Why has Apple increased app store prices in the UK?

Since payments to app developers are denominated in USD it does make sense to raise prices. The pound sterling has decreased in value relative to the USD by about 30 percent since May 2014, which I'm selecting as the arbitrary peak for comparison. So if they kept the same prices as before the recent slide, an app would make about 30% less relative to 2014. Even relative to Jan of 2016, the app would make about 14% less in USD.

As I'm involved with a few IT projects I can tell you that the marginal cost of production is never zero for a "finished product". In technology there is no such thing as a finished product. You can't put it in a box and deliver it and then say to the customer that it's their problem now (well you could but it wouldn't keep you in business very long).

I think it is perfectly reasonable to adjust app prices to reflect what appears to be a new normal exchange rate.

Edit 1:
On the marginal cost of an Apple Store app:

The question states that the cost of one additional sale is zero for an app. I disagreed in my above answer, but much too strongly and incorrectly.

The marginal cost of some apps is equal to or close to zero. Usually, these apps take the following pricing structure: a free version with advertising, and a paid version. Then there is another class of apps that benefit so strongly from economies of scale and maturity that their MC is near zero (Facebook, Spotify, ect.)

They do make up the vast majority of apps. This means that for all intents and purposes, the marginal cost of an app is zero.

But I would argue that some apps with small but enthusiastic user bases likely do have some marginal costs and set their prices near zero anyways because of consumer expectations. Each user generates bug reports, which generates cost, and small development teams are disproportionately affected by this situation.

It is one reason why pricing software is so difficult.


Does a more even distribution of wealth result in lower taxation?

EDIT: The research I found looks at income inequality, not wealth as the question asks.

Specifically with respect to social welfare spending (not necesserily taxation), the following articles from 2003 seem to suggest the opposite is true.
de Mello and Tiongson:

The political economy literature suggests that redistributive spending is higher in unequal societies due to median voter preferences.  Alternatively, it can be argued that unequal societies may spend less on redistribution because of capital  market imperfections.  Based on different  data sources, the cross-country evidence reported in this paper suggests that more unequal societies do spend less  on redistribution.

Moene and Wallerstein:

The empirical relationship between inequality and social-insurance spending as a share of GDP in advanced industrial societies differs across policies. For many policies--pensions, health care, family benefits, poverty alleviation--spending is largely uncorrelated with the inequality of wages and salaries. But for a significant set of policies that constitute roughly 30 percent of the welfare benefit--unemployment insurance, active labor-market policies, sickness pay, disability insurance, and occupational illness and injury--spending is significantly more generous in countries with a relatively egalitarian pretax distribution of wages and salaries.

But your question is framed somewhat more broadly. Inequality may potentially lead to greater spending in other areas like healthcare, police, and so on. I'm not seeing any really comprehensive studies that try to take account of all that. In terms of deficits, Larch (2010) finds that "a  more  unequal distribution of income can weigh on a country's fiscal performance." This does not necessarily show that net taxes are lower in more unequal societies, but Larch does hypothesize that "an  increase  in  the  inequality  of  income  tends  to  soften  the  relative  preference  for  fiscal  discipline". So I would say to answer your broader question, more research may be needed.


The monopolist solves,

$\max \pi=R(Q)-C(Q)=P(Q)Q-C(Q)$

The general solution is (if the problem is globally concave)

$MR(Q^*)=P'(Q^*)Q^*+P(Q^*)=C'(Q^*)$.

Since $P'(Q)<0$, notice that $MR(Q)<P(Q)$ and, therefore, $Q^*<Q^{C}$.

In this example, we have $P(Q)=3-\frac{Q}{16000}$ and $C(Q)=Q$. Hence,

$MR(Q)=-\frac{1}{160000}Q+(3-\frac{Q}{16000})=1=MC(Q)$.

Rewritting, we get
$3-\frac{Q}{8000}=1$ or $Q=16000$. To compute the price simply substitute in the demand function and get $P(16000)=3-1=2>1$.


What are the economic justifications of the size premium?

The following are NOT my words. Found this online. Please check full article here:

http://www.multifactorworld.com/Lists/Posts/Post.aspx?List=3b285530-ccfa-416a-ba3b-de55213abe17&ID=59

Hope this helps...

Size Premium: Why Does it Exist?
There seems to be general agreement that the size premium is a reward for bearing the risk that small companies may do poorly in bad economic environments. For example, compared to larger companies smaller companies may not be able to easily obtain credit in such environments. Bankruptcy risk is also typically a bigger risk for smaller companies in tough economic environments and in general. It could also be that some portion of the size premium is a reward for bearing the liquidity risk of small stocks compared to large stocks.


"1." is wrong and "2." is correct, because (see e.g. Wolfram):

$$A \setminus B = \{x: x \in A \text{ and } x \notin B\}$$

Say for example we have $S = \{a, b, c\}$ and $T = \{b,c\}$.

Then we'd write $S \setminus \{a\} = T$. We do NOT write $S \setminus a = T$.

And so in general, for any set $S$, $S\setminus \emptyset =S$.

We write $2^{X}\backslash \{\emptyset\}$ because $\emptyset \in 2^X$. In contrast, we have $2^{X}\backslash \emptyset = 2^{X}$.

Example. The set $S$ contains four elements: the empty set, an element called $a$, the set containing the empty set, and the set containing $a$.

$$
S=\Bigg\{\emptyset,a,\{\emptyset \},\{a \}\Bigg\}.
$$

Then we have:

$$ \begin{align} S \setminus \emptyset & = S
\\ S\setminus\{\emptyset\}& = \Bigg\{a,\{\emptyset \},\{a \}\Bigg\},
\\ S\setminus\{\{\emptyset\}\}& = \Bigg\{\emptyset,a,\{a \}\Bigg\},
\\ S\setminus\{a\}& = \Bigg\{\emptyset,\{\emptyset \},\{a \}\Bigg\},
\\ S\setminus\{a,\{\emptyset\}\}& = \Bigg\{\emptyset,\{a \}\Bigg\},
\\ S\setminus\{\{a,\emptyset\}\}& = S.
\end{align} $$

The last equation is true because $S$ does not contain the set $\{a,\emptyset\}$.

See Enderton, in particular p.3, top paragraph -- it's well worth spending a little time if only on the first chapter.


This link to a 43 page paper: Ricardian equivalence: an empirical application to the Portuguese economy CarlosFonseca Marinheiro
https://core.ac.uk/download/pdf/6978949.pdf
Introduction on page 3:

The fiscal policy may be used with a stabilising role if the government finance decisions are able to influence private consumption (i.e., aggregate demand) and saving. This influence depends on the degree to which consumers treat government debt as net wealth.

If Ricardian equivalence holds then the government cannot stimulate aggregate demand by increasing spending and/or reducing taxes. So history of economic debate over the effectiveness of fiscal policy includes this theory called Ricardian equivalence.

According  to  the  Keynesian  consensus  consumers  treat  government  debt  as  net  wealth. Therefore, a substitution of debt for taxes has a positive influence on private consumption and  aggregate  demand.  However,  the  consequent  decrease  in  private  and  national  saving, implies  an  increase  in  the  real  interest  rate,  which  crowds  out  private  investment.  The reduction  in  the  capital  accumulation  then  leads  to  a  reduction  of  the  long-term  growth prospects of the economy. This negative long-run effect offsets some of the positive short-term effects of the government deficit.

The  Ricardian  thesis  has  a  complete  opposite  view.  It  states  that,  for  a  given  expenditure path, substitution of debt for taxes has no effect on aggregate demand nor in interest rates.The  government’s  inter-temporal  budget  constraint  implies  that,  for  an  unaltered  level  of government outlays, a tax cut now implies a tax increase in the future. As borrowing only postpone  taxes  for  the  future,  consumers,  who  are  simultaneously  taxpayers,  anticipating the increase in future taxes, do not consider the current tax cut and the consequent increase in  disposable  income  as  being  permanent.  Their  inter-temporal  budget  restriction  is  left unaltered.  Therefore,  consumption  is  also  unaffected.  The  increased  disposable  income  is entirely saved.

Then there is another historic economic debate over the concept called "crowding out."


When and why did governments impose restrictions on money creation

In any healthy economy, the central bank is independent of the executive government. (Since the central bank can be thought of as part of the 'government', I want to be specific about which parts of government I am referring to—by "executive" I mean the body of people who are in power and who administer policies.)

This separation is in order to separate fiscal policy from monetary policy, as this makes it easier to achieve monetary policy objectives. Otherwise, the executive could enforce money creation merely to achieve some objective, such as increased government spending or a decrease of local-currency denominated debt, but this would send inflation skyrocketing. Instead, the executive (through the Treasury) sells bonds on the open market. Any entity is free to buy those Treasury bonds, and the price is determined by the market. (An executive that controls the central bank and forces them to buy Treasury bonds is not really independent of the central bank—it is just a show.) So, the Treasury selling bonds is how the executive raises money for itself, but it also increases the demand for money, so the central bank needs to be part of the group of entities that buy the bonds, in order for interest rates to remain unchanged. Needless to say, when the central bank buys bonds, it is increasing the supply of money.


How much should it cost to close the beach?

The best way to find out the competitive value of the temporarily privatised beach would seem to be to auction licences, charge a tax based on the self-assessed value for holding them, and require the holder to relinquish them at the self-assessed value. This is basically the idea of Eric Posner & Glen Weyl’s Radical Markets (2018).

Given that the organiser of the contest has to commit to a suitable date (presumably during the right season, at the weekend, etc.) she will be willing to pay a tax for holding the licence based on a self-assessed value high enough to deter other (groups of) surfers from buying out her licence for that particular date. The other surfers will likely have less propensity to pay the assessed value of the contest organiser than to pay nothing for using the beach the following day.

That said, the public authority would likely need a mechanism for guaranteeing that the beach remains public in principle. This could be achieved by limiting the number of days per year that the beach could be privatised, or requiring qualitative criteria (such as an event) for issuing a licence. In any case the public condition is mutually exclusive with privatising the beach for dwelling.


Simultaneity problem: use of Instrumental Variables and test

To clarify notation, and treating summarily the other variables we have a system

$$y = a_0 +a_1x + \mathbf w_1'\gamma_1 + u_1 \\
x = b_0 + b_1y + \mathbf w_2'\gamma_2 + u_2$$

If $b_1 \neq 0$, then solving for $x$ we obtain

$$ x = b_0 + b_1(a_0 +a_1x + \mathbf w_1'\gamma_1+ u_1) + \mathbf w_2'\gamma_2 + u_2 \\
\implies x = c + \mathbf w_1'\frac {\gamma_1}{1-a_1b_1}+\frac {b_1}{1-a_1b_1}u_1 + \mathbf w_2'\frac {\gamma_2}{1-a_1b_1}+\frac {1}{1-a_1b_1}u_2$$

and  $x$ is correlated with $u_1$ in the first equation, and so it is endogenous (we would obtain an analogous result if we solved for $y$ -it would be endogenous in the second equation). Historically, this is the "original" endogeneity problem, the one realized back in the '20s when looking at market equilibrium, where the assumption of market-clearing created the endogeneity. Namely it is not the usual "omitted variables" endogeneity problem, but the form of endogeneity due to "simultaneity".

So endogeneity of $x$ hinges on $b_1\neq 0$. But running OLS on the second equation and testing whether $b_1=0$ is not valid, because, as said, $y$ is endogenous in the second equation, so the OLS estimation is unreliable, being inconsistent.
This is a bit subtle. We want to test $b_1 =0$, so under the null hypothesis $y$ does not belong in the second equation. But inference will use $y$ itself. This means that if the null is rejected, the rejection is not reliable, because if the null is not true, then $b_1 \neq 0$ and automatically the estimator, t-tests etc become inconsistent.

The question asks to test for endogeneity in the first equation, so the evident thing to do is to search for an instrument, or instruments, for $x$ and then, for example, apply the Hausman test in the first equation.

Assume that we applied the Hausman test in the second equation, as the OP suggested. This would mean searching for instruments for $y$. Here if the null is not rejected, it would mean that $y$ is not correlated with $u_2$. But from this it does not necessarily follow that $b_1\neq 0$, and so it does not guarantee that $x$ is uncorrelated with $u_1$. If the null is rejected, it means that $y$ is correlated with $u_2$. But this could happen for any other reasons, not necessarily due to the simultaneity.
So the endogeneity tests should be conducted on the first equation.

Note that in reality, the Hausman test  in IV estimation has as its null hypothesis that both the OLS and the IVE estimator are consistent. If the null is rejected, it only tells us that at least one of the two is inconsistent. It cannot tell us which one, and it cannot guarantee that not both are inconsistent: The Hausman test is more informative and specific in its conclusions when the null is not rejected, rather than when the null is rejected. (This is a point usually not clearly stressed in the literature, although side remarks and footnotes do exist to that effect).


Free source of large amounts of financial data

I assume that you are looking for macro financial data across countries. So, you can check out:

World Bank Open Data with a specific query on Financial Sector
IMF Financial Statistics or IMF Finances. Last time I used is a free registration was required .
Agricultural Exchange Rate Data Set. This data set contains annual and monthly data for exchange rates (important to U.S. agriculture). It includes both nominal and real exchange rates for 79 countries, plus the European Union (EU), as well as real trade-weighted exchange rate indexes for many commodities and aggregations. All series are updated quarterly. Data series start at the beginning of 1970 and run to the last available data point.


Is the steady state of $k$ enough to show $Y$ grows at the rate $n+g$?

Based on the steady state your production function is Cobb-Douglas.Taking logs and derivatives wrt time of $Y$, $\frac{Y}{L}$ and $\frac{Y}{AL}$ in the steady state yields the desired result: $K$ grows with $n+g$ on the BGP.

It would be interesting to know what the supervisor's objection was.


(If urns are vacancies and balls are unemployed, what distinction between unemployed workers does the Red/Green dichotomy reflects?)

Each ball has in front of it an identical box, each with the exact same lottery tickets, its ticket has a number on it, and each number corresponds to an urn.

We say "Go!" and each ball draws "randomly" (i.e. with equal probability) a ticket from the box in front of it. Note that the requirement that the probabilities are Uniform, impose the condition that, if we want to have proper distributions, the number of urns must be finite (not even countably infinite -see this post in statistics.se) Then:

RED BALL SOLO
From the point of view of a single Red ball, what is the probability that it will draw a number that no other Red ball draws?

Since each ball draws independently from the others, it is clear that we don't care at all about the existence of Green balls, to the degree that no upper limit of "number of balls in an urn" exists (if it existed the draws would not be independent). So Green balls (their number and what they draw from their boxes) can be kept out of the picture.

Since Red balls are indistinguishable, let's take $R_1$ as our hero. If it draws the number $1$ (i.e. conditional on) we want the probability

$$\Pr(R_j \neq 1 \mid R_1 =1)\;\;  j=2,...N_R$$
But draws are independent so we want simply

$$\Pr(R_j \neq 1) = \prod_{j=2}^{N_R}[1-\Pr(R_j = 1)] =  (1-1/N_U)^{N_R-1}$$

But $R_1$ can draw not just $"1"$, but any number, and each with probability $1/N_U$. So the probability of the event that we want, denote it $\Pr(\text{RBS})$ is (Law of Total Probability)

$$ \Pr(\text{RBS}) = \sum_{i=1}^{N_U}\left(\frac {1}{N_U}\right)\cdot \Pr(R_j \neq i \mid R_1 =i) = \left(\frac {1}{N_U}\right)\sum_{i=1}^{N_U}(1-1/N_U)^{N_R-1}$$

$$\implies \Pr(\text{RBS}) = \left(\frac {1}{N_U}\right) \cdot N_U \cdot (1-1/N_U)^{N_R-1} = (1-1/N_U)^{N_R-1}$$

...again.

This is intuitive: the greater the number of urns, the higher the probability. The greater the number of Red balls, the lower the probability.

Now, define job market tightness as usual by
$$\theta = N_U/N_R \implies N_R = \frac 1 {\theta} N_U$$

(here "U" denotes "urns" so vacancies). Then

$$\Pr(\text{RBS}) = \left(1-\frac 1{N_U}\right)^{\frac 1 {\theta}  N_U-1}$$

If $N_U$ grows "large" then the above is well approximated by

$$\Pr(\text{RBS}) \approx e^{-\frac 1 {\theta} }$$

The higher job market tightness, the higher the probability that an unemployed will be matched solo to a vacancy (again, ignoring what the Greens represent).


Are business cycle fluctuations usually studied in a (New Keynesian) DSGE model, or can they also be studied in a growth model?

Business cycle fluctuations can be studied in many ways including also growth models. An example of using growth model to study business cycle would be this 1995 paper by Cho & Cooley. This being said growth models are not as common for examining business cycles as for example DSGE models so saying they are usually used for that purpose is correct. To my best knowledge no statistics on the relative use of this or that model for explaining business cycles exist but anecdotally, speaking from attending macro conferences, DSGE or various non-stochastic (when we talk about pure theory) New Keynesian or to lesser extent real business cycle models remain popular.


Total variable cost would not fit on that graph: total cost has units of "dollars", while the axes of the graph are "items" and "dollars per item"

But you can multiply the two together, which implies that areas on the graph are measured in dollars.

For example, below the pale blue area is the total variable cost for the quantity shown, while the pink area is the total cost


So you want to find $\partial p / \partial k$

By implicit differentiation what they mean is let $p=p(k)$ so p is a function of k.

Take your equilibrium condition $Q_s-Q_d=0$ and now differentiate with respect to k.


Is inconclusive region in durbin watson a problem?

There are cases error serial correlation is a disaster. For example, if your model is $y_t = \beta_0 + \beta_1 y_{t-1} + u_t$, then serial correlation in $u_t$ means correlation of $y_{t-1}$ and $u_t$ (in general) and your OLS estimator is biased and inconsistent. In other cases, serial correlation does not cause endogeneity and OLS is still consistent. An example is the case the right-hand side variable is strictly exogenous. Then you can use OLS and heteroskedasticity and auto-correlation consistent (HAC) inferences, e.g., using the Newey-West standard errors.

So my answer is "it depends". If some regressors become endogenous due to the error serial correlation, you will probably want to change your model. Otherwise, you can use OLS and HAC standard errors.

If your model is dynamic ($y_{t-1}$ on the right-hand side), the chance is high that you need to seriously worry about serial correlation in $u_t$. This happens in many time-series and financial applications.

By the way, Farebrother (1980, JRSS Series C) proposed Pan's procedure for $p$ values for DW tests. Inconclusiveness can be avoided.


This article is from 2002, but it does a pretty good job of summarizing the pros and cons and has a section on normative decisions. Here is the abstract:

  In this paper, we review the economic effects of intellectual property rights and specifically address the economics of the patent system. The production and dissemination of new knowledge is fraught with market failures because knowledge is a public good. Patents provide a second-best solution to the resulting appropriability problem. We review the main benefits and costs of the patent system, focusing on the role that patents play in providing incentives for innovation, in promoting the dissemination of knowledge, and in helping technology transfer and commercialization of new technology. From a more normative perspective, we address the questions of what the features of an optimal patent system are and whether the patent system is socially desirable. We examine the problem of the optimal length and scope of patent protection, both for the case of a single innovation and for the richer case of cumulative innovations. Finally, we review the issues related to how the patent system influences the market structure and research and development investments.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1317&context=card_workingpapers


what is the interpretation of $\beta_{ij}$ in the translog production function?

You may interpret $\alpha_i$ as the output elasticity regarding input factor $x_i$, keeping other factors $x_j, j\neq i$ constant. $\beta_{ij}$ is an indicator for complementarities in production, i.e. how much does the addition of one percent of both factors $x_i$ and $x_j$ add to $y$. Since there is the factor 0.5, and since $\alpha_i$ and $\alpha_j$ are relevant too, the interpretation is not that straightforward.


How to relate real rate of return on capital to bond interest rate: Lagrangian

Ignoring the expectations operator, your lagrangean has two mistakes: first the constraint is per-period so it is also multiplied by the discount factor. Second,  the way you have wrote the time indexes is inconsistent, as regards their interpretation for capital and bonds. If $K_t$ denotes "capital at the beginning of period $t$" (as it does), you should also write $B_{t}$ instead of $B_{t-1}$, to denote bonds held at the beginning of period $t$. This takes with it also the time index on $Q$.

Also  I would advise to write your constraint in "normal form" (as they say in micro), namely with the constraint as "higher than or equal to zero".

Let's try this, up to a point.

$$\Lambda = \sum_{t=0}^{\infty}\beta^t \Big[U(C_t,L_t) + \lambda_t\big(W_tL_t+R_tK_t+B_{t}+D_t - (P_tC_t + Q_{t+1}B_{t+1}+ P_tI_t)\big)\Big]$$

$$ = \sum_{t=0}^{\infty}\beta^t \Big[U(C_t,L_t)\\ + \lambda_t\big(W_tL_t+R_tK_t+B_{t}+D_t - P_tC_t - Q_{t+1}B_{t+1}- P_t\big[K_{t+1} - (1-\delta)K_t\big]\big)\Big]$$

Then

$$\frac{\partial \Lambda}{\partial K_{t+1}} = 0 \implies -\beta^{t}\lambda_t P_t + \beta^{t+1}\lambda_{t+1}[(1-\delta)P_{t+1} +R_{t+1}] =0$$

and

$$\frac{\partial \Lambda}{\partial B_{t+1}} = 0 \implies -\beta^{t}\lambda_t Q_{t+1} + \beta^{t+1}\lambda_{t+1} =0$$

Equating we get

$$\frac {[(1-\delta)P_{t+1} +R_{t+1}]}{P_t} = \frac 1{Q_{t+1}}$$

Now, $R_{t+1}$ is the nominal gross return to capital, so a $P_{t+1}$ is implicitly in there. If we set $r_{t+1}\equiv R_{t+1}/P_{t+1}$ the relation becomes

$$\frac {[(1+(r_{t+1} -\delta)]P_{t+1}}{P_t} = \frac 1{Q_{t+1}}$$

Taking logs, we get the Fisher equation written for period $t+1$.


What is the difference between budget constraint and budget line?

One is an algebraic formula, the other is the geometric representation of that formula.

They are the same as much as a straight line and the equation describing said line are the same.


Why wouldn't other firms follow suit if an individual firm decides to cut its price?

There could be many answers, but the one that comes to my mind immediately is product differentiation. If you're selling exactly the same thing as everyone else, buyers are going to flock to the lowest price buyer, so you everyone will have to follow a price cut. But say you are selling ice cream cones on the boardwalk at the beach. Any little thing, such as a slightly better location, a better selection of flavors, or a cheerier clerk scooping the cones could be enough to let you keep a good chunk of your customers without cutting the price.
BTW, even a slight perceived difference among sellers would (technically speaking) make the market an oligopoly, not perfect competition. In perfect competition, by definition, there can't be any differences. That is why we call perfect competition an "ideal type," not a description of real-world markets.


Why are prices of stocks not very very high with very very low r?

Equities and some other asset classes have gone up in price.

However, just because interest rates are very close to zero, does not mean discount rates are very close to zero.

Discount rates account not only for the price of money, but also for future uncertainties.

The extended period of depressed demand in many nations has added considerably to the uncertainty.

Hence, some asset classes have increased in price, but not to extreme values.


How does Marginal rate of substitution change?

I don't think any conclusion is possible without more detail, but if we make a few assumptions we can see where they lead us.

Assume $a,b > 0$ and $z < 0$

Assume the consumer in question has well-behaved preferences and therefore their indifference curves must be monotonic and convex

Given these assumptions (which I think are fairly reasonable, although I'd want to see them explicitly stated if this was a formal question) then the answer should be that the post-exchange indifference curve is different from the earlier one.

Given that $a$ and $b$ are greater than zero, we know that the consumer has lost some of the first good and gained some of the second. Examining all possibilities they could be in one of three positions relative to their original indifference curve:

Above the original curve
On the original curve at a different point
Below the original curve

Neither Case 2 nor Case 3 are plausible. We know that the slope of the new curve at $(x-a, y+b)$ is three times shallower than the that of the original curve at $(x, y)$. The assumptions of convexity and monotonicity require that the original indifference curve become steeper as it approaches the y-axis. See below for a colourfully-amateurish representation.

Note that in the example pictured above the resulting indifference curve is impossible as it crosses the original indifference curve - a situation that can never arise.

That leaves us with Case 1 as the sole possibility, which means that $(x-a, y+b)$ must be on a new indifference curve as per the illustration below.


Other factors that affect the supply of money

I assume that this question is related to this question on money growth - link to question

The monetary base and velocity completely pin down the wider monetary aggregate. The behaviour of acceleration is slightly trickier, but I will attack it here.

I will firstly use continuous time for simplicity. I will also use my own notation, which is explained.

Let $M$ be whatever wide monetary aggregate you want to study, and $B$ be the monetary base. Velocity $V$ is defined:
$$
V = \frac{M}{B}.
$$
Importantly, this is a definition, and always holds. There is no reason to ascribe changes in velocity to any particular factor, such as preferences of banks.

Then, the rate of change of the money stock is:
$$
\frac{dM}{dt} = \frac{d(BV)}{dt} = B\frac{dV}{dt} + V\frac{dB}{dt}.
$$
Since we are interested in the percentage rates of change, we define growth rates as the time derivative of the variable divided by the variable. That is,
$$
G_M(t) = \frac{\frac{dM}{dt}}{M},
$$
$$
G_B(t) = \frac{\frac{dB}{dt}}{B},
$$
$$
G_V(t) = \frac{\frac{dV}{dt}}{V}.
$$

Then, we see that:
$$ G_m(t) = \frac{B}{M}\frac{dV}{dt} + \frac{V}{M}\frac{dB}{dt}.
$$
Since $\frac{B}{M} = \frac{1}{V}$ and $\frac{V}{M} = \frac{1}{B}$,
$$
G_m(t) = \frac{\frac{dV}{dt}}{V} + \frac{\frac{dM}{dt}}{M} = G_V(t) + G_B(t).
$$
Since you are interested in acceleration, that would be $\frac{dG_M}{dt}$, which is given by:
$$
\frac{dG_M}{dt} = \frac{dG_V}{dt} + \frac{dG_M}{dt}.
$$

In other words, to characterise the acceleration of the money supply, you need to look at the acceleration of the monetary base and the acceleration of the velocity. If we refer back to the question on the money supply in the 1990s, you would have had to calculate the annual percentage change of velocity, and not look at the level of velocity (which is what was displayed).

If we return to discrete time (and observed data are discrete time), the relationships above become approximations. You would have to take the first differences, to get the rate of change, and the difference of the rate of change to get acceleration. For annual data, that means we end up comparing data two years apart, which may break the continuous time approximation.

Since this question asked about other factors influencing the relationship between the base and wider money: there are none. The above equations hold by definition. The question why velocity would change is a more difficult question.


A zero-sum game is one in which the players' payoffs sum to zero in every possible outcome of the game.
The given payoff description is incomplete. But if we assume that in the outcomes where only one player guesses correctly, the other player (who guesses wrong) would have to lose an amount equal to the total number of fingers shown (denoted $X$, say, to use @Bayesian's notation), then the game is indeed zero-sum:

In outcomes where both guess correctly or incorrectly, they each get $0$, which sum to zero.
In outcomes where only one guesses correctly, the correct guesser gets $X$ and the incorrect guesser gets $-X$, which also sum to zero.


How to find mixed optimal strategies in this zero-sum game?

First, a caveat: I'm on the job market this year in the midst of the couple weeks when calls are rolling in. Hence, this seemed like a nice way to kill some time (semi-)productively. This is also a disclaimer in case I've made an error :)

Now, let's look at the one you're suggested to try for, where player $1$ chooses $0$ with $p$ and $1$ with $1-p$. Again let player $2$ choose $G$ with full support.

If player $1$ chooses $0$ and player $2$ chooses $y$, Pl $1$ gets $$\int_{0}^{1}yg(y)dy = \mu$$
and if he chooses $1$, he gets $$\int_{0}^{1}(1-2y)g(y)dy = 1 - 2\mu$$
where $\mu := \mathbb{E}_{G}[y]$
Hence, for indifference we need $\mu = 1/3$. Ah but we also need this to be robust to any other deviation. Namely, we need.
$$\frac{1}{3} \geq x\int_{0}^{x}(1-2y)g(y) + (1-2x)\int_{x}^{1}yg(y)$$
for all $x \in [0,1]$.

Knowing that $\mathbb{E}_{G}(y) = 1/3$ for $G(y) = \sqrt{y}$, let's guess that that is the solution. The right-hand side reduces to
$$\dfrac{2x^\frac{3}{2}-2x+1}{3}$$ which is maximized at $x = 0$ and $x = 1$ and equals $1/3$ there, as required.

Finally, for player $2$, she is indifferent over any $y$ if $$p(-y) - (1-p)(1-2y)$$ does not depend on $y$. This holds if $p = 2/3$.

Thus the equilibrium is given as follows: player $1$ chooses $0$ with probability $2/3$ and $1$ with probability $1/3$; and player $2$ chooses cdf $G(y) = \sqrt{y}$ on $[0,1]$. The value of the game is $1/3$ for player $1$ and $-1/3$ for player $2$.


Why is not tram popular in most countries?

My guess: high investment costs, low gains relative to alternatives such as buses. You need rails and electric lines, which are expensive. In the western hemisphere trams already existed before alternatives such as large buses came about, so the costs consist mainly of maintenance. In other parts of the world the infrastructure is not in place, and getting it is expensive. For buses the infrastructure is already there as all you need is a decent road.


Troubleshooting Utility Maximization with the Lagrange Method

You needn't use the Lagrange here. And in general, this will be true for additively separable, linear utility functions.

Note that the marginal rate of substitution between any two goods $i,j$ is a utility function of the form $u(x_1,x_2,...,x_n)=a_1x_1+...+a_nx_n$ will be constant at $MRS_{i,j}=\frac{a_i}{a_j}$

A consumer will buy none of a good $x_i$ whenever $\frac{u_i}{p_i}$ is below the maximum such ratio of the $n$ goods. A consumer will exhaust her budget on whichever good has the maximum such ratio. And if all such ratios are equal, the consumer is indifferent between any consumption bundle that exhuats wealth $w$.

For your case of two goods, indifference curves are linear. For more than two goods, they are hyperplanes.


A good site, with high quality online lectures, is the MIT Open Course Ware.

You will find courses going from Principles of Macroeconomics, Economic Growth, Macroeconomic Theory I, II, III to Advanced Macroeconomics I or II.


Obvious Dominance and Milipede Games (Question to Pycia and Troyan (2019)

It is a convoluted definition because condition 3:

"At all $h$, if there exists a previously unclinchable payoff x that becomes impossible for agent $i_h$ at $h$, then $C_i^\subset (h) \subseteq C_i^h (h)$."

means that at every history if there is a payoff that player $i_h$ could not have secured (or clinched), but was feasible at every previous history where some other player was taking decisions and now is not feasible, then the set of clinchable payoffs cannot decrease.

That is, the set of clinchable payoffs is not allowed to decrease, except when some special condition is met.

Figure 1 is an example of a millipede game because the set of clinchable payoffs never decreases for players $j$ and $k$. It only decreases for player $i$ when (it goes from $\{x,y,z\}$ to $\{x,y\}$ (from the second to the third move), but this is fine because in player $i$'s third move the only payoff that was previously unclinchable is $w$, but it has not become impossible at that history (since the only impossible payoff is $z$ that was taken by player $j$).

Hope this helps.


What prerequisite courses do I need in order to specialize in Mechanism/Market design theory?

The list you posted was for an undergraduate degree in economics. Mechaism design usually won't come up in any serious way until graduate school. Until then, my advice is to take as much as the following as possible:

microeconomic theory,
game theory (often covered within microeconomic theory),
mathematical methods/optimisation.

Mechanism design falls under the rubric of microeconomics theory, which at the advanced (i.e. postgraduate) level is highly technical. So achieving a high degree of mathematical sophistication in addition to a solid grounding in microeconomic theory is probably the best strategy.


How can perfectly competitive firms earn zero profits?

First, let's deal with the semantics and terminology aspect: what the word "profit" means in Economics, and what the word means in the everyday/business/accounting use, are two different things.

In its everyday business use "profit" is the surplus above all expenses including depreciation. So what business call "profit" the Economics discipline calls "net return to invested capital".

In Economics on the other hand, "profit", although never really "officially" defined universally, it essentially means "excess returns to capital above the market level", where the "market level", sometimes means "average level" sometimes it means "perfect competition level", or some other notion of equilibrium returns in the absence of exploitation of market frictions, market power and the like.

Perfectly competitive firms earn zero profits because perfect competition drives prices down to average cost - but where in "cost" we include what business call profits, because for us, it is payment to capital used in production.

Now let's consider the specific example of the OP, where "for simplicity" we have a single production factor, and where, the production function is strictly concave:

$$Q= F(L), \;F'>0, \;F''<0$$

The profit maximization problem here is

$$\pi = pQ - wL,\;\; \text{f.o.c.}: \;pF' = w \implies L^* = (F')^{-1}(w/p)$$

$$\pi^* = pF - pF'L = p(F-F'L)$$

For zero profits we should have

$$\pi^*=0\implies F - F'L = 0 \implies \frac {F}{L} = F'$$

or that average product in real terms equals marginal product. As long as the functional form of the production function is such that there is a positive level of $L$ for which this equality can hold, the argument of the theory is that competition will push the price to a level where the level of labour chosen will be such that all revenues go to payments for labor.

And if the production function does not allow for such an equality, then it simply is not compatible with a perfectly competitive market structure.

The OP should figure out how and why the diagram used in their post is misleading.


Textbook for macroeconomics (advanced undergrad)

Advanced Macroeconomics by David Romer, now in its fourth edition. Link contains TOC and a sample chapter.

The presentation consists of formal theory models, but with lots of intuition too, followed by light empirical applications.


Did JP Morgan mitigate the effects of the Great Depression?

Not really. While it's possible that the Morgan intervention softened the initial stock market decline, the vast majority of the ultimate collapse in securities prices, commodity prices, industrial production, and the banking sector took place after the purchases had been liquidated.

First some history: the bankers' pool intended to lift the stock market was assembled at the J.P. Morgan & Co. offices on October 24, 1929, "Black Thursday", the initial day of market chaos. They pooled their resources and directed broker Richard Whitney to place high bids on blue chip stocks at the NYSE to lift the confidence of the markets. This stemmed the crash on Thursday and led to a rally that continued the next two days. Indeed, the following Sunday's New York Times lauded the "carefully arranged" plan to boost the market, and mentioned that the inclusion of the First National Bank in the buyers' pool the next day confirmed to Wall Street that "the danger of panic had been averted". Some history of these events is provided in the Wikipedia article on the crash.

Alas, the Monday and Tuesday immediately following this confident appraisal - October 28 and 29, 1929 - are known as Black Monday and Tuesday, respectively. They experienced the second and fourth largest daily percentage DJIA declines in history, combining for easily the largest two-day percentage decline in history. Most newspapers at the time described the bankers' pool as waiting to intervene until late Tuesday, when they cut margin requirements and placed buying orders (see AP article). An AP article describing the crash on Monday opened with "powerful financial interests stepped aside today and let the stock market drop...". Finally, on Wednesday the bankers' pool entered with strong buying orders, leading to the third highest daily percentage DJIA increase in history. The next day the pool apparently stepped away from the markets once more.

It kept going like this for a little while, with some more volatility in markets over the next few weeks but eventual stabilization and partial recovery. Finally, the banking group quietly wrapped up its activities in February 1930, with press accounts mentioning that it "come out about even".

Given all this, it's possible that the bankers' pool offered some initial stabilization of the markets. (Although even that's not clear: from press accounts, it seems just as likely that volatility was increased by the erratic day-to-day decisions of the pool.) But again, any significant effect seems unlikely, because the vast majority of the carnage in the Great Depression took place after the pool had been liquidated in February 1930. To convey some sense of this, the following graph shows (in log points) the fall of the DJIA, industrial production index, and wholesale price index in the US starting in September 1929. As you can see, the initial declines in late 1929 and early 1930 - although substantial - are far smaller than the eventual ones. This is in line with the modern consensus view on the Great Depression, which emphasizes the role of the gold standard and its propagation of adverse monetary and banking shocks, rather than the initial market crash itself.

This is all an interesting contrast with the more famous and successful case of intervention, performed in the Panic of 1907 by J.P. Morgan himself. (The original J.P. Morgan was long dead by 1929, though his eponymous son and firm lived on.) In 1907, Morgan engineered a wide variety of rescue measures, making large deposits in struggling banks and large call loans to brokers. These measures are thought to have helped end the panic.

The key difference, I think, is that the 1907 panic was a fairly conventional acute crisis in the banks and money markets: recent nominal growth and seasonal demand for currency meant that interest rates were already high, and then a run on banks and trust companies led to more hoarding of currency and extremely high interest rates. Extremely expensive margin borrowing led to a collapse in securities prices, which endangered the banks further (they were more exposed to stocks, through various channels, than they are now), leading to more run pressure, and so on. In this environment, where banks and firms were sound - and money markets would loosen - if only the panic would stop, it was possible for a few decisive interventions to swing momentum in the other direction.

The Depression was much different. Thanks to the Federal Reserve, seasonal fluctuations in currency demand and (to an extent) panicked currency hoarding did not lead to a spike in money market rates, which actually fell drastically in late 1929 and 1930 rather than spiking as they did in 1907. Also unlike 1907, the banking crisis did not coincide with the initial stock market collapse - instead, it only started with the Bank of United States failure in December 1930, and in the initial stages was arguably due to fundamental insolvency rather than illiquidity. (Since both nominal real estate prices and food prices collapsed, mortgage default rates - residential, commercial, and farm - were all extremely high.)

Bottom line: the crisis of 1929-33 was different enough from the crisis of 1907 that similar measures no longer worked.


Do capital goods have to be valued so that (discounted) real expected return on capital goods equal current present value of capital goods?

Here is a somewhat standard presentation that might be taught in intermediate macroeconomics at US universities.

Let $p_t$ be the price of a unit of capital at the end of period $t$. Suppose that production takes place at the start of each period. After production takes place, depreciation occurs. Let us say that the depreciation rate is constant at $\delta$. Let us also say that the real interest rate across any two consecutive periods is $r$. Assume also that the price of each unit of output is equal to $1.

If so, we would expect that

$p_1=\frac{1}{1+r}[MP^k_2+(1-\delta)p_2]$.

This has a nice interpretation. The price of capital at the end of period 1 equals: The present discounted value of the additional output it can give us at the start of period 2, plus the after-depreciation resale value of that unit of capital at the end of period 2.

We can write down an analogous expression for $p_2$:

$p_2=\frac{1}{1+r}[MP^k_3+(1-\delta)p_3]$

Now substitute this latter expression for $p_2$ into our first equation above, to get:

$p_1=\frac{1}{1+r} \left\{ [MP^k_2+(1-\delta)\frac{1}{1+r}[MP^k_3+(1-\delta)p_3] \right\}=\frac{1}{1+r}[MP^k_2+\frac{1-\delta}{1+r}MP^k_3+\frac{(1-\delta)^2}{1+r}p_3]$

This last expression has again a nice interpretation. The price of capital at the end of period 1 equals: The present discounted value of the additional output it can give us at the start of period 2, plus the additional output it can give us at the start of period 3, plus the resale value of that unit of capital at the end of period 3.

If we keep going, plugging in expressions for $p_3$, $p_4$, ..., eventually we'll find that

$p_1=\frac{1}{1+r}[MP^k_2+\frac{1-\delta}{1+r}MP^k_3+(\frac{1-\delta}{1+r})^2MP^k_4+(\frac{1-\delta}{1+r})^3MP^k_5+\dots]$

This has the desired result: The price of a unit of capital today is simply equal to the present value of the future stream of income that this unit of capital will generate, appropriately taking into account depreciation.

This generalizes to a broader principle employed not just in standard macroeconomics but in the real world too: The price of any asset should be equal simply to the present value of the future stream of income that the asset will generate.


Price ratio determined by endowments only?

No you are not. Preferences determine the equilibrium, even if they are identical, because they determine the value of the endowments.

Consider two agents, one ("A") having ice cream only, and one ("B") having lava only.

Case 1: Both hate lava

Assume lava burns tongues and is useless (typical Economist). Then, the initial endowments of B have zero value, as no one wants them. The equilibrium allocation equals the initial allocation.

Case 2: Both find lava useful

Now, imagine lava being useful for lava lamps (whatever). They still have identical preferences, but now they attach some positive value to lava. The value of B's initial endowment has changed, and hence he can trade at least some of it for ice cream. The equilibrium allocation no longer equals the initial allocation.


I don't feel these two interpretations are mutually exclusive, they belong to different sides of the same problem - one is empirical the other is theoretical. The conflict you seem to see between these two methods is that the "population" interpretation is individual specific. But the population component is a statistical convenience. It is "everything else" that the researcher has no time to model, and it is variation on the level of the population of observations not people - so it is not an individual specific error.

In fact, one acceptable interpretation of this convenience is that conditional on non-random components individuals are the same, and the errors are such that if presented with the same decision again the individual would make a different choice. This is precisely equivalent to your neurological interpretation. Extra structure on the random component may change the degree to which this justification applies - if errors for an individual are assumed to be correlated over time it decreases the relative impact of a neuroeconomic explanation and starts to be about unobservable heterogeneity on the level of the individual. But random utility can still easily encompass both explanations as answers to different sets of questions. The residual error will include both neuroeconomic reasoning and any variables that have been omitted.

To the extent that the randomness is specifically designed to capture "everything else" talking about what causes it is something most empiricists don't want to focus on. This being said, Michael Woodford has written something recently about neuroeconomics and random choice. And the "stocastic neural functions imply stochastic choice" is apparently a fairly common assertion in neuroeconomics: (see here, here, and here though some of these references point out that neuroeconomics places constraints on the form of random choice). But given the variables omitted in an average econometric study, I would hazard a guess that the random term captures relatively more of this omission and less of neurological processes.


I don't know about the "Weekend Effect" but a paper by Scott R. Baker, Nick Bloom, and Steven J. Davis try to answer to the question What triggers stock market jumps?. They show that some news can indeed impact the stock market.

Drawing on next-‐day newspaper accounts, they develop new evidence about the forces that trigger large daily jumps in national stock and bond markets. They read and code next-‐day interpretations of 200 or more daily jumps per country in recent decades, yielding five main results.

The Global Financial Crisis (GFC) of 2008-‐09 exhibits very high counts of daily equity market jumps around the world. Looking back to 1885 for the U.S., the Great Depression is the only period with equal or greater jump frequency.
U.S. developments trigger equity market jumps across the globe, especially during the GFC. Jumps sourced to the U.S. are hugely more important than jumps sourced to Europe except for European countries, where the counts are similar.
Policy news triggers about 20-‐25% of equity market jumps in most advanced economies and a larger share in other countries (e.g., China=33%, and India=46%).
News about the macroeconomic performance and outlook accounts for 23-‐38% of equity market jumps in advanced economies, and less in other countries.
For U.S. government bond yields, news about the macroeconomy triggers 65% of the jumps; adding news about monetary policy as well accounts for 93% of the jumps.

See also Nick Bloom's CEPR Vox column.


What is it about certain business types that seem to restrict how much large they can grow?

That's a very interesting question and there are multiple potential reasons for this.

One reason is that production is simply not efficient at a larger scale. There are diminishing returns to scale: i.e. doubling inputs will result in less than twice the output. This means that two smaller production facilities can be more efficient than one large one.

One way to think about this are kids shoveling snow in winter. Beyond a certain number of kids with shovels they're just going to get in each other's way shoveling the same driveway. Now they could of course split up into two or more groups, these two groups could be one firm or these could be two firms. But what's the rationale for combining these two firms into one? They probably will not economize on any fixed expenses. They need to ring the doorbell, bring their shovels, and ask people whether they wish their driveway to be shoveled. Sure one kid could specialize in negotiating a price and ringing many doorbells and scheduling appointments, but
the gains from specialization are probably not such that it's worthwhile. It's not how that "industry" works. You ring, you shovel, and you move on. So there is no good reason for combining. The kid ringing and making appointments is an unnecessary overhead. It's better that he shovels too.

Another reason are the fixed costs of certain regulations. You're probably familiar with the fact that in some jurisdictions firms above a certain size need to provide certain services to their employees. If you're a small firm, then you will incur that fixed costs all at once if you expand over that threshold. You'll have to get your existing employees those services and the new people. These additional fixed costs might not make it worthwhile to hire extra people unless you can be certain that you can expand your business well beyond that.

A third reason is that many people do not want to expand their businesses. They enjoy the work that they do and the extra income is not worth it. They'd have to spend less time doing the work that they enjoy and more time on administrating the work done by others. Effectively this is a change of employment. So you might as well ask why do some people do work A and not work B?

These are three reasons. The reasoning is the same each time though. The benefits of expanding do not justify the costs of expanding in that particular situation. Given that there are many different situations there will be many different reasons why people choose not to expand their business.

Does this answer your question?


Optimal spending over several periods with log utility and uncertain lifetime

Hint
Generalizing from your 2-period exercise, let $N=m-1$ be the last possible period of being alive. The problem to solve is
\begin{equation}
\max_{\{c_n\}_{n=0}^N}\;u(c_0)+p(1)u(c_1)+\cdots+p(N)u(c_N)
=\max_{\{c_n\}_{n=0}^N}\;\sum_{n=0}^Np(n)\log(c_n)
\end{equation}
subject to a lifetime budget constraint: $c_0++c_1+\cdots+c_N=y$ where $y$ is the lifetime income.


Example of a (not quasi-linear) production function whose inputs are not perfect substitutes but are not asymptotic at the axes

You could, for example take the function $f: \mathbb{R}^2_+ \times[0,1] \to \mathbb{R}$.
$$
f(L,K, \rho) = L + K + (1-\rho) L K.
$$
For $\rho = 1$, we have $f(L, K, 0) = L + K$ which is a production function with perfect substitutes.
The marginal product (of say $L$) is given by:
$$
\frac{\partial f(L, K, \rho)}{\partial x} = 1 + (1-\rho)K,
$$
which equals $1$ if $\rho = 1$.


Positive interaction term with one negative component?

Ignore the other variables. Let $g$ be growth, $C$ corruption and $A$ aid. You look at

$$g = \alpha C + \beta A + \gamma A\cdot C$$

You ask, "does lower corruption increases the effectiveness of foreign aid?"

For this you have indeed to examine the cross-partial derivative

$$\frac {\partial g}{\partial A\partial C} = \gamma$$

This is "how growth is affected at the margin by Aid, if corruption changes"... So if lower corruption increases the effectiveness of Aid, $\gamma$ should be negative, not positive...

...if your corruption index is mapped as "the higher the value of $C$ the higher the corruption".

If your "corruption index" is mapped in reverse (and so in reality it measures "non-corruption"), the anticipated sign reverses also.


Where can one find Beta values for stocks?

Google Finance and  Yahoo Finance are both sites which provide free information on Stock prices and their variables.

I prefer google as it is easier to use, but both sites are pretty good.


Doubt regarding Walrasian equilibrium with complements for both agents

Since demand equals supply holds for every price $p$, this simply means that every $p$ is an equilibrium price. However, the equilibrium allocation that $p$ supports varies with $p$. To be precise, price $p$ supports the allocation in which 1 consumes $\left(\frac{1+3p}{1+p}, \frac{1+3p}{1+p}\right)$ and 2 consumes $\left(\frac{3+p}{1+p}, \frac{3+p}{1+p}\right)$.


Derive the demand functions: Hotelling-style Model

You are treating the absolute value wrong. You seem to have done it right the first time around, do it like that again. (Make a little drawing, think about which side of 10 and 50 the indifferent $X_i$ will be on, etc.)

Another thing: The demand function you derived for $Q_1$ only holds for certain ranges of $P_1$, $P_2$ and $P_3$, because if the prices of firms 1 and 2 are very high but $P_3$ is low, a customer indiffirent between firms 1 and 2 would go to firm 3.


Why Is Cointegration Important In Practice?

I think it is a very classical economic teaching problem - showing how something is relevant in the real world.

First, it solved a problem where linear regressions could lead to spurious results: https://en.wikipedia.org/wiki/Cointegration

  ...Before the 1980s many economists used linear regressions on
  (de-trended[citation needed]) non-stationary time series data, which
  Nobel laureate Clive Granger and Paul Newbold showed to be a dangerous
  approach that could produce spurious correlation, since standard
  detrending techniques can result in data that are still
  non-stationary. Granger's 1987 paper with Robert Engle formalized the
  cointegrating vector approach, and coined the term.

  For integrated I(1) processes, Granger and Newbold showed that
  de-trending does not work to eliminate the problem of spurious
  correlation, and that the superior alternative is to check for
  co-integration. Two series with I(1) trends can be co-integrated only
  if there is a genuine relationship between the two. Thus the standard
  current methodology for time series regressions is to check all-time
  series involved for integration. If there are I(1) series on both
  sides of the regression relationship, then it's possible for
  regressions to give misleading results...

Second, our "Sister site" actually does a nice job explaning the intuition behind, which can explain why it is important in the real world - are two different time series related? And can we test for this?:

https://quant.stackexchange.com/questions/219/what-is-the-intuition-behind-cointegration

I really like this one:

  "Think of a man walking his dog. He will go along and his dog will
  stroll along running back and forth. Man and dog are mathematically
  "cointegrated".

  As an investor you bet that the dog is coming back to his master or
  that the leash has only a certain length.


Why could boasting oil during the current drop in oil demand be good for Saudi Arabia?

The country's funding costs are very low in comparison:

at an average of  25 per barrel (in comparison: Russia  is funding for 45, the fracking, which is mainly carried out in the United States, costs $ 75 per barrel). So the earnings are above average anyway.

The USA, which has become a serious competitor, should give up market share. Today the United States produces almost as much oil as Saudi Arabia. The Middle East is fighting back with a crowding-out competition.

With state reserves of $ 750 billion, the desert state can easily cope with lower oil revenues over the years.


Nash equilibrium is a pair of strategies. So your answer needs to give two strategies. The informal definition is "pair of strategies that are best responses to each other".

So take an arbitrary strategy; let's say each player chooses 50. Can anyone gain by playing 51 or 49? No, so (50, 50) is a NE. Like your equation says, any two strategies that add up to 100 make up a PSNE. Because there can be no profitable deviation for any of the two player (the proof is trivial; show that playing less gives less, and that playing more, gives less for any pair of strategies adding up to 100)

Now about MSNE. A good way to start thinking about it is to take an arbitrary strategy and play around with it. You will find no MSNE this way.

The other way to look for them is to find a point where two strategies make players indifferent between their options and are also best responses. I could only find this in the symmetric case.

There are some MSNE if you play strategies $( a_1, a_2 )$ where $a_1+a_2 = 100$ and your mixing probabilities are $P(a_1) = a_1/a_2$ (assuming $a_1 < a_2)$. Example here would be playing $(10, 90)$ with probabilities $(11.111, 88.888)$. This makes the opponent indifferent between $10$ and $90$, and as such he could play the mirror strategy, with both being indifferent to any mixing and both being a best response to each other.

I did not find any other case of MSNE, even after some work.


How do the Government, Firms and the Foreign sector pay their debts in this circular flow of income?

For the households it would through the savings arrow. Borrowing from macroeconomic perspective can be and in most advanced textbooks (such as Romers advanced macroeconomics) is often treated as a negative saving.

Hence when the households and businesses borrow they incur negative saving and when they repay debts it can be viewed as a positive increase in saving (which for people who borrowed would be negative).

This can actually be clearly seen from the diagram since savings arrow goes to the financial market and from financial market you get the borrowing.

It’s not that the financial markets lend people money - people lend money to each other through financial markets and then pay it back from the savings.

Saving is the difference between your income and consumption so in order to pay any debt you have to save (if you consume all your income you cannot pay your debts).

When it comes to the government it technically pays back by public savings which occurs when taxes are larger than spending so there in principle you could argue there should be another savings arrow from the government. However, I think that Krugman decided to include this in the transfer arrow. If you buy a government bond and your government is paying you back interest you could consider it as transfer.


Examples of bounded, positive, inverse demand curves

You can find demand functions like this in textbook and exam problems. Just a random example from internet:
$$Q = \ln 4 - 0.5 \ln P$$
which is just a special case of $Q=−1/a \ln (P/b)$ where a is 1 and b is also 1 and there is some additional constant $\ln 4$. We just had an exam where I also saw one of the other examples. If I remember right it was $Q = \frac{100}{P}-1$. I also remember seeing more of them in textbooks but it would take forever to find them in thick book full of math problems.


Okay, in layman's words, devaluing yuan (against USD) means, Chinese made goods are cheaper in USD. On the other hand, American made goods are expensive in China as a result. China wants to keep its export competitiveness and may want to keep its 'world's factory' status. Well, South Korea may as well want to use a similar strategy to make domestically produced export goods competitive, similarly, other aisan governments considering devaluing their currencies. There would then be a currency war... hope this is as simple as you wanted to see.


Why do developing countries adopt 21st century technology, but stay behind in output per capita?

Probably if you go to almost any country in the world, however poor, you will find considerable use of relatively cheap cutting-edge consumer technology such as smartphones and some more expensive cutting-edge technologies used in manufacturing or service industries.  However, there is a big gap between 'adopting' cutting-edge technology at that level and the much more comprehensive adoption of such technology by developed countries.  The benefits of technology in developing countries may be limited by any of the following:

Lack of adequate supporting infrastructure, eg electricity supply
unreliable or subject to regular power cuts, unreliable internet service.
Lack of adequate skills to maintain and make best use of technology.
Limited adoption, eg of smartphones, so that network benefits are
not fully realized.
Lack of capital equipment, knowledge and skills to produce
cutting-edge technology equipment, so that all such technology has to be
imported using scarce foreign exchange; or, for slightly more
developed countries, production of high-technology goods is limited
to low-level assembly work within an international supply chain
where most of the value is added elsewhere.

Suppose, then, that (recognizing the above problems), a developing country tries to adopt cutting-edge technology in a comprehensive way, so as to quickly catch up with developed countries.  Such an approach is heavily criticized in Justin Yifu Lin's book The Quest for Prosperity, which argues, in outline, that developing countries should target industries in which they have a comparative advantage (or can achieve one via improved infrastructure) and which offer an opportunity for gradual upgrading of technology.  While countries differ, this will often mean labour-intensive manufacturing, and it is suggested that industries to be targeted should be those currently dominated by countries whose GNP per capita is about twice theirs.  By contrast, attempting to compete immediately with countries with the most advanced technology is likely to lead to a continuing need for subsidy, distortion of resource allocation, financial repression to channel scarce capital to the subsidised sector, and political problems in handling those profiting from or dependent for employment on subsidies.


Martingale, random walk and rational expectations

All concepts are used in Economics. Definitions (not stated in a fully rigorous manner):

Martingale : A stochastic process $\{X_t\}$ is called "martingale" if and only if it holds that

$$E(X_{t+1} \mid X_t,X_{t-1},...) = X_t \tag{1}$$

There are extensions like "sub-martingale", "super-martingale" but the basic definition is the above

Random walk : A stochastic process $\{X_t\}$ is called "random walk" if and only if

$$X_{t+1} = X_t + u_{t+1}, \;\;\;u_t \sim \text{White Noise} \tag{2}$$

and you can look up the definition of "White Noise".

Here too there are spin-offs ("random walk with drift" etc).

Comment: it follows that a random walk is (one example of) a martingale, but a martingale does not imply a random walk, the former being a much broader concept.

Rational Expectations : Originally, the Rational Expectations Hypothesis stated that the aggregate expectation on some unknown (usually future) value of an aggregate random variable (see also this post on the matter), equals the conditional expectation (in the rigorous mathematical sense) of this variable "given all relevant information at the time of forming the expectation

$$X^e_{t+k|t} = E(X_{t+k}\mid I_t) \tag{3}$$

Comment: note that the conditioning set in the case of a martingale contains only past values of the process. The conditioning set in the case of the REH contains "whatever is available and is deemed relevant".

In a representative agent model, we are forced, for internal methodological consistency, to apply the Rational Expectations Hypothesis at the individual level (something that has raised all sorts of valid objections as regards the availability of information and the information processing constraints of an individual).

So do the forecasts of an individual follow a martingale? Let's examine it for one-step-ahead forecasts only: our stochastic process is

$$\{X^e_{t+1|t},\,X^e_{t+2|t+1},...\} = \{E(X_{t+1}\mid I_t),\,E(X_{t+2}\mid I_{t+1}),...\} \tag{4}$$

To obtain the martingale property it must hold that

$$E[X^e_{t+1|t} \mid X^e_{t|t-1}, X^e_{t-1|t-2},...] = ?\; X^e_{t|t-1} \tag{5}$$
(I will name in the end what expectations does eq. $(5)$ describes)

$$E[X^e_{t+1|t} \mid X^e_{t|t-1}, X^e_{t-1|t-2},...] = E[E(X_{t+1}\mid I_t) \mid \{E(X_{t}\mid I_{t-1}), E(X_{t-1}\mid I_{t-2}),...\}]$$

As already commented, the outer conditioning set is smaller than the inner conditioning set. By the Law of Iterated Expectations (the Tower property) of Conditional Expectation, we then obtain

$$E[E(X_{t+1}\mid I_t) \mid \{E(X_{t}\mid I_{t-1}), E(X_{t-1}\mid I_{t-2}),...\}] = E[X_{t+1} \mid \{E(X_{t}\mid I_{t-1}), E(X_{t-1}\mid I_{t-2}),...\}] \tag{6}$$

The right-hand-side of $(6)$ is not necessarily equal to the right-hand-side of $(5)$, so we cannot say that the stochastic process of one-step-ahead forecasts under the Rational Expectations Hypothesis is a martingale.

Equation $(5)$ describes the following situation: standing at period $t-1$ we form the expectation $X^e_{t|t-1}$, and "the best we can say" (best in the mean-squared-error sense) about our subsequent expectation $X^e_{t+1|t}$ is that it will be equal to $X^e_{t|t-1}$. This is (sometimes) called "static expectations" but beware because the term has two totally different meanings in the literature: for some authors, eq. $(5)$ represents "static" expectations in the sense that the expectation itself remains (or is expected to remain) unchanged in value. But you will often find authors who write the term "static expectations" and mean something totally different, namely $E(X_{t+1}\mid I_t) = X_t$ ("what is, will be"). This in turn looks like the martingale property but it is at best an extended concept of it (because the conditioning set is bigger), and in any case, it is a martingale-like property as regards the actual process $\{X_t\}$, and not the forecasts-process $\{X^e_{t+1|t}\}$.


How is an spontaneous currency changing phenomenon called?

When the switch from national currency is to US dollar it’s called dollarization (this mostly happens for US dollar but there are exemptions like Montenegro using euros) and in general it’s called currency substitution.  It often happens when people lost the trust in the local currency or if it’s impractical to have local currency (for small states like Montenegro).

Past examples are Equador and El Salvador or Zimbabwe. This list is not exclusive but those are the major ones you want to look at.

Also note that in case that people still use both currencies the currency substitution can be only partial but from my understanding Venezuela is close to full substitution.


In a framework such as CAPM, agents are assumed to be rational, and usually to have identical preferences. All agents have access to the same information, and agree on the level of riskiness of every particular stock. As returns are a reward for taking risk, the price of every asset do not depend on the supply but on its riskiness. Thus, the demand curve is flat: no matter how much of asset A their is on the market, nobody wants to pay a premium which would lead to have a lower reward for a same level of risk.

This models have very strong and restrictive assumptions, which is why we end up with this weird infinitely price-elastic demand curve. More complex models, with less restrictive assumptions, might produce downward sloping demand curves.


Solving for profit function $\pi (w,p)$ given the output of production function $f(z) = \sqrt{2z_1 + 3z_2}$

The production function has a particular feature: the inputs are perfectly substitutable. One unit of input 1 can be substituted by 2/3 of input 2 to produce the same quantity of output.
Intuitively, a producer would optimally use only one output to produce.
Suppose the production plan is $(z_1,z_2)$. By choosing $(z_1-1,z_2+2/3)$, the firm produces the same quantity but profits are increased by $w_1-2/3w_2$. We can conclude: the producer would buy only input 1 ($z_2=0$) if $w_1<2w_2/3$, and they would buy only input 2 ($z_1=0$) if $w_1>2w_2/3$. From that point, you can solve the maximisation problem by distinguishing these two cases.
The question can also be solved as a constrained optimisation problem if you add the constraints $z_1\geq 0$ and $z_2\geq 0$ to the Lagrangian. If $\lambda_1$ and $\lambda_2$ denote the Lagrangian parameters of these two constraints, you obtain the two cases aforementioned when i) $\lambda_1=0$ and ii) $\lambda_2=0$.


Effect on revenue of changing price of unitary demand elastic product

This is an excellent question (and congratulations for spotting an apparent inconsistency in your problem set!) To answer it, we need to distinguish between two slightly different concepts: the arc elasticity and the point elasticity.

The arc elasticity of demand can defined as the percentage change in quantity demanded divided by the percentage change in price. (Other definitions are available.) For instance (to borrow your example), if initially $P = 2$ and $Q = 100$, and later $P = 2.02$ and $Q = 99$, then the arc elasticity of demand is:

$$\frac{\%\Delta Q}{\%\Delta P} = \frac{\frac{100-99}{100}}{\frac{2-2.02}{2}} = -1$$

Notice that to calculate the arc elasticity of demand we need two price/quantity pairs.

In contrast, the point elasticity is defined slightly differently: it is the limit of the arc price elasticity as the size of the change in price goes to zero. In other words, it is essentially the same as the arc elasticity, but for a infinitesimally small price change. As a result, the 'old' price/quantity and 'new' price/quantity are essentially the same, so to compute the point elasticity we need only one price/quantity pair.

A bit more formally, consider a small price change $dP$ that leads to a small change in demand $dQ$. The point elasticity is given by

$$\frac{\%\Delta Q}{\%\Delta P} = \frac{\frac{dQ}{Q}}{\frac{dP}{P}} = \frac{dQ}{dP}\frac{P}{Q} $$

These preliminaries out of the way, we are now in position to answer your question.

Under standard assumptions (downward sloping and differentiable demand), revenue is maximised when the point elasticity equals 1. (You may wish to verify this!)
However, at the revenue maximising point, the arc elasticity need not be one. Conversely, if the arc elasticity is one, revenue need not be maximised (as your example indicates).
Insofar as the point elasticity is a good approximation for the arc elasticity (i.e. when considering small prices changes), the arc elasticity should be close to 1 at the revenue maximising price/quantity.


Is there a theory about investors' behaviour during bubbles?

The investor has erroneously overvalued the value of the stock/commodity.

Here is a prominent paper that models irrational bubbles:

  This paper attempts to formalise herd behaviour or mutual mimetic
  contagion in speculative markets. The emergence of bubbles is
  explained as a self-organising process of infection among traders
  leading to equilibrium prices which deviate from fundamental values.
  It is postulated furthermore that the speculators' readiness to follow
  the crowd depends on one basic economic variable, namely actual
  returns. Above average returns are reflected in a generally more
  optimistic attitude that fosters the disposition to overtake others'
  bullish beliefs and vice versa. This economic influence makes bubbles
  transient phenomena and leads to repeated fluctuations around
  fundamental values.

Herd Behaviour, Bubbles and Crashes (Lux (1995))

The investor is aware that it's a bubble and is overvalued, but believes he can make a profit by selling before the bubble bursts.

Bubbles, rational expectations and financial markets (Blanchard and Watson (1982)) is a famous paper on this subject.

  This paper investigates the nature and the presence of bubbles in
  financial markets. Are bubbles consistent with rationality? If they
  are, do they, like Ponzi games, require the presence of new players
  forever? Do they imply impossible events in finite time, such as
  negative prices? Do they need to go on forever to be rational? Can
  they have real effects? These are some of the questions asked in the
  first three sections. The general conclusion is that bubbles, in many
  markets, are consistent with rationality, that phenomena such as
  runaway asset prices and market crashes are consistent with rational
  bubbles. In the last two sections, we consider whether the presence of
  bubbles in a particular market can be detected statistically. The task
  is much easier if there are data on both prices and returns. In this
  case, as shown by Shiller and Singleton, the hypothesis of no bubble
  implies restrictions on their joint distribution and can be tested. In
  markets in which returns are difficult to observe, possibly because of
  a nonpecuniary component, such as gold, the task is more difficult. We
  consider the use of both "runs tests" and "tail tests" and conclude
  that they give circumstantial evidence at best.

Here is another:

  We present a model in which an asset bubble can persist despite the
  presence of rational arbitrageurs. The resilience of the bubble stems
  from the inability of arbitrageurs to temporarily coordinate their
  selling strategies. This synchronization problem together with the
  individual incentive to time the market results in the persistence of
  bubbles over a substantial period. Since the derived trading
  equilibrium is unique, our model rationalizes the existence of bubbles
  in a strong sense. The model also provides a natural setting in which
  news events, by enabling synchronization, can have a disproportionate
  impact relative to their intrinsic informational content.

Abreu and Brunnermeier (2003)

[I]s investor behaviour during bubbles well studied, and are there models explaining it?

Yes, this is a huge literature with probably thousands of papers and hundreds of books written about it. Here is a technical book on the subject:  Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding,
 These books are more historical than theoretical: Manias, Panics and Crashes: A History of Financial Crises, This Time Is Different: Eight Centuries of Financial Folly, Irrational Exuberance.


What assurances are there that the Federal Reserve does not give money to banks?

According to this official link

Does the Federal Reserve ever get audited?

  Yes, the Board of Governors, the 12 Federal Reserve Banks, and the
  Federal Reserve System as a whole are all subject to several levels of
  audit and review:
      The Government Accountability Office (GAO) conducts numerous reviews of Federal Reserve activities.
      The Board's financial statements, and its compliance with laws and regulations affecting those statements, are audited annually by an
  outside auditor retained by the Office of Inspector General (OIG).
      The Board's OIG audits and investigates Board programs and operations as well as those Board functions delegated to the Reserve
  Banks. Completed and active GAO reviews and completed OIG audits,
  reviews, and assessments are listed in the Board's Annual Report.
  (Before 2002, the reviews were listed in the Board's Annual Report:
  Budget Review.)
      The financial statements of the Reserve Banks are also audited annually by an independent outside auditor.
      Each week, the Federal Reserve publishes its balance sheet and charts of recent balance sheet trends, as well as provides an
  interactive guide to the Fed's balance sheet. The balance sheet is
  included in the Federal Reserve's H.4.1 statistical release, "Factors
  Affecting Reserve Balances of Depository Institutions and Condition
  Statement of Federal Reserve Banks."
  In addition, the Reserve Banks
  are subject to annual examination by the Board. The Board's financial
  statements and the combined financial statements for the Reserve Banks
  are published in the Board's Annual Report.
  See our audit page for
  more information on all of the above audits and more information on
  the accounting, financial reporting, and internal controls of the
  Federal Reserve Board and Federal Reserve Banks.


Intuition behind fixed effects estimator

I asked exactly the same question on math.stackexchange:

https://math.stackexchange.com/questions/1470490/fixed-effects-estimation

In short, the answer is yes, it can be viewed as running separate OLS regressions- the weights, however, are not arbitrary. It is a weighted average of the separate OLS regressions.


What are key economic mechanisms explaining long term damage during Great Recession

Big recessions produce mass unemployment and for long periods. People can get demotivated, and lose skills and training. As the economy evolves, new skills are needed and unemployed people might lose that. That is a long term damage. Think of all the youngsters in Spain, coming out of the university, ended up working in bars and other low skill jobs. That's not only demotivating but also bad for their human capital.

That's one example I can think of.


In the short run, why do higher interest rate lower inflation?

As you say:

  "Higher interest rates put less borrowing power in the hands of consumers and businesses. And when they spend less, firms are not selling everything and prices naturally falls."

There's also the effect on the velocity of money and the money supply that less borrowing brings to provide further downward pressure on prices.

I also think your concerns about supply falling in the short term are unfounded. Firms are slow to adjust their production/supply due to greater borrowing costs. Instead, they will attempt to keep up production and clear their inventories by lowering prices in the hopes that the next season will show them a success and bail them out. Any such effect from interest rates would be a much longer-run effect.


Disruptions in Server / Data Center Market

Institutions like banks for many years now keep  back-up archives of their data in physical locations owned and operated by other legal entities. This is a standard case of outsourcing to specialists a requirement of a business process that is not a directly value-added activity. So the "sole custodian of one's own data" has been proven in the market not to be strong enough against cost-reduction considerations (or against price/value optimization if you wish).

Cloud computing is comparatively still new. The cost incentives to adopt it are indeed very strong. In practice, I believe we will observe, as in the past with other innovations, a certain degree of selectiveness: as the risks and rewards of cloud-computing will become more well understood (and measured), companies may select certain parts of their activities to keep off the-cloud, while enjoying the cost reductions for all the rest.

This by the way, can be squarely put in the framework of the, dear to us economists, marginal optimization - it is not a "yes-or-no" decision but a process where each company will adopt the Cloud up to the degree where marginal benefit equals marginal cost.


Pure and Mixed Nash Equilibrium algorithm gives different results

Mixed strategy Nash equilibrium cannot involve strictly dominated strategies. In particular, Cooperate is strictly dominated for player 1 ($6<8$ and $3<4$). Therefore, no $b\in[0,1]$ can make player 1 indifferent between Cooperate and Non-cooperate. You made a mistake by trying to solve for $b$ by equating player 1's expected payoffs from his two pure strategies.


Deflate level and flow variables using PPI and CPI

Level (also called stock) variable, is a variable that measures something at a given point of time. For example, measuring GDP in 2017 would be a level (stock) variable.
Flow  variable is a variable that measures the rate of change in the stock over time. For example, if GDP growth (which measures flow of GDP) was $10\%$ in 2017 that means that the level/stock of GDP increased by $10\%$.
Wikipedia actually has surprisingly excellent explanation on differences between level/stock and flow/growth variables that you can read here (see the general use part).
There is no reason why you would always want to apply CPI to flow variables and PPI to level variables. This must be something that is specific to the research you are reading and there must be some other reason behind it.
If you want to deflate a level variable using some particular deflator you can use the following formula:
$$Real X = \frac{Nominal X}{deflator/100}$$
If you want to adjust flow variable then you can first calculate the change in deflator and subtract that change from the flow variable that is:
$$ g_r  = g_n - \pi$$
where $g$ is the real ($r$)/nominal ($n$) flow/growth rate respectively and $\pi$ inflation rate which is the change in deflator. Also the above is a first order approximation. The exact formula is given by $1-g_r = \frac{1+ g_n}{1+\pi}$ but this exact formula is not often used.


When solving game theory problems, how can you "assume" that the equilibrium strategy will lead to something happening?

This is a concept that I've seen confuses a lot of people when they first encounter it. However, it is completely perfect to assume that something will happen in equilibrium, use that fact to characterize equilibrium, and then check your assumption indeed holds. This is sometimes called guess-and-verify. Remember that an equilibrium has a very precise definition, it is a set of strategies that satisfy some conditions. If you find them, and they satisfy the conditions it doesn't really matter how you found them.

It might sound silly to just be guessing properties of an equilibrium and be hoping for the best. But actually, doing good guesses is hard; so there is merit to it. So long you check that your equilibrium satisfies the assumed (or guessed) property, then your original assumption is verified.

The alternative would be, in principle, to solve a game by computing for each player their best response as a function of all possible actions of every other player. And then find the intersection of these best responses. This can be an extremely hard thing to do, even for simple games like a Hotelling model where N firms set locations and prices and then consumers choose from where to buy. So making informed guesses can be a great shortcut to find an equilibrium.


Government spending and revenue statistics

As you want data for OECD countries have a look at the OECD's Statistics portal. You will find the data in the "National Accounts" as section as well as in the "Public Sector, Taxation and Market Regulation"

As an alternative or as a complement you can also have a look at the Eurostat database on Government Statistics. They do have data for Switzerland as well as for many OECD countries.

Finally, as you seem to be focussed on Switzerland, you can also try to find the data via the Federal Statistical Office.

You can obtain time series by following the links in the "additional information" column.


Basic New Keynesian Model: Why variables are defined by integral equations

First of all, this is not something special to "the basic New Keynesian Model". It is a commonly used technique when modeling an economy with competitive environment.

In your case, the author wrote that there is a continuum of differentiated goods represented by the interval [0,1] and each firm is producing one of the goods. You can imagine that there are uncountably many different kinds of good in the market. As a result, the size of each firm is negligible (just a point on the [0,1] interval) compared to the entire economy. Each firm indexed by $i$ solves its individual profit maximization problem to obtain its labor demand $N(i)$. Then, the aggregate labor demand is simply the "sum". In this case, the sum is replaced by the integral because the variety of different firms is the whole unit interval.


I think a situation in which corner solutions may occur would be more fruitful. Cobb-Douglass ensures an interior solution, so they can use the expenditure share trick pretty easily, for any number of goods. Perhaps a really simple CES function, like $x^.5+y^.5+z^{.5}$? Or, if you still want an interior solution, pick something that's not a well-known function, but where marginal utility becomes infinite as either good approaches 0, like $\ln(x) - \frac{1}{\sqrt{y}} - \frac{1}{z}$ You'll still get tidy derivatives, but substitution won't be possible.


Is there an index that describes how differentiated a product is?

For an industry, one indicator used by competition authorities is the Herfindahl–Hirschman Index measured as $\displaystyle \sum_{i=1}^N s_i^2$ where there are $N$ firms in the sector and the market share of the $i$th firm is $s_i$.

This gives a value of $1$ when there is a single firm and $\frac1N$ when the $N$ firms have equal market shares, but also deals sensibly with cases where different firms have different shares.

For a particular product, you could look at something else like comparing its price elasticity of demand when other similar products do not change price, perhaps comparing this with the price elasticity of demand for all the similar products together.  An undifferentiated product should have an almost infinite price elasticity as all demand will shift if there is any relative price change.


General mathematical model of incomplete contracts?

Just a model that can be used to state how (in)complete a particular contract is, whatever the reason.

I remember a debate at the end of the 90's on Incomplete Contracts: Where do We Stand? by Jean Tirole and Foundations of Incomplete Contracts by Hart and Moore, where they develop a model that provides a rigorous foundation for the idea that contracts are incomplete, followed by Complexity and Renegotiation: A Foundation for Incomplete Contracts by Segal.
Check also this book The Impact of Incomplete Contracts on Economics (2006) by Aghion, Dewatripont, Legros, and Zingales, including Comments on the Foundations of Incomplete Contracts by Maskin.
An interesting paper by Tirole Cognition and Incomplete Contracts: Thinking  about  contingencies,  designing  covenants,  and  seeing  through  their implications is costly. Parties to a contract accordingly use heuristics and leave it incomplete. The paper develops a model of limited cognition and examines its consequences for contractual design.


Why should Greece destroy their cash euros?

Using Google translate I have read some of the articles you referred. It seems to me that the problem the Swedish economists are talking about is that Greece will need its own currency (this is debatable). However it takes a long time to get currency in sufficient amounts.

  Before the 2003 invasion of Iraq, citizens in many parts of the country began to ditch their own currency, believing that it would soon be worthless as all notes carried a depiction of Saddam Hussein, the country’s president. De La Rue, the world’s largest printer of banknotes, began work on new cash. “In the case of Iraq, the issues may not have been dramatically different from any other currency change; but the timescale was,” the company said. “Working out the details … is all very well when you have a year or two to think about it. Make that a month or two, and things start to get interesting.”

Source

Hence the recommendation is to use what they have at hand: euro bills. But if their euros would be the same as other euros, then the exchange rate between a Greek euro and a 'normal' euro could not go over 1, because you could always take a Greek euro abroad and use it there. So if the government wants the exchange rate to change (this would increase competitiveness and this is the biggest advantage Greece could get out of a new currency) they will have to modify the existing euros they have. For example they can stamp holes in them. You can read about such speculation in more detail

  During a “euro pause” the Greek Central Bank is granted a three-year licence by the European Central Bank to create euros that are non-exchangeable for other euros.
  The euros in Greek ATM machines have “GT” (Greek Transactions) stamped prominently on them. In Greece, these euros circulate normally but a Greek individual planning a transaction outside Greece will need to convert their GT euros to non-GT euros at a market rate.

here.

It is debatable whether having a national currency two months ahead of time is worth the cost of 80 billion euros. However it might also be that the ECB will demand 80 billion euros from Greece if it leaves the Eurozone. (I am not sure whose property the cash allocated to Greece is.) In this case, stamping the euros may satisfy the ECB and give Greece their temporary currency.


Question about maintaining fixed exchange rates

In my textbook, borrowing from abroad is listed as one of the method to increase the value of a currency

The textbook is wrong. Borrowing from abroad tends to impact the exchange rate in a way that purchase power of the domestic currency decreases. A lower purchase power is the opposite of increased value.


Economic Reasoning for Structural change vs Cyclical change

It is not possible ex-ante, it's all about:

economic interpretations, i.e., economic schools of thought
expectations on central banks' next moves,
political/ideological views,
the (hidden and/or unconscious) emerging willingnesses to induce self-fulfilling prophecies

... hence the debates that take place here and there.

Whether a recession is cyclical or due to structural changes can only be assessed ex-post, i.e. with data and (well driven) studies.


Does scarcity apply only to commodities?

For completeness, you might want to include the definition of Fungibility as well:  "In economics, fungibility is the property of a good or a commodity whose individual units are essentially interchangeable, and each of its parts is indistinguishable from another part."

If you have a collection of objects, and each object is (in some relevant sense) interchangeable with any of the other objects, then they are fungible and thus a commodity.  If there aren't very many of them, they may be scarce.

However, if you have a collection of objects, none of which are interchangeable with any others in a relevant sense, then they are essentially unique objects.  You could still say that they were scarce (there is only one of each) but not that they are commodities... though in practice, people typically don't refer to unique objects as scarce.


Since there are infinitely many numbers in $[0,1]$, agent $i$ has infinitely many probabilities to randomize over actions in their action space $A_i$. So far this has nothing to do with the game being Baysian.  In a Baysian game, a strategy $\sigma_i: \Theta_i \rightarrow \Pi(A_i)$ (or $s_i$ instead of $\sigma_i$) maps a type into a probability distribution over (pure) actions. You can call the space of all such functions $\Sigma_i$ (or $S_i$). There are many other notational conventions. So you always have to take care of careful definitions.
Just because there are infinitely many ways to randomize over the actions (pure strategies) does not mean that you have to compare all of them one-by-one in an inequality like yours. You can invoke monotonicity arguments or simply maximize over the probabilities.
In a similar vein, you can find an extremum of the function $x^2$ for $x\in[-1,1]$ although there are infinitely many numbers $x\in[-1,1]$.


Does the Italian national debt really go all the way back to Mussolini's excessive prewar spending?

Debt Crises are the inability to pay interest on the debt a country has already taken up to that point. There are many theories, about how such crises can develop, however all have one thing in common. You need debt to have a debt crisis.

What is debt? Debt is when a country spends too much compared to how much it earns (in case you are familiar with simple econmic terms, that is G (gov. spending) exceeds T (gov earning — $ \Delta M$ could be taken into account, is however negligible for most countries, as money printing is usually not the way to finance debt.)

"War is expensive", means that during war-time there are HUGE governmental expenses to production of weapons and what not.
However, to say that the debt crisis is due to WW expenses one would have to look at how debt levels of Italy have developed after the world wars.

I have found a graph in my book and saw that in the 1970 Italy had a debt ration $(\frac{B}{Y})$ of 40 percent, which in today's terms is not much, rather below average.

In the years 1980 to 1990 Italy took up a lot of new debt, peaking at a debt level of ca. 120% in the early years of the 90's. From there debt has been 'steady' with increases to 130% in 2014. Thus, one can say. No. The debt crisis is not due to war expenses, when looking at it from this purely expense-tracking perspective...


Rationalizability is a solution concept that is robust to small misspecifications about the information players have, as well as the payoffs of their actions. I do think it provides reasonable predictions in situations where Nash equilibrium requires coordination or precise beliefs about other players actions or beliefs, especially when players have not had enough opportunity to learn the game. This is a long way to say that I agree with you regarding the chicken game. Though I believe that the most common outcome is that all players decide to wait longer than necessary, that's the main coordination problem that signs, lights, and rules solve, I  believe.


Profit maximizing, Quantity and Price with Dominant Firm and Competitive Fringe

Your answer for part (a) is correct.

For part (b) you need to first determine what's called 'residual demand', which can intuitively be thought of as 'leftover' demand given how much the competitive fringe will produce. Denote residual quantity demanded as $Q_r$.

Marginal cost for the competitive fringe is:
$$MC_f=\frac{d}{dQ_f} C_f = \frac{Q_f}{2}.$$
To find the supply function for the competitive fringe set $MC_f=P$
$$Q_f=2P.$$

To find residual demand, subtract fringe supply from direct market demand (not inverse market demand). Direct market demand is given by
$$Q=1005-P$$
so residual demand is
$$Q_r=Q-Q_f=1005-P-2P$$
$$Q_r=1005-3P.$$

Rearranging residual demand gives inverse residual demand of
$$P=335-\frac{Q_r}{3},$$
which gives residual marginal revenue of
$$MR_r=335-\frac{2Q_r}{3}.$$
First find the profit-maximizing output for the leader firm by setting $MR_r=MC$:
$$335-\frac{2Q_r}{3}=5$$
$$Q_r=495.$$
To find the leader's profit-maximizing price, plug $Q_r$ into inverse residual demand to get
$$P=335-\frac{Q_r}{3}=335-165=170.$$

Finally, we can find the quantity supplied by the fringe by plugging the leader's price into the fringe supply function
$$Q_f=2P=2*170=340.$$


Could a hike in interest rates cause a country to be unable to pay back it's debt burden?

The short answer is no, rate hikes have no effect whatsoever on whether a government can service its debt and repay maturing issues.  Here's why---

A government that issues debt in its own currency can always print currency to make its debt payments.  This is an extremely important point, despite that certain political actors would like you to forget it.
Central banks set rates at the short end of the Treasury curve.  They do not have an effect on the long end of the curve, which is set by market participants' expectations of long term inflation and real growth.  What's more, central banks today are data dependent---they state explicitly that they will react to data that the market already has priced.  For example, this month the U.S. Fed's FOMC will likely increase rates due to stronger economic data (GDP revisions upward, better than expected inflation, decline in unemployment and jobless claims). However the Treasury curve has already steepened massively over the last month in response to these data releases.
Many governments issue debt in foreign currencies like USD and EUR (e.g., Brazil, Mexico, Ukraine, and almost every other emerging market).  Because of uncovered interest rate parity, rate rises push up the value of the domestic currency relative to the currency in which external debt must be paid.  This actually decreases the cost of servicing the external debt.