Calculate GDP by three different methods

There are three ways to define GDP:

Expenditure approach: The sum of all expenditures on final products. The only final products in this economy are cars; steel, machines, and tires are intermediate products. By this definition, GDP is 5000. Of this amount, 1000 is exported (NX), so 4000 must be consumed domestically (C). There is no information about I and G, so we must assume they're zero.
Production approach (value added approach). The output of the steel, machine tool, and tire sectors is worth 6500 combined (4000 for steel, 2000 for machines, 500 for tires). All output of these three sectors is "consumed" by the auto sector, whose output is worth 5000. Total sales of all four sectors: 11,500. GDP is obtained by subtracting sales of intermediate products from total sales: 11500 - 6500 = 5000. The car sector appears to be making a net loss of 1500.
Income approach (GDI: gross domestic income, i.e., sum of wages and net profits). By definition, one entity's expenditures is another entity's income. We have no information about income shares (wages, profits, losses) of the various sectors, but in the aggregate GDI = GDP. Therefore, total GDI is also 5000. Total domestic consumption C on final products is 4000. Thus, aggregate saving S must be 1000 (since GDP = C+S). Of this aggregate, 2500 is net saving of the household, steel, machines, and tire sectors, and -1500 is the dissaving of the car sector. For the non-car sectors, the only additional piece of information we have is that whereas the machine sector "consumes" 1000 in steel, its output is 2000. Thus, the entire net saving in this economy is due to the machine sector.


Why is fiscal deficit always shown as a percentage of GDP?

In short: yes, it very well might be and this has been argued by some (see, for example: http://www.forbes.com/sites/jeffreydorfman/2014/07/12/forget-debt-as-a-percent-of-gdp-its-really-much-worse/#29b8530f6e0c).

The use of Debt/GDP is the current international norm, but Debt/Tax revenue might very well be a better, more concise option. Using Debt/GDP might be because it has been common to compare countries to each other with GDP or GDP/capita and having GDP as a factor relates the debt to these measures.

It can also be argued that in the end the whole population of a country is responsible for paying their country's debts, so in an extreme situation it's not just the government's income which might be used to pay off debts.


Finding the % of the total consummation by category using elasticity and MPC

Notice that the elasticity can be written as

$$\frac{\Delta C}{\Delta Y}\cdot\frac{Y}{C}$$

So for category $1$ you have $$\frac{\Delta C_1}{\Delta Y}\cdot\frac{Y}{C_1}=0.5$$ and $$MPC_1=\frac{\Delta C_1}{\Delta Y}=0.1$$

This gives you
$$0.1\cdot\frac{Y}{C_1}=0.5$$
which by reordering yields to
$$\frac{C_1}{Y}=\frac{1}5$$

Similarly $$\frac{C_2}{Y}=\frac{3}{10}$$

Finally notice that $C=C_1+C_2=1000$ and $Y=2000$, hence we must have that $\frac{C}{Y}=\frac{C_1+C_2}{Y}=0.5$ which is satisfied since
$$\frac{1}5+\frac{3}{10}=0.5$$

(Since we have that $2C=Y$ and $100\%$ of revenue comes from $C$ is straightforward to see that the $\%$ of revenue associated with each category are twice this numbers)


Confidence Intervals for Elasticity in Simple Linear Regression

If I understand correctly, what you are proposing is estimating:
$$y = \gamma_0 + \gamma_1 \cdot x + \xi $$
and then using the confidence interval of $\gamma_1$ to estimate the range of elasticities:
$$ [\frac{\gamma_1 - 2 \cdot \sigma{\gamma_1}}{y} \cdot x, \frac{\gamma_1 + 2 \cdot \sigma{\gamma_1}}{y} \cdot x] $$

This is wrong, essentially because of Jensen's inequality ($E[f(x)]\neq f(E[x])$). You should absolutely take into account the non-linear uncertainty when calculating the confidence interval. Fortunately, this is easy. If you run the regression estimate of $$\ln(y) = \beta_0 + \beta_1 \cdot \ln(x) + \epsilon$$
then the confidence interval of $\beta_1$ is the confidence interval of the elasticity.

In follow up discussion, the questioner asks if it is possible to do this with the linear regression coefficient ($\gamma_1$), along with $\bar{x},\bar{y}$ and the associated standard errors? Why would you want to know the elasticity at the average values instead of the average implied elasticity over all x-y pairs? The beta parameter is estimated on the full population so why not the elasticity too? Because all we only know the average values and the linear regression coefficient.

In principle, you can do a Monte Carlo simulation like the following:
$$ \hat{elasticity}= (\hat{\gamma} + SE_{\gamma} * N(0,1)) * (\bar{x}+ SE_{\bar{x}} * N(0,1)) / (\bar{y}+ SE_{\bar{y}} * N(0,1))$$
do a bunch of simulations, and use this to make a confidence interval around the elasticity estimate.

In simulations that I ran, this did result in 95% standard errors that included the true value, but that might simply be a function of the parameters I tried. Unsurprisingly, the resulting standard errors on the elasticity are much larger than in log specification.


preference convexity and existence of equilbria

Consider an economy with two commodities. Production is trivial, $Y=\{0\}$, there is a single consumer with endowment $(1,1)$ whose preferences are represented by the utility function given by $u(x_1,x_2)=\max\{x_1,x_2\}+1/2\cdot x_1 +1/2\cdot x_2$. These preferences are continuous, strictly monotone, but not convex. You can verify that there is no equilibrium in this economy by showing that for no price system, the demand of the consumer equals the supply of $(1,1)$.


What happens to Fed's balance sheet when its asset shrink in value?

When the Fed issues say 100k in liabilities (paper dollars) to buy 100k in assets (t-bills), then they will at the end of the year take their revenue from their assets, subtract their expenses and remit the profits to congress.  This is an obscure process but it does exist.  If you see a congressional budget, you'll see an item marked as miscellaneous revenue...this is a portion of if that.  Now they do not remit as much as they should but that is another story.

If the Fed had to write off say 50k in bad debt, then that would count against their income for their year (and mean they earned 50k less).  The 50k would actually be debited away from equity and not liabilities.

The Fed (that I know of), does not issue money directly to itself nor the treasury.  It can create money to buy investments...and then spend the profits from investments on their own expenses.  It can also sell equities for a profit (rarely done) to boost their equity.

A good question is what would happen if the Fed had to write off more in assets than they had in equity.  I doubt they would shrink the monetary base to balance the books, but instead would fabricate an asset to keep the illusion going (like say a 'good will from the treasury').


Why do different countries have different currency?

Two reasons: history and ability to control the money supply.

History: paper-money started by being issued by trustworthy agents from kings/governments, as a way to be able to transfer the ownership of gold without moving the gold physically. This looked something like "This paper is exchangeable for 10 gold coins at this location". This meant you had to go to that specific kingdom to get the paper (bill) converted into gold. These papers evolved into what we now call banknotes, and kept tied to a specific country.

Money supply control: having your own currency allows governments to fund themselves by producing more money, or stimulate exports by devaluing their currency. Look at the euro-zone difficulties with the government debt of Greece: this could be solved (with consequences, of course) if they had their own currency.


If I understand your question properly, it is correct that we do more ordinary statistical analyses such as linear regression to determine relationships between economic variables (e.g. price and demand.)  Utility is a tool which gives us a model of consumer behaviour based on very loose assumptions about how consumers behave (convexity of preferences, monotonicity, etc.)  While the demand curve can be derived from utility if we somehow knew the full preferences of all consumers, it can also be approximated from data by observing the real-world relationship between price and demand.  It is very difficult/impossible to measure utility in the real world, so using regular statistical techniques like linear regression to estimate the relationship is the logical thing to do.

Of course, the real-world demand curve may not be linear, so a simple linear regression on observed data in the form $p = \beta_0 + \beta_1 q$ may not accurately the demand curve.  Here, theory may help you decide on what nonlinear terms you should include in your model, but you would still need to do the regression analysis on real-world data to actually find the parameters.


Estimating CES utility (not production) function parameters

It may be interesting to exploit the homothetic separability of the CES utility function in $x$. It implies that
$$\frac{x_i}{x_j} = \left( \frac{\alpha_i}{\alpha_j}\frac{p_j}{p_i} \right)^\sigma $$
and after $log$-transformation:
$$\ln(x_i) - \ln(x_j) = \beta_{ij} + \sigma (\ln(p_j) - \ln(p_i)). $$
After adding a random term, this specification could be used to estimate the $\beta_{ij} \equiv \sigma (\ln(\alpha_i)-\ln(\alpha_j))$ and $\sigma$ parameters by OLS and in a second step identifying the $\alpha_i$ by minimum distance:
$$ \widehat{\alpha} = \arg \min_\alpha \Big(\widehat{\beta}-\widehat{\sigma}(\ln(\alpha)-\ln(P\alpha)) \Big)'\Omega^{-1} \Big( \widehat{\beta}-\widehat{\sigma}(\ln(\alpha)-\ln(P\alpha)) \Big), $$
where $P$ represents the adequate permutation matrix.
Some references estimating parameters of CES preference parameters include Diewert and Feenstra (2017) or Redding and Weinstein (2020), with a different approach, however, based on the unit expenditure function.
Most empirical contributions reject the validity of homothetic utility functions (like the CES). There are some propositions on how to built nonhomothetic CES production functions that could easily extended to utility functions, see Shimomura (1999) and the references therein. A simple extension of the above specification is:
$$\ln(x_i) - \ln(x_j) = \beta_{ij} + \sigma (\ln(p_j) - \ln(p_i)) + \gamma_{ij} M/p + \varepsilon, $$
where $p$ denotes a consumer and time specific aggregate price index (over all commodities). Notation $ij$ stands for commodities $i$ and $j$ and not for the observations (I skipped the $n,t$ subscripts for clarity).
This relationship is compatible with nonhomothetic preferences. The homothetic case is obtained for $\gamma_{ij}=0$ which can be tested.
Regarding the computer software... I fully switched to R few years ago, it would "easily" allow to code the minimum distance estimator, and the link between the utility and demand functions.
References:
Diewert, Erwin and Robert Feenstra (2017), “Estimating the Benefits and Costs of New and Disappearing Products,” mimeo, University of California at Davis.
Redding Stephen J and David E Weinstein (2020), "Measuring Aggregate Price Indices with Taste Shocks: Theory and Evidence for CES Preferences," The Quarterly Journal of Economics, 135, 503-560.
Shimomura, K., 1999, "A simple proof of the Sato proposition on non-homothetic CES functions," Economic Theory, 14, 501–503.


Determining subgame perfect Nash equilibriums

Just for sake of acknowledgement, please note that the game described in the question is a variation of the famous Ultimatum game. Knowing this can help you get a ton of literature on such games.
Further note that your professor has made an extremely important point that coming up with answer is sufficient, solving is not necessary. My answer is also limited to showing that a given action profile is equilibrium (whether NE or SPNE). Solving games (such as these) is different ballgame altogether (something in which I have no expertise).
For Nash Equilibrium:
To check whether a given action profile is an NE or not, it suffices to show that each player's response is the best response (BR) given other players' actions. Now consider the set of action profiles:
$$X := \{(x_1,x_2,x_3) \,\,|\,\, x_1 + x_2+x_3 = 100\}$$
For any $x \in X$, we can see that pay-off for each player is $200$ and no player can do any better by changing their action, given other players' action. Hence all action in $X$ are NE (note here that since this game is sequential and not simultaneous, we are not considering mixed strategy profiles).
The problem is that intuitively, this does not seem reasonable to us because for player 1, offering anything more than $0$ doesn't seem smart.
This is where the refinement of NE, Subgame Perfect Nash Equilibrium comes in:
In SPNE, the equilibrium should be an NE for each subgame of the game as well. This puts some restrictions and is thus a smaller set. In the above game, since player 3's best response is to play $100-x_1-x_2$ (for example, if player 3 insists that she'll play $0$ if others don't pay $33.33$ each, it is really a non-credible threat because it would be irrational for her to play like that), the best response of player 2 and player 1 becomes, $0$ for each.
Therefore, the only subgame perfect NE is $(0,0,100)$


Why do we treat demographic variables as being exogenous?

You are completely correct that mathematically endogeneity is defined $\text{cov(X,e)}\neq0$, which is true not only when it’s correlated with omitted variable but also with measurement error or reverse causation/simultaneity.

However, every single variable in economics can be subject to correlation with omitted variables or measurement error, so if you would want to use the word for any variable that potentially has this problem then it would be meaningless. So the word is reserved for those cases where this is actually a serious issue.

Furthermore, often endogeneity is used more narrowly as to referring just to simultaneity or reverse causality - variables determined by the model as opposes to variables outside the model - because that’s where the term originated.

Also some demographic variables can be definitely subject to endogeneity, but I would also say most of them aren’t. Consider race for example in a regression where you try to estimate let’s say how it affects the employment. Well you can’t pick your race (at least to my understanding technology is not there yet), so there is no possibility for simultaneity. Next race would be simple categorical variable so there is not much space for measurement error and also I am not biologist but from my understanding biologically there is only human race so social construct of race should be orthogonal on unobservables like innate ability. Similar story would hold also for sex (although you can have sex change that is usually not done on a whim but when person actually experiences gender dysmorphia so it can be treated as predetermined at birth) and age.

An example of endogenous demographics variable could be number of children in family that could have simultaneous relationship between wages/wealth and could be also correlated with some unobservables.

Sometimes it can be also context dependent. In model where you regress Y on X there can be endogeneity there but in a model where you regress S on the same X it might not be there because conditions are not satisfied. In your research you should always think hard about this issue and decide on case by case basis. But in general demographic variables  just happen to not be endogenous often when comparing to other types of variables.


What are some of the good textbooks on Modern Monetary Theory and the Development of Monetary Theory?

You ask for sources on MMT and then sources on development of monetary theory. I don't know about the former (but I know there are some users here who are fans of MMT so hopefully they will hook you up with some good sources) but when it comes to the latter then a classic text that covers development of contemporary monetary theory is Walsh Monetary Theory and Policy (The MIT Press).
It is probably one of the most comprehensive textbooks that cover monetary theory all the way  to the most recent advances. It unfortunately does not cover MMT because it is not an accepted theory in profession but has very good treatment of development of monetary theory all the way from canonical models to present day state of the art models.


How can I solve for total money spent if I use a non-constant MPC?

I don't know that the distribution you claim is wrong but it strikes me as having a couple of very undesirable features. For one, virtually the entire population all of the has >0.99 MPC, which seems very high for a distribution you think is centered around 3/4. For another, if you think that the income shocks in your model are transitory rather than permanent, my reading of the literature is that these are way too large. But that's all separate from your question about estimating things conditional on the actual MPC distribution and ordering.

Ordering from low to high MPC seems to take a strong stance on how trade works in the economy. Again, I don't know that is wrong but it doesn't strike me as true. If we think that MPC is caused by income then this is akin to saying the richest guy get the money first, spends some it at the second richest guy's store and saves the rest, he spends the spending of guy 1 at the third richest guy and so on. The economy strikes me as less neat than that and so a random ordering makes more sense to me in the absence of evidence about where high vs. low MPC people spend their money.

I tried messing around with simulations (in Python) to see how well we could approximate these random chains of consumption. In general, experimenting a few parameter values with uniform, beta, and your distribution above, distribution doesn't matter so much in determining total consumption effect. Using the geometric or arithmetic means of the MPC distribution give nice approximations of actual total consumption effect from the random chains of consumption.* On the other hand, if you order the households by MPC from high to low or low to high this makes a huge difference in the resulting total consumption effects.

* - That said, I explicitly ruled out distributions of MPC where MPC can be zero (or less). If that happens the geometric mean will be zero and the resulting approximation using the geometric mean quite poor.

Code:

# Python 3.x code for stack exchange question
# http://economics.stackexchange.com/questions/3123/how-can-i-solve-for-total-money-spent-if-i-use-a-non-constant-mpc
import numpy as np

# Parameters
households = 5001
sims = 10000
initial_shock = 5

# Alternative marginal propensity to consume populations
MPC_vec_lin = np.linspace(0.01, 0.2, households)
MPC_vec_beta = np.random.beta(2, 5, size=households)

# THis is the function Gragas asks for
nvec = np.arange(0,households)
MPC_vec_gragas = (nvec + 2) / np.sqrt(nvec**2 + 5*nvec + 7)

MPC_vec = MPC_vec_gragas

# Calculate geometeric and arithmetic mean MPC from the population
approx_geo_mean = np.mean(np.log(MPC_vec))
approx_long_run_eqiv_MPC = np.exp(approx_geo_mean)
avg_MPC = np.mean(MPC_vec)

# Assuming that all households had the same MPC (@ geo / arithmetic level), what is the total effect?
approx_total_shock_with_geo = initial_shock / (1 - approx_long_run_eqiv_MPC)
approx_total_shock_with_mean = initial_shock / (1 - avg_MPC)

print('Initial shock size:', initial_shock)
print('Approximate total effect with geometric mean:', approx_total_shock_with_geo)
print('Approximate total effect with arithmetic mean:', approx_total_shock_with_mean)

# Simulating many possible orderings, what is the total effect for each ordering?
total_shock_vec = np.zeros(sims)
for i, item in enumerate(total_shock_vec):
    MPC_vec_random_order = np.random.choice(MPC_vec, replace=False, size=households)
    random_products = np.cumproduct(MPC_vec_random_order)
    total_shock = np.sum(initial_shock * random_products) + initial_shock
    total_shock_vec[i] = total_shock
# Compare resulting average and std. of total effect from simulations
print('Average total effect from simulations:',  np.mean(total_shock_vec))
print('Std Dev of total effect from simulations:',  np.std(total_shock_vec))

# Conclusion, mean and geomean do a very nice job of approximating the resulting total effect without knowing the
# general specific ordering. However, the worst and best total effect ordering (low to high and high to low) are
# quite different.

MPC_vec_low_to_high = MPC_vec.copy()
MPC_vec_low_to_high.sort()

MPC_vec_high_to_low = MPC_vec_low_to_high.copy()
MPC_vec_high_to_low = MPC_vec_high_to_low[::-1]

product_low_to_high = np.cumproduct(MPC_vec_low_to_high)
total_shock_low_to_high = np.sum(initial_shock * product_low_to_high) + initial_shock

product_high_to_low = np.cumproduct(MPC_vec_high_to_low)
total_shock_high_to_low = np.sum(initial_shock * product_high_to_low) + initial_shock

print('Maximum effect ordering effect size:', total_shock_high_to_low)
print('Minimum effect ordering effect size:', total_shock_low_to_high)

Results:

Initial shock size: 5
Approximate total effect with geometric mean: 6023.35708426
Approximate total effect with arithmetic mean: 6154.37897874
Average total effect from simulations: 5999.77721767
Std Dev of total effect from simulations: 642.715838738
Maximum effect ordering effect size: 16675.8496051
Minimum effect ordering effect size: 777.157515589


Question on profit maximization with government taxation

ii.

I don't think the demand function in (i.) applies for this question. Here's what I did:

By Lerner Index, $\displaystyle{\frac{P - MC}{P} = \frac{-1}{E_d}}$

Since $E_d = -3$, we get $P = \frac{3}{2} MC$

We know before tax MC is $c'(q) = \alpha$, and after tax MC is $\alpha + 6$

In this sense, MC increased by 6, so P will probably increase by 9


Download original spreadsheet from Reinhart-Rogoff "Growth in a Time of Debt"

I know this was a while ago now but here is the excel file.

UPDATE: As somebody highlighted this is not the original 2010 file, but it does include the key raw data and some helpful instructions.

Here is the link to the video of where I go through the process step - by - step: https://www.youtube.com/watch?v=ItGMz0ERvcw

Hope this helps,
EBW.

Excel file


Does the Coase Theorem account for malicious externalities?

The problem you raise (you call it "malicious externalities") is perhaps better known as the extortion or blackmail problem. Medema (2014) reviews the history of this critique.

Those who raised the extortion critique.

Kneese (1964):

  an industrial enterprise may find it profitable to adopt processes which generate much waste in order to be able to accept payment for reducing discharge.

Wellisz (1964):

  any activity can be turned to profit as long as it is sufficiently annoying to someone else.

Kneese and Bower (1968):

  The administrative authority would have to stand ready to make payments to industrial plants which never do locate in the area but which would do so if a payment reflecting the costs their effluents would impose were not made. Payments on this basis would of course be an open invitation to extortion.

Rothenberg (1970):

  If external diseconomies against others can be expected to lead to bribes by victims to desist, then the production of negative externalities becomes a valid by-product of primary production. Profitability is enhanced whenever any firm can select from among its inputs and/or output alternatives those which cause substantial damage to third parties. Resource use will tend to become specialized toward much-augmented third-party interference. The new legal industry of selling protection against disturbance will be highly profitable.

Shoup (1971)

  Once it became apparent that polluters could extract payments for pollution reduction, there would be an incentive for entrepreneurs to threaten to undertake production activities with external pollution costs simply to be bribed not to undertake them.

Mumey (1971) makes the distinct point that contrary to some defenses of Coase against the extortion problem (below), in a frictionless world with costless communication, the problem of extortion or "malicious damages" is actually exacerbated.
Daly (1974) makes another distinct point:

  the simultaneous existence of competition and of small numbers of externally affected and effecting parties are mutually inconsistent phenomena. If the large numbers generally associated with competition are present there will exist the acknowledged prohibitive transactions costs. In the absence of such competition, however, market solutions are burdened with complex, game theoretic bargaining situations which involve a substantial likelihood of counterproductive (costly) activity and for which the achievement of efficient outcomes cannot be assured.

See also Daly and Giertz (1975, 1978).

Those who defended against the extortion critique.

Demsetz (1971):

  Under competition the price of a good, or of an agreement, is held to its cost. ... In the competitive regime, rivalry between “extortionists” would push to zero the price they receive not to undertake these excessive activity levels. Thus, competing “extortionists” would offer to refrain from extortionist activities for smaller, and ultimately for zero, payment.

Also:

  The concept of extortion is legal, not economic. All transactions are transfers of rights involving exactly what Shoup has called extortion with the possible exception of perfectly competitive ones, whatever they are. All transactions are preceded by attempts to gain information about the other party’s reservation price. A variety of techniques are used to do this. From the viewpoint of economic analysis, none of these differ. The use of the legal term “extortion” in economic analysis is likely to result in confusion and to hide implicit value judgments. There are some wealth transfer activities which, as citizens, we would seek to discourage through legal arrangements. But until we have formed a clearer notion of the criterion to be used to separate these from other “approved” wealth distributing activities, the use of terms such as “extortion” in economic discussions is laden with difficulties. Thus, if without any more valuable alternatives, Professor Shoup were to threaten to leave the University of Michigan unless he were paid more, thus threatening to impose real costs on that institution by depriving it of all the advantages of employing him, would that constitute an analytical use of the word “extortion”?

See also Demsetz (1972, 1978).

Zerbe (1971): extortion "does not occur ... because of high transactions costs, symmetrical possibilities of extortion [One man’s threat to raise chickens is countered by his neighbor’s threat to raise pigs], and the operation of the legal system in holding certain forms of extortion illegal [With costless bargaining, undoubtedly my neighbors would pay more for me not to grow pigs than I could make growing pigs. Hence, the law forbids my raising pigs in Chicago and in most cities]."

Zerbe also raises a separate point similar to Demsetz's second point:

  Extortion does occur, however, and it is not necessarily undesirable. The general economic rule governing extortion would seem to be that extortion rights should go to those who are willing to pay the most. This rule illumines the close relationship between extortion and market transaction. Indeed it is difficult to separate extortion from other production activities. Payments required to attract factors of production are also payments to ensure that these factors will not do something else. Consider Shoup’s case of the polluting oil company that threatens towns A, B, and C with location in them. This situation is similar to the case of a desirable company, say a computer company, also choosing among towns A, B, and C. The computer company chooses the town that offers the deal with the least amount of extortion money. In this way both end up in the right location. In this sense extortion is a method for achieving results and may or may not be desirable, depending on what other methods are available and whether or not extortion interferes with more desirable methods. In actuality, towns generally offer companies tax and other incentives and disincentives that reflect the town’s desire for the company. Here the cases become wholly symmetrical. Both the oil and computer companies choose the location for which the location price, ceteris paribus, is the least. At some price, the oil company becomes attractive to a town. It is not surprising then that there is a problem in distinguishing between legal and illegal extortion, and at this point an extensive legal-economic discussion of extortion is called for but is provided neither by this note, by Shoup, nor by my original article.

Jaffe (1975): With zero transaction costs, we will always arrive at the most efficient allocation:

  imagine that B can increase the destructive potential of his resource by physical investment. We further assume that the degree of destructive potential is an increasing function of physical investment. This function is known to both A and B.5 As B can now increase the potential of his resource, he may be able to exact an even greater bribe from A. However, applying the same rationality assumption used by Coase, no additional investment in the resource will actually be undertaken by B. Merely B's potential to create a more destructive resource should be enough to force A into a more generous bargain.

Coase (1988) himself:

  When I referred to the problem of blackmail in "The Problem of Social Cost," it was simply to show that, however rights were defined, there would always be opportunities for this type of blackmail. I pointed this out in the section of the article in which it was assumed that transactions were costless and in which therefore no resources were absorbed in bargaining. In such a world, it is not clear what the objection would be to this type of blackmail. The position is, however, very different if we make the realistic assumption that transaction costs are positive. It is obviously undesirable that resources should be devoted to bargaining which produces a situation no better than it was previously.

There is no such thing as the Coase Theorem.

Or as Coase stated repeatedly, including in this 2012 interview:

  "It's not about my work at all!"

Coase's goal was to illustrate that:

  institutions matter, and that understanding how and why they matter, and the implications of this for economic analysis and policy may require a re-orientation of the ways in which economists practiced their trade. Here, too, transaction costs occupy center stage. (Medema, 2014.)

Most of all, his goal was to direct economists' attention to the world of positive transaction costs.†

Ironically, economists became fixated with the simplistic world of zero transaction costs, where they could churn out a variety of clever "Coase Theorems" and avoid having to study the messier real world of positive transaction costs.

What I — and probably also Coase, Medema, and McCloskey — would say is this:

The extortion problem is worthy of our attention and analysis. However, we should study it not in the context of the economist's utopian world of zero transaction costs, but in the context of Coase's very real world of positive transaction costs.

We should study how and why, given the very real fact of transaction costs, institutions ameliorate (or fail to ameliorate) the extortion or blackmail problem. I think Coase (1988), Zerbe (1971), and Demsetz (1971) went some way towards initiating this discussion, but there is much more to be done.

† D. McCloskey has also repeatedly made the same point and stated the Coase Theorem (1985, p. 336) as:

  In the presence of transaction costs the location of a pollution tax or of other liability for damages does matter for efficiency.


Does GNI change once it has been published?

GNI is GDP plus net primary income from abroad

Any such numbers are subject to revision as more information is obtained; in many countries such numbers are typically revised every three months as part of a National Accounts and Balance of Payments update, with more substantial revisions annually.  Korea's latest update was published on 28 March

The fact that you did not see 2017 data the first time and did see 2017 data the second time was a pointer to these two readings being across such a revision point

Such numbers are typically never final, and potentially keep on changing, though after long periods revisions to numbers then tend to be affected more by methodology changes than by the arrival of new information


Monopoly firm's block pricing strategy making max. profit

The reason why the profit maximising condition is like that is because the monopoly is Price Discriminating between those who are willing to pay at  $P_1$ and $P_2$ for $Q_1$ and $Q_2$ respectively.

This increases the area of producer surplus to an area greater than what it would be by charging $P_1$ and $P_2$ alone.


Is it appropriate to change the base year of the real GDP to a more recent one?

Changing the base year of a time series is fine if you know what you're doing, check, for example the question: How do I change the base year of real GDP using the GDP deflator and nominal GDP?

I'd, personally, use the real GDP series - you won't gain anything substantial with the mathematical conversion. It might be an useful exercise if you, for example, were to compare GDP growth in two different countries and would like the base year to be the same.


Stable marriage problem with transferable utilities

There's two implementations that should somewhat cover your needs, matchingR which does reduced-form algorithmic matching and matchingMarkets, which estimates estructural matching models with some baysean tools to correct for endogeneity.

The problem is you're looking for some very specific models which may require some tweaks on these implementations.


How to deal with Prescott's formulation of time to build in his original RBC model?

By re-indexing all investment projects in the form of $s_{J,t}$, I get the following two equivalent constraints
\begin{align}
 c_t+y_{t+1}-y_t&\leq f(k_t)-\sum_{i=1}^\infty\varphi_j s_{J,t-(J-j)} % Resource Constraint
 \\
 k_{t+1}&=(1-\delta)k_t+S_{J,t-(J-1)} %Law of motion for capital formation
\end{align}
Hence, the Lagrangian for the maximization problem is
\begin{multline}
\mathcal{L}
 = \sum_{t=0}^\infty\beta^tu\left(c_t,1- \alpha_0n_t - \eta(1-\alpha_0) a_t\right) %Objective Function
 - \sum_{t=0}^\infty\lambda_{1t}\left(c_t+y_{t+1}-y_t+\sum_{i=1}^\infty\varphi_j s_{J,t-(J-j)}-f(k_t)\right) % Resource Constraint
\\ - \sum_{t=0}^\infty\lambda_{2t}\left(k_{t+1}-(1-\delta)k_t-S_{J,t-(J-1)}\right) %Law of motion for capital formation
 - \sum_{t=0}^\infty\lambda_{3t} \left(a_{t+1}-(1-\eta)a_t-n_{t}\right) % Law of motion for past leisure
\end{multline}
where the state variables at time $t$ are $a_t, k_t, y_t$ and the decision variables at time $t$ are $c_t, n_t, s_{Jt}, y_{t+1}$. It is easy to see that all constraints must bind.
The necessary conditions are
\begin{align}
 &\frac{\partial \mathcal{L}}{\partial c_t} =0\implies \beta^t \frac{\partial u}{\partial c_t}=\lambda_{1t}
\\ &\frac{\partial \mathcal{L}}{\partial s_{J,t}} =0\implies \sum_{i=1}^J\lambda_{1,t+J-i}\varphi_i=\lambda_{2,t+J-1}
\\ &\frac{\partial \mathcal{L}}{\partial k_t}=0\implies \lambda_{1t}\frac{\partial f}{\partial k_t}+\lambda_{2t}(1-\delta)=\lambda_{2,t-1},
\end{align}
which holds for all $t$.
Combining 14,15,16 and simplifying, we have
\begin{equation}
 \frac{\partial u}{\partial c_t}\frac{\partial f}{\partial k_t}
 + (1-\delta)\sum_{i=1}^J\beta^{1-i} \frac{\partial u}{\partial c_{t+1-i}}\varphi_i
 = \beta^{-1}\sum_{i=1}^J\beta^{1-i} \frac{\partial u}{\partial c_{t-i}}\varphi_i
\end{equation}
In steady state, $\frac{\partial u}{\partial c_t}=\frac{\partial u}{\partial c_{t'}}$, so we have
\begin{equation}
\frac{\partial f}{\partial k_t}
= \sum_{i=0}^{J-1}\beta^{J-i-1} \varphi_{J-i} \left(\frac{1-\beta}{\beta}+\delta\right)
\end{equation}
One observation is that since in steady state, no borrowing can occur between household and firm, the real interest rate $r=\frac{1-\beta}{\beta}$. We see that $\sum_{i=0}^{J-1}\beta^{J-i-1} \varphi_{J-i} $ is the price of the capital in steady state.


My understanding is that the gravity equation can be derived from different settings.

You can assume that each country

1/ produces only one product. This is knwon as the "Armington assumption" as in Armington (1969) and allows a simple derivation of the gravity equation as in Anderson and Van Wincoop (2003).

2/ is specialized in different products (or varieties of a product, given that each variety of a good counts as a distinct product). This is the case worked out by Feenstra, where products are differentiated by their country of origin. So, each country produces many distinct products.

Then the question is to know if the number of products is fixed, as in Armington (1969), or endogeneous and varies due to free entry, as in the monopolistic competition models.

References

ANDERSON, J. E., AND E. VAN WINCOOP (2003): “Gravity with Gravitas: A Solution to the Border Puzzle,” The American Economic Review, 93(1), 170–192.
ARMINGTON, P. S. (1969): “A Theory of Demand for Products Distinguished by Place of Production,” International Monetary Fund Staff Papers, 16, 159–178.


Tax revenue formula maximization- whats the name of this formula?

This seems to be just a spin on Ramsey's rule. By Ramsey's rule (derived through Lagrangian method assuming zero cross-elasticities) the optimal tax for a market where supply is not explicitly  modeled is given by:
$$\frac{t}{1+t}=\frac{1}{\epsilon}$$
Well technically it is $\frac{t}{1+t}=\frac{1-b}{\epsilon}$ but $b$ which is parameter that depends on marginal cost of public funds is often omitted/assumed to be zero in introductory problems for students.
The Ramsey rule above can be solved for $t$ as:
$$t = \frac{\epsilon}{\epsilon-1} \left( \frac{1}{\epsilon}\right)$$
The version you showcase is just derived with also explicitly taking into account supply as opposed to just from consumer problem where supply is not explicitly modeled showed above.
Given that I think it would still be called 'optimal Ramsey taxation'. To my best knowledge this kind of tweaking of the model would not have it's own special name, but I also tried  to do literature search on Ramsey models with supply elasticity which did not yielded any results, but absence of evidence is not necessarily evidence of absence.


What would be the economic effect of widespread instances of individuals paying off consumer debt?

This left me wondering what happens to the wider economy when people decide en masse rather than spending their disposable income on consumer goods/services, to instead pay down their debt and save/invest?

tl;dr: Answer depends on the situation/time horizon you are talking about. In long run increasing saving and investment will have no negative impact on economic activity. In fact it can even lead to higher long-run economic growth.  However, in short-run and in recession (especially one where economy is in a liquidity trap) it would lower everyone real incomes and depress economic activity.
Full Answer:
To see what mechanisms are in play here we can use little bit of math. This can be done with the standard goods market equilibrium model (see Blanchard et all Macroeconomics: a European Perspective)  Let us start with output/GDP definition for closed economy:
$$Y =C+I+G \tag{1}$$
where $Y$ is the economic output (which in economics must also be equal to people's incomes so I will be using output/income interchangeably). $C$ is consumption, $I$ is investment and $G$ is government spending. In order to get some meaningful answer from the above identity we have to specify what the consumption is. In order to make everything simple lets assume linear consumption function:
$$C= c_0 + c_1 (Y-T) \tag{2}$$
Where $c_0$ represents your autonomous consumption -  consumption you will consume regardless of your income, $c_1$ is your marginal propensity to consume, if $c_1=0.75$ that means you will consume $3/4$ of your disposable income and save the rest and finally $Y-T$ is income minus taxes which is the disposable income.
If we substitute $C$ back to the GDP definition and solve for $Y$, we get the goods market equilibrium:
$$Y = \frac{1}{1-c_1}\left( c_0+I+G-c_1T\right) \tag{3} $$
This shows that the level of economic output $Y$ will actually depend both on consumption and investment (and investment in turn depends on private savings as it is the sum of private and public saving). However, it also shows that if people's marginal propensity to consume increases (that is when $c_1$ becomes higher) the whole output multiplier $ \frac{1}{1-c_1}$ becomes higher and hence for any level of autonomous spending, investment or government spending the output will be higher.
Nonetheless the above is all just a short-run partial equilibrium analysis. Such analysis is appropriate when we are in recession and where investment is unresponsive because for example an economy is in a liquidity trap (a situation where increase in savings wont necessary fuel more investment). However, in long-run we have to acknowledge that investment is not just independent as the formula (3) suggests but also depends on income and interest rates as investment as mentioned above arises from private and public savings.
If we make just one very simple change to the model above and assume that investment is given as $I= I_0 + d_1 Y -d_2 i$ , where $I_0$ is autonomous investment, $d_1$ is the fraction of income that is invested, $d_2$ is parameter which determines how investment respond to interest rates and $i$ is an interest rate, then the goods market equilibrium will be given by:
$$  Y = \frac{1}{1-c_1-d_1}[c_0 + I_0 + G − c_1T] - \frac{d_2}{1-c_1-d_1}i \tag{4}$$
In this case you can see that the multiplier is given as $\frac{1}{1-c_1-d_1}$ so multiplier is given by spending not just on goods and services but also by investment spending - if people lower their marginal propensity to consume but increase marginal propensity to invest to offset that nothing will change. Furthermore, ceteris paribus increasing the amount of savings lowers the interest rate (which is the price for savings that can be used as investment).  So once we let investment to vary depending on interest rates and income we see that what matters is the amount of all spending no matter whether it is spending on goods and services or investing.
In addition the above is still missing explicit model for money-market and behavior of firms if I would add that in a result would show that in the long run all savings are transformed into investment either directly (which is modeled here) or indirectly through their effect on overall price level but an exposition of full model would be too complex so if you are interested you have a look at some undergraduate textbook (all models presented here are taken from the above mentioned Blanchard et all Macroeconomics but they will also appear in any standard macro book with perhaps different notation).
Furthermore, the discussion above does not factor in economic growth and saving and investment is crucial for economic growth. The main growth model used in modern economic literature are the Solow growth model in which increase in saving rate can result in short to medium term higher economic growth, but not really in  a long run per capita growth which will in the model be determined by the rate at which technology progresses. However,  for that to happen the economy still requires some level of saving - the model just shows that trying to increase the level of saving in the long run steady state wont make difference (See Romer Advanced Macroeconomics).
However, recently in literature endogenous growth models are becoming more popular (Romer even got his Nobel Prize in economics in  2018 for his work on them) and in such models savings rate actually can increase even long run economic growth (again see Romer Advanced Macroeconomics).

PS: This is little bit of an tangent and hence I included it as a post scriptum but actually this:

He cited the “simple fact of economics that high income earners are more likely to save or pay down debt with the tax cuts, rather than spend the extra funds to help stimulate the domestic economy”.

this is actually not entirely accurate - or at least taken out of context (probably journalist misunderstood statement about particular tax cut as a general one). You can always design a tax cut that specifically targets poor. For example suppose we have flat tax rate of $40\%$ we can always decide to cut tax rate to let's say $10\%$ only for people below some income threshold. Rather the issue is that taxes generally have lower multiplier attached to them and hence they affect output differently. From equation (3) you can see that if government spending $G$ increases by 1 the output will increase by $\frac{1}{1-c_1}$ but if the government decreases taxes output increases only by $\frac{c_1}{1-c_1}$ and on interval $0<c_1<1$ the former will be always larger as on that interval $\frac{1}{1-c_1}>\frac{c_1}{1-c_1}$. This also holds just in short-run though.


Understanding Classical Rationality at a Basic Level

After all,suppose person A knows they will derive satisfaction from honoring sunk costs. Then doesn't honoring sunk costs make them better off (it fulfills their desire for happiness), and therefore becomes rational for person A?

Utility experienced from honouring sunk costs is treated separately from honouring sunk costs themselves. ( section 2 of http://www.princeton.edu/~tkelly/papers/Sunk%20Costs,%20Rationality,%20and%20Acting%20For%20the%20Sake%20of%20the%20Past.htm )

If I feel good for rescuing my phone from a toilet, or feel bad, that’s separate from the phone being in the toilet.


Is a self-sustaining "loop" of economy a thing?

This is not a "sustainable loop" -you neglect the concept of value-added, created by the use of additional factors of production in each company.

Company $A$ has revenues $p_AQ_A$, with which it must cover all its expenses (including returns to capital).

These are only part of the costs of company $B$, because it also uses additional capital and labor to build the gearing system. So the revenues of company $B$, that have to cover expenses in order for $B$ to keep going, and given that it has purchased $Q_A$ units from $A$, is $p_BQ_B > p_AQ_A$.

These in turn, represent only part of the costs of company $C$. With the same reasoning, we end up with $p_CQ_C > p_BQ_B >p_AQ_A$. Not that the number of machines build by $C$ are the number of machines that $A$ demands in order to produce $Q_A$.

Under the described loop, company $A$ must now pay $p_CQ_C$ in order to do its job (plus any other costs for the labor and any additional capital it may be needed). But since $p_CQ_C > p_AQ_A$, this will mean that company $A$ will have losses, even before paying for the other factors of production it needs to hire. Hence, the loop is not sustainable.

A possible (misguided) objection ¨

  "But $p_CQ_C$ represents an investment for company $A$, it is not
  an operating expense"

Essentially this objection implicitly assumes that with $Q_C$ machines in hand, company $A$ will be able to produce more than $Q_A$ product units. But if this is the case, then after the first purchase, company $A$ won't demand more machines from $C$, hence $C$ won't demand additional gearing systems from $B$, hence $B$ won't demand any additional product units from $A$, and so any remaining productive capacity of the $Q_C$ machines will not be used.
If this objection does not assume the existence of this additional capacity, then the $Q_C$ machines will fully depreciate in order to produce $Q_A$ product units -and then, there may be initially classified as "investment", but at the end of the production cycle of $A$ they will be recorded as "depreciation charge", i.e. a bona fide running expense (which is what they are, since they fully depreciate).

I would suggest to study Input-Output models of an economy.


Why does one have to ask for a quote to get a price for some things?

I believe that prices are typically available for well-defined products that dont vary between customers (e.g. groceries, cars, etc.). Whereas some products are customer specific (more so on the business to business side) and depend on quantity and quality simultaneously. I think of the example of a landscaping job (a service) that would depend on how much time the job would take (labour inputs) as well as the type of machinery (capital inputs) that would be needed to get the job done. Another example would be buying paper for an office, the cost would depend on how much you are buying and what quality and every companies needs would be entirely different.


If Greece defaults, can it still stay with the euro?

The Greek banks will not be able to get any Euro to give out, if the IMF and the EU cuts of all the loans to Greece.   Therefore if Greeces remains in the Euro, there will be no banking system in Greece.


what is monotonicity and strict monotonicity in preferences?

The word monotonic means "always moving in the same direction", in our case, always going up.
Monotonic preferences mean that the customer always prefers more of a good.
This comes in two flavors:

Strictly monotonic: More of one good is always preferred to less of that good.
Weakly monotonic: More of one good is always equivalent or better than less of that good.


Is "management science" a subsection of economics?

The journal Management Science is a decent one. Many economists do publish on that journal. In fact, a quick look at its editorial board reveals many notable names, with the recent Nobel laureate in Economics Richard Thaler among the associate editors.

The field of management science is more interdisciplinary in nature. See Wikipedia's entry for it. To me, it's like operations research for business or applied microeconomics in the context of business decision-making. To call it a "separate science" is a bit too much, though.


Change in Consumer Surplus with Two Price Changes: Is it Path Dependent?

Your reasoning is correct (i.e. the book is wrong). First, let's follow the book's logic a little more carefully step by step, beginning with the case where p1 changes first:

a fall in p1 leads to a gain of A in market 1 and of H in market 2 (because D2 shifts).
a subsequent fall in p2 leads to a gain of B + C in market 2 (we are now using the new demand curve in market 2); moreover, it causes D1 to shift so the consumer gains G + D.
thus, the total gain is A + H +G + D + B + C.

If p2 changes first:

a fall in p2 leads to a gain of B in market 2 and of G in market 1 (because D1 shifts).
a subsequent fall in p1 leads to a gain of A + D in market 1 (we are now using the new demand curve in market 1); moreover, it causes D2 to shift so the consumer gains H + C.
thus, the total gain is A + H +G + D + B + C.

We see that the order doesn't matter.

Another quick way to verify that the order doesn't matter is to note that these demand curves are coming from a consumer who chooses the optimal demand vector for given prices. If the prices are the same then the solution to the maximisation problem will also be the same.


Will a Free Software Society ever become a reality?

Your questions asks:

  A society where all the players abide by the above FSF principles

It won't ever happen for all players. As a counter-example, consider an engineer who invents software that can predict when a jet engine needs maintenance. It's worth millions to the airlines, who will happily pay him for the software. He has no incentive at all to release this software as open-source. The same principle applies to all sorts of specialist software.

In contrast, open-source works well for common building blocks. For example, there are all sorts of companies who need an operating system for their product - cloud providers, home network kit, super computing, all sorts. While they could build their own, it is much more efficient to have a common pool, like the Linux kernel. This environment means that companies can put paid engineers on open-source projects, and still run a profitable business.


Where do Montenegro and Kosovo get their euros from?

Countries can get foreign currency from tourists, exports or intergovernmental aid programs. (And several other ways, but these should suffice here.)


The equation for the firm's demand for labour

You could set up the profit function as suggested by @Kitsune , but here I take the expenditure minimization approach (which is mathematically equivalent).

Set up the minimization:
$$\min_{L,K} wL + rK \\
s.t. \quad 25LK=Q$$

Using the Lagrange method:
$$\mathcal{L}=(-wL - rK) + \lambda (25LK-Q)$$
Giving us the First-Order Conditions:
$$(1)\quad \frac{\partial\mathcal{L}}{\partial L}=-w + \lambda 25K=0$$
$$(2) \quad \frac{\partial\mathcal{L}}{\partial Q}=-r + \lambda 25L=0$$
$$(3) \quad \frac{\partial\mathcal{L}}{\partial \lambda}=25LK-Q=0$$

Taking (1) and (2):
$$(4) \quad K=\frac{w}{r}L$$
From (3):
$$(5) \quad K=\frac{Q}{25L}$$

From (4) and (5):
$$\frac{w}{r}L=\frac{Q}{25L}$$

Solving for L:
$$L=\sqrt{\frac{rQ}{25w}}$$


Apparently it's deposit accounts, which I think means both savings and checking accounts.

No, brokerage accounts are definitely not it, as they do not represent a liability for the bank. The issue is that the 'bail-in" is a reduction in the debt the bank has towards its depositors. When you own 100 of deposits, it's the bank that owes you, but when you owe a brokerage account that owns 100 of stocks, the bank owes you nothing. So, you received a statement from the bank saying you had 200000 euros in your account, then they impose a 40% levy on the uninsured part (100000). So next month you receive a bank statement that says you have now have 160000 in your account...


What's unique about the Chicago School's Methodology?

I would submit that the premise is flawed. There is no methodological criteria that unified the Chicago school and separates it from the rest of economics. I think this is true even for Friedman, whose methodological paper you refer to. If you look at his macroeconomic work such as his Monetary History of the United States, Theory of the Consumption Function or his famous presidential address to the AEA,there is nothing in it which is different from the methodology of mainstream economics. I think the same can be said about Stigler's work on industrial organization.

The Chicago School was united by its institutional links and its faith in capitalism and not by a distinctive method.

And it would be wrong to think of the School as a homogenous entity. See for example this article by Patinkin on the diversity of views on the quantity theory of money:

http://stevereads.com/papers_to_read/the_chicago_tradition_the_quantity_theory_and_friedman.pdf


Generally, it's the place (e.g. NYSE) where the equity portion of company's capital structure trades publicly. The exchange creates a market for buyers and sellers of equity to meet and trade (think Ebay but for equity).

Equity is in contrast to the debt portion of a company's capital structure, which is often comprised mainly of term loans and revolvers provided by banks as well as as secured and unsecured bonds of the company purchased by investors.

To understand a stock market, it's important to understand priority of payments in a capital structure. Imagine a company goes bankrupt - this is the order in which stakeholders lose money (first to last): equity, unsecured bonds, secured bonds, 2nd lien term loan, 1st lien term loan.


Start with the second stage, this is just Cournot competition between firm 2 and firm 3. You can solve this for the Nash equilibrium by setting the first order condition for firm 2 and firm 3 and solving these two equations, taking $q_1$ as given. This will give you quantities $q_2$ and $q_3$ in terms of $q_1$ which you can then plug into the profit function of firm 1 and you can maximize (i.e. find which $q_1$ firm 1 should choose to make sure the Nash equilibrium in stage 2 will be the most favourable Nash equilibrium possible for firm 1). So, step by step:

Start with the second round, find the Nash equilibrium by solving the following two equations:
\
$$\pi_2'(q_1,q_2,q_3)=0\\
\pi_3'(q_1, q_2, q_3)=0$$
Which will give you:
$$q_2=q_3=f(q_1)$$
plug this into the profit function of firm 1 and maximize this expression which has $q_1$ as choice variable.


In the basic "Cobweb Theorem", the assumption is that suppliers must decide now next period production, and that they assume that next period market price will be equal to current period market price.

$$p^e_{t} = p_{t-1} \tag{1}$$

Now, we know that if price follows a random walk without drift, then the above a priori assumed expectations formation hypothesis coincides with the Rational Expectations Hypothesis:

$$\text {If}\;\;\; p_{t} = p_{t-1} + u_{t} ,\;\;\; u_{t} \sim WN \implies E_{t-1}p_{t} = p_{t-1} \tag{2}$$

But in the cobweb model, the actual price process is determined endogenously, given the a priori assumption $(1)$: we assume $(1)$ and we let the equilibrium of the model give us the model-consistent price process.

Assuming linear demand  (say, with a white noise preference shock), and supply function we have

$$D_t = a-bp_t +u_t,\;\;\; S_t = \gamma + \delta p^e_t$$

Imposing market clearing (an additional assumption no less) we have

$$D_t = S_t \implies p_t = \frac {a-\gamma}{b} - \frac {\delta}{b} p^e_t +u_t$$

and applying the a priori expectations formation assumption $(1)$ we obtain

$$p_t = \frac {a-\gamma}{b} - \frac {\delta}{b} p_{t-1} + u_t\tag{3}$$

$(3)$ is the model-consistent price process here (i.e. given $(1)$), not $(2)$. Only in the case that $a=\gamma, b=-\delta$ do we get that the price process emerges as a random walk without drift and so that the expectations formations hypothesis used in the model coincides with the Rational Expectations hypothesis.

The price that suppliers use is the current market clearing price indeed. But how they use it (hypothesis $(1)$), does not "reflect all available information", which is embodied in $(3)$ which suppliers ignore.


The fact that your outcome is measured at a later point in time than the treatment is not a problem, and not unusual either - this is true to some extent in all RD studies (if this were not the case, there couldn't be causation).

As for how to analyze your outcome, I suggest a couple of different specifications. First, you should use the continuous fine measure as an outcome. Second, you could use an indicator variable equal to 1 if they got a fine, and equal to zero if they didn't. You might be able to do similar dummy regressions for the more/less cases too, but that's a little more complicated. If you want to be a little more advanced, it would certainly be interesting to take the RD framework to a tobit model. Shouldn't be too complicated, I expect, but I haven't thought that much about it.


In the context of banking, how (if at all) do the terms 'capital', 'net worth', and 'equity' differ?

Consider this...

A bank's capital is used to finance the purchase of assets. Those assets are mostly loans made to households and businesses and governments. When a loan is made the bank is "buying" the future payments of the borrower. This suggests the whole right side of the balance sheet is capital because it takes the entire right side to finance the purchase of all of the left side. Some of the capital is the bank's own debt or equivalently, the liabilities, which is owed to depositors and bond holders. The remainder of the capital is called equity which is what the common and preferred shareholders own. The value of all the shares or of all the "stock" (as in stock market) is the equity. Stockholders and shareholders and equity owners are the same thing. Confusingly "the stock in a business" is the assets (in blue on the left) but the "stock" (also known as the shares) are the smaller part called equity (in teal on the right). In order to control all the assets (in blue) you have to own the equity (in teal) so that is why the word "stock" is used to refer to the teal part and also the blue part.

Net worth is assets minus liabilities so it is the same thing as equity. "Net worth" is the usual term when speaking of the equity of an individual or a household. When speaking of the equity of a company sometimes stock market participants call it "book value".

I don't think non-bank companies think of these terms any differently than banks. In any company there is the distinction between physical capital which is always on the left side, and financial capital which are the two parts on the right side. If the company has some cash or other valuable paper on hand then there is also some financial capital on the left side (held as an asset).

The above diagram is useful in the context of accounting and finance but less useful in the context of economics.

I think economists use the word capital to refer to an accumulation of value, in other words assets. In economics capital has the connotation of being an accumulation rather than a flow. For example savings are an accumulation of a flow of income. (Capital) accumulations can be land, structures, equipment, software, patents, copyrights, and human capital (skilled people and educated people), and other assets.


At time $(t-1)$, the investor buys some risk free bond, $B_{t-1}$ and some risky asset $X_{t-1}$ at price $P_{t-1}$, such that the budget constraint holds, i.e.
$$
W_{t-1} = B_{t-1} + P_{t-1}X_{t-1}
$$
At period $t$, one unit of risk free bond pays off one unit, so $B_{t-1}$ units of risk free bonds pays off $B_{t-1}$ units of wealth at $t$.

For a risky asset, one unit of $X$ is now worth $P_t$. Since the agent holds $X_{t-1}$ units of the risky asset, she effectively has wealth $P_tX_{t-1}$ from risky assets.

Adding the two gives us the budget at period $t$, i.e.
$$
W_{t} = B_{t-1} + P_t X_{t-1}
$$
And period $t$ portfolio choice satisfies
$$
W_t = B_t + P_t X_t
$$
Combining the two gives us the desired equation.


The tax increase is first measured in dollars. Since the dollar amounts are large, and not comparable across countries, they divide by the size of nominal GDP. This is standard practice in discussing macroeconomic policy. This way, we can quickly judge the relative size of a quantity without having to run to an economic database to see statistics on the size of the economy.

In this case, they are looking at a tax increase that is assumed to equal 1% of nominal GDP. That is, if nominal GDP were \$100 billion, that is a tax increase of \$1 billion.


In a context of globalisation and high wage inequality, why should governments help unskilled workers?

After the exchange in comments, the OP appears to ask "why government intervention if the market will take care of the situation?"

Well we know that government intervention is advised when markets "fail" or, more accurately, are not characterized, at least to a respectable degree, by the ideal conditions required by theory in order for them to function efficiently.

But I will not discuss what could be the "market failures" in the specific case the OP discusses, and whether government intervention could actually improve things.

Instead I will give an extra reason why in principle, government intervention may be required: the speed by which market forces operate. This matters a lot from a socio-political point of view, and it should also matter from a strict economic point of view, because, the slower the pace of adjustment, the longer the period of inefficiencies. So even if market forces will eventually tend to reduce the skills-and-wage gap, it is important how long it will take. And if it takes, say, decades, government intervention could in principle be argued in favor in order to speed up the process.


Is the equity lending involved in naked short-selling structurally equivalent to fractional-reserve banking?

There’s no inconsistency between allowing fractional reserve banking and banning naked short selling. Banks generally lend out money that they borrow (setting aside capital lent), and if one lends out a stock that one has borrowed, that’s not a naked short position, it’s just a plain short. What’s illegal about “naked” shorting (in the U.S.) is selling a stock that one hasn’t either borrowed or determined that one can borrow within the delivery timeframe (currently two days for equities).
The restrictions around money are actually in quite a bit tighter— if a bank were to enter into a transaction that required that it deliver funds and it failed to do so because it couldn’t borrow enough money, it would not be acceptable for the bank to say “oops we thought we could borrow the money sorry” and just pay a small fee (as with equities, where one pays a fails charge). It would instead be an event of default(1), which is more or less the end for a bank.
(1) “Generally, a bank is closed when it is unable to meet its obligations to depositors and others.”


I believe Marshallian demands are less steep than Hicksian demands because we reverse the y and x axis in economics. Thus a larger derivative of x with respect to p will be less steep since p is on the vertical and x is on the horizontal.

(image from here)


When does an economic theory become true?

It's just the same as in the physical sciences:

an economic theory, like any theory or model in physics or chemistry or biology, is wrong (H/T George Box), but it can be useful.

"True knowledge" is not a really a useful concept; not in physical sciences, not in social sciences. The important question is whether a model gives us useful insights, and whether it can constructively inform decisions.

An economic theory gains acceptance when it turns out to be useful.


Does a strong exchange rate always mean a cheap holiday regardless of where you are from and going?

If by "stronger exchange rate" you mean you get more units of the local currency for a dollar (or whatever currency your home country has) then no, not at all. In fact, in economics the term "stronger exchange rate" is not used in this context. People sometimes say that currency A is growing stronger compared to currency B, but this only means that the exchange rate is improving, it does not say if the number is large or small.

If you want to know whether you get more bang for your 'buck' then what matters is purchasing power: how many goods in the foreign country you can buy for the currency units you got for your dollar. Assuming an exchange rate of $E$ units of currency X per one dollar, and a foreign price $p^*$ for some good, you can buy $E/p^*$ units of said good. You can compare this number for different countries to see how far your money will go.

The comparison is made somewhat difficult by the fact that the ratio will probably differ for different goods. Economists usually construct price indices to go around this problem. However no price index is perfect, each one is made to suit a distinct purpose. Basically you would have to look at the price of the goods that you are likely to buy during your stay.


Forecasting quarterly EUR/USD exchange rate

There are many methods to forecast exchange rates- look at the series of influential papers by Meese and Rogoff for a primer (https://scholar.harvard.edu/files/rogoff/files/51_jie1983.pdf).

To answer your particular question- what you describe seems to be driven by  multicollinearity. For instance, returns on stocks will be correlated with the inflation rate if they are nominal,  CAB and FA balance will be related to both the GDP growth rate as well as the interest rates etc. To test whether your results are susceptible to multicollinearity in the variables, try Variance Inflation Factor type tests.


Infinite Horizon Transversality Condition

Your confusion comes from the fact that you treat the Transversality condition as a constraint, while it is a condition for optimality. So the formulation at the end of your question is wrong. What you do is you solve your model as usual, and then check if the solution (here the balanced growth path) satisfies the Transversality condition.

The Transversality condition (TVC) is

$$\lim_{t \rightarrow \infty} \beta^t u'(c_t)k_{t+1} =0$$

In your model, $u'(c_t) = c_t^{-\gamma}$ and $c_t = (2-\delta)k_t - k_{t+1}$

So the TVC becomes indeed

$$\lim_{t \rightarrow \infty} \frac {\beta^tk_{t+1}}{[(2-\delta)k_t - k_{t+1}]^{\gamma}} =0$$

What we want is to check whether the (unique) balanced growth path satisfies the Transversality condition. On the balanced growth path we have $k_{t+1} = (1+g)k_t$ where $g$ is a constant determined by the exogenous parameters of the model and the optimizing conditions. So when on the balanced growth path, the TVC becomes

$$\lim_{t \rightarrow \infty} \frac {\beta^t(1+g)k_t}{[(2-\delta)k_t - (1+g)k_t]^{\gamma}} = \lim_{t \rightarrow \infty}\frac {(1+g)}{(1-\delta-g)^{\gamma}}\beta^t k_t^{1-\gamma}=0$$

The constants are unimportant, so we want that

$$\lim_{t \rightarrow \infty}\beta^t k_t^{1-\gamma}=0$$

holds. Note that if $\gamma \geq 1$ (which is the consensus in the literature) the condition does hold, since capital goes in the denominator, and we're done.

For the case $\gamma <1$, we want the expression to hold at the limit only, i.e. eventually . "Eventually" implies that, if $k^*_s$ is the level of capital at time $s<t$ when the economy arrives at the balanced growth path, we will have $k_t = k_s^*\cdot (1+g)^{t-s}$. Then the TVC becomes

$$\lim_{t \rightarrow \infty}\beta^t \big[k_s^*\cdot (1+g)^{t-s}\big]^{1-\gamma}=0$$

Again, shed things that do not depend on $t$, to get

$$\lim_{t \rightarrow \infty}\beta^t (1+g)^{(1-\gamma)t}=0 \implies \lim_{t \rightarrow \infty}\big[\beta (1+g)^{(1-\gamma)}\big]^t=0$$

So the TVC will hold if and only if

$$\beta (1+g)^{(1-\gamma)} <1$$

In the other question the OP has obtained

$$1+g = [\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}}$$

Substituting, we want

$$\beta \Big([\beta(\alpha+1-\delta)]^{1/\gamma}\Big)^{(1-\gamma)} <1$$

This can be simplified a bit, and it is a restriction on the parameters that must hold in order for the balanced growth path to satisfy the Transversality condition (when $\gamma <1$). For baseline values of the other parameters one can see that the condition is not likely to be satisfied.


How effective is progressive tax at regulating monopoly?

Progressive income taxes on individuals is not very likely to have any real impact on market power, but progressive income taxes on corporations can certainly have this effect. The key idea is that in many industries (for example, the financial industry), there are geniune economies of scale: having a big balance sheet allows an investment bank to take advantage of various synergies, meet demand from customers at lower costs and therefore at lower prices. This means that there is a natural pressure towards concentration in the financial industry. High levels of concentration, however, allow firms to price out their competitors from the market due to their size and the aforementioned scale effects, and can lead to a scenario of oligopoly.

The government can try to deal with this problem by introducing negative incentives for corporations to get bigger. This can be in the form of an explicit limit on the size of corporate balance sheets rel to GDP, as advocated by Stiglitz, or it can be in the form of income taxes, or capital requirements, which are "progressive" in the sense that they impose higher marginal costs on larger and more complex firms over smaller ones. Such a measure can offset the natural incentive towards aggregation in the financial industry, for example. This is already done to some extent by Dodd-Frank (perhaps unintentionally) in the US, which raised capital requirements for SIFIs (which are well-integrated financial institutions with large balance sheets). The adverse effect is that by doing this, a country can effectively price its financial institutions out of international markets.

In turn, a policy which reduces market power can have positive effects on the income distribution, as it would lead to lower prices and higher wages. The problem is to find the right balance between the benefits (lower costs) and costs (market power) of concentration in the marketplace.


nash equilibrium public good - is this correct?

For the Nash equilibrium of any simultaneous-play game, you are looking for the point where each player is playing a best response to all other players at the same time.

So your steps to solving this game should be:

Determine player 1's best response function ($g_1$ as a function of $g_2$)
Determine player 2's best response function ($g_2$ as a function of $g_1$)
Now you have two equations with two unknowns. Solve this system of equations algebraically to get the Nash equilibrium $g_1^*$ and $g_2^*$.

So how can you get the best response functions? Take player 1 for example. Player 1 wants to set $MB_1 = MC_1$. You have $MB_1$, and $MC_1 = 4$. So set them equal to each other and solve for $g_1$ to get player 1's best response function $g_1^*$ as a function of $g_2$. Then similarly for player 2 to get $g_2^*$ as a function of $g_1$.

That's not quite all, though. In this game, the players (presumably) can't contribute negative amounts. So the real best response functions will be of the form

$$
g_1^*=
\begin{cases}
some.function.of.g_2 \text{ if } some.condition.of.g_2\\
0 \text{ otherwise}
\end{cases}
$$

So, first, try to solve your two best response functions together algebraically, assuming that both $some.condition.of.g_2$ and $some.condition.of.g_1$ hold so neither is 0. If you don't get a solution, or if that solution makes one of them negative or violates one of the $some.condition$s, then instead try setting $g_1 = 0$, find $g_2^*$, and see if that gives you a Nash equilibrium (i.e. if $g_1^*(g_2^*) = 0$). Then try setting $g_2 = 0$, find $g_1^*$, and see if that gives you a Nash.


You just have to write down some utility function that changes in each period, in the first there should be an additional term (shock) a constant multiplicative on the disutility of work that reduces it, and then in the second period this constant becomes one.
A dynamic model solution is a vector of $(l^*_1,l^*_2)$ of the equilibrium hours of work that the consumer chooses according to her preferences, and restrictions.


Positive Monotonic Transformations and Nested Functions

The value of $h(g(f(u(x,y))$ is not necessarily positive but the transformation $u \rightarrow h \circ g \circ f \circ u$ is indeed a positive monotone transformation.

To see this, take any $(x,y,z,w)$. We have the following chain of equivalences:
\begin{align}
u(x,y) \geq u(z,w) & \Leftrightarrow f(u(x,y)) \geq f(u(z,w)) \text{ since } f \text{ is increasing} \\
& \Leftrightarrow g(f(u(x,y)) \geq g(f(u(z,w)) \text{ since } g \text{ is increasing} \\
& \Leftrightarrow h(g(f(u(x,y))) \geq h(g(f(u(z,w))) \text{ since } h \text{ is increasing}
\end{align}
Thus, the transformation $u \rightarrow h \circ g \circ f \circ u$ preserves the ordering, i.e. it is a positive monotone transformation.

Another way to see it is to assume that all functions are differentiable and notice that
\begin{equation*}
\frac{d (h \circ g \circ f )(x)}{dx}= f'(x) g'(f(x)) h'(g(f(x))
\end{equation*}
which is positive since $f'>0,g'>0,h'>0$.


maximize profit of competitive firm in short-run

Is the cost given the marginal or cumulative cost? If it's the cumulative cost, then the profit is  pq-C(q): your price is money per item, so you have to multiply by the number of items to find the total gross revenue. The problem is then to maximize the profit, i.e. find $\max_{q} pq-C(q)$.

If C(q) is the marginal cost, then the marginal profit is marginal revenue minus marginal cost, i.e. marginal profit = p-C(q).

The FOC is that the firm should keep selling the product until selling more would cause their marginal profit to go negative. For a continuous function to go from positive to negative, it has to go through zero. Thus, the FOC is that their marginal profit be zero.

While the marginal profit going from positive to negative means that it's going through zero, going through zero doesn't necessarily mean that you're going from positive to negative; you still have to check which direction it's going. So the SOC is that the marginal profit be decreasing.

FOC: marginal profit = 0
SOC: marginal profit decreasing

If the marginal profit is decreasing, then when it hits zero, it's going from positive to negative.


Endogenous Growth: Balanced Growth Path with CRRA Utility

You have obtained

$$ \frac{c_{t+1}}{c_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}} \equiv 1+g$$

and

$$\frac{k_{t+1}}{k_t}=1+(1-\delta)-\frac{c_t}{k_t}$$

By equating you can show that there is a unique rule that maintains a balanced growth path

$$\frac{k_{t+1}}{k_t}=\frac{c_{t+1}}{c_t} \implies c_t =( 1-\delta-g)k_t$$

(too much consumption, by the way). This shows that the model has a unique balanced growth path.

If you want further to argue that the economy will indeed choose this path, you have to invoke the Transversality condition (which constraints the consequences that a chosen path should have  on capital accumulation), and maybe the Inada condition that your chosen utility function satisfies.


What does Smith mean by "price of work" in this context?

Smith was talking of "the price of work" as meaning the price of economic products bought from the employers and merchants, not the price of labour bought from the workers.
"Work" had this dual meaning historically, referring to both the activity of work and the result of work, whereas nowadays the latter meaning tends to be qualified (like a "work of art", or a "piece of work", etc.).
The distinction Smith makes is that if you put the wages of every worker involved up by 5%, the overall market price rises by 5%, because the 5% increase is by reference to their old wages.
Whereas if you increase the profits of a chain of merchants by 5%, the output price rises by much more than 5%, because each merchant in the chain takes 5% on top of all what the previous merchant took. The increase is by reference to the difference between input and output prices, not by reference to the merchant's old income.
Hence, in Smith's view, increases in wages result in "simple" percentage increases in output cost, whereas increases in profits result in "compound" increases in output cost.


Confused by the Term "Welfare Relative to Full Information Welfare"

What it means is the following:

Consider the same bargaining game played by two players that are fully informed (i.e. the buyer is also informed about the value of $q$). Then find the equilibrium of this game and calculate welfare. This is the ``full information welfare'' that the question refers to.

Then solve the model where only the seller is informed about the value of $q$ i.e. find the equilibrium of this game. Now calculate equilibrium welfare. How does it compare to the one you calculated for the full-information case? I would probably do this by calculating the fraction
$$\frac{\text{Asymmetric info welfare}}{\text{Full info welfare}}.$$

Lastly, you need to see how this fraction changes as the values of $\alpha$ and $\delta$ change.


The effect of Monetary Policy on Asset Price Inflation

My understanding of the effect of easing monetary policy on asset prices is that it can work through a more direct channel.

If markets are efficient and risk neutral then generally the value of an asset $i$ is equal to the net present value of the expected cash flows generated by that project:
$$ P_i = NPV(i) = \sum_{\tau=0}^{\infty} \frac{E[C_\tau]}{\Pi_{\ell=0}^{\tau}(1+r_{\ell})} $$
Monetary policy directly lowers the denominator of the the fraction in the sum, so it raises the value of the sum. In English, by reducing the rate at which we discount future investment the present value of the asset's cash flows become more valuable. If monetary policy can also improve the economy more generally, it can also increase the expected size of the cash flows, further increasing the value of the asset (it also raises the numerator).


The Fed may increase the rates. But will that make the government debt unsustainable?

You have to differenciate the debt that is already existing from the one that is going to be issued: whenever the US Treasury issue a bond lets say a 10% coupon 10yrs... for a 100USD. It's still going to pay 10USD to the holders during the 10yrs. When you see the yield going up or down it's usually on the secondary market. I say usually because sometimes the government issues bond with a floating rate which can be impacted by a rate increase.

As for the new debt that is going to be issued you are right to say that this going to be more costly !


What does it mean "non-Walrasian" or "quasi-Walrasian"?

Well non-Walrasian is an model that is not Walrasian so to explain that we should explain what  a Walrasian model is. Generally speaking in macro context when we talk about a Walrasian model it is a Walrasian general equilibrium model.
A Walrasian general equilibrium has the following characteristics see MWG pp 579:

Definition 17.B.1: Given a private ownership economy specified by
$$ \{(X_i, \succeq_i) \}^I_{j=1}, \{Y_j \}^J_{j=1},\{(\omega_i, \theta_{i1}, ... , \theta_{iJ})\}^I_{i=1}$$
an allocation $(x^*,y^*)$ and a price vector $p=(p_1, ..., p_L)$ constitute a Walrasian (or competitive, or market, or price -taking) equilibrium if:
(i) For every $j,y^*_j \in Y_j$ maximizes profits in $Y_j$ that is, $p\cdot y_j\leq py^*_j$ for all $y_j \in Y_j$
(ii) For every $i,x^*_i \in X_i$ is maximal for $\succeq_i$ in the budget set $\{ x_i \in X_i:p \cdot x_i \leq p \cot \omega_i + \sum_j \theta_{ij} p \cdot y_j^*\}$
(iii) $\sum_i x_i^* = \sum_i \omega_i + \sum_j y^*_j$

Slightly different definition holds for pure exchange economy, but for more detail you can have a look at the rest of the chapter in MWG. The gist of what is a Walrasian general equilibrium (and thus if model is Walrasian) is given by the authors in the brackets, it is a model characterized by competitive, or market, or price-taking equilibrium. A non-Walrasian model would be some general model that is not characterized by such equilibrium.
Following Calvo (1978)  quasi-Walrasian model is (emphasis mine):

These [quasi Walrasian models] are models
where agents perceive themselves as infinitesimal micro units, unable to exercise any kind
of "market power"; the agents set the variables under their control in order to maximize
their objective functions (utility, profit, etc.).
In contrast to the so-called "disequilibrium"
or "fix-price" models (see Allan Drazen)
there is no exogenous rationing or price
inflexibilities. However, they differ from standard Walrasian models (for example, see
Kenneth Arrow and Frank Hahn, chs. 1-5) in
that agents are not necessarily price takers
and, typically, markets are incomplete

The explanation above is pretty much self explanatory.
Lastly, New Keynesian general equilibrium models are quasi-Walrasian precisely because they allow for market power (in fact that is what in some NK models creates price rigidities/stickiness), or incomplete markets and so on. You can see nice overview of NK models in Woodford Interest & Prices, you will see that virtually all of them have some twist where there is market power, or similar market imperfection.


It is important to remember that figures are not proofs. Minor mistakes in the figure are hard to detect but can change the outcome. Algebra is much more reliable.

Right now in your figure the low risk type consumers have indifference curves that are upward sloping somewhere. It seems though that is not critical here.

The visualisation of subsidies seems to be the bigger issue.  You need G and H to be the same distance away from their respective original zero profit lines. Thus the 'new' isoprofit lines should run parallel to the old ones, and at the exact same distance.

It seems to me that it is impossible to place GH on these lines while preserving incentive compatibility because of the relative slopes. However, this is also not proof. If you outline an exact GH pair and type up the equations that you think show that they constitute an improvement while being incentive compatible, you may get better answers (as well as a better understanding of the model).

EDIT

It seems to me, that in your numerical example the subsidies do not cancel each other out. It is not enough to change the rates by the same percentage, because the different types do not buy the same amount of insurance.

It costs $8\% \cdot 90 = 7.2$ to subsidize the high risk types, but only $8\% \cdot 15.4242 \approx 2.78$ is collected from additional fees on the low risk types. If there are the same number of people in both types, this will not work. But it seems to work if there are for example three times as many low risk people. I actually find this very surprising. I checked both IC constraints and all the utilities, and everything seems to check out.

Your new offers will definitely not be an equilibrium though. As long as one insurer offers the high risk insurance, other insurers can offer just the low risk one. With no high risk customers to subsidize, they will make a profit. By moving the high risk offer a bit more to the right they can siphon off all the low risk consumers from the cross-subsidizing insurance company, who will then suffer losses. So I guess the RS equilibrium is a kind of long run equilibrium.


is it possible to find PPP monthly data?

That doesn't exist. Gdp is quarterly.

There are no short term indicators like that on a global levels.


Sovereign rates in developed countries represent the credit risk-free curve. What maturity on the curve you want to choose is an analytical choice.
The natural rate is in reference to a risk-free curve, and so should align with the sovereign curve. Also,  “Natural rate” is somewhat dated at this point, the preference is to use a more neutral term like r*.


Basic question about why supply/demand should move to the equilibrium price?

What you write is true in a monopoly situation, but not when there are multiple firms on the supply side.

However, when equilibrium is discussed in this context people usually mean competitive equilibrium, that is equilibrium in a market where many firms compete. So at price \$1.80 several suppliers want to sell an aggregate amount of about 680 gallons. However demand is only 500 gallons. This would prompt the suppliers who are worried about not being able to unload all their stock to offer it at a slightly lower price, say \$1.79, undercutting their rivals' price. The idea is that by lowering the price somewhat you can greatly increase your market share, and unload your stock, to the general dissatisfaction of your rivals, who are now stuck with their own surplus gasoline.

Thus \$1.80 is not an equilibrium price, as suppliers have incentives to unilaterally push their own prices down.


influence of growing foreign demand for local products on the net exports in a fixed exchange rate regime

Yes, $IM$ increases in the equilibrium, but, no, $IM$ cannot increase more than $X$ does.

Logical explanation: "$\Delta X < \Delta IM$ in the equilibrium" means that $A$'s GNP decreases. When GNP decreases, we have $\Delta IM < 0$ while $0 < \Delta X$ so we can't have $\Delta X<\Delta IM$. Contradiction! We can never have $X-IM$ decreasing in the equilibrium.

Mathematical illustration: Consider the model $Y=C+I+G+X-IM$, $C=a+bY$, $IM=c+dY$. Then the equilibrium is $Y = (a-c+I+G+X)/(1-b+d)$. When $X$ increases by $\Delta X$, $\Delta Y = \Delta X / (1-b+d)$ and $\Delta IM = d\Delta X / (1-b+d)$. Thus,
$$\Delta X - \Delta IM = \left(1 - \frac{d}{1-b+d} \right) \Delta X = \left( \frac{1-b}{1-b+d} \right) \Delta X > 0.$$


Conditional independence and no correlation

[2] does not imply [1]. [2] and $E(u)=0$ imply $cov(u,x)=0$, which is about linear independence. [1] is stronger, as it refers to any type of dependence.

The classic counterexample to show this is $x=u^2$ over a symmetric domain. These are dependent yet linearly independent.

The R code below shows this:

set.seed(1)
u <- runif(100, min = -1, max = 1)
e <- rnorm(100, mean = 0, sd = 0.1)
x <- u^2 + e

plot(u,x)
abline(lm(x ~ u)) # Yields an R^2 of 0.006539

cov(u,x) # Yields 0.01206663

The plot is, where the black line represents the regression line:

Read more about this here.


Why did the Greek bailouts "fail" whereas the Portuguese bailouts succeeded?

Greece had more debt, less growth and higher budget deficits than Portugal throughout the crisis.

As a result, the Greek bailout should have been even bigger, but this was politically untenable, while the Portuguese bailout proved sufficient in size.

The lower budget deficits in Portugal, for example, also made it easier to comply with the austerity imposed by international organisations.

The Portuguese economy was also better able to adjust by virtue of being more competitive. It was recently ranked 38 on competitiveness vs Greece ranked 81.

http://reports.weforum.org/global-competitiveness-report-2015-2016/competitiveness-rankings/


Efficiency in a market that is both a monopoly and a monopsony

Efficiency in general means that the item being traded should be given to the party that values it the most.
In a competitive market, this means trade should continue as long as a consumer's value of a good, as captured by the demand curve, is greater than a seller's value of that good, captured by the supply curve.
In a monopoly, seller's value is captured by its marginal cost curve. But since price is greater than marginal cost in equilibrium, the units for which a consumer's value is higher that MC but lower than equilibrium price don't get traded. Hence there is inefficiency: those units of the good are not in the hands of the party (i.e. the consumers) that values them the most. The story for monopsony is similar.
In a bilateral trade setting with one seller and one buyer, we need to discuss two cases. In the first case, the traders' values are common knowledge. In this case efficiency can be attained, though price depends on the relative bargaining power of the two parties. In the second case, suppose neither trader knows the other party's value, but they do know that gains from trade are possible but not certain. Then the Myerson-Satterthwaite theorem tells us that no bilateral trading mechanism that is individually rational and incentive compatible can guarantee efficiency in the ex post sense; that is, the party with a lower value for the good being traded may end up having the good.


Market demand independent of distribution of income

I take homogeneity of preferences here to mean that preferences are homothetic; that is, preferences can be represented by a utility function that is homogeneous of degree one.

For an arbitrary good $x$ and a vector of prices $\mathbf{p}$, let individual demand be $x_i(\mathbf{p}, M_i)$ for each consumer $i$ with income $M_i$. Thus market demand is $D = \sum_{i} x_i(\mathbf{p},M_i)$. Since preferences are identical,

$$\sum_{i} x_i(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},M_i).$$

Then, by homogeneity,

$$\sum_{i} x(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},1) M_i = x(\mathbf{p},1) \sum_{i} M_i = x(\mathbf{p},1)M,$$

where $\sum_i M_i = M$ is aggregate consumer income. Thus $D$ is independent of the distribution of income and is solely a function of aggregate income across consumers, $M$.


intuitive interpretation of the marginal return/productivity of capital being less than one

Several thoughts in this area:

1) Capital stock lasts over time. I may trade 1 euro for 1 unit of capital. This capital returns 0.01 euros every period forever.  It pays itself off eventually.

2) If one has an incredible high rate of return and a sufficiently low discount rate, one might have divergent consumption paths, ex: I spend all of my money on capital, each unit of capital returns \$2 for the next period. I repeat forever, but never consume anything, because any consumption represents a loss of \$2 tomorrow.  After 1000 periods, I have a huge amount of capital but no consumption.

3) Cobb-Douglass is normalized so the exponents sum to one. This led to some off-the-cuff work. It has derivatives:

$MPK =\alpha_k * (\frac{L}{K})^{1-\alpha_k}=\alpha_k k^{-(1-\alpha_k)}$

$MPL= (1-\alpha_k) (\frac{K}{L})^{\alpha_k} = (1-\alpha_k) k^{\alpha_k}$

In optimal long-run state, these two will be equal.

$\alpha_k k^{-(1-\alpha_k)} = (1-\alpha_k) k^{\alpha_k}$

$\alpha_k = (1-\alpha_k) k^{\alpha_k --(1-\alpha_k)}$

$\alpha_k = (1-\alpha_k) k^{1}$

$k^{1} =  \frac{\alpha_k}{1-\alpha_k}$

Plug this back in and you get:

$MPL^{*} = \alpha_k \frac{\alpha_k}{1-\alpha_k}^{-(1-\alpha_k)}$

$MPL^{*} = \alpha_k \frac{\alpha_k}{1-\alpha_k}^{\alpha_k-1}$

$MPL^{*} = \alpha_k \frac{1-\alpha_k}{\alpha_k} * \frac{\alpha_k}{1-\alpha_k}^{\alpha_k}$

$MPL^{*} = (1-\alpha_k) * \frac{\alpha_k}{1-\alpha_k}^{\alpha_k}$

$MPL^{*} = (1-\alpha_k) * \frac{\alpha_k^{\alpha_k}}{(1-\alpha_k)^{\alpha_k}}$

$MPL^{*} = \frac{\alpha_k^{\alpha_k}}{(1-\alpha_k)^{\alpha_k - 1}}$

$MPL^{*} = \alpha_k^{\alpha_k}*(1-\alpha_k)^{1-\alpha_k}$

This looks like it's some sort of fraction. A quick screenshot on wolfram alpha suggests indeed that is the case.


Johansen cointegration will usually yield more than one cointegrating vectors (cointegration regression coefficients). By definition, for a cointegrating vector, residuals must be stationary. And a stationary process can take any stationary ARMA form. So, adding AR term is cheating. What you shall do is to test for stationarity of residuals obtianed from cointegration vector. I do not mean to be rude, but you must be doing something wrong. Perhaps, your original variables are not I(1), or you need to include a determinsitc or stationary trend in your stationarity test, and the like. Just for comparison purposes, how do your Johansen results compare to Engel-Granger test. This might shed some light.
Best,


Hint: I will talk through a simpler case, and let you extend it to the generic case.

Let's say there's only two bidders and only the top 1 bidders win. Whoever wins pays whatever the loser bid. This situation is basically a second price auction.

You, player 1, like the object up for auction with a value of $v_1$, but the consolation prize of $w$ is also nice. The solutions you gave say that the optimal bid is $b_1^* = v_1 - w$. Let's confirm this.

If you win this auction with this bid, that means the other bidder bid less than you, $b_1^* > b_2^*$. The total value of winning is $v_1 - b_2^*$. If you lose, then the other bidder bid higher than you, and you receive $w$. So the difference between winning and losing is

$$v_1 - b_2^* - w > v_1 - b_1^* - w = v_1 - (v_1 - w) - w = 0$$

So hooray! There is value in winning compared to taking the consolation. :)

What happens if you unilaterally deviate from this proposed strategy (so your opponent doesn't change their bid) and bid higher up to  $b'_1 = v_1 + \epsilon - w$? Well, if you win when you were going to win anyway compared to the above case...you still get $v_1 - b_2^*$ as the total reward. If you lose when you were going to lose anyway, then you still get $w$. The interesting part is when you bid high enough to then win when you would have previously lost.

That implies we would have $b_1^* < b_2^* < b'_1$. You win and receive $v_1 - b_2^*$, so seems like nothing is different...or is it?

$$v_1 - b_2^* - w < v_1 - b_1^* - w = v_1 - (v_1 - w) - w = 0$$

So now you've won...but you should've just taken the consolation prize. :(

You can do a similar thought experiment for bidding lower. If you bid lower and win, then what you pay doesn't change from bidding the proposed strategy. If you lose when you would've lost anyway, then nothing changes. If you lose when originally you would've won, then you're taking a relative loss again.

So there is no outcome that makes you better off by unilaterally deviating from the proposed strategy of bidding $b_1^*$. You can extend this to realize it is a dominant strategy for all players to follow this line of reasoning.

What about with three players with two winners? Three players with one winner? This question is really just a weird extension of a second price auction. If you understand that, that will help, but otherwise, try following the intuition above and re-reading your solutions given to you.


Limits to Quantitative Easing Programmes

1) In principle, there is nothing material to stop the central banks. Apparently, during the great recession in the 1930's in the US, the fed bought even stranger stuff. They could go domestic bonds, equities, houses, etc, and then they could go to foreign assets.

2) No, they cannot literally go bankrupt because they can always print money. Some banks have in fact begun to consider printing money as a way to forcefully lower the value of their currencies.

3) My guess is that the issue is really the opposite one. Banks are doing too well: they buy assets, and those assets do well, then the bank is actually backing up each unit of currency with more value than before. It's hard to create inflation like that.

4) The practical limits are twofold: it might be politically unsustainable. A) Sooner or later the public thinks this is useless and they elect a new central banker or ask the president to appoint a new one. B) It might create all kinds of strange incentives. For example, you might want to bribe the central bank reserve's managers to buy your company's assets.


Why is the definition of opportunity cost "the benefit of the best alternative choice" but not "the benefits of all alternative choices combined"?

This is because it does not make sense to compare chosen options to all other alternatives that are mutually exclusive. For example, if you have 5 choices A, B, C, D, E, you can choose one, and C is the most preferred option it does not make any sense to compare picking C to jointly picking A, B, D and E. Only one of the other options can be pick so you should use best of them for comparison