weierstrass theorem.
An example of a sufficient condition for $Z$ to exist is to assume that $f(0) = 0$ and that there exists an input level $z_0$ such that for all $z > z_0$, $p f(z) - w z < 0$. In words, there is an input level $z_0$ such that having a higher input level will generate negative profits. Then we can set
$$
Z = \{z \in \mathbb{R}^{K-1}_+: z \le z_0\}.
$$
Notice that $Z$ is compact. In order for this to work we need to show that the PMP has an optimal solution if and only if it has an optimal solution in $Z$.
To see that this is true, first notice that $z = 0$ is a feasible solution for the PMP and it is also in $Z$. As such, the optimal solution to the PMP will always generate a profit that is larger or equal to zero, which means that no solution $z$ to the PMP will be outside the set $Z$.


Equilibrium price and quantity - consumer and producer surplus

Equating supply and demand we obtain the equilibrium

$$P^* = 75, Q^*=100$$

The corresponding diagram is

Consumer Surplus is the area of triangle $B-E-C$ so

$$CS = \frac 12 \cdot (100-75)\cdot 100 = 1250$$

Producer Surplus is the area of the triangle $B-E-A$ so

$$PS = \frac 12 \cdot (75-20)\cdot 100 = 2750$$


The $\beta$-$\delta$ model, otherwise known as the quasi-hyperbolic discounting model, is introduced as an alternative to the traditional exponential discounting which suffers the problem of being inconsistent with empirical evidence.

Exponential Discounting

Traditionally, economists use exponential discounting to capture the fact that things in the future has less value compared to things at present. Such a model presumes a constant discount factor $\delta$, which is used to weight utilities in different periods. For instance, the present value of one's utility from consumption over the next $T$ periods can be written as
\begin{equation}
u(x_0)+\delta u(x_1)+\delta^2u(x_2)+\cdots+\delta^Tu(x_T)=\sum_{t=0}^T\delta^tu(x_t),
\end{equation}
where $x_t$ is the consumption/investment decision made at time $t$. If we let $u(x_t)$ be the amount of money earned in period $t$, then the above formula is also the usual way to calculate the present value of a stream of income. From a theoretical perspective, the exponential discounting model has the desirable feature that it produces dynamically consistent choices. That is, choices deemed as optimal today will still be considered optimal at arbitrary points in the future.

Issue with Exponential Discounting

Despite the theoretical appeals of exponential discounting, evidence suggests that this model does not measure well against actual human choices. The classic example involves two sets of choices:

Getting \$100 today vs. Getting \$101 tomorrow
Getting \$100 30 days from today vs. Getting \$101 31 days from today

When presented with these two choices, most people would choose to get \$100 today but would choose to wait one more day for the extra dollar. This is a clear violation of exponential discounting. If people are exponential discounters with a constant discount factor $\delta$, then choosing \$100 today implies
\begin{equation}
u(\$100)>\delta u(\$101).\tag{1}
\end{equation}
But choosing \$101 31 days from today implies
\begin{equation}
\delta^{30}u(\$100)<\delta^{31}u(\$101)\quad\Rightarrow\quad u(\$100)<\delta u(\$101).\tag{2}
\end{equation}
The incompatibility between $(1)$ and $(2)$ suggests that exponential discounting is not a very good descriptive model of people's time preference.

Quasi-Hyperbolic Discounting

The above example illustrates that people have a strong "bias" toward current consumption, or receiving a reward today as opposed to any future date. In other words, the values of all future ($t\ge1$) rewards/utility are downplayed relative to the reward/utility of the current period ($t=0$). Therefore, based on the exponential discounting model, the quasi-hyperbolic discounting model introduces an extra parameter $\beta$ that captures this bias towards the present. The mathematical formulation is as follows:
\begin{equation}
u(x_0)+\beta\bigl(\delta u(x_1)+\delta^2u(x_2)+\cdots+\delta^Tu(x_T)\bigr)=u(x_0)+\beta\sum_{t=1}^T\delta^tu(x_t).
\end{equation}
The parameter $\beta$ is the present bias parameter, as it measure the extent to which future outcomes are weighed relative to the current one. Usually $\beta$ is assumed to be less than $1$, so that future utilities weigh less than the present one. But it is easy to see that if $\beta=1$, then we get exponential discounting as a special case where there is no present bias. With an appropriately chosen $\beta$, we can therefore reconcile the inconsistency in equations $(1)$ and $(2)$.


In a bank-based economy, how do banks determine how to split the economic pie with the businesses?

I would say that if you assume away competition among banks, then the interest rate would be equal to the rate of profit in the industry or firm considered. If there is only one bank in a position of monopoly as a supplier of loans, then it will set the highest possible rate of interest, and that is the rate of profit : firms can not pay higher interest without bankruptcy. Then of course you can argue that there is a decreasing marginal efficiency of "capital", therefore the rate of profit would depend on how much money firms borrow from our single bank, and things would be more complicated. But I think it might still be a good approximation for practical cases. And if there is no available experience on a new industry, then our bank has to guess about the future rate of profit.

If you assume perfect competition among banks, then the question is about the lowest possible interest rate, and that would depend on risk of default : banks have to gain some profits on most loans to compensate for the defaults. Again if you don't know about risk you have to estimate it.


Why Law of Supply indicate direct relation between price and supply?

You are confusing producer behavior and equilibrium.

If producers can sell at a price $p$ some of them will do so. Other producers cannot do so profitably at this price so the do not. At a price $p'>p$ more producers can produce profitably including everyone who was already selling at $p$ so total supply increases. In this story the price was given exogenously. It was simply stated that the price is $p$ or $p'$ and we did not look at the reasons behind this. We simply stated what producers would do when facing certain price levels.

If given a price $p$ supply is larger than demand $p$ is not an equilibrium price. (An exception: it can be, if $p=0$. Would you like to buy some air from me?) You can say that this price does not occur, or at least not for a long time. As you state prices will probably decrease. In this story we looked at whether a price was actually likely given the market conditions (demand and supply). We did not simply assess what each side would do facing a given price but we tried to see if the actions of the two sides constituted an equilibrium.


What exactly people mean when they say "X is a $Y billion industry"?

Considering the economic of world as a whole it could simply refers to that total spending by people on yoga might be \$8 billion. As in an economy total spending is the total revenue of a sector. For this particular scenario it simply means that yoga as an economic variable have transaction of \$8 billion.
Regarding the calculation we can de assured that no exact and absolute calculation could be made on national level how could it be calculated exactly at global level. However the estimated aggregated spending on yoga sector nation could be determined. Which further on processing might result to provied aggregate global spending on yoga as a figure, which is tentative not exact figure.


Forecasting vs econometrics, how is regression used differently?

In forecasting, we just want to predict what will happen. So assuming that there is a certain degree of stability in the relation, so that the relation is valid outside the sample, we estimate how past sales co-vary with education level (or whatever), and we say "so if we move into New Area "X", and apply the same sales strategy, since in "X" the educational level is "Y" we predict that our sales will be "Z", given the estimated co-movement relation".

In an econometric analysis, we certainly are interested in estimating the same relation, but if we want to perform also causality analysis, identifying assumptions become necessary and specific models arise, (like the "Differences-in-Differences" approach for example), designed to detect and assess causality.

By the way the first sentence in the "A" part of the question is "wrong in the other direction": the phrase "is not likely to represent a causal effect" is inaccurate: it is "as likely" to represent one, as it is not, since basic regression is "agnostic" about causal effects, it is just about detecting and estimating correlation.


Is economics a science based more on statistics (observations) rather than rigorous reasoning?

At the very foundation of sciences like Physics, rest relations and properties that they just are, no explanation can be given, but they have been found directly or indirectly to continue to hold, or at least to change at an inconsequential rate compared to the time-span (or space-span) of humans.

Moreover, although Physics laws are also idealizations of reality (and they hold under "normal circumstances", too, as regards for example temperature or pressure), it has been found that a) "normal circumstances" abound, b) these idealized versions suffer negligible variation in the real world for any of (or most) practical purposes and c) what happens under "special circumstances" is also stable, a complementary law in essence.

For Mathematics, the foundations are also arbitrary, but not being of this world but purely mental concepts, they can be whatever we define them to be -no reality will come crushing in to refute them (it is a different matter when we want to map abstract Mathematics to the real world).

In the Social Sciences, to which Economics belongs, the foundations contain relations and properties that have been found to change over time and over space, at rates much higher than physical properties change, and certainly within the time-span and space-span of humans. Plus, the observed variation around their idealized formulation is much larger.

This is what makes Social Sciences much more treacherous and harder than the Hard Sciences, and this is why progression towards more universally valid results is so much slower, or even, the arrival at a bouquet of results for which we will know with enough certainty to which observed relations do they correspond.

  "It seems to me that results in macroeconomics cannot be rigorously
  argued like the ones in mathematics or even physics."

Results in Economics can be fully rigorously argued, ever since Economics adopted mathematical symbolism to express itself. The issue is not with the process of argumentation, but with the premises one starts from.


How much money was distributed during quantative easing?

The easiest way to estimate the total amount of quantitative easing is to look at the change in the balance-sheet of the Federal Reserve because every bit of money put in through quantitative easing involved the purchase of assets which grew the central bank's balance sheet.

You can see in the graph above that the total growth is about \$3.6T. There are non-quantitative easing reasons for these reserves to grow over time so this should be treated as an approximate figure.

Additionally, if the proceeds of the QE are used to purchase still more assets those asset purchases might not be universally agreed on as QE. However, the quantity of QE was decided on as the amount of purchases to do taking into account these other issues. If the other considerations had lead to lower asset growth it is possible that the FRB would have done more QE, leaving the balance sheet unchanged. If so, the total change in assets during active QE could be regarded as the intended level of QE.


Most significant factors in the price of a goverments bonds

1) The following quote is a poor description of how to look at bond markets.

  Inflation is a bond's worst enemy. Inflation erodes the purchasing
  power of a bond's future cash flows. Put simply, the higher the
  current rate of inflation and the higher the (expected) future rates
  of inflation, the higher the yields will rise across the yield curve,
  as investors will demand this higher yield to compensate for inflation
  risk"

The statement that higher expected inflation will raise nominal bond yields is outright incorrect. It assumes that real yields are constant. We can look at the inflation-linked market to see that real yields are highly variable, and in fact have moved more than inflation expectations in countries like Canada and the United States in the past couple of decades.

Trying to change that to assuming that inflation expectations rise and we assume that real yields are constant is just a way of saying that higher nominal yields generate capital losses for bonds. Everybody knows that already.

A proper analysis needs to look very carefully at what we mean by "inflation."

Is it observed inflation? Which measure - e.g., headline or core (ex-food and energy)?
Is it breakeven inflation, which can be extracted from inflation-linked bond prices?
Is it a survey of inflation expectations? However, we need a survey of bond investors, not economists or consumers - and such surveys are not available.

However, we can look at any of these definitions of inflation, and find cases where nominal bond yields fell, yet the inflation measure rose. For example, an oil price spike can be associated with a recession, which causes the central bank to cut interest rates.

As was pointed out in a comment, one could argue that poor inflation performance could raise term premia. (Essentially, adding an inflation risk premium on top of existing term premia.) This is a plausible argument. However, it is not exactly in line with the quoted text. It is not just “inflation” itself that is the problem - we already have inflation - rather, it has to be a significant burst of unanticipated inflation. However, most central banks would probably react violently to a big burst of inflation, and the curve would likely be fairly flat. Hard to distinguish between a term premium and rate expectations in that environment. In any event, that story is quite different than the quoted text from the external article.

2) The statement:

  the funds rate is set to control inflation, the inflation expectations
  should be pretty much the same as short term rate expectations due to
  this.

is incorrect. Once again, it assumes that the real rate is constant. This is certainly not the case.

We can look at the Fed Funds rate (link to FRED database) and see that it has varied in a range from 0-5% since the early 2000s. Meanwhile, core PCE inflation (the Fed's preferred measure of inflation) has been stuck between 1-2.5% over that period. (link to FRED). Surveys of inflation have been similarly far more stable than nominal rates.

The only way to phrase this is to say that the central bank sets nominal interest rates at a level to keep inflation near target. Bond market pricing is heavily influenced by the expected path of those nominal rates -- although a term premium exists. However, if the inflation targeting is effective, there will be no link between observed inflation and nominal yields -- since real yields are being adjusted to keep inflation stable.

Take a look at the post-1995 history in Canada and the United States (and working from memory, the U.K., Australia, New Zealand...). Just plugging in "2%" (or sometimes 2.5%, depending on the inflation series) was a very effective "forecasting algorithm," and probably outperformed a lot of human forecasters. This did not mean that nominal yields were constant.


Is there a super term for goods/products and services?

I'm not aware of any such "technical" term in economics. Goods and services are referred to as goods and services in economics, period. From the example you give in the comments, it looks like you're trying to write an introduction/review of a business. In that case, not using a technical jargon, be it existing or not, would probably be better anyways. By the way, X = "offering" sounds pretty good to me.


Endogenous Growth in the case of externalities (Learning by doing)

You don’t explain what $w$ even is but I will assume it’s wage. The wage increases with the capital accumulation for several reasons. First, as the capital becomes more abundant the labor inputs become relatively more scarce and hence more valuable. Thus, also the price of labor relatively to capital should increase through higher wages.

Second, in most commonly used production function capital needs to be used with labor for example with standard Cobb-Douglass production function $AK^{\alpha}L^{1-\alpha}$ if you only use capital and no labor output is 0 no matter how much capital you employ. Hence using these kind of production function imply that when the capital stock increases, demand for labor increases as well putting more pressure on wages.

Another way how to think about it more intuitively is to think about labor productivity. It’s well established that wages vary positively with labor productivity and working with more capital makes people more productive hence increasing their wages.


You're on the right track here. You need to check every outcome for its potential to be a NE. You're correct in stating that outcomes (5,10) and (10,5) are NEs however you didn't identify that (5,5) is also an NE.

(5,5)
If player one deviates he receives a payout of 5.
If player two deviates he receives a payout of 5.

Therefore no player has an incentive to unilaterally deviate and (5,5) is a Nash equilibrium.

You're also correct that (5,5) is the dominant strategy solution. Just keep in mind that the dominant strategy solution will always be a Nash equilibrium but not all Nash equilibria are dominant strategy solutions.


How do Sharks make money on Shark Tank through equity alone?

Getting cash out is no different for sharks on Shark Tank than any other investors in private equity: they can make money back from dividends that are yielded by profits; or by selling on the shares to someone else; or by asset-stripping the companies - i.e. selling off their assets, and closing down the trading activities of the company.

As with most other equity investors, they own shares in a mix of companies. So to make money overall, they don't need to make money on every single company. They can accept that some will do badly (often losing them their whole investment), and some will do well, but overall, they expect to make a profit on the portfolio.

Indeed, in general, for investing in small companies with private equity, the expectation might normally be that most investments will yield the investor a 100% loss, but the few that do make a profit, make such a big profit that the overall bottom line is positive.

However, the sharks do have the huge advantage that every one of their companies has received significant free tv advertising, and has had the public endorsement of a supposed "smart" investor, which tilts the odds in their favour.


Purchasing Power Parity (PPP) and Law of One Price (LOP)

If PPP holds, so does the LOP.

The law of one price (LOP) is an economic concept which posits that "a good must sell for the same price in all locations". The law of one price constitutes the basis of the theory of purchasing power parity and is derived from the no arbitrage assumption.


Calculate the elasticity of substitution of Epstein-Zin preferences

$
\newcommand{\dd}{\, \mathrm{d}}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
$
Here is my solution. Let me know if anybody has a cleaner/clearer approach.

Consider a fixed (non-random)
 consumption sequence $C = (C_0, C_1,...)$.
Then, the elasticity of intertemporal substitution (EIS) is defined
as
$$
\text{EIS} = \left| \frac{\dd \ln(C_s/C_t)}{\dd \ln MRS_{s,t}}\right|
 = \left|\frac{\dd \ln \left(\frac{C_s}{C_t}\right)}
 {\dd \ln \left( \frac{\partial U/\partial C_s}{\partial U/\partial C_{t}}\right)}
 \right |.
$$
To compute this figure, let's start by
computing
\begin{align*}
\pd{U_t}{C_t} &= f_c(C_t, q_t(U_{t+1}(C_{t+1}^+)))  \\
  &= \frac{1}{1-\rho}
  \left( (1-\beta) C_t^{1-\rho} + \beta q^{1-\rho} \right)^{\frac{\rho}{1-\rho}}
  (1-\beta) (1-\rho) C_t^{-\rho} \\
  &= (1-\beta) f_t^{\rho} C_t^{-\rho}.
\end{align*}
Also,
\begin{align*}
\pd{U_t}{C_{t+1}} &=  f_q \cdot \frac{\dd q_t}{\dd U_{t+1}}
  \cdot \pd{U_{t+1}}{C_{t+1}}.
\end{align*}
It will be easier to compute these pieces in parts. First,
$$
f_q = \beta f^{\rho} q_t^{-\rho}.
$$
Next, consider $\frac{\dd q_t}{\dd U_{t+1}}$. This is simplified
by the fact that $C$ is non-random,
$$
\frac{\dd q_t}{\dd U_{t+1}} = q_t^\gamma U_{t+1}^{-\gamma} = 1.
$$
Finally,
$$
\pd{U_{t+1}}{C_{t+1}} = (1-\beta) f_{t+1}^{\rho} C_{t+1}^{-\rho},
$$
which follows from our earlier calculation. Thus,
\begin{align*}
\pd{U_t}{C_{t+1}} &=  f_q \cdot \frac{\dd q_t}{\dd U_{t+1}}
  \cdot \pd{U_{t+1}}{C_{t+1}} \\
  &= \beta f_t^{\rho} q_t^{-\rho}
  (1-\beta) f_{t+1}^{\rho} C_{t+1}^{-\rho},
\end{align*}
where $f_t = f(C_t, q_t)$ and $q_t = q(U_{t+1}(C_{t+1}^+))$.
Now, we can calculate
\begin{align*}
\frac{\partial U_t/\partial C_{t+1}}{\partial U_t/\partial C_t}
  &= \frac
  { \beta f_t^{\rho} q_t^{-\rho}
  (1-\beta) f_{t+1}^{\rho} C_{t+1}^{-\rho}}
  {(1-\beta) f_t^{\rho} C_t^{-\rho}} \\
  &= \beta q_t^{-\rho}
   f_{t+1}^{\rho}
   \left(\frac{C_{t+1}}{C_t}\right)^{-\rho}\\
\end{align*}
Now, let
$$
\frac{p_{t+1}}{p_t} = \frac{\partial U_t/\partial C_{t+1}}{\partial U_t/\partial C_t}.
$$
Then, taking the differentials,
$$
\dd \left(\frac{p_t}{p_0}\right) = \frac{\partial U_t/\partial C_{t+1}}{\partial U_t/\partial C_t}
  \left(\frac{C_{t+1}}{C_t}\right)^{-1} (-\rho) \dd \frac{C_{t+1}}{C_t}.
$$
So,
\begin{align*}
\frac{\dd\left(\frac{ C_{t+1}}{C_t}\right)}{\dd \left(\frac{p_{t+1}}{p_t}\right)}
\cdot \frac{\frac{p_{t+1}}{p_t}}{\frac{C_{t+1}}{C_t}}
  &= \left(
\frac{\partial U_t/\partial C_{t+1}}{\partial U_t/\partial C_t}
  \left(\frac{C_{t+1}}{C_t}\right)^{-1} (-\rho)
  \right)^{-1} \cdot \frac{\frac{p_{t+1}}{p_t}}{\frac{C_{t+1}}{C_t}} \\
  &= -\frac{\frac{C_{t+1}}{C_t} \frac{p_{t+1}}{p_t}}
  {\frac{p_{t+1}}{p_t} \frac{C_{t+1}}{C_t}} \rho^{-1} = -\rho^{-1}.
\end{align*}
Now, plugging this into the definition of the EIS,
\begin{align*}
\text{EIS} &= \left|\frac{\dd \ln \left(\frac{C_s}{C_t}\right)}
 {\dd \ln \left( \frac{\partial U/\partial C_s}{\partial U/\partial C_{t}}\right)}
 \right | \\
  &= \left|
\frac{\dd\left(\frac{ C_{t+1}}{C_t}\right)}{\dd \left(\frac{p_{t+1}}{p_t}\right)}
\cdot \frac{\frac{p_{t+1}}{p_t}}{\frac{C_{t+1}}{C_t}}
  \right| \\
  &= \rho^{-1}.
\end{align*}


Is there any economic explanation why industrialization first occured in Britain?

A detailed and thoroughly researched analysis of why Britain industrialized first and other countries only later may be found in Robert Allen's The British Industrial Revolution in Global Perspective.  The following quotes are from my own summary of his arguments in a blog post.

Had I been asked beforehand for answers to these questions, I might have listed the following features of 18th century Britain:

Large and accessible coal deposits.
Ingenuity of its inventors, exploiting the scientific knowledge of the enlightenment.
Relatively good governance by the standards of the time.
Rural land reform (enclosure) facilitating higher food production to support growing cities.

According to Allen, however, 1 and 2 were not unique to Britain, and 3 and 4 are dubious.  His analysis focuses instead on relative prices.  And his main conclusion is this: in 18th century Britain, a combination of relatively high wages and cheap energy from coal made it profitable to substitute steam power for labour, even though the steam engines of the time were very inefficient.  Allen presents detailed evidence showing that the wages of labourers in 18th century Britain were much higher than those in most of Europe, India and China, and comparable only with those in the Netherlands and parts of North America.   Only Britain had significantly exploited coal at that time, and the price of energy in British coalfield regions was the lowest in the world.

How had this situation come about?  The reasons are complex and Allen’s explanation goes back several centuries.  Prominent in his account are: Britain’s success in exporting woollen cloth, supported by improvements in sheep farming; its economic gains from mercantilism and empire; the growth of London beyond the point at which its energy needs could be met at reasonable transport cost by wood; the development (via what Allen terms ‘collective invention’ by London builders) of houses designed to be heated by coal; and the consequent stimulation of coal mining around Newcastle, from where London was supplied by ship.

The power obtained from steam engines per ton of coal increased tenfold between 1730 and 1850.  Steam power became profitable at progressively lower wage / coal-price ratios, and around 1850 was rapidly adopted in other European countries and the US.  These countries had not been without coal deposits, inventors or entrepreneurs, but profitability at their prices determined the timing of adoption. Since moreover they were able to adopt the latest and most efficient technology, they did not need to waste resources repeating Britain’s long experimentation with early steam engines.


Why does the government need to control interest rates?

why do we need government to control interest rates? Assuming banks
  are rational and want to make money, wouldn't banks tighten credit
  rules in the midst of an overheating economy?

I assume that by government you mean central bank. Otherwise, I agree with you by saying that governmental intrusion in monetary policy often creates inefficiencies and distortions.

The purpose of any [private] financial entity is to maximize its profits rather than the stability of the domestic currency. Thus, a central bank's primary goal is (or should be) to procure that currency stability, which it pursues by "printing money" and issuing bonds. The conventional term "control [of interest rates]", however, is a overstatement/misnomer from the standpoint that financial entities are not prohibited to transact with each other (or with the public) at lower or higher rates than the ones pursued by the central bank.

That being said, a central bank's pursuit of other goals in addition to --or in lieu of-- currency stability tends to impair its ability to achieve that primary goal (see also last paragraph of this answer). See, for instance, the Federal Reserve's assertions of its dual mandate of maximum employment and price stability. Notwithstanding Ben Bernanke's renown inflationary profile, his repeated calls in the midst of the Great Recession for Congress to enact reforms reflects his awareness that his monetary policy of essentially printing more money was improper, insufficient, and/or ineffective to address the core issues of the labor market. The vulnerability is more severe in those economies where a government exerts greater control on that nation's central bank.

  You could argue that banks are greedy, but at the same time, they're
  smart and have a long-term view. If a bank loaned money in an
  overheating market, they wouldn't be in business for too long.

Some financial institutions negligently or recklessly incur excessive risk, thereby causing severe repercussions when unexpected market conditions occur. Consider the Barings Bank. This entity was founded in the XVIII century, yet the lack of scrutiny of a strategy performed by one of its hitherto "star traders" caused its collapse in 1995.

My allusion to the Barings Bank diverts from your central inquiry, but it illustrates that an entity's longstanding prevalence in the economy does not disprove the necessity for a [purportedly disinterested] regulator or central bank to scrutinize anomalies that are potentially catastrophic.


What does it mean when we talk about "same basket" exactly for both GDP and CPI?

Same basket means same goods and services at factor cost or market price. Weighting method could be different.


For this question, its useful to work off of the equation of exchange.
Let's start with the equation:

MV=PY

M = the money supply, V = Velocity of Money, P = Price level, Y = real output

Log both sides

ln(MV)=ln(PY)

Split the terms up according to log properties

ln(M) + ln(V) = ln(P) + ln(Y)

Take the derivative of each term so that we have

dM/M + dV/V = dP/P + dY/Y

to define each of these, dM/M = the rate of change of the money supply(nominal), dV/V = the rate of change of the money velocity, dP/P is the inflation rate, and dY/Y is the real GDP growth rate.

Real Money growth is but a simple algebraic manipulation by doing as such:

dM/M - dP/P = dY/Y - dV/V

Real Money growth = dY/Y - dV/V

Real money growth must be correlated with real GDP growth. The money supply can only be said to have increased in "real" terms if the quantity of goods and services that the money supply can purchase has increased.

So we can see that if we wanted to have 0% real money growth, we would simply need to have the rate of change of the money supply be equal to the inflation rate, or alternatively the real GDP growth rate be equal to the rate of change of the money velocity.

However, we can also see clearly that a nominal money supply increase of 0% is likely undesirable. If dM/M is just 0, then a positive value on the right side, which is to mean that real GDP is expanding faster than money velocity is increasing, the result will be deflation.

As to policy implications, central banks usually no longer target monetary aggregates, as they tend to be unreliable and volatile. Central banks these days target interest rates or alternatively there is some interest in targeting a certain growth of the nominal GDP, which is just the MV of the MV=PY equation (or P*Y if you like). The main disadvantage of interest targeting is that the central bank can be said to have "run out of ammo" in a sense when it pushes interest rates down to the zero lower bound. This is to mean that the target rate can not go below zero, so the stance of the Fed cannot become more expansionary past this point. This is important as the stance of the Fed is derived from the position of the target rate relative to the natural rate of interest. If there is a shortfall in spending in the economy, and the natural rate of interest falls to zero, then The Federal Reserve can be said to have no power in this situation. Even if it were to drop interest rates to zero, the effect would be a neutral stance, which is highly undesirable in a recession. Recommended reading on the subject: https://www.brookings.edu/blog/ben-bernanke/2017/04/12/how-big-a-problem-is-the-zero-lower-bound-on-interest-rates/


Macroeconomics - Fiscal/Monetary Policy and the Price level

I believe that tax policies refer to the supply side (AS). Since no information is given about government expenses in a, b and c, I would only shift AD in response to the monetary policy actions in a and c, and the explicit increase in goverment spending in option D.


How can a free market be operated in a way that significant social and environmental costs are intrinsically covered?

As far as we can tell, no, there is no way to do it using only a free market.

Coase postulated that within a free market, agents would negotiate private contracts to internalise the externalities.

In reality, this didn't happen. It turned out that humans aren't perfectly informed hyper-rational beings. And as Coase himself had inferred, the transaction costs are typically too high.

There are well-documented ways in which markets have been adjusted for externalities: sometimes taxes or subsidies are sufficient; other times, prescription or proscription has been necessary.

There are even well-documented ways in which the commons have been protected, meaning that the destruction of them is no inevitable tragedy: Elinor Ostrom won the Nobel Economics Prize for her work showing how empowered stewardship by a group of custodians had repeatedly protected the custodians' commons.

So the relative efficiency of market mechanism versus planned mechanism is nowhere near clear enough to claim that "It would be far more efficient and sustainable if the tendency to externalize costs were managed by market mechanisms ... ". Each of the alternatives has inherent inefficiences: planned mechanisms have dead-weight costs. market-mechanisms have transaction costs and failures due to imperfect information and bounded rationality. There's a lot of evidence, particularly when it comes to the wicked problems you describe, that the market inefficiencies are far worse than the planning inefficiencies.


Notation of a Cobb-Douglas function printed in 1989

Seems to be the second one, so $$ Q_t = A_t*(K_t^\alpha N^\beta_t T_t^\rho). $$

Two clues:

This is the usual specification.
On the top of page 14 it is written that
$$
w_t = \beta Q_t/N_t.
$$
Given a competitive labor market you have
$$
w_t = \frac{\partial Q_t}{\partial N_t}.
$$
If you assume the production function above then you indeed have
$$
\frac{\partial Q_t}{\partial N_t} = \beta Q_t/N_t.
$$


In general a best response function returns a set of best responses. This can be seen in much simpler games than Cournot.

To give a degenerate example, if a player is always indifferent between their strategies, their best response function will always return the set of all their strategies.

When you have best response functions that give sets with more than one element, Nash equilibrium happens when each player's play is contained in the other player's best response set.


Unemployment and the Minimum Wage---what are the main counter-arguments to Card and Krueger?

Isaac Sorkin, a grad student at Michigan, has addressed this. Here is Miles Kimball blogging it, Isaac Sorkin: Don't Be Too Reassured by Small Short-Run Effects of the Minimum Wage, Jonathan Meer and Jeremy West: Effects of the Minimum Wage on Employment Dynamics, and The Economist—Destination Unknown: Large Increases in the Minimum Wage Could Have Severe Long-Term Effects. The main argument there is that previous work measures short run elasticities, which are less responsive than in the long run. You can surely find more by checking Sorkin's citations.


Does US gold reserves includes other counties' gold reserves?

No.

The $8,133.5$ tonnes of gold, i.e. $261,498,926.230$ fine troy ounces, of the U.S. Government owned gold reserve is detailed in a monthly status report produced by the US Department of the Treasury, with the latest figure being for August 31, 2016.

The weight and notional value of $\$11$ billion (based on a statutory rate of $\$42.2222$ per Fine Troy Ounce) have change by very small amounts in recent years, I would guess related to selling gold coins and buying stock to make them.  For example the total in January 2012 was $261,498,899.316$ fine troy ounces, a difference of about $27$ troy ounces from the current figure.

Gold belonging to other countries are stored in the US, particularly in New York as being convenient for the market and for transport. But this is not counted in the US Treasury report.


What is the name for a market where everyone is basically selling the same thing?

Such markets are said to be homogeneous goods markets or commodities markets. And the goods traded within them are sometimes known as commodities. From Wikipedia:

  "In economics, a commodity is an economic good or service that has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them. Most commodities are raw materials, basic resources, agricultural, or mining products, such as iron ore, sugar, or grains like rice and wheat. Commodities can also be mass-produced unspecialized products such as chemicals and computer memory."


How huge is the multiplicator of social welfare in current societies?

Keynesianism (what you're apparently basing this on) is incoherent and relies on arbitrary eigenvalues such as "autonomous consumption". In real life the multiplier is small and inconsequential.

However, the multiplier may appear close to one or slightly above simply because GDP increases the instant the government approves spending in public-subsidised enterprises, due to the way GDP is defined. For example, let's say the government spends $1 to subsidize Fannie Mae and this is counted as a consumption item (for whatever reason). GDP immediately increases by a dollar. Now Fannie Mae spends this on some irrelevant bureaucratic expense (buying mortgage derivatives, legal fees, financial services etc). GDP increases by a dollar again. So we just achieved a multiplier of two while having absolutely no effect on the real world.

As you can see, measured GDP is subject to all sorts of random problems that make any attempt to calculate the multiplier pointless.

Furthermore, strictly speaking, the "multiplier" is only technically supposed to apply to government consumption, and not transfers as you are asking about.


Is there a specific definition of short run and long run?

The actual length of time does not matter. Simply, in the short-run capital is fixed and labor is variable. In the long-run both capital and labor are variable (which gives firms enough time to, for example, build a new factory to match an increase in demand). It would be really fortunate if there was a general time frame, but that would vary from industry to industry. The long-run for, let say, Ford is a pretty long-time because they are very capital intensive. However the long-run for, lets say, a Clown Rental Service, is very short because it is entirely labor dependent.

Hopefully this answer helps.


Dynamic programming, optimal consumption-savings (finite horizon) problem

Your value function is as follows:
$$
V_t[w] = \max_{c_t \in[0,w]} \left\{u(c_t) + \frac{1}{2}V_{t+1}[\alpha(w_t - c_t)] + \frac{1}{2}V_{t+1}[\beta(w_t-c_t)] \right\}
$$
with the terminal condition
$$
V_{T}[w_T] = \max_{c_T \in [0,w_T]} u(c_T)
$$

So, we can solve this via backward induction. Clearly, at the final period $T$, since $u$ is monotonic, we consume everything, so $V_T[w_T] = u(w_T) = \ln(w_T)$.

Let us now consider one period previous, so period $T-1$. The value function is:
$$
V_{T-1}[w] = \max_{c_{T-1} \in [0,w_{T-1}]} \left\{u(c_{T-1}) + \frac{1}{2}V_{T}[\alpha(w_{T-1} - c_{T-1})] + \frac{1}{2}V_{T}[\beta(w_{T-1}-c_{T-1})] \right\}
$$
We already know what $V_T[\cdot]$ is, so substituting,
$$
V_{T-1}[w] = \max_{c_{T-1} \in [0,w_{T-1}]} \left\{u(c_{T-1}) + \frac{1}{2}u(\alpha(w_{T-1} - c_{T-1})) + \frac{1}{2}u(\beta(w_{T-1}-c_{T-1})) \right\}
$$
considering the case $u(\cdot) = \ln(\cdot)$,
$$
V_{T-1}[w] = \max_{c_{T-1} \in [0,w_{T-1}]} \left\{\ln(c_{T-1}) + \frac{1}{2}\ln(\alpha(w_{T-1} - c_{T-1})) + \frac{1}{2}\ln(\beta(w_{T-1}-c_{T-1})) \right\}
$$
Take first order conditions, and we can get optimal $c_{T-1}^*(w_{T-1})$.

Then we have solved what $V_{T-1}[\cdot]$ is! We can then do the same procedure by considering $V_{T-2}$. Repeat until we get to $V_0$.

EDIT to reflect comment.
You now know that $c^*_{T−1}(w_{−1})=\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1}$.
Plugging back into $V_{T-1}$, we have solved for the value of $V_{T-1}$,
\begin{align}
V_{T-1}[w] &= \ln(\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1}) + \frac{1}{2}\ln(\alpha(w_{T-1} - \frac{\alpha\beta}{1 + \alpha \beta}w_{T-1})) + \frac{1}{2}\ln(\beta(w_{T-1}-\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1})) \\
&=\ln(\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1}) + \frac{1}{2}\ln(\alpha(\frac{w_{T-1}}{1 + \alpha \beta})) + \frac{1}{2}\ln(\beta(\frac{w_{T-1}}{1 + \alpha \beta}))
\end{align}

Finally, lets now go to $T-2$. The value function is
$$
V_{T-2}[w] = \max_{c_{T-2} \in[0,w_{T-2}]} \left\{\ln(c_{T-2}) + \frac{1}{2}V_{T-1}[\alpha(w_{T-2} - c_{T-2})] + \frac{1}{2}V_{T-1}[\beta(w_{T-2}-c_{T-2})] \right\}
$$
We just solved what $V_{T-1}$ is! Plug it in, and repeat.

This expression will probably blow up quite quickly, so it will be a pain to solve analytically.


It is my impression that the correct expression for "Dixit-Stiglitz" preferences is

$$U=\left(\int_{0}^{1}c(\omega)^{\rho}d\omega\right)^{\frac{1}{\rho}}$$

which then can be seen as a continuous incarnation (in [0,1]) of, say

$$\left (\sum_{i=1}^na_i\omega_i^{\rho}\right)^{\frac{1}{\rho}}$$

with $c(\omega_i) = a_i^{1/\rho}\omega_i$.

In other words, a definite (Riemann) integral is indeed conceived as a sum of infinitesimally small rectangles, but it can also be seen as the continuous incarnation of a sum.

A formal link between an integral and a sum is provided by the Euler-MacLaurin formula.


How is project value enhanced by uncertainty in market payoff?

In general, if you own an option, holding fixed the expected value of the underlying asset and the strike price of the option, the greater the variance in the distribution of the prices of the option the (weakly) more valuable the option is. Here is a graphical example of a call option at maturity. They grey line indicates the value of the option for each value of the underlying asset where the strike price is \$5. The blue diamonds indicate the payoffs for values of the underlying at 4 and 6. If each are equally likely (p=0.5) then the underlying has an expected value of 5 and the option of \$0.5 (the lower green square). This distribution as a standard deviation of 0.5. Increase the standard deviation to 1.0 but hold the expected value constant at 5 by  changing the two underlying outcomes to \$3 and \$7 (orange squares). The expected value of this option is 1 (higher green square), so the increase in variance of the distribution of the underlying asset has increased the value of the option even though the expected value of the underlying is unchanged.

The reason for this is essentially Jensen's inequality. We can model the value of an option as a convex function $f(x)$ where $x$ is the underlying. The value of the option is $g(x) = E[f(x)]$ This implies $g(x)$ is increasing in the variance of $x$. We can see this because when we do the Taylor expansion of the function f : $$g(x) \approx E[f(x_0) +  (x - x_0)\cdot f'(x_0) +  (x - x_0)^2 \cdot f''(x_0)]$$

Because of the convexity, the second order term $f''(x_0)$ will be positive. The additive part $E[(x - x_0)^2 \cdot f''(x_0)] = E[(x - x_0)^2] \cdot f''(x_0)$ is increasing in the variance $E[(x - \bar{x})^2]$ so the value of the option is increasing in the variance.


How does the money supply behave when bank loans are repaid?

The money is removed when the loan principal is repaid. The actual point in the loan this occurs depends on the loan terms. For a typical compound interest rate loan, this means a small portion of the principal is repaid every month, and a matching liability deposit (money in the customers account) is removed.

For an interest only loan, this occurs at the end of the loan - assuming all the principal is repaid then.

Interest payments essentially circulate through the monetary system. They're deducted from the customer's account, recognised as income by the bank, and then paid out as some form of expense, e.g. salaries, rent, taxes, etc. They may also be moved into an internal account to provide required loss provisions on loans.

Loan defaults are also an expense, and this is the achilles heel of the banking system. If defaults on loans exceed the bank's loss provisions and profits from interest, then the bank will have to write off against its capital - and this interferes with the regulatory controls on lending, causing the money supply to shrink. In practice, central bank or government intervention is inevitable if this occurs.

As far as the relationship of principal to the money supply. Essentially the banking system relies on new lending always being sufficient to replace the money being removed. In most banking systems, new lending is typically in excess of loan repayment, and so we see the money supply more or less continuously expanding.


Can somebody explain to me why exactly people necessarily had to starve under "communist rule"?

I would like to just add to +1 answer of @fizz.
It is completely true that dictators often are self interested but the problem with communism is not only this but the fact that many ideas that are according to self proclaimed communists at the core of communism are are inherently inefficient.
For example, many communists want to get rid of private property (See the history of economic thought by Grant and Brue).
But a problem is that private property is an irreplaceable institution necessary for economy to prosper (see any economics textbook for example Mankiws Principles of Economics). In fact the famine in communist Russia is often attributed to farm collectivization which took the ownership of the farms from farmers and put them into hands of state where people had little incentive to take well care of them. This being said some level of economic output is attainable even without property rights, pre-historic hunter-gatherer societies existed without them, but it is generally agreed that they are necessary precondition for sustained economic development (See Acemoglu & Robinson Why Nations Fail). Also, lack of property rights is actually the most cited cause of our current environmental problems i.e. over-pollution, over-fishing etc  (again see Mankiw's Principles of Economics) as absence of property rights creates the tragedy of commons.
There reason why lack of property rights leads to loss of output and efficiency is that without secure property rights people cant be sure they will be able to use their output in any way. A farmer would not produce any output if all crops are expropriated.  Furthermore, due to tragedy of commons in absence of property rights people are locked in prisoner dilemma where everyone is incentivised to overuse resources before anyone else has chance to already collect them.  There are also other issues and this is only short and surface level exploration of importance of property rights as full explanation would be way beyond the scope of this answer but I encourage you to explore it more fully in the sources I provided and works cited therein.
Moreover, communists often believe in radical egalitarianism (Again see Grant & Brue). As Marx himself wrote “from each according to his ability, to each according to his needs.” Now there is nothing wrong with believing inequality is bad and trying to reduce it but what many self proclaimed communist fail to grasp is that inequality is what gives people incentive to put in extra effort, and hence any society that wants to increase its standards of living has to have some inequality. If someone works hard and gets the same pay as someone who slacks of then what’s the point? People soon learn it’s better not to put extra effort - and consequently many communist countries eventually result to some sort of forced labor which is again much less efficient than free labor where people are properly incentivized. This does not mean that every single action in an economy needs economic incentives (narrowly defined) there can obviously be instances of altruism because peoples' utility can take into account the well-being of others (as famously alluded to already by Adam Smith in his Theory of Moral Sentiments). Moreover, people also can intrinsically value some work in itself - if you have any hobby where you produce something that would be an example of that.  But it is generally agreed upon by profession that no large economic system can function without a system that provides an economic incentives for people to produce goods and services optimally.
Even the goal of the proletariat dictatorship, in the Marxist-Leninist tradition is supposed to be true communism that is property-less, moneyless (i.e. based on barter and communal exchanges), classless (i.e. there are no workers no owners no classes at all) and stateless (Marxism is in its hart anarchist movement as state is only seen as temporary apparatus that dies out once the true communism is achieved).
However, as already mentioned above property, inequality (which implies some sort of class) are important. Money is also important in economics as it provides a solution the issue of double coincidence of wants which otherwise wastes precious time and other resources, and economists agree state is necessary for provision of public goods and very helpful in solution to many problems that are based on externalities. So even if we could achieve the true communism it would probably fail to deliver high material welfare standards.
In fact the only time in human history where the sort of society that Marx thought communism would be was actually in existence, was in prehistoric hunter-gatherer societies which were always living at subsistence level.
Hence often unappreciated fact about communist regimes is that they would often not work even if their leaders would be angels. To give you analogy, no matter how a good person you are if you want to cure people using homeopathy you won’t help them.
However, you should also note that communism as mentioned by Giskards +1 comment is badly defined. In economics communism does not even have a definition - you won’t find the term in many modern economic textbooks. The term communism is so often used and misused that it is meaningless. Trying to define it is as trying to define a deity - nobody will probably ever agree on single definition. Here I use the term communism as a system that would describe the ideologies and economic goals of Soviet Union or Maoist China.


What is the main cause of differences in the costs of living?

The United States has had a common currency for over two hundred years.  The cost of living in rural Georgia has not equalized with that in New York City during that time.  If anything, they've diverged.

Land in New York City is more expensive than land in rural Georgia.  As a result, rent is more expensive in New York City.  This has a cascading effect.  Because residential rent is more expensive, wages need to be higher.  Because commercial rent and wages are higher, prices of products and services need to be higher.  Which makes wages need to be higher again.

Note that commodities will often have very similar prices even in these conditions.  This is especially true of things that can be ordered from outside the region.  Even something like a haircut might be similar just because of economies of scale.  In New York City, there are more people needing haircuts, so a moderate price may still produce more revenue.  And of course, there are places that charge much more for a haircut.

Theoretical differences can be caused by something as simple as preferences.  Or by history.  Or by a single industry.  Or by some combination of those.  For example, New York City has historically been a financial center and had the kind of night life that a large city can.  That's history, preference for an advanced night life, and a special industry.  The reasons in your hypothetical countries are also likely to be complicated and probably not entirely understood.

In the end, the most we can say is that people are willing to pay more to live in New York City than rural Georgia.  If you went around and asked people in New York City, they might give a bunch of different reasons.  One person has always lived there.  Another came to work on Wall Street.  Another tried several places and just prefers it.  Someone else came to be on Broadway.  Unfortunately, it's unlikely that we can point to a single economic reason unless we pick something bland like, "because they could."

Given two actual countries, we might be able to do a little better.  That's why I've been picking on the New York City/rural Georgia comparison.  That at least gives something to discuss.  The big point that I wanted to make is that your assumption that historical differences would disappear is incorrect.  Historical differences could persist.  Prices don't have to equalize in a common currency area.


What will happen to the world economy if the price of oil/Coal/natural gas/electricity and all energy sources drops to lowest?

There would be a short-term boost to profits, as a significant cost driver has dropped. There will be some substitution, as energy replaces other factor inputs (probably human labour in particular). In theory, overall, everyone collectively is better off, but redistribution effects will be significant and unpredictable.

There would be a massive shift in the geopolitics of energy: countries that are currently powerful primarily due to their energy supplies (e.g. Saudi Arabia) could become insignificant on the international scene.

If it's a fossil fuel, then there would be a catastrophic medium and long-term reduction in total global wealth owing to accelerated anthropogenic climate change. A significant chunk of global economic activity would have to be diverted to adaptation to a rapidly changing, highly random, highly variable, highly unpredictable climatic period. International refugee migrations would be of the order of a billion people or more. There would be a non-trivial probability that human civilisation would essentially cease to exist in any recognisable form.

If it's not a fossil fuel then, well, we're close to that point now: PV is around 2 (\$/€)cent/kWh for a big proportion of the world, and is expected to halve again over the next decade or so. So all being well, we'll live to see this. Note that the benefits will not be confined to one country, though: the resource is globally distributed.


How much heath expenditure is due to conditions that people are born with?

After reviewing the two sources linked to by the CDC (a Rand report on the impacts of comorbid chronic conditions and a brief breakdown of US health expenditures) the exact methodology behind the 90% figure is not at all clear to me.

What you are asking for would be a considerably more complex calculation. The most directly relevant study I'm seeing so far is an old one, "An Epidemiological Model for Health Policy Analysis" G. E. Alan Dever (1976). In the following table (p. 462), the author attempts to break down how four different causes (one of which is "Human biology") contribute to mortality from various kinds of diseases:

Notice that even for "congenital abnormalities", the contribution of human biology is only weighted at 79%. The rest is accounted for by the health care system, lifestyle and environment. The average contribution of human biology for all causes of deaths is estimated at 27%. This is not directly relevant to the question, but I think it illustrates how the author is thinking about the contribution of different factors to disease.

Taking in to account the relative prevalence of different diseases, the author uses these estimates to further project that about 7% of federal healthcare expendatures are the direct result of human biology, independent of other factors.

There may be more up-to-date studies out there with better methodologies, but I hope this is helpful as an example of how an expert might approach the question.


Why is ATC tangent to Demand in a Monopolistic or Mopolistically Competitive Firm Making Zero Profit?

A profit maximiser will aim to seek to maximise the difference between total revenue and total cost, providing this is non-negative

If there is a point at which this is positive, then this is equivalent to a point at which average total cost is below the demand curve, for example slightly to the right of $Q^*$ in your second diagram. At this point the firm will be making a profit, contrary to your assumption of zero profit.  So your second diagram does not illustrate the zero-profit case
Alternatively, if the demand curve is always strictly below the average total cost curve then all situations lead to a loss, so not a zero profit
That leaves the case where the demand curve touches the average total cost curve but does not cross it, as in your first diagram.  At this point there is zero profit and no opportunity to increase it.  If the curves are smooth enough, you might describe this as the average total cost being tangent to demand


Do general equilibrium models include money?

While many general equilibrium models do not need to model money to approach the questions that they would like to answer, there are many models that do include money to address questions that need money to be a relevant feature of the model. These models do it in a variety of ways -- some might be more relevant than others. I will try and describe two of the approaches briefly.

Cash-in-advance

When I think of a cash-in-advance (CIA) model, the first models that come to mind are Lucas (1982) and Lucas Stokey (1987). Models that take this approach have agents saving in assets and money, but they must finance any consumption solely through their money holdings (basically assets are less liquid than money which isn't the worst approach). There are many critiques of the CIA of which I find Wallace (1998) to be one of the more thoroughly presented.

New Monetarist

More recently there has been a push to use models of money that are less "ad-hoc" about how money works. Some foundational papers of this field are the Kiyotaki Wright papers (1989, 1991, 1993). These papers build a very basic model of money and show what role it plays in economics -- There is a good discussion on Randall Wright's Wikipedia page under the sub-title "Research contribution." While these models confirmed some of the reasons for money to exist in a model, they were quite simplistic and lacked many other features of the real economy. A monetary course I took labeled these types of models as "Generation 1 models of money." There was a group of papers that were known as "Generation 2 models of money, but these shared many of the simplifications made in Generation 1 (so I won't discuss them here).

Since this time Lagos Wright (2004) introduced a more complete framework (we referred to these as "Generation 3 models of money." One of the issues faced by these new monetarist models is that in order to solve the model, one must keep track of the entire distribution of money -- This is computationally infeasible unless the authors introduce a simplification in a different area of the model (in the Lagos Wright model, they simplify the form of utility in order to be able to compute the distribution of money next period). There is on-going research in this area and, in my opinion, is an open area for improvement in the literature.

References

Nobuhiro Kiyotaki and Randall Wright. "On Money as a Medium of Exchange." Journal of Political Economics. 1989.
Nobuhiro Kiyotaki and Randall Wright. "A Contribution to the Pure Theory of Money." Journal of Economic Theory. 1991.
Nobuhiro Kiyotaki and Randall Wright. "A search-theoretic approach to monetary economics." American Economic Review. 1993.
Ricardo Lagos and Randall Wright. "A Unified Framework for Monetary Theory and Policy Analysis." Journal of Political Economy. 2005.
Robert E. Lucas Jr. "Interest Rates and Currency Prices in a Two Country Model." Journal of Monetary Economics. 1982.
Robert E. Lucas Jr and Nancy L. Stokey. "Money and Interest in a Cash-In-Advance Economy." Econometrica. 1987.
Neil Wallace. "A Dictum for Monetary Theory." Federal Reserve Bank of Minneapolis Quarterly Review. Winter 1998.


What do you call a system in which the payer is not the decision maker?

This is the principal-agent problem or agency problem.  It's one of the most studied problems in economics over the last 40 years.  The Wikipedia entry offers a thorough summary.  One topic of research is mechanism design, which is how to design institutions to avoid or minimize agency problems.


How do interest rates affect the distribution of wealth?

Good question! But difficult to answer.

A) Typically, economists think of most interest rates as being set by markets. Under that assumption, you can't change rates by themselves because the market would bring them back to a value that equilibrates demand and supply of savings. Economists try to model how everything is in equilibrium with "general equilibrium" models. Of course, in reality, the fed does set an interest rate...

B) In the "life cycle model" of an individual's life, people borrow when they are young and need to study and create businesses, save when they are middle aged and draw down their savings when they are old. In that context, young people suffer from a high rate and old people gain from a high rate. However, its often the case that people invest in long-term bonds and so it's not really clear that they benefit from an increase in interest rates, because it pushes down the price of their bonds (and their houses too).

C) If instead the idea is to figure out what happens when the equilibrium interest rate is low, then it's good to try to imagine the scenarios when it is indeed low. Economists think that the real interest rate is low when economic growth is low and when there is little demand for capital from firms or a lot of supply of capital by savers. These two things have happened together many times, but they don't have to. For example, a disaster that wipes out a very productive asset, like port infrastructure, will lead to a lot of investment but not a lot of net growth. Similarly, an extra supply of savings should lead firms to invest more and should lead the economy to grow. But now in 2016, we are in a low growth, relatively low investment scenario.

D) It doesn't seem like the interest rate is the biggest determinant to the wealth distribution out there. Some evidence to think this way is that capital  markets have become very open, making interest rates very similar everywhere, but inequality remains very different across countries: Commonwealth + Europe+ Japan (low) vs. Russia,+ US +China + Latin America +Africa(high). The distribution of income seems to depend more on the link between political power and economic power that is the norm in many countries, some because of corruption, others by design. It also seems to be more dependent on the quality public education and other public services.

Some similar ideas are discussed in a sort of reasonable Forbes article here: http://www.forbes.com/sites/jeffreydorfman/2015/05/12/the-feds-low-interest-rates-are-increasing-inequality/2/#65fdfe4679cc


Question regarding to Government Debt in Williamson Textbook

Given the clarification from the OP in the comments, there appears to be a mix-up here.

We learn that the author explains as follows:

  This is an open market operation wherein the fiscal authority
  issues debt, and the monetary authority purchases some of this debt by
  issuing new money.

This sentence describes a situation where the monetary authority is separate from government. But if this were so, we could not have in the same budget constraint both inflows from bond issue and from increase of the money supply.

Namely, the budget constraint describes a different situation, one where the government is a single entity that has three options to finance itself:
1) Collect Taxes from the private sector
2) Issue bonds that are purchased by the private sector
3) Issue new money by fiat

In such a case, with fixed outlays in the left-hand side, more money issuance allows indeed for less debt.

I note the the author writes "the monetary authority purchases some of this debt". This complicates things a bit, it means that part of the debt is bought by the private sector, and then we could have a partial substitution effect, but then we would have to distinguish between "debt held by the monetary authority" and "debt held by the private sector", etc.


The utility function taken for estimating demand systems depends on several parameters:

The nature of theoretical models and definitions of the variables used in these models are quite different from the nature of data that is available to us for estimation. Standard microeconomic theory tells us that demand is obtained as a result of some form of utility maximization. Naturally, therefore, the structure of demand function would depend heavily on the structure of utility function used to specify the consumer's preferences. Therefore, when we are trying to estimate a demand system, we look at the data we have at hand and the structure of the data, the variables available etc. give us an idea of what kind of demand system would be suitable, and hence, what kind of utility function we should start with.

Sometimes, the nature of analysis we would like to conduct can only be done only with a particular demand system. And therefore, it is natural to start with a utility specification that yields that particular demand system. On other occasions, we might just want to "keep things easy" and therefore choose utility functions which yield "tractable" demand functions.  Stone-Geary is a good example of this. Stone Geary utility function yields what is known as the "linear expenditure systems". The analysis can then be carried out in terms of "product-wise expenditures", which is usually what is available to us in the form of data. CES-utility function is another such example.

Aggregation is also an important consideration. We might want aggregate the demand system and be able to say something about the properties of a more macro kind of economic entity. In that case, we don't really have many options, since aggregation requires very strict requirements. We then stick to some usual homothetic utility function (like Cobb-Douglas or Stone-Geary).

Finally, there are demand systems which have all the above listed nice properties. This is indeed what is usually done these days. Deaton and Muellbauer's (Linear/Quadratic - approximated) Almost Ideal Demand System (AIDS) is the state of the art demand system, inspired by the work of Stone and Geary on expenditure systems, as well as Gorman's work on aggregation. It's "easy" to estimate, requires lesser assumptions, has nice(est) aggregation properties, and can be made to fit usually available household-level survey data.

Hope it helps.


Does money neutrality mean that nominal GDP (output) changes following changes in money supply?

I looked at “Monetary Economics: Theory and Policy” by Bennett T. McCallum. On page 95, he describes what he refers to as “the Classical Model”, where the money supply is set outside the model (exogenous). In the Classical Model, if the money supply is changed, nominal values in equilibrium change, but ratios like wage/prices and real values do not. (The implication being that nominal GDP changes proportionally to the money supply, but real GDP is fixed.) He describes this as money neutrality, and it matches the description in the question.

I do not know how money neutrality can be defined for models where the money supply is determined within the model (endogenous).


How should one expand the information set in VAR models?

My recommendation is that you read through this paper from Bernanke, Boivin, and Eliasz. This paper introduced something called factor-augmented VAR. The gist is that you can use principle component analysis to do exactly what you're after. This method allows you to include hundreds of time series in the conditioning information. And FYI, this does allow one to solve the price puzzle with solid economic rationale instead of the ad hoc inclusion of a single time series.


Why are there infinitely many indifference curves?

Utility is constant for all points $(q_B,q_A)$ on an indifference curve. So there is a number $u_1$ such that
$$
\forall (q_B,q_A) \in IC_1: \ U(q_B,q_A) = u_1.
$$
Similarly there is a number $u_2$, such that
$$
\forall (q_B,q_A) \in IC_2: \ U(q_B,q_A) = u_2.
$$
If $IC_1 \neq IC_2$ then $u_1 \neq u_2$.


Pricing a European call option while absence of arbitrage is violated

I believe there is not a unique price. Say, instead of buying the option  you spent 0.5 on a half a unit of the asset $S^2_1$ This asset pays out $[0.4, 0.6, 0.8]$ which first order stochastically dominates the option. So, no matter your probability beliefs about the states,  in that setting you'd never pay $0.5$ for the option which pays less in every state. This suggests the right price is less than $0.5$. Similarly, buying $0.25$ units of the $S^0_1$ asset or $0.167$ units of the $S^1_1$ asset would likewise stochastically dominate the option. In fact, because for $0.375$ units of asset $S^1_2$, which costs on $0.375$, you can still have an asset that pays out $[0.3, 0.45, 0.6]$, it seems unlikely that the price could even be as high as $0.375$. Asset 0 implies a price below $0.4$ and asset 1 below $0.45$

Some python code to solve:

import numpy as np
S0 = np.array([[2],[3],[1]])
D = np.array([[1,2,3], [2,2,4], [0.8, 1.2, 1.6]])
X = np.array([[0.0],[0.2],[0.6]])
phi = np.dot(np.linalg.inv(D.transpose()), X)
print('The weights of the portfolio that replicates payoff X are: \n', phi)
P_X = np.dot(S0.transpose(), phi)
print('With a price: ', P_X)
print('Normalizing to pay a fixed price P_X for each of the three assets, what payoffs can you get?')
D_norm = D/(2*S0)
print(D_norm)
print('Notice that all three first order stochastically dominate the option for a price of: ', P_X)
print(D_norm - X.transpose())
print('Using each of the base assets, what\'s the minimum quantity that dominates?')
D_relative = X.transpose() / D
print(D_relative)
Min_dominating_fraction = np.max(D_relative,axis=1)
print('Minimum fraction of each of the assets that dominates X\n', Min_dominating_fraction)
P_Min_dominating_fraction = S0.transpose() * Min_dominating_fraction
print('At prices of: ', P_Min_dominating_fraction)
print('Therefore the option price should be less than: ', np.min(P_Min_dominating_fraction))

This code doesn't spell out the price of the option, it just show my calculations for the paragraph above. I believe the real price of this option would actually be zero if shorting is permitted. If you buy three units of asset $S^2_0$  and short one unit of $S^1_0$ you get an asset with payouts $[ 0.4,  1.6,  0.8]$. This position costs nothing to take, has positive payouts for all states, and first order stochastically dominates the option itself. Since it is possible to make a better than replicating portfolio at zero cost the price should be zero. Oh the insanity at work when arbitrages are present!


Can a rescue package be considered as a subsidy to the industry?

It can be considered subsidy, especially if you are asking for the quoted passage. Although looking at it directly the subsidy seems to de jure be on the consumption (this being said some or maybe even most of the benefit will be most likely captured by carmakers through the economic incidence of the subsidy).

For example, following Mankiw and Taylor, economics 3ed subsidy is defined as:

  payment to buyers and sellers to supplement income or lower costs and which  thus encourages consumption or provides an advantage to the recipient

Under this definition the policy you explicitly mention above would definitely qualify as a subsidy.

In different textbooks wording will change but not in any drastic way (to the best of my knowledge).

However, a rescue package itself is not an economic term. It is possible that the plan also includes policies that could not be considered subsidies as well.


Can we model risk with only probability?

I don't think it even makes sense to talk about risk without specifying the payoffs.
Take your two examplary gambles and suppose that $a=b=c=d=e=0$. In that case, there is no risk involved at all. It is like flipping a coin and if it lands Heads, nothing happens, and if it lands Tails, nothing happens. This would not even be considered a gamble.
Alternatively, let $a=b=c=d=0$ and let $e>0$. Then gamble $B$ gives you the possibility of free money. Similarly, let $a=b=c=d=0$ and let $e<0$. Then gamble $B$ simply is a possibility to burn money with some probability. However, these two gambles $B$ are very different.
Alternatively, suppose that all $a,b,c,d,e$ are events. The first four are associated with some payoff and in event $e$ simply gamble $A$ is played. Then gamble $A$ is in fact gamble $B$. Without a specification of the events, you cannot compare the two.


Are Microeconomics limited to one country?

It is my impression that this would  fall in Intrernational Microeconomics, which is one of the two main subfields of Interrnational Economics. Essentially it is the modern "International Trade", where the prevailing features of the modern era (less trade barriers so less room for state policy, larger trade flows so increased degree of activity at firm-level, large private global players) make it acquire more "micro" aspects compared to the older International Trade field with its preoccupation with customs and tariffs (although these still exist of course).

It cannot be categorized as a "macroeconomic" subject, because macroeconomics always looks things from a general equilibrium (and sometimes disequilibrium) perspective -but a general one, not partial. And a cross-section of markets of the same good is a fragmented but single market of the world economy.

You may be talking about whole countries, like OPEC countries, but you will essentially consider them as producers of a specific product, not as national economies.


Is the loan principal ever delivered in pieces over time?

Handing out the principal amount of debt gradually, in increments, is standard practice in investment loans extended by a bank to a corporation.

The rationale is clear : the corporation wants to make an investment, say a new factory. The whole plan is laid out and the cash flows of the pre-operational, construction period are also detailed, based on forecasts or specific agreements with the suppliers (contractors etc) that will be involved. The bank checks and accepts all these, but it also wants to see the corporation deliver (both the corporation's part in the financing but also as regards the actual build up of the investment).

So specific milestones are agreed between the two parties, and to each milestone a round of financing is agreed (these milestones become contracutal obligations, legally binding). Milestones usually include specific technical phases of the build up to be accomplished (say, the building's foundations are complete, or the power infrastructure has been installed, etc).

In this way risk is reduced without hurting the necessary cash flow for the project. It also reduces the interest burden for the firm.


Exam question implies that a perfect labour market can have wage differentials?

The question (implicitly) assumes that there are relevant skill differences in the population. As you say, it is true that in a perfect labor market there are no wage differences if workers can move freely across sectors.

If your course defined perfect labor markets as having homogeneity of labor and perfect mobility, then the given answer is wrong. However, typically these two factors are not necessarily a feature of perfect labor markets. That usually just means there is no market power on either side.

As the question asks about the case of different labor supplies, there can be no homogeneity of labor nor perfect mobility. Different labor supply is only possible if not everyone can do each type of work, meaning there are more available workers for certain jobs, thereby implying not all workers have the skills to do all jobs. Hence, to the extent that the question assumes supply differences are possible, it also assumes that workers cannot move freely across sectors. That is the reason wages do not equalize according to the answer.


You may be confusing money velocity with money supply. From Wikipedia, the velocity of money is "measure of the number of times that the average unit of currency is used to purchase goods and services within a given time period." There is indeed more M2 than M1 but the velocity of M1 is often higher than that of M2. The shared portion of M1 and M2 will have the same velocity: this is the same thing as saying the velocity of M1 is the same as the velocity of M1. However, the portion unique to M2 will have a lower velocity: for example, saving deposits are usually moved around less than credit in demand deposits (checking accounts). Hence, overall M2 will often have a lower velocity than M1.


Set the inequality $P>Par$, substitute the bond price equation, and then manipulate the inequality to find your result:

$$P>Par$$
$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\left(\frac{Par}{(1+y)^T}\right)>Par$$
$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)>Par\left(1-\frac{1}{(1+y)^T}\right)$$
$$\frac{c}{y}>Par$$

Let $c=r_cPar$ so that $r_c$ is the coupon rate (this is the fraction of the Par value that the coupon represents. Then the last equality is equivalent to

$$r_c>y$$

Therefore, the first inequality is true if and only if all of the following inequalities are. In particular, the last inequality gives the result you are looking for.


Why did Federal Reserve buy already-issued corporate bonds, ETFs, and agency commercial mortgage-backed securities?

Federal Reserve | Published July 10, 2020 | Full text published under the following link (with info-graphic):
https://www.clevelandfed.org/en/newsroom-and-events/infographics-library/smccf.aspx
FULL TEXT

The Fed’s Secondary Market Corporate Credit Facility, explained

America needs a bridge—a bridge that will get households, communities, and businesses over the unanticipated challenges created by the COVID-19 shutdown. In helping to build that bridge, the Federal Reserve, with authorization from Congress, has created and revived a number of rare lending programs, each providing targeted assistance to the needs of those impacted.

The challenge: Investors buy corporate bonds, which are essentially IOUs that companies issue when companies need money. While on the investors’ books, the bonds generate income as issuing companies pay interest and, after some period of time, the original amount financed. In the event an investor wants to sell off some of its bonds before they are paid off by the borrower, they typically can do so because many other investors are willing to buy corporate bonds. Current conditions have made investors less willing to buy such bonds, leaving those that want or need to sell the bonds without buyers. At the same time, the pandemic-related shutdown has caused lost revenues for many companies, some of which invest in corporate bonds, and that means more investors may want to sell off investments such as corporate bonds to turn them into cash they can use to pay their bills.

The response: The Fed is buying corporate bond-related assets so that investors that want or need to sell them off are better able to do so and get the cash they need.

The details: This work is intended to stimulate the marketplace in which corporate bonds are bought and sold after they have been issued. Corporate bonds are first issued by large businesses to raise funds. Investors buy the bonds, lending businesses the cash they need, and expect to be paid interest in addition to the original amount borrowed. In the secondary market, those initial investors sell the corporate bonds to other investors. The Fed’s purchases are intended to allow investors to buy and sell corporate bond investments in the secondary market as needed. Should losses occur, the US Treasury has invested $25 billion to cover losses arising from this lending.

The goal? To help ensure a reliable market for investors that need cash in order to pay off debt and sustain themselves until economic conditions normalize.

The Fed is committed to transparency and makes periodic reports and updates related to this lending available.

For more information: https://www.newyorkfed.org/markets/secondary-market-corporate-credit-facility

Federal Reserve | Published July 10, 2020


Can I use percentage change in CPI to calculate fare rate schedule?

Without anything better than a CPI, a CPI would approximate it. That said, I don't think it would be accurate. A better gauge would be a measure of the cost of living for the effected people. Vehicle maintenance and the like would be much higher than a typical person.


What does economics say about “starting from” pricing?

Glenn Ellison's "A model of Add-On Pricing" seems closely related to this issue.

He addressed the question 'why don't firms compete more aggressively to attract consumers by lowering the initial price?' The answer that he offers is that doing so induces an adverse selection problem because lowering the advertised price will disproportionately attract "cheapskates" who will not pay the add-on price for a more expensive version of the goods.


If I find the GDP deflator (i.e., price level) for a given time series, can I then find real consumption given nominal consumption?

GDP deflator deflates GDP. If you want to deflate consumption, you'd most likely want to use the consumer price index (CPI) to deflate. GDP deflator would take into account, for example, the price level faced by producers (PPI).