Is clustered standard errors the best way for all regressions that uses clustered data?

Using clustered standard errors in Stata doesn't impose any additional restrictions on your coefficient estimates. The coefficient estimates are independent of the vce() option you choose.
If you have reason to believe your standard errors should be clustered, it means you think that there is some association between the cluster variable and your outcome. If that's the case, then you should absolutely include the cluster variable in the regression itself, at least as a fixed effect. Typically, it's most efficient to do that using xtreg orareg. Both allow the vce(cluster clustvar) option.
If you only include your cluster variable as a fixed effect, then you are implicitly assuming that the coefficients across clusters are the same but allowing for different intercepts - that's not a consequence of clustering the standard errors, but of the underlying parametric model. If you want to allow for varying coefficient estimates between clusters, you could include interactions between your independent variables and your cluster variables.


This set of lecture notes cites a bunch of great studies.

Effectively, this lecture said that fiscal consolidation (austerity) might have been expansionary (good). The Federal Reserve, however says:

  fiscal consolidation is not likely to be expansionary today.

The deciding factor seems to be whether Monetary Policy can offset short-term effects, thogh the Fed goes on to close with:

  In summary, economists have not definitively answered the question of whether and when austerity is likely to be beneficial. Much of the debate ignores the question of whether implementing austerity is feasible in nations with nonexistent records of long-run fiscal consolidation or that rely on foreign sources of deficit financing. The economic volatility of recent years provides a ripe area of future study regarding the true effects of fiscal contractions.

Or, "We'll see..."


Maximize utility given an arbitrary number of goods and the condition that exactly X number of items must be bought

Utility maximization problem is
\begin{eqnarray*} \max_{x_1, x_2, \ldots, x_n} && u(x_1, x_2, \ldots, x_n) \\ \text{s.t.} && \sum_{i=1}^n p_ix_i \leq M \\ && \sum_{i=1}^n x_i = X \\ && x_i\in\mathbb{Z}_+ \ \ \forall i\in\{1,2,\ldots, n\} \end{eqnarray*}

If we ignore the budget constraint, the equation $\sum\limits_{i=1}^n x_i = X$ has ${X+n-1\choose n-1}$ solutions. List them, sort them from high to low by utility, and then identify the highest one in the sorted list that also satisfy the budget constraint.

If $u$ is strictly increasing, differentiable and quasi-concave on $\mathbb{R}_+^n$, then one can solve the utility maximization problem in the usual way by ignoring the $\sum_{i=1}^n x_i = X$ constraint, and then inspect the integral solutions of $\sum_{i=1}^n x_i = X$ that surrounds the solution of the utility maximization problem, and choose the best amongst them to get the final solution.


Applying the Law of Iterated Expectations on the defining property of a sub-martingale $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$ we have that

$$E\Big[E(X_{t+1}|X_1,\dots,X_t)\Big]=E(X_{t+1}) \geq E(X_t)$$

So a sub-martingale has also its unconditional expected value non-decreasing. Therefore any economic variable that appears to consistently increase in value is a candidate for being a sub-martingale, since the defining property here conditions only on the variable's past, not on any other factors that may influence it. National Output/GDP could be reasonably modeled as a sub-martingale, since it historically "always" increases, bar catastrophic events.

In general, any economic variable that can be modeled as a random walk with non-negative drift,

$$X_{t+1} = \alpha + X_{t} + u_t,\;\; \alpha \geq 0$$

is a sub-martingale.


How much interest rate would make his earnings the same

I don't know how you got 48% (and it would have been appropriate to include your work in your question), but think about it intuitively. Take your second to last step. If the landowner lends to the villagers and on average gets 60 dollars plus interest back, then if the interest rate was 80%, you'd get 60 + 48 = 108 dollars, the same as depositing to the commercial bank.

Algebraically, here's how you'd set up solving for $r$:

$$(100 \cdot (1+r))0.6 + 0(0.4) = 108$$

Where you have a 60% chance of being paid back 100 with interest. Then you just solve.


How long can I live in the United States with my German monthly income?

This is tricky to answer precisely because governments vary a lot in the services they extent to visitors. There is also variation in the mix of consumption goods across countries as well as their relative prices. You might respond to local conditions by changing your consumption bundle more or less than the average person and that would change the economics for you relative to an average calculation.

The simplest approximation is to take GDP (or the C component of consumption), measured using exchange rates rather than PPP, and divide it by the population.

Think about it this way. Nominal GDP per capita in the US is about \$62,606 and in Germany is \$48,264. Assume it is safe to either ignore taxes or taxes are comparable in the two countries. If you have have saved the salary of the average German then you can afford the spending of the average American for about 9 months.

Another way to do this is to look at destination country household final consumption expenditure per capita. As of 2016 this was about \$36,373 in 2010 dollars, probably about \$43k in 2019 dollars. According to Wikipedia, average German salary, net of taxes, is about \$32,000. So if you have a pension that pays net of taxes the salary of a typical German (also net of taxes), you can afford about 75 percent of an average American's spending.

A third way is to use expatriate cost of living indices, although I've been told that these are more accurate for affluent urban households. For example, expatistan says that the cost of living is 53 percent higher in NYC than in Frankfurt. This suggests you can live for eight months in NYC for the same income that buys a year's worth of consumption in Germany.


Generally ln is used to represent the natural logarithm.  I know of no notation system that uses it to represent differentiation.

Natural logarithm:

$$\frac{\ln{y} - \ln{y_0}}{t}$$

Differentiation would look more like

$$\frac{dy}{dt}$$

or

$$f'(t)$$

It is common to take logarithms of growth functions because they involve exponents and logarithms can simplify equations with exponents.  In particular,

$$ \ln{(1+g)^t} = t\ln{(1+g)}$$

is a basic property of logarithms.

A Maclaurin expansion is an alternative way to write the equation.  See Taylor series and Maclaurin's series for more information.  A Maclaurin expansion does in fact take advantage of differentiation.

This logarithm is referred to as natural because its base arises from normal mathematical calculations rather than being arbitrary, much like $\pi$.


Demand correspondence is both upper and lower hemi-continuous; is the preference continuous?

I think that you should proceed by contradiction assume D is continuous, but $\succsim$ is not, then for a bundle either the more preferred than or the less preferred than sets are not closed. Choose the problematic set and choose a set B appropriately to get a contradiction of the existence of a maximum (remember that not closed sets might not admit a maximum), this will give you the desired contradiction. Once preferences are continuous, remains to check completeness of preferences to use the classic theorem that if preferences are complete, transitive and continuous, then a utility function representation exists.


Why do prices of consumer products seem to rise independent of their supply?

To make it simple: let say your coconuts production is constant over time but the population is growing. It's a constant supply but an increasing demand, resulting in an increase of coconuts price.

If the prices were constant, the demand would be higher and thus not fully satisfied.


Why income elasticity of demand of luxury good in greater than 1?

As for your first question: income elasticity of demand is just a percentage change in quantity demanded divided by a percentage change in demand. If you divide two things that are equal you get one: $\frac{a}{b}=1 \iff a=b$ (as long as $b \neq 0$). Same thing goes for income elasticity of demand, $1$ is not just some random value that was chosen to separate between goods. When $\%\Delta$ in quantity demanded is greater than $\%\Delta$ in income, then $\mathrm{YED}>1$ just as in terms of letters $a$ and $b$: $\frac{a}{b}>1 \iff a>b$ (as long as $b > 0$).

As for the second question: income elasticity of demand that is greater than one can be translated to "as the income grows, proportionally more is spent on luxuries". Which is exactly what you have in mind.


Does education have any value for most people's work?

Yes, this debate actually exists in economics. There are two main arguments why education increases wages:

Theory of Human Capital: This is theory that posits that education is an investment that is virtually analogous to investment in regular capital goods. You pay for your education (pay for capital), education makes you more productive (and wages critically depend on productivity) hence afterwards you get higher wages (return to your human capital). This theory was first developed by Becker (see Becker, 1980), it got a bit more nuanced since then but since you state you are non-economist I will not go more deeper into the arcana.  This is basically 'your' view of education.

Signaling/Screening Theory: This theory posits that education is a sort of screening device (signal) for employers about your innate ability and does not necessarily builds up your human capital. If you are unable to pay attention, memorize stuff, be able to coherently reason you will not get good grades and graduate and likely not be a good employee either. This theory was developed by Spence (1973) and later expanded Stiglitz (1975) - also the same caveat as above applies. This is basically 'your friends' view.

Now the above theories are not mutually exclusive as they can be both true at the same time. Education can be both a screening device and productivity enhancer. In fact empirically most of the time they both will be present. This being said it is an empirical question which one dominates and answers will differ from field to field. I think it is fair to say that most empirical evidence shows that education increases wages via human capital accumulation, but this is not true for every field.
For example, Hussey (2012) shows empirical evidence that when it comes to MBA degrees most increase in graduate wages was due to screening not human capital (although returns to human capital are present too). Again I think most economists would say that education contributes more to wages via human capital accumulation but the example above illustrates that this is not universally true and has to be empirically tested. It might very well be that your friend is just from an industry where screening is more important than human capital.


How can I prove $U(x) = [_1_1^+_2_2^]^{(1/)}$ is equal to Cobb-douglas Utility function when $\rightarrow0$

Hint: find $\ln(U(x))$ and then use L'Hopital's rule to find the limit as $\rho\rightarrow0$.


Economic theory / papers on sovereign debt?

Carmen Reinhart from Harvard Kennedy School works extensively on these questions. She has a recent paper forthcoming  in the Journal of the European Economic Association, 2016, entitled "Sovereign Debt Relief and its Aftermath" with Christoph Trebesch. They study sovereign debt relief in a long-term perspective, using two distinct samples: 1920-1939, and 1978-2010. They published a short Vox-CEPR column on this research.

In their book, with Kenneth Rogoff, "This time is different" they address this question. They have also other interesting and good papers:

Reinhart, Carmen, and Kenneth Rogoff. "Dealing with Debt." Journal of International Economics 96 (July 2015): S43-S55.
Reinhart, Carmen, and Kenneth Rogoff. "Financial and Sovereign Debt
Crises: Some Lessons Learned and Those Forgotten." Financial Crises:
Causes, Consequences, and Policy Responses. Ed. Stijn Claessens, M.
Ayhan Kose, Luc Laeven, and Fabian Valencia. International Monetary
Fund, 2014.

Check also:

Reinhart, Carmen M., and M. Belen Sbrancia. "The Liquidation of
Government Debt." Economic Policy 30.82 (March 2015): 291-333.


The US Census Bureau American Community Survey: https://www.census.gov/acs/www/data/data-tables-and-tools/


What is the definition of social welfare in monopoly?

By social welfare your professor probably means total economic surplus which is the sum of consumer's surplus and producer's surplus.


When should a receiver randomize across actions in a signaling game?

I think this cannot happen with risk averse senders, risk neutral receiver, and $A$ rich enough.

For example, and to stick to the canonical signaling model, suppose that $A$ is the positive real line and senders' utility $u$ is increasing in $a$ while receiver's have linear utility decreasing in $a$.

(Admittedly, this is only a partial answer as the framework is much less general that the one in your question, so it might not be satisfactory to you. I still provide an argument in case you were ok with these assumptions)

To derive a contradiction, suppose that at an equilibrium $\sigma^m_R(a') > 0$ and $\sigma^m_R(a'') > 0$ for some $a' \neq a'' \in A$. Let

$$a''' \equiv \frac{\sigma^m_R(a')}{\sigma^m_R(a') + \sigma^m_R(a'') }  a' + \frac{\sigma^m_R(a'')}{\sigma^m_R(a') + \sigma^m_R(a'') } a''.$$

By risk aversion

$$ u[ a''' ] >  \frac{\sigma^m_R(a')}{\sigma^m_R(a') + \sigma^m_R(a'') } u(a') + \frac{\sigma^m_R(a'')}{\sigma^m_R(a') + \sigma^m_R(a'') } u(a'').$$
$$ [\sigma^m_R(a') + \sigma^m_R(a'')] u( a''' ) >  \sigma^m_R(a') u(a') + \sigma^m_R(a'') u(a'').$$

Under some continuity assumption, there must also exist

$$ a '''' < a''' $$

such that

$$ [\sigma^m_R(a') + \sigma^m_R(a'')] u( a'''' ) =  \sigma^m_R(a') u(a') + \sigma^m_R(a'') u(a'').$$

So consider $\sigma^m_R{'}$ constructed in the following way

$\sigma^m_R{'}(a') = \sigma^m_R{'}(a'') = 0$,
$\sigma^m_R{'}(a'''') = \sigma^m_R(a'''') + [\sigma^m_R(a') + \sigma^m_R(a'')]$
For all other $\tilde{a}$, $\sigma^m_R{'}(\tilde{a}) = \sigma^m_R(\tilde{a})$

Receivers would prefer $\sigma^m_R{'}$ over $\sigma^m_R$ if it did not alter the signals sent by the senders, because it involves lower expected compensations. But by construction, senders are indifferent between $\sigma^m_R{'}$ and $\sigma^m_R$, so they should send the same signals as in $\sigma^m_R$.  Thus $\sigma^m_R$ cannot be an equilibrium which shows that we cannot have two different actions played with positive probability at an equilibrium.


What are the determinants of the natural rate of interest and how market rate converges to it?

What determines interest rates?
Let's try to understand the emergence of interest rates from first principles. We'll imagine an island inhabited only by Robinson Crusoe and Friday. For sustenance, they pick coconuts - the only food source available to them. Each coconut can either be consumed immediately or saved up for future use.
One day, Crusoe does not manage to pick any coconuts and having none saved up asks frugal Friday to loan him one from his stockpile. In return, Crusoe promises to give Friday more than one coconut at some later time. Let's consider what factors would affect how many future coconuts Crusoe must promise Friday in order to persuade him to issue the loan.

If Crusoe is hungry right now, he will have a high time preference (=low patience). Therefore he will be willing to promise a large number of future coconuts for one present coconut.
If Friday has a large number of coconuts saved up, say 100, he will be more willing to part with one of them for a modest fee. If he only has 10 saved up, he will demand more future coconuts if he is to loan out 10% of his savings. How many coconuts Friday has saved up determines the marginal utility of 1 coconut and thus how willing he is to loan it out.
If Friday thinks there is a high risk that Crusoe will struggle to pick enough coconuts in the future to pay back his loan, he will be less willing to issue a loan without being compensated for it.

If more inhabitants are added to the island, we can imagine a market for "coconut loans". Lenders will compete to offer customers an interest rate that correctly balances time preference, risks, etc. Those who offer too high an interest rate will get few customers, while those who offer one that is too low risk going bust.
Note that there is no such thing as "the interest rate". There are only particular interest rates on particular loans made between particular people. Like many other things, talking about the "natural" interest rate and how lenders converge to it can sometimes be a helpful abstraction, but the true, underlying mechanisms should always be kept in mind.
In reality, the picture is more complicated since it is possible for central banks to artificially affect it in a way that is not constricted by supply and demand. I can recommend this article for a discussion of the subject.


Deriving long-run cost functions from production function

Don't get stuck with math.

Look again at the production function. Perhaps draw out isoquant curves (for the same output, how could you vary $K$ and $L$). How are the isoquants for this production function different from those of standard Cobb-Douglas production function?

Hint: perfect substitute.


Investment = Saving relation in an open economy

First it should be clear that this is an (ex post) national account equality:

$Y=C+I+G+NX$, the private saving is $S_p=Y-C-T$ and public saving is $S_g=T-G$
thus you have $S_p+S_g-I=NX$.

Later you see in this book that the net exports, which depend on the exchange rate, are exactly equal to the Net Capital Outflow. Why? because an export is like a capital import (and vice versa).


Pooling equilibrium in akerlof's lemon market with certification cost

We are looking for a pooling equilibrium where both types choose to certify their cars.
The certification has two outcomes $\{success,\;failure\}$.
The buyer faces three possible types of histories: (1) (type $h_1$) a successful certification, (2) (type $h_2$) an unsuccessful certification, (3) (type $h_3$) no certification. (each of these histories are followed by a standard lemons market probem with a price offer from the seller and a buy/not buy decision from the buyer).  Then, by Bayes rule (wherever possible), the interim beliefs following history $h$ are:

$Pr(good|successful\; test) = \frac{2}{3}$
$Pr(good|unsuccessful\; test) = 0$
Off path Belief: $Pr(good|no\; test) = 0$.

Game following history $h_1$:
Buyer's maximum willingness to pay is $\frac{2}{3}100+\frac{1}{3}20 = \frac{220}{3}>50$. Hence the equilibrium price in this game $p(h_1)=\frac{220}{3}$ and both types sell their cars.
Gross Payoff for either type: $\frac{220}{3}-C$

Game following history $h_2$:
Buyer's maximum willingness to pay = 20 <50. In this case, the equilibrium price is $p(h_2)=20$ and hence only the lemons sell.
Gross payoff to lemons: 20-C
Gross payoff to good cars: 50 - C

Thus the expected payoff to Lemons if they choose to certify: $\frac{1}{4}\frac{220}{3}+\frac{3}{4}20 -C = \frac{400}{12}-C$.
The expected payoff to Good cars following certification: $\frac{220}{3}-C$.

Off-path choices (history $h_3$):
To sustain this equilibrium, neither type should deviate to no testing. Given off path beliefs, the buyer's maximum willingness to pay is 20<50. Hence only lemons are sold in this scenario, yielding payoffs 20 and 50 for lemons and good cars respectively.

Pooling can be sustained when:

$\frac{220}{3}-C>50 \implies C<\frac{70}{3}$, and
$\frac{400}{12} - C > 20 \implies C<\frac{160}{12}$

Hence pooling is an equilibrium as long as $C<\frac{40}{3}$


How do central banks regulate the money supply to avoid inflation?

The answer provided by Mick is pretty good, but I'll put my own spin on this.

1st: What is inflation?

Roughly speaking it is when the ratio of money to the things money buys increases.  A bumper food crop can reduce inflation.  Crop failure can increase it.  But certainly the most volatile component of inflation is the supply of money itself.  Economists love to exaggerate the importance of the former (productivity) to cover-up the mess they are making with the latter (money supply).

2nd: What is the supply of money?

A difficult answer, as money is whatever we accept.  But the most common forms of money however are classified into large aggregates by economists.

MB: Aka the monetary base in the USA (all money created by the
government...coins, dollar bills, us notes fed deposits)
M1: Bank demand depots (promises of MB) plus cash outside of the banking system
M2: M1 + interest bearing deposits and other checking substitutes

Because private banks can get away with promising more money then they have...and we accept this...then banks actually create money.  And because the higher aggregates like M2 are much larger than MB, it is plain to see that banks are more important variables to the money supply equation and the supply of government money (aka MB).

3rd: How does the Fed regulate the money supply?

It is confusing because the Fed directly dictates the amount of base money in the economy, but also indirectly dictates how much bank money is in an economy.

The primary means by which MB changes include discount loans to banks, buying and selling repos from big banks, and special bailouts.

The primary means by which the Fed controls the supply of bank money is through the open market.  The idea is if the banks have more base money, they will be able to pyramid more deposits on top of that, and vice versa.  To facilitate this the Fed will buy repos from a primary dealer ("special banks") with fabricated money.  The primary dealer than resells the new reserves to lesser banks which then allows them to create more bank money.

Historically, controlling the supply of bank deposits through a reserve ratio was a technique used (and also capital requirements), but these techniques are not considered fashionable among entrenched interests.

Has the Fed been doing a good job?

This is a complicated question.  QE did not create massive inflation because the banking sector was in collapse and a large amount of bank deposits had been destroyed (meaning we had less money).  Whether private for-profit interests should have had privileged access to government resources is a very good question.  Certainly the problem of 2008 was created by the banks themselves. Simply put, they made promises to depositors and creditors they couldn't keep.  Banks operate by mismatching short term/low yield debt to long term/high yield debt.  This is a very profitable but risky practice in which banks are not held proportionately responsible when things go wrong.

Classifying inflation is also very tricky because of international trade.  The dollar is a popular currency overseas, so foreigners help sponge up the damage from inflation.  Trade also obfuscates inflation. If you double say the number of dollars, not all prices will increase proportionately.  International commodities can be brought in multiple currencies, whereas local products (like healthcare and college education) can't and can only be purchased with dollars so they increase much more rapidly.

For a fascinating look at the politics of CPI/inflation, checkout: http://www.shadowstats.com/article/no-438-public-comment-on-inflation-measurement

The most paramount question unasked is if the Fed is seeding the conditions by which another future crisis will happen, to which I fear the answer is yes.


A system is said to be in steady state if certain variables do not change over time (and where "certain variables" depend on the context and ought to have been clearly specified by the writer).

Say we define a system to be in steady state if inflation does not change over time. Then if inflation is constant at 2%, we say that the system is in steady state even though prices are changing. (This is contrary to what you've suggested in your post.)

(The term steady state was, like the term equilibrium, appropriated from the sciences for use in economics. However, although the term equilibrium is often ill-defined, the term steady state is usually amenable to precise definition, provided the writer makes an effort.)


What is the difference between the international Fisher effect and uncovered interest rate parity?

They are related and have very similar implications but they are not the same. I will try to provide simple explanation here for more nuance you can have look at for example Mogaji (2019) and sources cited therein.
Uncovered interest rate parity (UIP) says that the ratio expected exchange rate $E_t[S_{t+k}]$ to spot exchange rate $S_t$, has to be equal to nominal interest rates between home $i$ and foreign $i_f$ country.
$$   \frac{E_{t}(S_{{t+k}})}{S_{t}} = {\frac  {(1+i)}{(1+i_{f})}} \implies E_t [\Delta S] \approx i - i_f$$
This is based on the idea that there should be no possibility of arbitrage (riskless profit) in the market resulting from different nominal interest rates.
For example, if interest rate in US would be $10\%$ and interest rate in UK would be $5\%$ and if exchange rate would be exactly $1{\\\$}=1£$, anyone who could would stop saving in UK and just transfer all their savings into US. However, to do so they have to go through forex market and by creating new demand for USD they should change exchange rate sufficiently enough to eliminate the arbitrage opportunity.
The UIP simply  posits that due to this possibility the ratio between expected exchange rate and spot exchange rate must be equal to the ratio of interest rates otherwise people would start entering forex market and making trades that would force the above parity hold true.
International Fisher effect (IFE) is based on the idea that ultimately exchange rate must reflect differences in price levels between two countries the expected change in exchange rate must reflect the relative differences in expected inflation rates between two countries ($E_t[\pi]$ and $E[\pi_f]$), since if change in price levels between two countries would not be balanced there would be arbitrages in goods market and violations of the law of one price.  Hence we should have:
$$E_t[\Delta S_t] \approx E_t[\pi] - E_t[\pi_f]$$
Now in turn by Fisher equation the rate of inflation is approximately equal tot he difference between nominal ($i$) and real $(r)$ interest rate $e_t[\pi] \approx i-r$. Furthermore, IFE assumes that due to perfect capital mobility the real interest rate must be everywhere equal $r=r_f$. Well given these assumptions we will find out that change in the exchange rate should be given by change in nominal interest rates since:
$$E_t[\Delta S_t] \approx E_t[\pi] - E_t[\pi_f] \approx  i-r - (i_f-r_f) \approx i - i_f$$
Hence there are subtle differences between these theories as the mechanisms in UIP and IFE are different. UIP is based on an idea of impossibility of arbitrage in money market. IFE is based on the idea that there cannot be arbitrage in goods market.
Sometimes two different theories might predict the same outcome, this is known in science as observational equivalence. Observational equivalence does not necessarily mean there is no difference between two theories as two different mechanisms can sometimes give rise to same outcome.


Should we get rid of the Penny?

No. That would be as foolish as suppressing a symptom instead of addressing the illness.

The reason why "a penny does not buy a penny" --or being more accurate, why it no longer buys a penny-- is that currency depreciation has been policymakers' answer for too long and for too many issues.

The permanent, excessive printing of money inevitably weakens the purchase power of the USD. Metal prices, like everything else, go up where economic problems are "addressed" by issuing more bonds, giving more subsidies, ignoring the rapidly changing demographics, and pretending that welfare state still is sustainable.

The weakening of a currency by inflationary measures is not limited to rendering coin metals out of reach. At some point, even the cost of paper on which money is printed can also exceed the purchase power of the denominations printed on that paper. History has many examples of this.


A game where the payoff function $F_1,F_2$ are such that for any two strategy profiles $s,s'$
$$
F_1(s) > F_1(s') \Leftrightarrow F_2(s) < F_2(s')
$$
is called an antagonistic game. Sounds like what you want is a symmetric antagonistic game?


Dynamics of interest rates, exchange rates and bond prices within the UIP

One possible flaw in your reasoning is your assumption about what happens to bond prices when there are capital inflows.  For example , suppose that the Bank of England has fixed short term interest rates at 1%, and that the yield of 5yr gilts is also 1%.  Then there is a sudden unexpected rise in the Bank of England rate to 1.5%, resulting in capital inflows to Sterling investments and thus a rise in the FX value of the pound.  However, the short term interest rate is still 1.5% because it is fixed by the BoE, and the 5yr gilt yield is likely to settle at a yield much higher than 1%, depending on the reason for the interest rate raise.  If the BofE says there will be a series of hikes, this yield could settle at 2% or higher.  If the BofE says it is a ‘one and done’ hike, it could settle lower, say 1.25-1.50%.  These numbers are all made up, the point being that it is unlikely that the bond yields stay at 1% when the BofE has raised its short term rate, which appears to be an assumption you made.


Is there a reference that plots different countries' debt-to-GDP ratios over (long) times?

Data for the United States 1789 - 2019.
The IMF has the Historical Public Debt Database. It has data for almost every country with varying starting dates depending on country.

From my brief analysis France, Spain, Belgium, and Italy all appear to have exceeded 100% debt to GDP ratios before 1944.


Indifferent choice assumption in economics

Because otherwise you would not have subgame perfect equilibria. Suppose I have an item you need and the item is worthless to me. I would be happy to give you the item for any positive amount of money. If you give me no money for it, I am indifferent, both giving you the item and not giving you the item is an optimal choice for me. Just to be clear: we are not bargaining. You set I price and I will take it or leave it.

If you offer price 0 and I give you the item that is an equilibrium. No one could have made a better choice given the other person's choice.
 If you offer price 0 and I do not give you the item that is not an equilibrium, because I would give you the item for any small positive payment and you would be willing to pay that, so you could have made a better move.
 If you offer price $2a>0$ I will give you the item. But this is not an equilibrium. You could also have offered me $a$, which is also positive, and I would also have given you the item then. So you could have saved some money, your original move was not optimal.

Because there is no smallest positive number, the only equilibrium in such cases is when one side is indifferent.


How do you read currency exchange rates in terms of domestic and foreign currency?

The currency in the denominator is always the currency that is being priced. For example $\$/€=x$ means that the price of euros is $x$ dollars. That is, if you want $1$ euro, you have to pay $x$ dollars (Keep in mind that in practice the euro will have a bid and an ask price, but in both cases, the euro will be in the denominator). In fact, the notation $\$/€$ should be read "dollars per euro". In contrasts, $€/\$$ reads "euros per dollar", so the currency that is being priced now is the dollar (of course in terms of euros).

With this in mind, the statement online that says "E$\$/€ = 1.50$, [...] would mean that $1\$$ would buy $1.50€$". was either misunderstood or simply wrong. Such notation means that $1.50\$$ is equivalent to $1€$.

Finally, keep in mind that, in practice, countries decide to write their exchange rate in terms of the dollar according to convenience of the numbers. For example, the exchange rate for the Mexican peso is almost always written as peso/dollar. Technically speaking such notation is giving the price of dollars, but note that it is more convenient to say that E$(peso/dollar)=24$ pesos rather than saying E$(dollar/peso)=0.041666$ dollars. This creates no problem since both quotes are equivalent.


What does a "sell-off" mean if every stock transaction is balanced?

The Cambridge English dictionary has a definition here: link to definition

I will paraphrase their definition: a sale of equity shares that causes its price to fall.

As the question notes, that appears to make little sense, if for every buyer, there is a seller. If stock prices were negotiated by individual verbal contracts, the implication was that the seller was more morivated to sell than the buyer to buy, and so the price had to be adjusted lower.

However, the common usage of “sell off” in the financial press is different: it is a synonym for a stock price (or stock index) experiencing a price fall on a day versus the previous day. The implication is that all the sellers were motivated than buyers, and so the price fell, like in the verbal contract.

However, if the observed price on the exchange is lower, we only know two things:

For every buyer, there is a seller.
The price of the transactions on the current day is lower than the previous day.

There’s any number of ways the price can end up lower, and short of questioning everyone involved in trading that day, we have no idea what really happened.

I will now give an example how intuition about supply and demand can be misleading.

Imagine that Acme Inc. traded yesterday at \$10. The next day, market makers set the price around that price, with a bid-offer spread. There is no trading.

Then imagine that at 10 am, Acme announces a successful lawsuit against it by the plaintiff Wile E. Coyote. The market makers are shocked, and drop the mid of the bid/offer prices to \$5, and still there is no trading.

Meanwhile, retail stock traders who did not know about the lawsuit saw the new price, and rushed to buy. 100 customers buy 100 shares each for the market makers, who are happy to sell out of inventory.

From the persepctive of an outside observer, we saw 100 retail traders rush to buy shares - and the price fell by half.


University course references for Market Structure/Industrial Organization

The standard, if somewhat aged textbook for graduate study of industrial organization is

The Theory of Industrial Organization by Jean Tirole.

Tirole has the merit of tacking issues around market structure in a quite direct fashion. A slightly more modern text is

Industrial Organization: Markets and Strategies by Paul Bellflamme and Martin Peitz

A less advanced text that can be downloaded for free is

Industrial Organization: A Strategic Approach by Jeffrey R Church and Roger Ware

It's worth mentioning that Industrial Organization is typically not encountered as a stand-alone course until the late stages of an undergraduate degree or until graduate level, so books on this subject will typically be more advanced than a standard econ101 text.


Does free capital mobility invalidate the basic comparative advantage argument for free trade in goods?

Daly actually does explain further what he means by that in footnote 7 of his paper:

  Capitalists are interested in maximizing absolute profits and therefore seek to minimize absolute costs. If capital is mobile between nations, it will move to the nation with lowest absolute costs. Only if capital is internationally immobile will capitalists bother to compare internal cost ratios of countries and choose to specialize in the domestic products having the lowest relative cost compared to other nations, and to trade that good (in which they have a comparative advantage) for other goods. In other words, comparative advantage is a second-best policy that capitalists will follow only when the first-best policy of following absolute advantage is blocked by international capital immobility. For more on this, see Chapter 18 in Herman Daly and Joshua Farley, Ecological Economics (Washington, DC: Island Press, 2004).

But IMHO, the absolute conclusion that  international trade must be regulated as following from that seems silly. After all, one can regulate capital flows too negating that first-best policy (argument). In fact there are plenty of barriers to capital flow, some implicit and some explicit.

Having said that, I cannot quite find the ultimate conclusion stated in those absolute terms by Daly, although he does say (we should) "Move from free trade and free capital mobility to balanced and regulated international trade." Which is frankly pretty vague. I mean doesn't the WTO do at least some of the latter?

The whole writeup of Daly smacks of a certain social agenda. Later he says

  Restore the US Full Employment Act of 1945 and its equivalent in other nations in order to make full employment once again the end, and economic growth the temporary means.

I think Bernie Sanders also said he wants something like that. Daly also says

  Un/under-employment is the price we pay for growth from automation, off-shoring, deregulated trade, and a cheap-labor immigration policy.

And finally, Dally says (we should)

  strengthen the original Bretton Woods vision of interdependent national economies, and resist the WTO vision of a single integrated global economy

That's basically a slogan in disguise, to put it diplomatically. You could just as well phrase it as "take back control of our economy/tariffs/borders/immigration". So his conclusions are more or less socio-political goals; it's hard to take them at face value as pure economic arguments. So there will undoubtedly be disagreement on such statements.

Perhaps to save his "ecological economics" creds Daly does say that in contrast

  climate change and arms control require global institutions

To use a political science neologism, that's an ambivalox with demanding no world-wide regulatory institutions for any other economic issues, like trade.

But to come back to the more serious economic topic, glossing over Daly's black-and-white approach, capital mobility as partially substituting for goods exchange is hardly a new discovery. See Nadel (1971) for example.

  It has long been recognized that commodity movements and factor movements [e.g. capital, my note] are, to a degree, substitutes for each other in international exchange [...] Yet, until recently, the dominant theory of international trade, the Heckscher-Ohlin model, had been rather thoroughly recognized under the rigid assumption of the immobility of factors. Only in 1957, with the publication of Robert Mundell's important article, was capital mobility in the H-O model explored.

(That important article being: Mundell, Robert, “International Trade and Factor Mobility,” American Economic Review 67
(1957):321–35. It has around 2800 citations in Google Scholar.)

What is somewhat novel (in ecology-related economics) is that under some models of pollution, capital movement is preferable to goods movement:

  Using a Heckscher–Ohlin model, this paper re-examines Robert Mundell’s famous thesis that free trade and
  unimpeded capital mobility are perfect substitutes. Under very general conditions which, according to many
  economists, have caused international convergence of factor rewards,we show that in a polluted environment
  free trade is inferior to free international investment.This happens even though commodity prices and factor
  rewards are the same with both policies.The practical side of our thesis is that the world will be better off by
  reducing the volume of trade while removing all barriers to foreign direct investment that at present hamper
  the service industries.

That paper Batra and Belladi (2012) does also say that

  Daly (1996), Batra (1992), and Batra
  et al. (1998) argue that transportation is highly pollution intensive, specially relative to production

That seems to be a cornerstone assumption of the model. It is [to me at least] an interesting separate question if economists outside of this narrow field of "ecological economics" agree with these premises.


Why are real median household incomes stagnant?

The metric of median household income is also used by others to argue the presence of income inequality:
https://en.wikipedia.org/wiki/Income_inequality_in_the_United_States#Causes
However, it seems that it is not only the median but also the mean that stagnates:

(I used family instead of household income because I could not find a time series for mean household income.)
If the problem was only one of inequality then the mean income should still increase.
I think there are several reasons why median household income seems to stagnate.
First of all I think there is a visible upward trend that is only somewhat masked by the 2008 financial crisis. Granted this trend is still smaller than the increase of productivity. Why median household income grows at a slower pace than before is still a question.
I think the problem arises because neither the number of households nor their distribution is constant.
Changing size of households
One thing that I believe has an effect is that [average household size has decreased over the period you are looking at][3].

(I had some trouble opening the Census Bureau's xls files.)
So while total household income has in fact increased it is now divided between more numerous, smaller households making both the mean and the median smaller.
Changing composition of the US labor force
Another reason might be immigration. According to [this site][6] (I don't know if the statistics are reliable, there are a lot of links on the site I consider to be politically biased) the mean immigrant income is lower than the mean 'native' income. This is not surprising, immigrants need time to adjust and build the social networks necessary to get good jobs. Depending on the measure of productivity you use this can mean that while productivity in any given industry increases the weight of low-productivity industries increases as well. Thereby immigration may decrease mean and median income.
Note that this does not imply anything about the effect of immigration on the welfare of 'natives'. I am not claiming that the median household income of 'natives' is increased or decreased, I am pointing out that the weights used in the calculation are shifting.


The margin with 1 bag was \$25 (revenue) / \$15 (labor + materials), or ~166%. The margin with 2 bags, assuming no additional labor costs, is \$50/$25, or 200%.

When this business plans out expenditures on labor and materials for the third time around, it has \$50 cash, enough to pay $5 for labor and up to of 4.5 bags of candy.


Let's start by writing down the profit functions:
$$
\pi_1 = p_1 q_1 - c_1 q_1 = (p_1 - c_1) q_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2)
$$
and similarlily:
$$
\pi_2 = (p_2 - c_2)(a_2 + b_{21} p_1 +b_{22} p_2)
$$
In a simultaneous equilibrium, both firms simultaneously maximize their profits taking the strategies of the players as given.
Doing this will give you a best response of player 1 that will be a function of $p_2$: $p_1 = p_1^\ast(p_2)$. Similarly, the best response of player 2 will be a function of $p_1$: $p_2 = p_2^\ast(p_1)$.
In a Nash equilibrium, you are looking for a solution for both equations simultaneously:
$$
p_1 = p_1^\ast(p_2),\\
p_2 = p_2^\ast(p_1).
$$
In a sequential equilibrium on the other hand, you assume that one firm, say 1, chooses $p_1$ first and next firm 2 decides in $q_2$, after being informed about the value of $q_1$ chosen by firm 1.  Firm 1, however, is smart enough, such that it can predict the optimal action that firm 2 is going to take in stage 2.
As such, we solve the game by backward induction, first solving for firm 2, giving $p_2 = p_2^\ast(p_1)$ and then we substitute this into the profit function of firm 1.
$$
\pi_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2^\ast(p_1))
$$
This substitution conforms to the fact that player 1 knows that by choosing $p_1$ it will influence the price of firm 2. So firm 1 takes the future behaviour of firm 2 into account when choosing its optimal level of $q_1$.
Maximizing $\pi_1$ with respect to $p_1$ will give an optimal value $p_1^\ast$ (which will no longer be a function of $p_2$).
Then, to compute the equilibrium, you substitute this value into the best response for firm 2:
$$
p_2 = p_2^\ast(p_1^\ast).
$$


How do you define the market cap of an ETF?

An ETF’s assets will fluctuate based on both changes in the value of
  the underlying securities and the creation of new shares or redemption
  of existing shares. It is worth noting that there may be a difference
  between an ETF’s market capitalization and the net asset value (NAV)
  of its underlying securities. The difference results in either a
  discount or a premium in the trading price of the ETF.

ETF DB: Largest ETFs: Top 100 ETFs By Assets

The market capitalization at time $t$ is therefore  the number of shares at $t$ times the price of those shares at $t$. The net assets are the total value of all the assets in the fund at 't':
$$ Net\ Assets_t = U_{t} \cdot NAV_t = U_{t} \cdot \sum_{i} (S_{i,t} \cdot weight_{i,t})$$
where $S_{i,t} $ is the value of a share of stock $i$ at time $t$, $weight_{i,t}$ is the fraction of the index invested in stock $i$ at time $t$, and $U_{t}$ is the number of shares of the index.
Or equivalently:

$$ Net\ Assets_t = \cdot \sum_{i} (S_{i,t} \cdot q_{i,t}) $$
Where $q_{i,t}$ is the number of shares of stock $i$ at time $t$ held by the ETF. This sounds simpler but it is often easier to find index weights than exact share holdings measured in number of shares.

But for open-ended funds that allow creation and redemption or have minimal expenses these NAV and market price based approaches will give similar answers.


When can the fiscal multiplier be negative?

The paper How Big (Small?) are Fiscal Multipliers? by Ethan Ilzetzki, Enrique G. Mendoza, Carlos A. Végh (2010)should give you useful information.

  We contribute to the debate on the macroeconomic effects of fiscal
  stimuli by showing that the impact of government expenditure shocks
  depends crucially on key country characteristics, such as the level of
  development, exchange rate regime, openness to trade, and public
  indebtedness. Based on a novel quarterly dataset of government
  expenditure in 44 countries, we find that (i) the output effect of an
  increase in government consumption is larger in industrial than in
  developing countries, (ii) the fiscal multiplier is relatively large
  in economies operating under predetermined exchange rates but is zero
  in economies operating under flexible exchange rates; (iii) fiscal
  multipliers in open economies are smaller than in closed economies;
  (iv) fiscal multipliers in high-debt countries are negative.


Gold and loss of faith in central banking

I think @DornerA gave a good answer for reasoning behind the mechanism.

For a historical example, look no further than the fiscal crisis and the rapid increase in gold price from 2008-2011.

Of course a lot happened during those years with house prices collapsing in the U.S. Faith in central banking is closely associated with the value of money and the belief of whether a central bank can keep the economy steady or not and in a situation where this belief is low, people gravitate towards widely accepted stores of wealth.


With a European view:

The CEPR VOX EU: an economics blog created by the Centre for Economic Policy Research, which promotes research excellence and policy relevance in European economics. It covers a diverse range of research based analyses on policy and economic issues.
The Economist a global scope with a European perspective.
EconomicsUK: personal website of The Sunday Times' Economics Editor. A perfect blog to learn about European economics.
Alpha.sources: personal website of a Eurozone Economist. Covers macroeconomic issues.
Thomas Piketty's Blog: Economist views on current social and political issues in France, Europe and the world.


How does a cash deposit change the M1 measure of the money supply?

It's best to always check the exact definitions for M questions, because they can vary a little between countries.  I'll use the US Federal Reserve's here, viz. "M1 is defined as the sum of currency held by the public and transaction deposits at depository institutions (which are financial institutions that obtain their funds mainly through deposits from the public, such as commercial banks, savings and loan associations, savings banks, and credit unions)."

The reason M1 doesn't increase is essentially semantic. Before the deposit occurs, M1 is the sum of currency held outside the banking system ("held by the public") and deposits at banks etc. Notice that currency held by the banks is not included in the M1 definition.

When the deposit occurs, the accompanying double entry book keeping is [debit cash, credit customer deposit]. Now the cash is held by the bank, so not included in M1, but the customer deposit that was created by the cash deposit is - so there's no change. This is the best way to handle this btw., since once the cash is deposited at a bank, it's no longer playing an active role in the money supply.

As far as the potential impact on the reserve requirement, and multiplier effect. As you're quite rightly suggesting, the question is trying to avoid that by saying "immediate". However, if you're in the USA, reserves no longer control the money supply the way the text book tries to describe, so there would not be expected to be a long term impact either. (reserve requirements only cover a fraction of bank deposits, and capital requirements actually dominate in terms of regulatory impact.)


TL;DR You have defined a de facto currency with that bank as a central bank.  It is generally considered better to have everyone using the same currency.

In theory, what you suggest can indeed work, and has been used for period of time in various places.  Legally speaking the Bank of England doesn't mandate a minimum reserve, and so in theory a bank could operate with zero reserve (The EU mandates 2%, the US 10%).  In practice each bank must agree their own reserve ratio with the BoE, and the bank would likely laugh if someone said "zero".

You system in essence works, and all you have done is created a currency tied to that bank (lets call it Bank Z).  The problem is that no-one can actually pay anyone who isn't a customer of Bank Z (for more on electronic transfers see: https://economics.stackexchange.com/a/6170/119).  Paying someone who banks with Bank Z is easy, just move the money from your account to theirs, but if they don't have an account we can't give them any money - because we don't have any.

What we could do is give them an IOU - "if you open an account with us, we will credit it with Z\$X".  This would be a bank note for Bank Z.  Of course the value in that note is tied to our faith in that bank.

Suppose now we imagine Banks X, Y and Z operating in our economy.  What happens when Xavier (who banks at Bank X) want to pay Yvette (who banks at bank Y)?  Well neither bank can change the customer account of the other, so both customers are given IOUs (bank notes) and take them to the other bank.  Should the other bank accept these notes and increase they account accordingly?   Well that depends...

If both banks are both well used then sure, why not.  But suppose instead that 98% of people bank with Bank X, and 1% with bank Y - whose IOU would you rather have?  It is much easier to spend Bank X money, than Bank Y money, so that in itself could cause bank X money to command a premium.

If X\$100 = Y\$101 then clearly I will charge you more Y\$ for fixing your roof that X\$.  This can quickly spiral and before you know if Y\$ are worthless - they have suffered hyper inflation and been inflated away.

You have effectively created a state with multiple currencies, and in such situations, one currency would be expected to fall into disuse and become worthless.  Historically countries have often actively defended their currency from the use of external currencies by imposing limits on how much currency can be brought in.  It can be a real problem if your people are trading using dollars instead of your national currency, for the same reason.  If dollars are more useful, then why use local currency, and so it decreases in value.

Your set up is particularly fragile, because the banks are merely lightly regulated corporation, not the government.  The government can mandate that all government workers are paid in local currency - in effect it can force itself to have a large number of customers.  It is not a coincidence that the central banks are the bankers to the treasury of their respective countries.

As a government this is also a pain - you have multiple currencies, and you don't control any of them.  Your economy is not in your control, which has cause problems in the past.

Gold backed?  Assets?

I deliberately haven't talked about this part, as you don't really address it, but I suspect many will think first about it.  The backing and assets of the bank are really secondary to this discussion.  Their is nothing to stop all our banks saying that their dollars are pegged to the value of gold.  This simply means that you can, on demand, exchange your IOU for gold.  So long as nobody does this then you are fine.  Equally, even if a few do, you can buy they gold with your own IOUs and so that is still fine.

Any assets your bank has, can be used to "back" your currency and add strength to it, even though you have no "reserve" with any central bank.  This will help you.  Bank Y could respond by advertising that it has tons of gold in the vault and owns lots of houses and farmland, so it will be more likely to be able to honour it's IOUs that Bank X can.  This could turn the tide back in your favour, as people lose faith in X\$ and switch to Y\$.

Remember, even if your Bank Z starts with zero assets, it will make a profit in Z\$ over time, which can be used to buy things and so it will accumulate assets with time.  None of these need become liquid reserves.

Examples?

Your case is a bit too extreme, but near examples are worth a look

The biggest example that springs to mind is the Free Banking Era (aka "Wildcat Banking") in the US from about 1837-1862.  Andrew Jackson had "defeated" the Second US central bank and believed that banks could just be allowed to run things on their own.  The average life of a bank in that era was 5 years, and each bank's currency was worth a different amount, depending on the regard with which that bank was held.  These banks did not run without reserve, for the reasons given about, but they were free to choose how much and how to do it.  The only reserve requirement was having enough to convince your customers to use you...

A more modern example is "points".  Plenty of businesses award "points" for various activities, which can be redeemed for goods and services.  If you think about it, these points are like a constrained version of exactly your situations.  Usually points can only be exchanged with the issuer, but if you added the possibility to transfer points from one person to another, then you have a basic currency operating as you described.  The key feature here, is that these points are not backed by anything, and cannot be converted to gold or "hard currency", only airline flight, or groceries or whatnot.


Why would males' shorter pension's savings produce more, and not less, income?

I suspect your confusion is merely because you don't know what an annuity is. An annuity pays you a regular stream of income until your death (e.g. $10,000 every year until you die).

So say there's a man and a woman. Each is aged 65 and each has \$100,000 in pension savings. Each person uses their pension savings to buy an annuity.

Suppose the financial institution (usually an insurance company) selling them the annuity estimates that the man's life expectancy is 20 years, while the woman's is 40 years. Then the actuarially-fair payout would be \$5,000 per year for the man and \$2,500 per year for the woman (if we ignore interest rates, discount rates, and similar complications).

Hence, for the same amount spent on the annuity (\$100,000), the man gets a highly yearly income (\$5,000 per year) than the woman (\$2,500 per year). This is what your quoted passage means.

(More on how the annuity works: If the man/woman lives longer than the expected 20/40 years, then good for him/her. He/she keeps receiving the agreed-upon yearly payout even past those 20/40 years.

Conversely, if the man/woman lives shorter than the expected 20/40 years, then too bad for him/her. He/she gets the agreed-upon yearly payout only until he/she dies.)


Concave production function implies convex cost function

Given the fixed input price $w$, the cost function can be written as
$$
C(q)=f^{-1}(q)\times w
$$
where $f^{-1}$ is the inverse of the production function $f$. From the discussion here, one can conclude that the inverse of a concave strictly increasing function is convex. Thus, $C(q)$ is convex as well.

Going back to your approach, you might like to have this clearly stated. Let $q''=\alpha q + (1-\alpha)q'$, $f(z)=q$, $f(z')=q'$, and $f(z'')=q''$ Then
$$
\begin{align}
f(z'')&=&q''\\
&=&(\alpha q + (1-\alpha)q')\\
&=&\alpha f(z)+(1-\alpha) f(z')\\
&\leq&f(\alpha z +(1-\alpha z'))
\end{align}
$$
Then $f(z'')\leq f(\alpha z +(1-\alpha z'))$ implying that $\alpha z +(1-\alpha z')\geq z''$ since $f$ is (strictly) increasing. Hence,
$$
z''w=C(q'')\leq \alpha C(q)+ (1-\alpha) C(q')
$$


Game with an equilibrium in pure, but none in mixed strategies?

As I said in the comment, Nash's theorem shows the existence of a Nash equilibrium (possibly but not necessarily in mixed strategies). If you are interested in Nash equilibria is proper mixed strategies, i.e., NE in which all players play at least two actions with positive probability, you can easily show the following impossibility result: No such NE can exist if one player $i$ has a stricly dominant strategy, i.e., a strategy that gives a higher payoff regardless of the other players' strategy. Since the strictly dominant strategy is $i$'s unique best response against all strategies of the others, this pure strategy must be a part of each equilibrium. Each other strategy by $i$ can be improved upon by deviating to the dominant strategy.
In your case 1 has a strictly dominant strategy, $B$ - so even if 2 always plays $L$, 1 would not be indifferent between $A$ and $B$, but always choose $B, 9>8$. Without indifference, mixing is never optimal and therefore never part of a NE.
Even if $B$ was only weakly dominant, it would always be the unique best response to every strategy in which player 2 plays $L$ and $N$ with positive probability.


The only assertion here that goes beyond self-evident matrix algebra and substitutions, is that

$$P_XP_1 = P_1$$

Since the columns of $X_1$ are in $X$ it follows that

$$P_X X_1 = X_1$$

because in general it holds that

$$P_X X = X$$

Namely the projection matrix operates as the identity matrix for the columns of the matrix from which it has been constructed. But then

$$P_X P_1 = P_XX_1(X_1'X_1)^{-1}X_1' = (P_XX_1)(X_1'X_1)^{-1}X_1'$$

$$=X_1(X_1'X_1)^{-1}X_1' = P_1$$


Repeatedly playing an equilibrium in a repeated game

Suppose all players play unconditioned Nash equilibria, except player $i$ who diverts and plays another strategy. The other strategy of player $i$ may depend on history. We have to show that player $i$'s payoff in the repeated game does not increase.

For every time-period t, define the following random variables:

$X^{t}$ - the utility of player $i$ in time t when all players, including player $i$, play the basic Nash equilibrium.
$Y_{h}^{t}$ (where $h$ is a history vector of length $t-1$) - the utility of player $i$ in time $t$ when player $i$ diverts and plays the alternative strategy while all other players continue to play the basic Nash equilibrium.
$Y^{t}$ - the utility of player $i$  in time $t$ when player $i$  diverts while all other players continue to play the basic Nash equilibrium:
$$Y^{t}=\sum_{h}Prob[h]\cdot Y_{h}^{t}$$

By definition of a Nash equilibrium, for every time $t$
and for every history $h$:

$$E[Y_{h}^{t}]\leq E[X^{t}]$$

Hence also:

$$E[Y^{t}]\,=\,\sum_{h}Pr[h]\cdot E[Y_{h}^{t}]\,\leq\,\sum_{h}Pr[h]\cdot E[X^{t}]\,\leq\,E[X^{t}]\cdot\sum_{h}Pr[h]\,\leq\,E[X^{t}]$$

Summing over the entire series, and using the fact that the expectation of a sum is the sum of expectations:

$$E[\sum_{t=1}^{T}Y^{t}]\leq E[\sum_{t=1}^{T}X^{t}]$$

So the player cannot gain by deviating.

Thanks to @denesp for verifying this proof!


Understanding agents selecting among markets

If a market has free entry, then profit will tend to zero in the long run. Monopolistic competition and perfect competition are both characterized by zero long-run profit. Of course a firm would prefer to enter a market in which long run profits can be earned, but such markets are characterized by barriers to entry.

To better understand this, consider a market in which firms can enter with entry cost $c$. Then, in long-run equilibrium, the profit that will be earned by incumbent firms cannot exceed $c$, otherwise potential entrants would have an incentive to enter the market.


You can simply apply what you know about perfect complements. If we have $C_i$ for $i=1,2,...,n$, where $C_i$ is consumption in the $i$th state of the world, each state of the world occuring with some probability, the utility function $U=\min\{C_1,C_2,...,C_n\}$ will ensure that the consumer will choose to have equal consumption in every state of the world.


Why does Walmart (SYMBOL:WMT) have its Q1 2015 end April 30 instead of March 30

A December fiscal year end, which gives a first quarter of three months ending March 30, aligns the fiscal and the tax year. This can be very convenient and in the United States is sometimes required. In addition, some regulated firms like banks are required to prepare documents on calendar quarters regardless of the month of their fiscal year end, and it is often cheaper and easier to have them be synchronized rather than close the books additionally for regulators.

Firms may be motivated to alter their fiscal years so that they can better serve their customers or investors. For example, some banks have a fiscal year end not in December so that customers needing to do financial transactions at the end of their year could do so without the bank simultaneously having to close their own books. For many years Morgan Stanley did this. Walmart may be doing this because December is the Christmas shopping season in the United States and management wants to focus on that critical shopping period without the distraction of accounting issues. Additionally, if some shopping comes in after the holiday (say if Hanukkah comes late or because of a rash of belated Festivus shopping), it might split a single shopping season into different fiscal years which could be undesirable for investors trying to compare retailer performance.

I don't want to lean too hard on any of these explanations for Walmart's timing. If I recall correctly from looking in Compustat a few years ago more than 80 percent of US listed firms have a December FYE and this didn't vary a huge amount across industries.

UPDATE:
I went into Compustat and I redid the table of retail, banking, and other firms showing the months of fiscal year end the last 12 months (8/2014-7/2015).
I was surprised to see that retail was indeed vastly (>20 times) more likely to have a January FYE than a typical publicly listed firm in 2015.  The results appear in the figure below.

Source: Compustat

This gives credence to the hypothesis that as a retailer with significant holiday sales, Walmart may have a January FYE to highlight those results.


The textbook assumes that assets are real assets (inventory, equipment, etc.) that are falling in nominal value along with the price level. However, this will not be the case for financial firms, as their assets are almost always fixed nominal loans. So there is an embedded assumption that this only refers to non-financial firms, which is not specified in the quote provided. (Note that “borrowers” does not imply non-financial, since financial firms are major borrowers.)

However, this is a questionable reason to explain a financial crsis. No firm revalues assets or liabilities based on changes to the price level. The real concern is that falling prices imply falling nominal revenues, which are matched against fixed nominal debt payments. So it’s not really the balance sheet value, rather the prospective value based on future profits that are shrinking. In other words, the problem shows up in the income statement, not the balance sheet.

On paper, accounting values are supposed to reflect prospective profits, but in practice, continuous revaluations are not part of accepted accounting procedures. An asset has to be obviously impaired before being written down. Therefore, balance sheet revaluations (other than loan losses) are not the driver of a financial crisis.


Models with Learning by Doing and Knowledge Spillovers - Barro, Sala-i-Martin (2003)

We can show this by adding some public good to the model that will be financed by lump-sum taxes (which is also discussed in Barro & Sala-i-Martin (2004). Economic Growth 2nd ed. ch 4.4.1). So suppose Cobb-Douglas is given like in Barro 1990 as:
$$Y_i=AL_i^{1-\alpha} K_i^{\alpha}G^{1-\alpha} \tag{1}$$
Now for any given $G$ profit maximizing firms will equate the marginal product of capital to the rental price $r+\delta$ and this will give us:
$$\alpha A k_i ^{ -(1-\alpha)}G^{1-\alpha}= r+\delta. \tag{2}$$
Since firms are homogenous they will all choose some optimal $k_i=k$ and hence we get:
$$Y= AL^{1-\alpha}K^{\alpha}G^{1-\alpha} \implies G = \left(\frac{G}{Y}\right)^{\frac{1}{\alpha}} (AL)^{\frac{1}{\alpha}}k \tag{3}$$
Now we have to assume government will pick some constant $G/Y$ and using this combining (3) and (4) gives us:
$$\alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha}   = r+ \delta \tag{4}$$
Now because $G/Y$ and $L$ are constant also marginal product of capital will be constant with respect to time. As a result:
$$\frac{\dot{c}}{c} = \frac{1}{\theta} \left( \alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha}   -  \delta - \rho \right) \tag{5} $$
Now the above also happens to be optimum chosen by a benevolent social planner in this case because the planner would choose $c$, $k$ and $G$ to max:
$$\int^\infty_0 e^{-\rho t}\frac{c^{1-\theta}-1}{1-\theta} dt \tag{6}$$
which is the household's utility, subject to the constraint:
$$ \dot{k} = A K^{\alpha}G^{1-\alpha}-c-\delta k - G/L \tag{7}$$
we combine (6) and (7) by setting Hamiltonian:
$$H =  e^{-\rho t}\frac{c^{1-\theta}-1}{1-\theta} + v\left(  A K^{\alpha}G^{1-\alpha}-c-\delta k - G/L \right) \tag{8}$$
which will give us the following FOC's:
$$ e^{-\rho t} c^{-\theta} = v \tag{9}$$
$$  A(1-\alpha)k^{\alpha}G^{-\alpha} = \frac{1}{L} \implies \partial Y/ \partial G =1 \tag{10} $$
$$ - \dot{v} = v \left(A \alpha K^{\alpha-1}G^{ 1-\alpha} - \delta \right) \tag{11}$$
and we also have to impose the transversality condition.
Now actually the FOC given in equation (10) by implying that $ \implies \partial Y/ \partial G =1$ tells us that in the optimum $G/Y=1-\alpha$ (which is where the value of investment tax credit comes from).
Finally we find that when $G/Y=1-\alpha$ social planner would choose:
$$\frac{\dot{c}}{c} \bigg\vert_{\text{social planner}}= \frac{1}{\theta} \left( \alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha}   -  \delta - \rho \right) \tag{12} $$
which is exactly the same as a decentralized equilibrium given by 5. However, the assumption of lump-sum taxes is important for this result and generally using some distortionary tax we won't get the same result.


Free Disposal: the production set less the vector space of positive reals

So the way that $Y$ is defined is such that negative values are  inputs and positive values are outputs. For example, if I used 3 units of labor, and 2 of capital to produce 10 pencils, that will be reflected as a single vector in $\mathbb{R}^3$ equal to $[-3;-2;10]\in Y$.

Notice that subtracting 1 unit to either the first or the second element of that vector will mean that I used more inputs to produce the same output, for example, $[-3;-3;-10]$. Similarly, if I subtract one unit to the third element, it would mean that I produced less output with the same inputs. For example $[-3;-2;9]$.

Both of these operations, respect the idea of free disposal (so long as you are subtracting units, not adding to them). Moreover, any notion of free disposal can be represented by one of these operations: i.e producing less with the same inputs, or using more inputs to produce the same.

Therefore you can conclude that a production set, $Y$, satisfies free disposal if and only if $Y-\mathbb{R}^L_+\subset Y$.


In what sense is economic growth cumulative?

The basic idea is simple. If you grow 1% each year for five years you end up with a GDP that is (1.01)^5 of what you started: the 1% growth of each year accumulates over time.


The shadow price is formally not the increase in the objective function for relaxing a constraint by a single unit, but by an infinitesimal relaxation. In the world of linear programming it can be valid for up to a unit but it doesn't have to be. In your example problem it is actually not.  This becomes clearest when we use pictures.

I am going to assume that you were maximizing, not minimizing your objective function and furthermore that you have two additional constraints that you hadn't actually stated:

$K\geq0$ and
$S\geq0$

That would result in the following graphic with K on the y-axis and S on the x-axis, and the optimum at the red dot. The black dotted lines are isoquants of Z, for Z=100, Z=150 and Z=200.

The feasible region is made up of the x and y axis, the blue line from the y-axis up to the crossing, and the orange line from that point on to the x axis.

Let us now relax your constraint $K+2S\leq6$ by one unit, as you propose. In essence this means that the constraint becomes $K+2S\leq7$ and when we draw the new constraints we get:

Turns out that the new constraint no longer plays a role in determining the optimum! The optimum is now determined by the constraints: $S\geq0$ and $2K+3S\leq10$.

The answer to your question should now be clear: the shadow price is valid up to the point where that particular combination of constraints continues to determine the optimum. In this case that means that you can increase 6 to $6\frac{2}{3}$, for a corresponding increase in the objective function of $10*\frac{2}{3}=\frac{20}{3}$  After that point the combination of constraints no longer determines the optimum.

As a side point let me add that at least the Excel solver (and I expect therefore many other solvers too) report up to which point you can relax the constraints before the optimum is determined by another set of constraints.


What happens to the money that the government is in debt from?

This does not work like that. If you want to add-up all the debt in the world you will not end-up with 0.

Think about a situation in which I owe you \$100, you owe someone else \$100, and this someone else owe me \$100. Total debt would be \$300, even if another guy is debt-free.

Moreover, country debt (called sovereign debt) is not only owned by other countries but most of the world-debt, is owned by people, mostly through pension funds (and other investments). So even in a single-country economy, the country may issue some debt that consumers or firms can invest in. And the global amount of sovereign debt will be strictly positive.

Again most of the world-debt is privately owned and it may even be the case that those five countries you're talking about don't hold a single debt contract.


Why are European countries happy to be outside Eurozone?

I largely agree with mathtastic, but I think it is necessary to add something.

When joining a monetary union a country does indeed lose its power to conduct monetary policy. However simply not being able to reduce the interest rate was not the main issue that Greece and other countries faced because of being in the Eurozone. This is because the ECB lowered interest rates (almost) down to the zero lower bound.

To adequately address your question we have to distinguish advantages of a country from having its own currency while in a crisis, vs. advantages from always having had its own currency in the first place.

Reasons why it is helpful to have your own currency when a crisis has already hit:

The issue in this case was the impossibility of devaluation of the currency. One way to understand the Greek crisis is as a current account crisis. Not only was the government over indebted, but the country as a whole. That is to say that there was a too large trade deficit for too long. The trade deficit creates a sort of debt that the whole country owes to other countries in total and was a main point of focus in the debates about Greek debt. You might have heard the issue of competitiveness being cited as a factor in the Greek crisis, which simply means the impossibiliy of the Greek economy to export sufficiently. You may have also heard about "living above their means" or "BMWs going to Greece financed by German money". All this refers to the trade deficit, i.e. the current accuont crisis.

To help itself out of this current account crisis, Greece would have to reduce its trade deficit, i.e export more and import less. If Greece would have had its own currency, a natural reaction to the crisis (without any government intervention even required) would be that Greece's currency loses value. Hence its exports are cheaper which leads to more exports and its imports are more expensive, which leads to less imports. In this way Greece would have an easier time dealing with the crisis and could also "grow its way out of the crisis" - policy advice often heard in this whole debate.

In addition to reducing the trade deficit, a currency devaluation would have helped Greece's economy grow and help with debt service, because the private and public sector were over indebted and not in a position to borrow. This means that domestic demand was very low, so foreign demand- i.e. exports- would have to pick up the slack.

Now, you might say that Greece could export more by simply reducing its prices even without currency devaluation. This is in theory true and is indeed what was recommended and what was attempted in Greece. You can look this up under the term "internal devaluation". However there are several problems with this. First of all it is easier to simply devalue the currency (one price adjustment) than to reduce prices of all export goods (many price adjustments). Furthermore, price decreases mean deflation, which brings with it further problems and is hard to combat later on, when it is no longer desired. A term to look up for more information on this would be "deflationary spiral". Third and perhaps most importantly is the issue of "downward nominal rigidity". This is the empirically observed phenomenon that nominal prices tend to not go down as much as they should or at all, i.e. are rigid in going downwards. This come perhaps from money illusion. It is easier to inflate away workers' wages than to actually reduce them, because the former doesn't cause any protest, while the latter does. Although this is not rational, it happens. Hence it is very difficult to reduce prices and export more (i.e. be more competitive) without an own currency.

This is the reason many countries are likely happy to have their own currency. As to why they are happy despite possibly being "pegged" to the Euro, is because they can break that peg much more easily than Greece if they wanted to.

Reasons why it is helpful to have your own currency before a crisis hits:

For information on how it could have come to such a loss of competitiveness in Greece I can recommend looking up the "Balassa-Samuelson effect", although this may not be the whole story.

Personally, I am not totally convinced by the conventional wisdom, especially popular in European media, that if Greece were to return now to the Drachma that things would be better. This is because their debts may then legally still be denominated in Euro and after the inevitable devaluation of the Drachma against the Euro (the whole point of having the Drachma now) would cause the debt burden to increase even further. For more on models where such effects are present see so-called "third generation currency crisis models".

However had they not had the Euro in the first place, like the "happy" European countries you are mentioning, then their debts would have likely never been denominated in Euro in the first place and could more easily pay that debt through a currency devluation / inflation.

Another reason countries are happy to not have had the Euro in the first place is because the Euro created interest rates for Greece's public debt that were too low than what they should have been. This means the government became over-indebted, which lead to the problems we see today. Had the interest rates been (correctly) higher the government would have perhaps borrowed less and not too much. This is to say that due to the Euro (and the false positie expectations caused by it) the "price signal" was not working so well. Other countries that never adopted the Euro did not experience such a loss of information from the price signal.


Equations of "rational mechanics" and general equilibrium

Pareto believed economics could be studied with the same mathematical rigor as physics, so it's quite possible he was speaking to the similarities between mathematical economics and classical mechanics (the latter of which was just coming into its own in the early 1900s).  I'm a physicist by training too, and those similarities helped me rapidly cover a great deal of ground my colleagues in economics had to learn fresh, so your intuition makes sense to me.


Why is Engel curve a straight ray through the origin if $D_wx$ = x(p, 1)$?

The function $x(p,1)$ is constant in $w$. If you integrate it wrt $w$, you end up with a straight line (wrt $w$), whose equation is ...


Insect farming as future of protein nutrition: reality check

Currently, the problem lies in the genetic selection to make insects suitable for mass processing.

Let's compare this to chicken farming

Human society already select chicken that will grow lots of meat for many years
There are machinery available to "process" the chicken
There are existing demands and supply chain.
Currently chicken is cheap

In industrial scale, it is all about ease of processing.  Currently, insects like crickets is not a good competing candidate. However, it is just a matter of time for the food scientist to discover a suitable insect.


Utility functions and positive monotone transformations

Note that
$$ln(x_1)+2*ln(x_2) = ln(x_1)+ln(x_2^2) = ln (x_1 * x_2^2),$$
and note that
$$MRS_v = \frac{g'(x_1 * x_2^2) x_1 * 2 x_2}{g'(x_1 * x_2^2)   x_2^2}$$
such that the derivative of $g$ cancels out.
For more intuition, see here.


Why do low-budget films charge the same amount at the box office as super-high budget films?

Opportunity Cost of the Seats

Once the movie is made the cost of production is sunk and irrelevant to the proper pricing of tickets. Only the marginal costs of serving an additional customer and the opportunity cost of showing a different film would enter into ticket pricing. Since cinemas should be setting the number of screens for each movie so that the opportunity costs of the seats are equated across films, this makes them want to charge the same for all films. In support of this idea, I offer that movies do vary in price by time of day. This is a rational response to the perishability of seats (once the movie starts an empty seat for that show is worthless) and time (of day) varying demand for film-going which varies the opportunity cost of the seats by time of day. The theater can't easily equate the opportunity cost across time the way it can across films.

At the Movies: The Economics of Exhibition Contracts (Filson, Switzer, and Besocke (2004)) provides the following explanation:

  Practitioners provide several explanations for inflexible ticket
  prices. Exhibitors want to avoid menu costs and eliminate consumer
  uncertainty about what the movie will cost. Exhibitors do not increase
  prices of hits because they are engaged in repeat business with local
  consumers, and the potential loss of goodwill from increased prices
  outweighs the potential gain. Charging different prices for different
  movies at multiplexes necessitates employing monitors to ensure that
  consumers see the movies they pay for. Even offering mid-week
  discounts may lead to more time shifting than new demand. Not all
  analysts or practitioners agree that inflexible prices are optimal
  (see Orbach and Einav 2001), although it seems unlikely that such an
  easy-to-exploit profit opportunity would persist. Some practioners
  have experimented with non-uniform prices in  the U.S. in the recent
  past but inflexible prices remain the norm.


How does a mortgage-backed security issued by a bank allows it to borrow cheaper and make more loans?

Here are two ways by which selling mortgage-backed securities (MBS) benefited banks:

The MBS is less costly (or equivalently, more valuable) in the hands of the investor than in the hands of the bank.
The investor is misinformed/misled about the MBS and overpays for it.

Analogy for #1: When you buy car/fire/health/whatever insurance, you transfer the risk to another party (the insurance company) for whom that same risk is less costly (for example because they are able to pool/diversify risk across large numbers). This is the traditional, orthodox "markets are good" view. In this view, MBS's were a good development that further and more efficiently spread out risks and benefited everyone.

Analogy for #2: You suspect you have cancer and go buy some expensive health insurance. So you transfer this toxic asset (or risk) to another party (the insurance company) who, not knowing what you know, incorrectly judges the magnitude of this risk. This is to a large extent what actually happened.

(Note though that #1 and #2 are not mutually exclusive possibilities. Some of both probably did occur and continue to occur. But unfortunately, leading to the financial crisis, more of #2 and less of #1 occurred than was ideal.)


Unemployment and Labor Force MacroEconomics

A short note: please try to show your math work in the original post, and maybe learn some basic Math TeX commands. It makes people much more willing to help with homework questions.

So first, we have our unemployment for the year 2000. Let $U_t, u_t, L_t$ denote the unemployment rate, number of unemployed, and number of laborers at time $t$, respectively.

$$U_t = \frac {u_t}{u_t + L_t}$$
$$U_{2000} = \frac {2,000,000}{2,000,000 + 16,000,000} = \frac{1}{9} \approx 0.111$$

So in the year 2001, 800,000 people turn 16, but half do not enter the labor force. So the question is how many people have to find a job for unemployment to fall below 8%.

$$U_{2001} = \frac{2,000,000 + (400,000 - n)}{2,000,000 + (400,000 - n) + 16,000,000 + n} = 0.08$$

where $n$ is the number of those who found jobs, and $400,000 - n$ is the net change in unemployment numbers. Solve for $n$.

$$\frac{2,400,000 - n}{18,400,000} = 0.08$$
$$\implies 1,472,000 = 2,400,000 - n \implies 928,000 = n$$

So what does this tell us? Even if all 400,000 new 16 year olds found jobs, that would not drive unemployment down to 8%. You need quite a few more people to find jobs.


Are really 90 % of all foreign exchange trades dealt in USD?

The actual rate for the most recent year(2016) we have data is 88%.  This comes from the Bank for International Settlements.

  Why does every electronic transaction of USD legally take place on American territory? What laws/processes are responsible?

This isn't true.  The US handles about 20% of all FX trades and the UK about 40%, most of which are dollar trades.  However, it is probably the case that most trades in dollars could be considered regulated by the United States.  This is a practical result of several sets of laws and not one specific law.

For example, the US asserts jurisdiction over the entire world for its citizens.  If an American engages in some action that is legal in the country they are standing in, but not legal in the United States, then it is a crime in the US.  If they commit a crime in another country it can also be enforced in the United States.  For corporations, if they do any business in or with the United States then the United States acquires legal and regulatory jurisdiction.  Since the banks which trade foreign currency generally have offices in the United States, they are subject to US jurisdiction.

Additionally, while there are several wire and exchange firms, they end up being required to follow US law and to interact with the Federal Reserve System if they have any nexus with America.  As such anything that is fraudulent, that uses any form of wire, becomes liable under US wire fraud laws.

The statement isn't probably literally true, but since people don't have any physical control over which networks the institutions use, it is safe to assume the US has jurisdiction over the transaction along with any other countries that are directly or incidentally involved.

  What about non-electronic transactions in USD?

Non-electronic transactions in paper USD do not create US jurisdiction unless one of the parties has a tie to the United States.  All American fall under US jurisdiction at all times, everywhere.  The only paper transactions which would have automatic US involvement would be in counterfeiting.  If a Bolivian and an Argentian are standing in London and one buys a hamburger from another for a paper dollar, the US has no standing if a crime were committed.


elasticity of income if price changes by 10 percent

Try work it out step by step and don't just concentrate on the math.

Suppose Arista has income of $w$, and the price of whatzits is $p$. She spends 10% of that on whatzits, so how many whatzits does she buy? Let's call this $q$.

Now suppose income is now $x \cdot w$ (so if her income increases by 3% then $x$ is 1.03.) The price of whatzits is still the same, $p$. How many whatzits does she buy now? Let's call this $q'$.

The income elasticity of [quantity] demand[ed] is simply the percentage change in quantity demanded divided by the percentage change in income:

$$e = \frac{\frac{q' - q}{q}}{\frac{xw - w}{w}}.$$


Can someone please explain how personal savings and fiscal deficit are related in this context?

The private sector is households and companies.  The public sector is the state and national governments.  When the author says "private investment" they presumably mean non-financial investment like homes and educations for household members or business processes and software and factories for companies.
The money the private sector wants to save can go into "private investment" or a loan to the government. The more the government wants to borrow the higher the rate of return required by lenders. That rate is the interest rate.
In actuality a household buying a government bond can also be described as making a "private investment" but then it would not make sense to suggest there is "not enough left over for private investment". In order to make a sensible interpretation it is necessary to believe the "private investment" the author intends is non-financial (like a home or factory).


Is Marginal Revenue not always equal to the price?

No, marginal revenue is not always equal to the price. It would be if marginal revenue were defined as only the amount paid for the last sold unit, which is equal to the price by definition. However, selling one more unit of a good might have  implications on the market price itself, thus impacting the revenue as a whole.

Consider a monopolist selling a quantity Q1 in a given period at a price of P1. It is a monopolist so it is selling the maximum amount demand can bear at this price. I.e. Qd(P1)=Q1.
Suppose he is now deciding whether or not to sell an additional unit and prices for all goods he sells have to be uniform (i.e. no price discrimination). Since at the current price P1 no additional unit can be sold, the price has to drop to P2 so that: Qd(P2)=Q1+1.

What is the marginal revenue of this decision? It is not equal to P2 since this would ignore the effects on the revenue overall. Instead, the effect is : MR=P2+(P2-P1)*Qd(P1). So on the one hand there is the price of the last unit sold, but on the other hand there is the effect on all other units sold because the price itself has changed.

You are right that a demand curve theoretically speaking is not a smooth function, and shouldn't have a slope but instead be some kind of step-function technically speaking. However if this step function for example implies that every increase of 1 in the price means a decrease of say 2 in quantity demanded, then nothing is lost in making it a linear function of the form Qd=a-2P.

So yes your last definition is correct.


An agent's expected utility depends only on mean and variance

In order to understand this problem, I will work through the generic case. Say that a user had generalized quadratic (Bernoulli) utility, similar to your problem:

$$u(x) = \beta x^2 + \gamma x$$

and suppose that there is a distribution for the outcome of $x$, denoted $F(x)$. Thus, utility over this distribution is equal to

$$\begin{align}
\int u(x) \text{d}F(x) & = \int (\beta x^2 + \gamma x) \text{d}F(x) \\
& = \beta \int x^2 \text{d}F(x) + \gamma \int x \text{d}F(x) \\
& = \beta \int x^2 \text{d}F(x) + \left[ - \beta \left(\int x \text{d}F(x)\right)^2 + \beta \left(\int x \text{d}F(x)\right)^2 \right] + \gamma \int x \text{d}F(x) \\
& = \left[\beta \int x^2 \text{d}F(x) - \beta \left(\int x \text{d}F(x)\right)^2 \right] + \beta \left(\int x \text{d}F(x)\right)^2  + \gamma \int x \text{d}F(x) \\
& = \beta\cdot(\text{variance of F(x)}) + \beta\cdot(\text{mean of F(x)})^2 + \gamma \cdot (\text{mean of F(x)})
\end{align}$$

So utility is determined by the mean and variance of the payoff distribution. If you understand the above work, then it should be easy to do the specific case you have given us. Try it for yourself.

Considering your specific case:

$$U(Y_s) = -\frac{1}{2}(\alpha - Y_s)^2$$

implies

$$U(L) = \sum_{s=1}^{S}\pi_s (-\frac{1}{2}(\alpha - Y_s)^2)$$

You can technically work with either finding $\int U(L) \text{d}L$ or $\int U(Y_s) \text{d}L$. Best of luck with your work.

Edit: Working in the discrete case will mean using expected values may prove helpful.


Finitely repeated Prisoner’s Dilemma with switching cost

A couple hints.

Regarding the lower bound on $\epsilon$: What happens if deviation occurs at stage $T$? In other words, there is no opportunity for your so-called "punishment stages".
Regarding the upper bound on $\epsilon$: Suppose player 2 deviates at stage $T-1$ but player 1 does not. What must be true about $\epsilon$ in order for player 1 to have an incentive to switch to $D$ at the very last stage?

In general, to show that the given strategy profile is subgame perfect, you need to argue that playing $C$ after $(C,C)$ and $D$ after either $(C,D),(D,C),(D,D)$ are optimal.


How does excessive consumption in an area decrease tax revenue?

Actually the text does not state that consumption leads to a downward trend in taxation, but actually the other way around--

According to the text, anti-growth ideology says that low taxes do indeed lead to development, mainly focusing on population increase, but that this population growth will strain local institutions, meaning public services like emergency response, sewage services, etc.

Greater fiscal expenditure would be required to expand these services to accommodate the increased population.

During its explanation of the anti-growth ideology the article is not referencing increasing consumption of private goods, such as at local businesses, but rather of public goods as explained above.


What effect would raising reserve requirements while printing money have?

During the global financial crisis of 2008-2009, both the US and UK undertook quantitative easing (QE) in conjunction with a policy to fortify banks against risk by increasing reserve requirements.  So there are recent cases which illustrate what happens.

A rise in reserve requirements constrains a bank's ability to lend.  So it seems to counteract the idea of QE.  So why did both the US and UK implement these two seemingly antithetical policies?

Because the two policies also accomplish other objectives.

An increase in capital requirements for banks was thought to be necessary to restore investor confidence in a badly shaken credit system (raising reserves was not done to counteract inflation).  Investors did not have confidence in the quality of assets on bank balance sheets, so requiring banks to retain cash deposits was one way of ensuring that banks had (a) a clear base of assets that were not obfuscated (e.g. subprime); and (b) limitations on how much risk (via lending) they could undertake.  So investors could feel safer about their deposits and the fundamental strength of the banking system.
Quantitative easing was aimed at injecting liquidity into an economy paralyzed by loss of confidence and liquidity, resulting in low spending and investment levels.  In order to offset the negative effects of raising capital reserves, the QE programs needed to be very large, which is why vast amounts of money were pumped into circulation in both countries.

The second part of your question is around follow-on economic effects.  Using the dynamics above you can infer the likely effects.

Raising capital requirements had the effect of exactly that.  It also had the effect of requiring larger volume of QE to counteract the drag on lending (in effect, money velocity or money multipler) cause by the higher reserve requirements.
Had QE not been sufficient to overcome the higher capital  reserves, you would have seen bank balance sheets swell while lending remained low....which is exactly what happened in the early to mid stages of QE while banks shored up their balance sheets before timidly starting to lend at risk again.