You made the right start by calculating the utiltiy at $p_1=p_2=1$.

When we change a price, two things happen. Firstly, one of the goods becomes relatively more expensive, so people substitute away from that good.

Secondly, since the total amount of goods someone can afford is lower when a price increases, it is as if their income went down.

To find the substitution effect, we need to shut down the second of these effects and focus on the first.

It sounds like you are after what is more properly known as the Hicksian substitution effect. To calculate that, we need to compensate the consumer for the aparent loss of income. In other words, we need to answer the following question: "Given $p_1=2$ and $p_2=1$, what income would be needed to achieve the same utility as before (i.e. $U=14$)?" Call the income level that answers this question $\widetilde{Y}$.

Then, all we need to do is calculate the optimal consumption bundle when $p_1=2$, $p_2=1$, and income is $\widetilde{Y}$. The substitution effect is the the difference in $q_1,q_2$ between this new bundle and the first one you calculated.

Hopefully, this is enough to help you figure out the solution.


Remember that the M in MSB standard for marginal. In other words, the marginal social benefit is the extra benefit we would get by increasing the activity by one unit and the marginal social cost is the extra cost we would get by increasing the activity by one unit.

Suppose, at the status quo, the MSB is 5 and the MSC is 3 (so MSB > MSC). That means if we increased the activity by one unit we would get 5 units of extra benefit and 3 units of extra cost, so the net change in welfare is $5-3=+2$. Thus, the status quo can't be optimal because we could increase welfare by adding an extra unit of the activity.

Suppose, at the status quo, the MSB is 3 and the MSC is 5 (so MSB < MSC). That means if we reduce the activity by one unit we would loose 3 units of benefit but save 5 units of cost, so the net change in welfare is $-3+5=+2$. Thus, the status quo can't be optimal because we could increase welfare by reducing the activity by one unit.

Only if MSB=MSC is it the case that we can't increase welfare by either increasing or reducing the activity.


Amplification Effects: Negative Denominator

$\beta$ appears to be a decision variable, at firm level. Then its value should be determined optimally, under some criterion. So what is the criterion, what is the objective function which should be optimized over the value of $\beta$?

It appears that we should maximize the value of the option $V(\beta)$

$$V(\beta) = \frac{-c + q(\theta)(1-2\beta)J}{\rho - q(\theta)\beta}$$

The first derivative is (I will dispense with $\theta$)

$$\frac {\partial V(\beta)}{\partial \beta} = \frac{-2qJ\cdot[\rho - q\beta]+q\cdot[-c + q(1-2\beta)J]}{[\rho - q\beta]^2}$$

$$\implies \frac {\partial V(\beta)}{\partial \beta} = \frac{-2qJ\rho -cq + q^2J}{[\rho - q\beta]^2} = \frac{q}{[\rho - q\beta]^2}\cdot (-2J\rho -c + qJ)$$

$$\implies {\rm sign}\frac {\partial V(\beta)}{\partial \beta} = {\rm sign}\big\{(q-2\rho)J-c\big\} \tag {1}$$

So $V(\beta)$ is monotonic in $\beta$. Therefore, depending on the values taken by $J,\rho, c, q$, it will be optimal to set $\beta$ equal to either $0$ or $1$.

Assume now that $\partial V(\beta)/\partial \beta >0$. This requires that

$$(q-2\rho)J-c > 0 \implies q > \frac {c}{J} + 2\rho \implies q > \rho \tag {2}$$

Then $\beta^* =1$ and

$$V^*|_{\beta =1} = \frac{-c - qJ}{\rho - q} = \frac{c + qJ}{q-\rho } > 0$$

given $(2)$.

In turn, assume that $\partial V(\beta)/\partial \beta <0$. This requires that

$$(q-2\rho)J-c < 0 \implies q < \frac {c}{J} + 2\rho  \tag {3}$$

Then $\beta^* =0$ and

$$V^*|_{\beta =0} = \frac{-c + qJ}{\rho} $$

To have $V^*|_{\beta =0} >0 $ we need

$$\frac {c}{J} < q < \frac {c}{J} + 2\rho \tag {4}$$

to respect also $(3)$. So here a necessary restriction on the parameters is imposed, in order to obtain a positive value for $V$.

Combining, if we impose the restrictions

$$q > \rho, \;\;\; q > c/J$$

we allow for either $\beta =1$ or $\beta =0$ (it will depend on the exact parameter values), while at the same time guaranteeing that $V>0$ in all cases.
This is not unusual in economic models -i.e. the need to restrict the parameters in such a way so as for the model to provide solutions that are economically meaningful. And we need $V$ to be positive, because otherwise the vacancy would not exist in the first place, as I understand the set up.


Why is capital often not included in New Keynesian models? Is there a reason other than modeling difficulty?

Capital is included in all the big estimated New Keynesian models (Smets-Wouters, Christiano-Eichenbaum-Evans), etc. But you're absolutely right that the stylized core NK model does not have capital - which is hard to defend on empirical grounds, since capital investment is a very important part of business cycle fluctuations and the response to monetary policy. Ultimately, the reason does basically boil down to the "modeling difficulties" that you mention.

First, there is an obvious way in which capital makes the NK model more complicated: at an absolute minimum, it introduces at least one additional backward-looking state variable $K$. In contrast, the two core equations (the intertemporal Euler equation and New Keynesian Phillips curve) of the ordinary log-linearized NK model are completely forward-looking. Adding $K$ to the mix eliminates this nice analytical feature. Still, on its own, this is not such a compelling reason to leave $K$ out of the standard presentation of the model, since the increase in complexity would still be tolerable and possibly justified by the added realism.

The additional complications that make capital much more difficult to include are the following.

Capital adjustment costs are needed to avoid absurd results. Suppose that there are no capital adjustment costs, and that firms rent capital each period on competitive markets. Suppose also that there are no shocks today. The real rental cost of capital today will be approximately $r+\delta$, where $r$ is the real interest rate that was expected yesterday and $\delta$ is depreciation. For each firm $i$, we have $MC(i)=(r+\delta)/MPK(i)$: real marginal cost equals the real cost of producing another unit of output by increasing capital.

Now, suppose for simplicity that firms are identical and there is no price dispersion. (Moving away from this, everything I say will still be approximately true.) Then we can eliminate the $i$ and just write $MC=(r+\delta)/MPK$. Furthermore, real marginal cost is just the inverse markup, so we can rewrite this in terms of markup $\mathcal{M}$ as $MPK=\mathcal{M}(r+\delta)$. Finally, if we assume that production is Cobb-Douglas with capital share $\alpha$ this becomes
$$\frac{K}{Y}=\frac{\alpha}{\mathcal{M}}\frac{1}{r+\delta}$$
Pick some reasonable parameters: say, $\alpha=0.3$, $\mathcal{M}=1.33$, $r=0.04$, and $\delta=0.06$. Given these we have $K/Y\approx 2.26$. Suppose that we exist in this steady-state world for a while, and then the Fed pushes down the projected path of interest rates such that the expected real rate declines to $r=0.02$. Then leaving $\mathcal{M}$ constant, the value of $K/Y$ given by the expression above increases to $K/Y\approx 2.82$. The NK model implies that this value should hold in the period after the shock.

This is an immense increase in $K/Y$, and the model tells us that it should happen in one period. If our period is a quarter, and $Y$ was not expected to take a sudden dive, then we'd have to invest well above the entire usual level of GDP to accomplish this. (In the continuous time limit, it becomes simply infeasible.) And although the assumption than $\mathcal{M}$ is constant is not right ($\mathcal{M}$ is determined endogenously in the NK model from the relationship of sticky prices to costs), relaxing this could easily make the puzzle more extreme: if $r=0.02$ is expected to prevail for a while, then it will imply higher-than-usual output $Y$ and lower-than-usual markups $\mathcal{M}$, which raise the implied $K$ even further.

The model simply doesn't work in this form: you need some form of capital adjustment costs. And these, of course, make the model still more complicated. (By the way, the problem here isn't so much the NK model as the fact that an assumption of no adjustment costs is generally absurd: seemingly small changes in the real interest rate must be accompanied by massive swings in the capital-output ratio, which we never see in practice. The NK model simply brings this absurdity, which is found in the basic RBC model as well, into sharper relief because exogenous interest rate shocks are such an important feature of the NK environment.)

Firm-specific capital is needed for strategic complementarity. Even if we fix the problem above by including capital adjustment costs of some form, we run into another awkward feature of NK models: taken alone, the Calvo price rigidity is not plausibly large enough to make the NKPC as flat as we think it is.

The most popular fix is some form of strategic complementarity, where firms try not to set prices too far from the aggregate price level. And the most popular way to get strategic complementarity is to assume that firms face both a high elasticity of demand and a steeply upward-sloping marginal cost curve. That way, for instance, any firm that sets its price too far below the average price will receive a flood of demand that causes its marginal cost to spike - and this discourages the firm from setting such a low price in the first place. (Yes, this sounds a little ridiculous, but it's how the models work.)

When the model excludes capital altogether, it's easy to just write a declining-returns-to-scale production function for each firm with labor as the only input. This makes each firm's marginal cost curve slope upward. But when we include both capital and labor in the model, the firm's production function probably should be much closer to constant-returns-to-scale. And this means, if the firm can rent any amount of capital from a competitive market on demand, that the firm's marginal cost curve is much closer to flat. This limits strategic complementarity.

To get around this, you need to dispense with the assumption of a common rental market for capital, and start talking about firm-specific capital accumulation. But then the model becomes much more complicated, and you're reduced to opaque quantitative exercises like ACEL.

Given all this, you can imagine how economists in insight-building mode often just dispense with capital altogether.


What are the effects of liquidity crisis?

Lack of liquidity depresses the output and can lead to recessions because it forces otherwise healthy firms to go bankrupt and it also undermines the financial infrastructure of an economy.

I think good way of understanding the problem is to explore the difference between illiquidity and insolvency. When firms go bust it is often mainly due to one of the two following reasons:

Firm is insolvent - when firm is insolvent it means that it is economically not viable. The insolvency means that the firm simply does not generate enough revenue and consequently also profit to meet its obligations.
Firm is illiquid  - when firm is illiquid it generates enough profit to meet its obligations but not enough cash. This is because the customers of the firm might not pay for the products on time (which during the crisis often happens - for example in the current corona virus crisis some people might pay their rents late - some countries even consider canceling them/postponing them). But this the hurts the ability of that firm to pay its employees suppliers and creditors in general. Consequently even though firms business might be economically healthy there might not be able to generate enough cash in short term to pay its own creditors.

(Note: I am using the words insolvent and illiquid in broad economic sense - in finance and accounting those words have little bit more narrow definitions)

You want to let insolvent firms to fail and be replaced with more efficient firms - its the part of Schumpeterian creative destruction. However, firms with liquidity problems are fundamentally healthy- especially when the iliquidity is caused by shock and is not chronic. Hence when illiquid firms fail it is more or less pointless loss. Unfortunately, in practice it is hard to distinguish which firm is actually insolvent and which illiquid and hence it is not possible to just have some regulation that would effectively shield illiquid firms from the bankruptcy (such regulations exists but unfortunately they often turn determining whether firm is insolvent or illiquid into legal battle that has less to do with economics). As a result most economists stress preventing liquidity crises in the first place.

Furthermore, in the above I was talking just about firms in general but liquidity is actually extremely important to the financial sector. Banks assets are essentially illiquid because they offer loans, and they cant just ask the people they lent money to repay it back before maturity. However, banks liabilities are liquid because they consist mostly of deposit accounts from which people can withdraw cash any time they want. Now most countries have deposit insurance - a regulation that helps prevent people frantically withdrawing cash due to fear of bank going bust (i.e. bank runs). However, this does not extent to other financial institutions and shadow banks which can experience runs by creditors who worry that they wont get their money back, and since financial sector is all interconnected falling of these institutions can spread to all other financial institutions.

Any problem affecting financial institutions will eventually hurt the real economy because financial intermediaries have first order impact on the economy, because many businesses relay on them for financing - especially startups, but they are important overall (see Levine, 2005 for discussion of that).

So to sum up a liquidity crisis often leads to recession and has negative impacts on economy because it damages the financial infrastructure and forces otherwise healthy businesses to go bankrupt.


How can power/electricity prices be negative?

Yes, customers who were exposed to day-ahead wholesale market prices were paid to use electricity.

It is a combination of several different factors that make the day-ahead wholesale electricity market special.

Firstly, very few electricity consumers participate in it directly. So the demand side is very illiquid, with very low short-run elasticity - and it quickly becomes perfectly inelastic.

Secondly, there's very little time-arbitrage. The market for electricity at 0500 and the market at 1500 are two separate markets: there are very few arbitrageurs who can buy at 0500 and sell at 1500. That's because until recently, there's been very low rewards for doing so (more on this, below).

Thirdly, there are several suppliers for whom reducing supply would incur expense: so, for market-clearing purposes, they have negative short-run marginal costs. There are two ways that can happen.

very inflexible conventional generators (usually nuclear) just can't turn their output up and down every hour or two, without incurring maintenance costs.
generators with contracts that enable them to sell all their output at a pre-specified positive price: they would need wholesale prices to go negative by the more than that amount, to make it worthwhile for them to reduce their output.

Germany now has such a high combination of those two kinds of generators, that there are times when their combined output is greater than domestic consumption, and that's when prices go negative.

Fourthly, although Germany shares a grid with many other continental European countries, the opportunities for spatial arbitrage are very limited, because total German generation is much much larger than the interconnection power capacity to the rest of the grid: so that interconnection capacity strictly limits the amount of energy that can be arbitraged with other countries within a given trading hour.

Finally, though it seems strange, these negative prices are a good thing. They're a temporary condition that will only last for a transition period, as electricity markets decarbonise. At the moment, we've got transmission networks and market structures that were designed for, and built around, excessively-polluting suppliers such as coal and gas. We have to stop that pollution - the cost of it way exceeds the benefits. So, the industry has to change structurally, and markets and networks will have to change too.

Coal- and gas-fuelled generators have been very helpful for in-day markets, because they can rapidly change the amount they supply. So the near-absence of participation from the consumption side hasn't been a problem at all, until recently - and why the rewards for arbitrageurs were low. But now, there is an increasing number of hours when those very flexible generators no longer dominate the market.

As we move to a grid that no longer has that supply-side responsiveness, we need responsiveness from new arbitrageurs, and from the demand-side. Period of negative pricing provide strong incentives for those new market participants to come forwards. And, as more and more of those new participants enter the market, the negative prices will become rarer and rarer.

So this isn't just Germany. It has happened, and will happen, elsewhere too.


Terminal Condition for American Put Option

I don't think this boundary condition has anything to do with the optimal exercise price and it should therefore hold for both European and American style options. It is simply a terminal condition which allows us to rewrite the PDE in closed-form.

The relationship you show above in which:

  $\mathop {\lim }\limits_{t \to T} V(S,t) = \max \left\{ {X - S,0} \right\}$

$\to$

  $\mathop {\lim }\limits_{\tau  \to 0} V(S,\tau) = 0$

simply means that the extrinsic value of the option (i.e., the value in excess of $X-S> 0, \forall S<X)$ tends towards $0$ as $t \to T$.

Dropping the "$max$" function is simply a restatement of the payoff condition in terms of the Heaviside function. The Heaviside function is essentially equivalent to the maximum function except that it enforces the following boundary at $t=T$:

${\displaystyle V(S,\,\tau)=0\quad \forall \;\;S<X}$,

Though the difference seems subtle, it is important since $V_t$ need not be finite at S = 0, or even defined for that matter, which allows us to more easily perform the substitution of variables required to express the contingent pay-off in terms of the (heat) diffusion equation which is the general solution to the Black-Scholes model.

I hope that the intuition of this boundary condition is clearer from this explanation.


How much leverage has the Federal Reserve taken on today (Jan 2016)?

Page 4 of this document has the balance sheet as of November 2015: http://www.federalreserve.gov/monetarypolicy/files/quarterly_balance_sheet_developments_report_201511.pdf

Their assets are 4489 liabilities are 4431. Thus Equity is 58, L/E = 76 (approx).

It would not matter even if Fed had infinite leverage ratio. However, as far as I know congress has agreed to keep Fed solvent (so the ratio can't go negative). Although, the institution is not even solvent, as some of it's assets are marked mark to model, thus having much lower fair value than presented in the balance sheet.

The balance sheet reflects created money (growth in base money) and not debt. Created money generates inflation, but significant amount of bank credit was destroyed which also constituted money people transacted with. Thus the newly created Fed money has just replaced the bank credit and there has been no significant inflation, as the M2 money supply has only raised slightly. Some of the currency has also been exported overseas as USA continues to have a balance of payment / trade deficits.

Based on the quote, it seems the author has confused two issues. If you want to investigate the debt and liabilities of the government, you need to look at the government debt and it's other liabilities (not merely the Fed). He is correct that the debt is at a quite high historical level (only during world war 2 was it higher), while other liabilities (such as social security and medicare liabilities) I believe are at an unprecedented historical level.


Why didn't the money printing by the US Federal Reserve since 2008 lead to inflation?

Because you can't push on a piece of string.

As in the Great Depression of the 1930s, we've collectively got massive personal debt, depressing demand, moving us towards deflation, which pushes up debt, and so it goes.

We've also known since the 1930s that monetary expansion in and of itself cannot stimulate demand under these conditions: it can only create space for demand to expand into.

Add in the international fashion for fiscal contraction, a demand-suppressant, and that's how we get QE without broad-money expansion and without inflation.

We do have potential inflation. And at some point that potential will be realised. But not while there's this huge overhang of personal debt. It might get worn down over a long period of deflation, through defaults. Some other inflationary shock might inflate some of the debt away. And enlightened policy-makers might engage in a large exercise in personal-debt cancellation.


If you believe the efficient market hypothesis, then the answer is no;

At the moment relevant information (say, which allow the company to be added to the index reveal, the stock price would automatically reflect the price as if it was already added to the index, as otherwise there would be arbitrage closable by buying the stock at this moment and selling it when the ETFs are getting in.

In practice, I've tried it a few times myself but it seems that in general the market already takes future ETF investments into pricing quite well.


Balance of payments and the free floating exchange rate system

The Balance of Payments are a form of double-entry bookkeeping and so in theory should always balance overall

If official reserves do not change because the currency is floating freely and the Central Bank or Treasury is not intervenening, then a country's current account balance should be offset exactly by the financial/capital account balance.  But measuring this is difficult, which is why you often see a "statistical discrepancy", for example in row 100 of page 10 of the US Balance of Payments

But the individual flows matter, and it might then be wrong to say "the BOP position is irrelevant", at least in the medium to long term. A country with a persistent current account deficit will need to see net inward financial flows, which may be desirable if it leads to increased real investment and later to increased output, but may be less desirable if it merely leads to greater indebtedness and the need to pay for that debt

Meanwhile a country with a persistent current account surplus is effectively continually exporting more than it needs to pay for imports, so reducing its consumption and standard of living; this may make sense if it later intends to increase consumption with the income from its investments abroad, but otherwise is unnecessary and potentially wasteful and can lead to a failure of these unbalanced investments, as happened with German banks' speculative assets in the USA during the financial crisis


Perfect complement preferences in an exchange economy

Set of Pareto Efficient Allocations consists of feasible allocations $((x_J, y_J), (x_D, y_D))$ satisfying $y_J=\displaystyle\frac{x_J}{2}$.

Competitive Equilibrium is the price $(p_x, p_y=1)$ satisfying the following conditions:

Budget Requirement: $p_xx_J+ y_J = 2p_x + 2$ and $p_xx_D+ y_D = 4p_x + 1$
Equilibrium Conditions: $\displaystyle\frac{2y_J}{x_J} = p_x$ and $y_D=\displaystyle\frac{x_D}{2}$

Solving the system of equations we get equilibrium price vector as:
$(p_x, p_y) = (1,1)$
and the equilibrium allocation is
$((x_J, y_J), (x_D, y_D)) = \displaystyle \left(\left(\frac{8}{3}, \frac{4}{3}\right), \left(\frac{10}{3}, \frac{5}{3}\right)\right)$


What type of an income distribution would allow the fastest growth and/or highest sustainability

I would like to get some help figuring out the basic concepts and definitions in order to address these type of questions. Namely, how to distribute income among population in the best way for the collective.

There is actually no single best way of distributing income for the 'collective'. This is because what is best depends on normative judgement. What is best for utilitarian might not be best for Ralwsian or libertarian.
This being said there are actually ways of how to study what an optimal (re)distribution and taxation should look like depending on income distribution. To be more specific these models try to answer the question that given what your moral/political beliefs are how you can redistribute income in most optimal way.
If you are looking for general models then Mirrlees (1971), Diamond (1998) and Saez (2001) are good place to start. These papers consider optimal income taxation in static case but they are probably good starting point, even though due to being static models they are  not really good at answering growth related question but given that the dynamic models of taxation are infinitely more complex you would want to probably start here. A simplified version of the optimal income tax formula is given:
$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_{n0})}$$,
with $b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $.
This formula tells you how the marginal tax rates $\frac{T'(z_n)}{1-T'(z_n)} $ that govern both how much people are taxed and how much redistribution transfers people receive (since a transfer is just negative tax) will depend on the efficiency parameter $\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$  which is given by elasticity of labor supply to income taxes and which tells you how much taxation 'hurts' the economy.
The second part $\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b_n$  which depends on both utility of consumers and the societal utility function (i.e. is our society Ralwsian, Utilitarian, Libertarian ...?- depending on different philosophy people are assigned different welfare weights - which you can interpret as telling you how much society  ‘values’ given person e.g. under Ralwsian min-max principle poorest person gets weight 1 and anyone else 0). Furthermore $f(z)$ and $F(z)$ are the density and cumulative distribution function of the income distribution respectively.
Finally $\frac{1-F(z_n)}{z_nf(z_{n0})}$ is the part that captures the relative magnitude of distortions created by this taxation. It gives us indication about how 'broad' tax base is at given point in income distribution and on margin how many people are affected.
Again if growth is an integral part of your question you will have to go beyond static models into dynamic ones (see short literature review on those in Stantcheva 2020) but I recommend starting with the static case mentioned above.
Furthermore, if by sustainability you mean environmental sustainability that is a question that is actually, generally speaking, across economic literature divorced from any distributional question. The reason for this is that the main point of government intervention in this area is to solve some environmental issue such as $CO_2$ over-pollution due to lack of private property for clean air and these problems usually exist regardless of any income distribution considerations. Usually these sustainability issues involve Pigouvian taxes or subsidies so that is a key word you can use in search for those kind of models.
There are also few papers that look at interplay between Pigouvian taxes and redistribution systems. On of the best papers I read on this issue is probably Jacobs and  De Mooij (2015) and this issue was also explored by Lange, & Requate (2000), but this being said there are not that many papers on this topic especially when you want to also add growth into the mix.

Per request in comments I am adding also my response comments to question if Pareto distribution is good for the economy.
The answer depends on how do you exactly define good and in what context. For example, empirically most ex ante (ex ante redistribution that is) income distributions around the world can be characterized as a log-normal distribution with Pareto tail. This is almost universally true. Now whether that’s good or bad depends on definition of what is good, for a Ralwsian such distribution is not good as a Ralwsian would in ideal world where there are no costs to redistribution prefer uniform distribution. This being said since in real life redistribution generally introduces inefficiency we can’t just choose arbitrary distribution of income we want. When you plug some real life distribution in the optimal redistribution models usually the ex-post distribution of income does not have different shape altogether rather it’s compressed or stretch here and there for example a Ralwsian social welfare function would stretch the log-normal distribution with Pareto tail at its left tail and compress it at the right Pareto tail but the shape does not change drastically. For other commonly used welfare functions same would hold but I suppose there could be some exotic one for which it would not hold.


Practice question on Correspondences and maximization

I think the easiest way is to notice that since the problem is increasing in both arguments, we can assume $px_1 + x_2 = 1$ the budget binds (at least for $p \not =0$).
Substituting our constraint into the objective function, we have:
$$
\max_{x_1} x_1 + 5(1 - p x_1) = x_1(1 - 5p) + 5
$$
If $1 - 5p<0$, we choose the smallest possible $x_1$, so $x_1 = 0$ and thus $x_2 = 1$.
If $1 - 5p = 0$, any $x_1$ would maximise the above.
If $1 - 5 p > 0$, you choose the biggest $x_1$ possible, so $x_2 = 0$ and $x_1 = \frac{1}{p}$.
If $p = 0$, there is no constraint on $x_1$, and your objective function is unbounded in $x_1$, so you would choose $x_1 = \infty$, so no solution exists.


Prove the sample variance is an unbiased estimator

For a shorter proof, here are a few things we need to know before we start:

$X_1, X_2 , ..., X_n$ are independent observations from a population with mean $\mu$ and variance $\sigma^{2}$

$\mathbb E(X_i) = \mu$ , $\mathbb{Var}(X_i)= \sigma^{2}$

$\mathbb E(X^2) = \sigma^{2} + \mu^{2}$

$\mathbb{Var}(X)=\mathbb E(X^2)-\mathbb [E(X)]^2$

$\mathbb E(\bar{X}^2) = \frac{\sigma^2}{n} + \mu^2$

Let's try to show that $\mathbb E(s^2)= \mathbb E\left(\frac{\sum_{i=1}^n (X_i - \bar X)^2}{n-1}\right) = \sigma^{2}$

To make my life easier, I will omit the limits of summation from now onwards, but let it be known that we are always summing from $1$ to $n$.

$\mathbb E\left(\sum (X_i - \bar X)^2 \right) = \mathbb E\left(\sum X_{i}^2 - 2 \bar X \sum X_i + n \bar X^2 \right) = \sum \mathbb E(X_{i}^2) - \mathbb E\left(n \bar X^2 \right)$

$\sum \mathbb E(X_{i}^2) - \mathbb E\left(n \bar X^2 \right) = \sum \mathbb E(X_{i}^2) - n \mathbb E\left(\bar X^2\right) = n \sigma^2 + n \mu^2 - \sigma^2 -n \mu^2$

This simplifies to $(n-1) \sigma^2$

So far, we have shown that $\mathbb E\left(\sum (X_i - \bar X)^2 \right) = (n-1)\sigma^2$

$\mathbb E(s^2)= \mathbb E\left(\frac{\sum (X_i - \bar X)^2}{n-1}\right) = \frac{1}{n-1} \mathbb E\left(\sum (X_i - \bar X)^2 \right)$

$\mathbb E(s^2) = \frac {(n-1)\sigma^2}{n-1} = \sigma^2$

We have now shown that the sample variance is an unbiased estimator of the population variance.


Why is it that when a bank provides loans to people, it is considered a 'liability' on the bank's balance sheet? (Khan Academy Video-Related)

A checking account is considered a liability because bank essentially owes the amount on the checking account to the owner of the account.

If a bank loans someone money that is considered an asset on the banks balance sheet but if that person decides to open checking account at the same bank checking account becomes liability. The loan will still be recorded as an asset on balance sheet but the new checking account is still considered a liability because the money in it belong  to the borrower.

Also purpose of accounting is to track all asset and liability and equity changes. Even if a company would borrow money to company it itself owns  - which is more or less economically just juggling money around, it needs to be recorded in accounting.


Why doesn't anyone talk about GNW (Gross National Wealth)?

Historically, the distinction between GDP and GNW goes back (at least) to Adam Smith. Before Smith, it was quite common to measure a country's economic power by its wealth. For example, this was very characteristic of the Mercantilistic school in the 17th century. Following the motto "you are what you have", mercantilists suggested that countries should try to accumulate as much wealth (for them: gold) as possible. This was to be achieved by a combination of an export oriented economy and political measures to limit imports such as tariffs. Selling goods abroad and keeping one's currency (again, gold) they hoped would result in having more money and thus being more wealthy. This idea is reasonable for the middleages where the difference between having and not having money can decide about which country is able to hire more mercenaries and has the potential to decide wars.

For more modern societies a country's wealth (read assets) becomes less important.  Adam Smith was the first to realize that the "Wealth of Nations" is not so much about how much money they have but rather about how much economic welfare they can generate. This is measured in the number of goods and services produced in a given year, nowadays known as GDP.

From today's perspective, why do we use GDP rather than GNW:

Actual production is much more important to determine a country's productivity than its wealth
Accumulated wealth (GNP) measures past wealth while this years production of goods and services (GDP) is a good indicator of future wealth
There is probably also some path dependency. Given that hardly anyone ever hears about GNP, I agree with you that (despite all arguments against it) GNP is slightly underrated.


The relationship between profit and wages

At the end of the day, currency (such as $) is just a means of purchasing real goods and services in an economy. Roughly, utility corresponds to consumption. Since you assume the capitalist isn't allowed to consume, there is nothing for the capitalist to gain even if they capture all the currency.

Here is a slightly modified thought experiment.

Suppose there is one employer and one employee in an economy. Both are consumers. There is only one good: tomatoes. In other words, every credit is spent on purchasing tomatoes, and the real GDP of this economy is the number of tomatoes produced. If the employee earns less wages, then the employer has a larger slice of the revenue, and the employer now can purchase more tomatoes.

Now what if the employer tries to pay the employee almost nothing? The employee might just quit: no tomatoes are produced, and everyone is worse off as no one gets tomatoes. Even if the employee doesn't quit, at some point the employee will be less productive because they are starving from not being able to consume tomatoes. In other words, for the employer that wants to consume as many tomatoes as possible, there is a balance between taking as large a share from the existing pie of tomatoes and increasing the pie by allowing the employee to have enough to be productive.


Constant Elasticity of Substitution: Special Cases

We know that if $u$ represents $\succeq$ on $X$, then for any strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, then $v(x) = f(u(x))$ represents $\succeq$ on $X$

($X$ in this case is $\mathbb{R^n}$)

Consider $v(x, \rho) = \ln(u(x, \rho)) - \frac{\ln\left[\sum^n_{i=1}\alpha_i \right]}{\rho}$, which is strictly increasing.

$$v(x, \rho) = \frac{\ln\left[\sum^n_{i=1} \alpha_i x^\rho_i \right]}{\rho} - \frac{\ln\left[\sum^n_{i=1}\alpha_i \right]}{\rho} =
\frac{\ln\left[\sum^n_{i=1} \alpha_i x^\rho_i \right] - \left [\ln\sum^n_{i=1}\alpha_i \right]}{\rho}$$

The limit of this as $\rho \rightarrow 0$ is indeterminate, $\frac{0}{0}$. So we can use L'Hopital's Rule and take the derivative with respect to $\rho$ of the numerator and denominator.

$$\lim_{\rho \rightarrow 0} \frac{\ln\left[\sum^n_{i=1} \alpha_i x^\rho_i \right] - \left [\ln\sum^n_{i=1}\alpha_i \right]}{\rho} =
\lim_{\rho \rightarrow 0} \frac{1}{\sum^n_{i=1} \alpha_i x_i^\rho} \cdot \left(\sum^n_{i=1} \alpha_i x_i^\rho \ln x_i\right)$$

by the Chain Rule.

$$= \lim_{\rho \rightarrow 0} \frac{\sum^n_{i=1} \alpha_i x_i^\rho \ln x_i}{\sum^n_{i=1} \alpha_i x_i^\rho} = \frac{\sum^n_{i=1} \alpha_i \ln x_i}{\sum^n_{i=1} \alpha_i} = \frac{1}{\sum^n_{i=1} \alpha_i} \cdot \ln\left(\prod^n_{i=1} x_i^{\alpha_i}\right)$$

Consider $w(x, \rho) = \mathrm{e}^{(\sum^n_{i=1} \alpha_i) \cdot v(x, \rho)}$, which is another monotonic transformation, strictly increasing. So $w$ still represents the same preference as $u$.

$$\lim_{\rho \rightarrow 0} w(x, \rho) = \mathrm{e}^{(\sum^n_{i=1} \alpha_i) \cdot \lim_{\rho \rightarrow 0} v(x, \rho)} = \prod^n_{i=1} x_i^{\alpha_i}$$

which is a Cobb-Douglas function.

$\square$

To show the second point, it is sufficient to show that

$$\lim_{\rho \rightarrow -\infty} u(x) = \left\{x_k \ \forall j \neq k \mid x_j \geq x_k \right\}$$

$$u(x) = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho} = x_k \left[(\sum^n_{i=1, i \neq k} \alpha_i x^\rho_i) + \alpha_k \right]^\frac{1}{\rho}$$

$(\frac{x_j}{x_k})^\rho \rightarrow 0$ as $\rho \rightarrow -\infty$ if $x_j > x_k$

$(\frac{x_j}{x_k})^\rho \rightarrow 1$ as $\rho \rightarrow -\infty$ if $x_j = x_k$

So

$$\lim_{\rho \rightarrow -\infty} x_k \left[(\sum^n_{i=1, i \neq k} \alpha_i x^\rho_i) + \alpha_k \right]^\frac{1}{\rho} = x_k$$

since $1/\rho \rightarrow 0$ and a constant to the zeroth power is 1.

Construct a similar argument for any $k$. Thus $\lim_{\rho \rightarrow -\infty} u(x) = \min \left\{x_1,...,x_n \right\} $

$\square$


Why did Greece not solve its debt problem by selling treasury bond to its own people?

there are a few reasons:

Directly selling bonds isn't a common method to get money from your people. Raising taxes works more effectively. A promise of future spending would be roughly equivalent to selling bonds.
why not sell bonds to everybody? not just the Greeks? To a large extent, this has already been done. You don't get to 158% Dept to GDP without a lot of borrowing. Borrowing to pay interest is common. It isn't a long term solution.

The Greeks have been negotiating the terms of the bailout for many months now. Liquidating assets would have been considered and I expect will happen as part of these new measures.


There is a good Planet Money episode on ticket scalping; I recommend it.

The reason for banning ticket scalping has nothing to do with economic harm, and everything to do with making the arts (or sports, whatever) accessible to people of more-modest means. Consider the fact that artists could, if they wanted, just auction off all the seats to their shows, capturing all the surplus and putting scalpers out of business.*

If artists did this, however, lots of people wouldn't be able to afford to see popular artists perform. Many artists (the Planet Money episode uses Kid Rock as an example) want to try to make sure that their fans have a reasonable shot at attending, which means that they want to price the tickets low... but the only way this can work is if resale is illegal.

It's been suggested that artists (again, the same goes for sports teams) might benefit from making low-priced tickets available to fans for two reasons: because they plan on selling stuff to people once they're inside, and because an artist-fan relationship is hopefully not just a one-shot transaction. The idea is that fans who can attend the occasional show might be more likely to buy recordings, merchandise, and tickets in the future, providing the sort of long-running support that can keep an artist viable over many years.

  [P]erformers who undercharge their fans can paradoxically reap higher profits than those who maximize each ticket price. It’s a strategy similar to the one employed by ventures like casinos and cruise ships, which take a hit on admission prices but make their money once the customers are inside. Concert promoters can overcharge on everything from beer sales to T-shirts, and the benefits of low-priced tickets can accrue significantly over the years as loyal fans return.

*For the most part— there's some amount of resale that is sort of frictional, which arises from people not being able to attend a show and unloading tickets so they don't just eat the full cost, etc.


What conditions characterise "cheap-talk" games

Your general question is in my opinion far more interesting than the course problem, so I will answer that.

What conditions characterise “cheap-talk” games?

As you have noted yourself, the talk needs to be cheap, meaning signals/messages should have no direct effect on the players' utility. Indirect effects are allowed. Even in cheap talk games it is possible that the beliefs of a player are updated by a cheap message, hence his response changes altering the outcome. Simple examples include coordination problems.
A second necessary condition is that all messages are available to all types. hence there are no type specific message spaces $M_i$, only a shared message space $M$ from which all types of the sender can freely choose their message. Without this, a signal, though costless, could carry information, as certain types are incapable of sending it.

It's a cheap-talk game, so what?

If these two conditions are met then the game is a cheap-talk game. Why is that important?

Let us call a version of the cheap-talk game where we leave out the signal round the simple variant. For any equilibrium of the simple variant there is a payoff equivalent equilibrium of the cheap-talk game. Similarly, for any
equilibrium of the cheap-talk game there is a payoff equivalent equilibrium of the simple variant. Hence it seems that the ability to signal did not fundamentally alter the set of equilibrium outcomes. The reason for this is that the signal did not carry any information, as it was easily imitable by other types, it was 'cheap'.


A strategy is rationalizable if it survives an iterated elimination of strictly dominated strategies. That is first you eliminate all strategies that are never a best response independent what action the other player chooses. Then you only consider the strategies that survive and repeat this step: eliminate all strategies that are never a best response independent what action from the set of surviving strategies the other player chooses.
The set of rationalizable strategies is then a set that only contains best responses to some rationalizable strategy. Therefore this is a larger set than Nash equilibria, which are best responses against each other.


How do I calculate quantity to minimize long-run average total cost?

Your are right. You have to minimize the average cost.

$$c(Q)=\frac{C(Q)}{Q}=6000 +40Q+Q^2$$

Calculate the first derivative and set it equal to zero:

$ c'(Q)=40+2Q=0 $

Solve this equation for $Q$. Denote the optimal value as $Q^*$. $Q^*$ can be a local maximum or a local minimum

If $c''(Q^*)>0$, then you have found the local minimum.

The local minimum is also the absolute minimum, because

$$\lim_{Q \to \infty } 6000 +40Q+Q^2=\infty$$

$$\lim_{Q \to -\infty } 6000 +40Q+Q^2=\infty$$


What is the rate of new house construction in Auckland, New Zealand?

If you're really serious about researching this data, I'd suggest contacting the Real Estate Institution of New Zealand: https://www.reinz.co.nz/reinz/public/reinz-statistics/reinz-statistics_home.cfm

As long as there are reasons to move to Auckland (more work etc.) and the supplied housing is not high enough to match the demand, the prices will continue to rise. This effect is visible in many major cities where the supply of housing is (for some reason like goverment rules or slow building) inelastic.


Why do increase in firms profit margins cause cost push inflation?

You can think on prices like $p = (1+m)w$, which means that prices are the value of inputs ($w$) multiplied by the margin ($1+m$) of the firms. So it is clear that raising firms margin would increase prices. This is cost-push inflation because the margin are part of the firm´s costs, it is a payment to a factor, like capital or entrepreneurship.

A raise in money supply don´t cause cost-push inflation directly, but causes demand-push inflation, which can increase also increase the prices of productive factors and inputs to production (w) as well, if firms can pass this increases in their cost to produce to their price, then you have a channel of cost-push inflation.


How much would the EU budget reduce by after Brexit?

Part A: how much does the UK contributes to the EU Budget

This depends on whether you use fiscal year or calendar year estimates. The figures for the 2015/2016 fiscal year are here. The key is the following graph:

The calendar year estimates are here. The following Table summarizes the data:

Naturally, the net contribution is what matters for analysis the case of "EU budget without UK". However, as both report point out, the UK receives other contribution, which come from an "allocation of funds via competition". Think of the EU having a fund for, say, improvement in infrastructure. Countries apply to this fund, and some countries get it. Well, as the first report says, the UK got £1bn extra through that mechanism in the 2015/16 year. Should this be counted as part of the exercise? Probably not, because those extra funds would have been allocated to another country anyway.

So, given that the total budget of the EU in 2015 was €141,280 million, and taking an average EUR/GBP of 0.73, the percentage of contribution of UK to EU budget in 2015 was:

$$ \frac{8,473}{141,280 \times 0.73} = 8.2\% $$

Regarding 2017, table in page 11 shows only the gross payments, including the rebate, but not the public rector receipts. As such, the effective contribution of the UK is not going to be 12.5% as the document notes, but lower. Given historical levels of receipts, you could estimate that the contribution of the UK to the EU budget would be perhaps around 9%. In any case, as far as I know, this cannot be known ex-ante, reason why it is not included in the official budget document.

Part B: how would the EU budget change after Brexit

Since this relates to a future event, it is impossible to tell. This is for several reasons:

Budget are negotiated annually among all members, and contributions vary accordingly. As such, if countries expect less revenues, they might either lower expenditures, or increase demand for contributions to other countries. For instance, immediately after the Brexit vote, speculation arose that Germany would cover any hole left by the UK.
An important component of budget (31% in 2014) does not come from individual countries contributions. As such, how these evolve affects particularly how much countries contribute.
Brexit might induce an adjustment on the so-called correction mechanisms.
Countries contribution vary over time depending on many national circumstances, including the size of their economy, which then depends on economic growth, uneven across the EU.
The UK is expected to pay the EU before it leaves, known as a "divorce" or "exit" bill, which depends on the outcome of negotiations.
It depends on what deal the UK and EU have after Brexit. A Norway style of deal means the UK would have to pay to the EU.

In conclusion, it is simply impossible to tell. The calculations you make are a simple ceteris paribus exercise of removing the UK from the _current EU budget. That might be illustrative, but given the huge uncertainty on the issue, it's probably a rather brute calculation.


Yes. See for example this passage from Choice under Uncertainty by Jonathan Levin:

2.1  Prizes and Lotteries
The starting point for the model is a set $X$ of possible prizes or consequences. In many economic problems (and for much of this class), $X$ will be a set of monetary payoffs. But it need not be. If we are considering who will win Big Game this year, the set of consequences might be:
$$X=\lbrace\text{Stanford wins}, \text{Cal wins}, \text{Tie}\rbrace.$$
We represent an uncertain prospect as a lottery or probability distribution over the prize space. For instance, the prospect that Stanford and Cal are equally likely to win Big Game can be written as $p=(1/2,1/2,0)$. Perhaps these probabilities depend on who Stanford starts at quarterback. In that case, there might be two prospects: $p$ or $p'=(5/9,3/9,1/9)$, depending on who starts.

(not sure what I was thinking yesterday evening. It seems like I completely misunderstood the question)


Can the US dollar collapse and what are steps that I should take in the event that this happens?

The reason the US dollar is not destined to collapse anytime soon is that the United States has the most powerful military in the world, and one of the strongest economies.

To put it simply, we produce sufficient food and clean water to feed our people, and we have sufficient housing and infrastructure to ensure that basic needs are met for the vast majority of our citizens. That isn't likely to change in the near future. This means that our currency in a way (though not directly) is backed by those resources (food, water, shelter, electricity, etc.) despite not being tied directly to an indicator like gold. This being the case the only other possible downfall of our currency would be theft or extortion by foreign powers that would deprive us of those resources. As I first noted, our military ensures that this will not happen.

Often, a point of concern for many people is our debt. We have $20 Trillion in debt at the moment. However, just like you wouldn't go try to collect the debt owed to you by "El Chapo," Al Capone, John Gotti, or other violent groups with more guns than you, other countries are deterred from attempts at aggressive collections on the US debt. This is where the intertwining of our economies becomes important. With those economic relationships in place, these countries ultimately get things of value from the United States. Why would they cede those things to the detriment of both the US and that country unless they had an extraordinary reason? This paradox helps to ensure the stability of the currency as well.

That said, if it's still something you're worried about gold and bitcoin are both good investments, as are foreign currencies. A fundamental principle in investments is diversification. The more you spread your money into reasonable investments the less likely it becomes that all of your investments will fail. Thus you will always have something to fall back on.


Conforming mortgage loan limits by county and year

The high-cost area conforming loan limits within the contiguous United States were created by HERA (the Housing and Economic Recovery Act of 2008) and therefore do not exist prior to 2008. For periods prior to the enactment of HERA, just use the historical nationwide conforming limits. There was also a temporary bump to the conforming limits enacted a few months before the passage of HERA, in the Economic Stimulus Act of 2008.

The high-cost areas that you identify (Alaska, Hawaii, Virgin Islands since 1992, and Guam since 2001) had, by statute, high-cost area loan limits that were capped at, and as a result effectively always, 1.5 times the conforming loan limits for the contiguous US states. Somewhat amusingly, the legal text responsible for the cap at 1.5x is actually still in the law; the authority granted to the Director of the FHFA to modify conforming loan limits was just tacked on to the end. You can find this text at 12 USC § 1717(b)(2)(C).


How might the Swiss currency floor be unwound?

Central banks have a lot of leeway from a balance sheet perspective to do these things. They can simply stop the practice of buying foreign currency when the CHF appreciates which would let it appreciate beyond 1.2. If it keeps appreciating they will book losses, but unless they are enormous, these typically don't make a dent in a central bank's balance sheet.

If it depreciates, they can use the foreign currency to buy CHF, or not. If they do not use it, then they book a profit, as the comments suggest.

In many cases, the legal implication of large loses or gains in the CB's balance sheet is a transfer to or from the country's treasury. Here's a recent example from Mexico Bloomberg Article


What are the general charateristics of matching markets that make then different from regular markets?

The deviance from a "normal" market boils down to how that market clears. Matching markets are markets in which prices alone don't clear the market, as thought in mainstream economic theory, and so the market is clear using the help of other institutions.

Take it from Alvin Roth's informative EconTalk interview:

"...labor markets are like these--they are among the markets that I've studied. And in labor markets, as in other matching markets, you can't just choose what you want. You also have to be chosen. So, labor markets have all these other institutions about applications and interviews and offers and acceptances. And sometimes those are used for competitive purposes in various ways. So, for example, right now if you knew someone who was just graduating this month from a prestigious law school and was going to take his first job as a clerk for an appellate judge, there's a good chance he would have gotten that job two years ago, when he was a beginning second-year law student. And he would have gotten that job with an exploding offer--which meant that he had to say yes or no when the offer came, before he could find out what other offers might be coming...And so that's a market that doesn't work the way we normally think a well-functioning market should work, where you can consider lots of alternatives. But it's one of the ways that judges compete with each other. And that's a market where the judges can't compete on price because Congress sets the wage for clerks. But it turns out that other markets work that way, too. When private equity firms are trying to hire young investment bankers, they do the very same thing. When law firms which pay salaries very freely, try to hire new associates, they do the same thing--often with competition through summer associateships while people are in law school. So that new lawyers in general will often, as they go to their jobs after finishing three years of law school, have been committed to those jobs for years."

In some other instances, consider trying to work for Stackexchange, where you can't just choose to work here: you have to be hired. Also this company doesn't actually try to lower it's price in order to clear the market like with commodities, but it will set a wage wide enough to be attractive to many people, in which it gets the opportunity to choose whomever it believes to be the best fit for their company. Or if we want to reverse the situation where you're a well sought after worker (perhaps a sports player, or a consultant) firms will express their strong interest in hiring you, but you can't just be hired because you have the opportunity to pick which of the employers you want to work for. I hope this helps.

Oh, and Consider Roth's book on the subject matter. Cheers.


Is egyptology a pyramid scheme: Self-perpetuating university programs

Partial and tangential answer, but no other answers so far, so here goes:
For the first part of the definition (people who complete the university program do not find placement in the field), data from the Survey of Earned Doctorates may be useful, particularly "Postgraduation plans (e.g., work, postdoc, other study or training) - Type and location of employer". Papers that use this dataset may compare by field.
The "no benefit" part could be studied by regression discontinuity around the admission cutoff, but the small sample size of fields like egyptology makes it low-powered.


Labour markets and supply-side policies: too much stress put on the WS push factors?

I think it depends on the category of literature you're reading. The category of innovation literature has a lot about supply-side cost changes, and about related policies - you'll find lots of work on learning curves, technology diffusion, harnessing economies of scale. Frank Geels of Manchester does a lot of work on the theoretical side of this.

And there's broader literature on industrial policy and regulation too.


How are petro states able to maintain gasoline subsidies without black market export smuggling?

There has been rampant smuggling from Venezuela to Colombia - see links below. This is the basis for my question.

http://www.wsj.com/video/colombia-black-market-gas-trade-/BC14C5B7-69A0-4A0E-84B2-5377A7FDF79B.html

http://www.wsj.com/articles/venezuela-pays-price-for-smuggling-1402271308

http://www.wsj.com/articles/venezuela-to-reopen-colombia-border-1470962864


Difference between Unit Banking and Direct Banking

Direct banking is done online. Unit banking is just normal banks. Instead of working with some franchise direct banking allows you to go straight to the computers with the ones and zeroes and do banking. Whereas unit banking just offers you physical branches as well.

You already defined all this in your post. There is no question to answer here.


What types of identification are there in Econometrics?

I think to answer this it is best to first go over definition of identification.

Following Stachurski (2016) identification or identifiability (I omit formal description since its also in that wikipedia article you provided in your comment):

  means that the parameter vector associated with unknown distribution can eventually be distinguised from the data.

Hence identification is more or less what people usually call estimation. For example, in OLS ($y= X\beta+e$) where the $\beta$ coefficient is:

$$\hat{\beta} = (X' X)^{-1}X'y$$

it can be proven that $\hat{\beta}$ can only be identified when the $X'X$ matrix is invertible otherwise $(X'X)^{-1}$ is not defined and you simply wont be able to calculate the $\beta$ or R or Python or Stata would give you error message, like for example where you have perfect multicolinearity.

Every model you can think of has some identification conditions - hence its not really appropriate to talk about types of identification, identification means the model can estimate the parameters and every model has its own conditions for identification of the parameters.

Consequently the process of estimating any coefficients is usually called identification strategy. If by asking about different types of identification you mean different models/ identificaiton strategies, a good-all-purpose textbook is Pesaran (2015). Verbeek (2008) is good intermediate introduction textbook, if you look only for time series analysis Hamilton (1994) is classic although bit dated now. For treatment evaluation I recommend Angrist & Pischke (2008). I wont go over all possible types of models as nowadays you have such a variety its impossible to make exhaustive list in answer on SE.

Next when we talk about weak or strong identification we mean by weak identification that the estimators and test statistics are not well approximated by their standard asymptotic limits because of limited information in the data. Basically, the point is that identification in itself does not guarantee the coefficients are consistent and unbiased only that they can be estimated and usually the term weak identification is applied when the aforementioned are not guaranteed and strong identification when they are. Also I mostly seen these terms being applied to the IV, GMM or other instrument using estimator where weak identification is often used as a synonym for the fact that the first stage or instruments are weak. But I dont see people denoting weak or strong as a type so I dont know if this is what you mean by that. To see when the coefficient estimates are consistent and unbiased you can again for each separate model see in the literature I recommended above.

References:

Angrist, J. D., & Pischke, J. S. (2008). Mostly harmless econometrics: An empiricist's companion. Princeton university press.

Hamilton, J. D. (1994). Time series analysis (Vol. 2, pp. 690-696). New Jersey: Princeton.

Pesaran, M. H. (2015). Time series and panel data econometrics. Oxford University Press.

Stachurski, J. (2016). A primer in econometric theory. Mit Press.

Verbeek, M. (2008). A guide to modern econometrics. John Wiley & Sons.


If a retailer consumes a good he bought from the producer, does that get accounted into the GDP of a nation?

Good question, which goes into the heart of the definition of GDP.

The answer is that some of the value of the drink is accounted irrespective of what the seller does with the drink. Recall that GDP accounts for the value added of final goods, value which is "added" at each stage of production (and equivalent to the final price of the good).

As such, when the retailer buys the can of coke from the wholesaler or producer, a portion of the total value has already been recorded in GDP. It is only the value added by the retailer which is not counted if the seller drinks the coke.

Naturally, this is true only if the seller does not sell the coke to herself. Otherwise it makes no difference at all.


I don't get it, you say you have a household ID. If it is the same across waves, then it is the same household over waves. Most likely the people who collected the data will be able to answer the question of whether this is the same household better than you.
The exact definition of who constitutes a household is necessarily arbitrary and varies to some degree between countries. However, the large surveys tend to gravitate around a definition that includes everybody who has lived in the same "building" (whatever that may be) and at least one meal together for more than 6 months in the last year. Then they tend to narrow this down e.g. excluding workers being paid while living there etc. You can check out a questionnaire from a large survey, for instance Living Standard Measurement Survey by the World Bank, at the beginning of the household roster they usually have a section describing who is part of the household.
Households will be given a unique id the first time they are interviewed and followed using this id over the subsequent waves (there is no kind of probability-based matching going on). To re-identify the same households, surveys will use address / GPS, name, and contact details.
In sum: There is really not much point worrying about this. Just group based on hhid and spend your time on other problems in the data (which likely there will be enough of ;))


Show that $W_t - \int_0^t \xi_s ds$ is forward-measure-Brownian

(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.)

General Fact

Let $W$ be standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$. Consider $(L_t)_{t \in [0,T]}$ defined by
$$
\frac{dL_t}{L_t} = \psi_t dL_t, \; L_0 = 1.
$$
In general, $L_t = e^{\int_0^t \psi_s dW_s - \frac12 \int_0^t \psi^2_s ds }$ is a super
martingale. Under some conditions (e.g. Novikov's condition), $L_t$ is a martingale and
one can define a probability measure $\mathbb{Q}$ by
$$
\frac{d \mathbb Q}{d \mathbb P} = L_T.
$$
Under $\mathbb{Q}$, the process
$$
W^{\mathbb{Q}}_t = W_t - \int_0^t \psi_s ds
$$
is a standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$.

An informal indication why this is true is as follows. Consider $W^{\lambda}_t = W_t + \int_0^t \lambda_s ds$. By Bayes theorem, $W^{\lambda}$ is a $\mathbb{Q}$-martingale if and only if $L W^{\lambda}$ is a $\mathbb{P}$-martingale. Since

\begin{align*}
d L W^{\lambda} &= L d W^{\lambda} + W^{\lambda} dL + dL dW^{\lambda}\\
                &= L (\psi + \lambda) dt + (\cdots) dW,
\end{align*}
we must have $\lambda = - \psi$, for $W^{\lambda}$ to be a $\mathbb{Q}$-Brownian motion.

Discounted Price as Probability Density

The implicit assumptions are that there is a underlying asset whose price $S_t$ follows
$$
\frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t
$$
under the risk neutral measure $\mathbb{P}$. The short rate $(r_t)$ and volatility
$\sigma_t$ processes are adapted with sufficient regularity so that the integrals exist. (For this to be true, the Brownian filtration generated by $(W_t)$ under the risk neutral measure has to be the same as that generated by the physical Brownian motion under the physical measure, so that the Martingale Representation Theorem applies.)

In this Brownian filtration setting, for any time-$T$ claim $X_T$, the risk-neutral dynamics of its price $X_t$ takes the form
$$
\frac{d X_t}{X_t} = r_t dt + \psi_t dW_t.
$$
The process $(\psi_t)$ is the volatility of return of $X_t$, under both the physical and risk-neutral measure.

In other words, the risk-neutral dynamics of the discounted price
$M_t = e^{- \int_0^t r_s ds} X_t$ is given by
$$
\frac{d M_t}{M_t} = \psi_t dW_t, \, M_0 = X_0.
$$
(The discounted price of any $T$-claim must follow a martingale under risk neutral measure, by no arbitrage.)

If Novikov's condition holds, then $L_T = \frac{M_T}{M_0}$ defines a Radon-Nikodym density
$$
\frac{d \mathbb Q}{d \mathbb P} = L_T.
$$
Under $\mathbb{Q}$, the process
$$
W_t - \int_0^t \psi_s ds
$$
is a standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$.

In other words, the discounted payoff $ e^{- \int_0^T r_s ds} X_T$ of any $T$-claim $X_T$, normalized by its time-$0$ price $X_0$, can be considered as the Radon-Nikodym density of a measure $\mathbb Q$. Under $\mathbb Q$, the risk-neutral Brownian motion now has drift given by the volatility of return $\frac{d X_t}{X_t}$.

If $(Y_t)$ is the price of a traded asset, then
$ e^{- \int_0^t r_s ds} Y_t$ is a $\mathbb P$-martingale. This implies that $(\frac{Y_t}{X_t})$ is a $\mathbb Q$-martingale.

Forward Measure

The forward measure is a special case of above where $X_t = P(t,T)$ is the time-$t$ price of the zero coupon bond maturing at $T$.
In particular, $X_T = P(T,T) = 1$.
In the expression
$$
\frac{dP(t,T)}{P(t,T)} = r_t dt + \xi_t dW_t,
$$
$\xi_t$ is the volatility of return on zero coupon bond.

(If $(r_t)$ is deterministic, then $\xi = 0$, and the forward measure is the same as the risk neutral measure. The zero-coupon bond is a risky asset only when the short rate is stochastic.)

The corresponding measure $\mathbb Q$ is defined by
$$
\frac{d \mathbb Q}{d \mathbb P} = \frac{e^{- \int_0^T r_s ds} P(T,T)}{P(0,T)} = L_T.
$$
Since
$$
\frac{d L_t}{L_t} = \xi_t dW_t,
$$
it follows from general discussion above that, under $\mathbb{Q}$, the process
$$
W_t - \int_0^t \xi_s ds
$$
is a standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$.

(In the question posted, the martingale $M_t$ should be $\frac{e^{- \int_0^t r_s ds} P(t,T)}{P(0,T)}$. It's the discounted asset prices that are martingales under risk-neutral measure.)

Empirical Comments

The forward measure $\mathbb Q$ has the property that the forward prices form a $\mathbb Q$-martingale.

Suppose $F(t,T)$ is the forward price of the forward contract entered at $t$ with maturity $T$. By no-arbitrage (spot-forward parity, in this case)
$$
F(t,T) P(t,T) = S_t
$$
which, after discounting, is a $\mathbb P$-martingale. So $F(t,T)$ is a $\mathbb Q$-martingale.

Since the forward price
$$
F(t,T) = \frac{S_t}{P(t,T)}
$$
moves inversely relative to the $P(t,T)$.
The forward measure shifts probability mass toward states where
the discounted return of zero coupon bond
$$
\frac{d \left( e^{- \int_0^t r_s ds} P(t,T) \right)}{ e^{- \int_0^t r_s ds} P(t,T)} = \xi_t dW_t,
$$
is high, in such a way that counteracts the movement in $P(t,T)$ and keeps the (conditional) expectation constant.


What's the term for consumers not being able to pay their mortgage?

Not paying a mortgage is first known as delinquency (when payments are missed) and later, default (when the loan is written down and the lender seeks foreclosure). But an inability to pay has no particular term of art attached to it, because an inability to pay can arise either out of illiquidity (debts exceed liquid assets) or insolvency (debts exceed liquid assets and ability to borrow against future income).


There are many such utility functions. Most commonly:

\begin{equation}
u(x)=L-\mathrm e^{-ax},\tag{1}
\end{equation}
where typically $a>0$. Or
\begin{equation}
u(x)=L-(x-a)^2.\tag{2}
\end{equation}

Both of these are uniformly bounded below $L\in\mathbb R$ and both tend to $L$ as the functions approach their respective maximum.


Characterising a set of outcomes containing the collection of pure strategy Nash equilibria

Condition 3 can be written in the following way: $a^\ast \in B$ iff for all $i$ and all $j$
$$
a^\ast_{i,j} = 1 \iff u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i})
$$
As there are no ties, this can be written as:
$$
a^\ast_{i,j} = 1 \to u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i}),\\
a^\ast_{i,j} = 0 \to u_i(a_i^\ast + \{a_{i,j} = 0\}, a^\ast_{-i}) \ge u_i(a_i^\ast + \{a_{i,j} = 1\}, a^\ast_{-i})
$$
If $a^\ast$ is a Nash equilibrium (i.e. $\in A$), then  for all $i$ and all $a_i$:
$$
u_i(a_i^\ast, a_{-i}^\ast) \ge u_i(a_i, a_{-i}^\ast)
$$

Is $A \subseteq B$?

Yes. Let $a^\ast \in A$. We want to show that $a^\ast \in B$. To show that $a^\ast \in B$, we only need to look at deviations where $a_{i,j}^\ast$ changes for a particular $j$. Towards this end, define $a_i = (a_{i,\ell}|\ell \le J)$ such that:
$$
a_{i,\ell} = \left\{\begin{array}{ll} a^\ast_{i,l} &\text{ if } \ell \ne j\\ 1- a_{i,j}^\ast &\text{ if } \ell = j \end{array}\right.
$$
So $a_i$ differs from $a_i^\ast$ only because $a_{i,j}$ switched value (from 1 to 0 or vice versa). To show that $a^\ast \in B$, we show the two conditions.

If $a_{i,j}^\ast = 1$ , we have:

$$
u_i(a_i^\ast, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 1\}, a_{-i}^\ast),\\
\ge u_i(a_i, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 0\}, a_{-i}^\ast).
$$
The inequality follows from the definition of a Nash equilibrium. The equalities follow from how we defined $a_i$.

If $a^\ast_{i,j} = 0$, we have:

$$
u_i(a_i^\ast, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 0\}, a_{-i}^\ast),\\
\ge u_i(a_i, a_{-i}^\ast) = u_i(a_i^\ast + \{a_{i,j} = 1\}, a_{-i}^\ast).
$$
This shows that $a^\ast \in B$.

If $B$ is empty, then $A$ is empty?

This follows from $A \subseteq B$. If $B = \emptyset$ and $A$ is a subset of $B$ then immediately $A = \emptyset$.
Notice that the inclusion can be is strict, i.e. there are cases where $A \ne B$. To see this, consider the case where $J = 2$ and the payoff table is the following.

(00)
(01)
(10)
(11)

(00)
4,4
1,1
2,2
4,0

(01)
2,2
2,3
1,4
1,0

(10)
1,4
3,2
4,3
2,0

(11)
0,4
4,0
0,2
3,3

We have that $(11,11)$ is in $B$ but not in $A$ as it is not a Nash equilibrium. The only Nash equilibrium is $(00,00)$.


Recommended tests do detect breaks in time series

For a multivariate regression:

If you're testing for a break at a known date, say at a historic event or change in policy regime, I would recommend that you use a Chow test (a special case of an F-test). This will give you an idea of whether the coefficients are constant on each side of the proposed break.

On the other hand, if you want to test for a break at an unknown date, you have three options. You could try a Quandt, Mann-Wald, or Andrews-Ploberger statistic. The choice is up to you, but I like Mann-Wald. To me, it's more intuitive than the others because it's basically the Mean Value Theorem evaluated using a Riemann sum of the F-test at each potential break. Of course, the asymptotic distributions necessary to use any of these methods are beyond my knowledge, but you don't need those if you're modeling in a nice software package with pre-loaded diagnostic tests.

Lastly, testing for "breaks" (as in multiple breaks) is a computational nightmare. You probably have deeper issues with your model (like omitted variables or some sort of misspecification) if you need multiple breaks.


What are some examples of modern economic research that integrate economics and computer science?

I can try to give a bunch of examples about specific branches in economics that have been reshaped by computer science over the last decade, but I will leave this to the ones that are more knowledgeable, and far better writers.

That being said, I really enjoyed Hal Varian's take away of it:

https://www.aeaweb.org/articles?id=10.1257/jep.28.2.3

And this video where Susan Athey elaborates upon the role of machine learning in the study of causal effects and their connection with impact evaluation literature:

https://www.youtube.com/watch?v=VeTl4mIdqWA

Good luck in your studies!


How can nominal interest rates be negative?

There was only one reason to ever think that nominal interest rates couldn't go negative, which is that the nominal return on both forms of base money (electronic reserves, and paper currency) had a floor of zero -- and investors wouldn't accept a below-zero nominal return when they could get a higher one by holding base money.

But for electronic reserves, there's certainly no practical need for a floor of zero: the central bank can easily pay negative interest on reserves. (For instance, the Swiss National Bank is now paying -0.75% on most "sight deposits", which is their term for electronic bank reserves.) It's quite easy: they just tell the computer to charge a certain amount of interest on reserve accounts, just as they would normally tell it to pay interest.

The potentially serious problem is paper currency, which by its very nature generally pays a nominal interest rate of zero. (If you have a \$5 bill, it will still be worth \$5 in a year - zero nominal interest.) The concern is that if we try to set nominal interest rates below zero, people will just massively switch to paper currency paying 0%, and the traditional financial system would disappear and be replaced by paper.

This is where we reach a very important point, which is that return isn't all that matters in an asset. Other factors, like convenience, play a role as well; different assets are convenient in different ways, and aren't perfect substitutes for each other. This is why in normal times, people are willing to hold paper currency even though it pays less than electronic accounts (because paper is useful in some ways that electronic accounts aren't); inversely, it's why in these abnormal times, when Swiss cash pays more than Swiss deposits, investors don't flock exclusively into cash.

Hence, when the SNB pays -0.75% on electronic reserves, rates in Swiss-denominated money markets (like the SARON overnight rate or LIBOR CHF) also drop to near -0.75%, and not everyone converts their electronic assets into paper. Indeed, there has been very little response on that front: if you look at SNB balance sheet column 16, banknotes in circulation, it hasn't budged very much thus far since negative rates began in December and the rates fell to 0.75% in January. Again, this is just a matter of imperfect substitutability: paper currency simply doesn't have the same kind of transactional and liquidity value that electronic accounts do. (And then there are the costs and risks of paper currency storage, as mentioned in the article you cite.) This is the feature of the world that wasn't adequately captured, before recent events, in the simple zero lower bound models used by many economists.

The one quasi-mystery, to me, is why banks haven't stockpiled paper. After all, even though an electronic account is more useful than paper to an end user, this doesn't apply to a bank that's already overflowing in electronic reserves. (Indeed, conceivably banks could be large-scale intermediaries here, stockpiling paper to back electronic deposits held by their customers - effectively transforming paper into a more convenient asset.) Yet the quantity of banknotes held by Swiss banks has remained low (see the 4th column at the bottom of the 2nd page here).

There are two possibilities here. One is that the logistical costs of holding cash are still higher than the 0.75% spread, so that it's still not worthwhile for banks to hoard. Another is that banks could make a profit by embarking on some large-scale cash hoarding program, but they know that the central bank would quickly get angry and create new rules (like a 0.75% tax on excess cash holdings by banks) to stop this, so they don't bother -- and indeed, maybe there has been under-the-table pressure from the central bank already!

Ultimately, when the government has the authority to regulate financial institutions, it can always disallow organized cash hoarding schemes that threaten the negative interest rate policy, so none of this is a long-term threat. The real issue is cash hoarding by end users - and there, the disadvantages of cash apparently still loom large at -0.75%.

(One complication, by the way, is that I think banks are generally not charging negative interest on small-scale retail deposits. At least superficially, then, the relevant interest rate isn't negative for the typical consumer; but I suspect that banks are making up the difference by charging fees, etc., which is already necessary to recover their costs even when interest rates are at 0%. For large-scale deposits, on the other hand, banks are charging clients to hold cash - and, of course, this must be true, since money market rates wouldn't be negative if it was always possible to find some bank that would give you a zero rate.)


Each player seeks to maximize his or her revenue. As you said, in the Cournot model, the player varies the output. In the Bertrand model, the player varies the price.

So we first solve for $Q_{2}$ as a function of $P_{2}$:

$$Q_{2} = 200 - 2P_{2} - 0.8Q_{1}$$

We then substitute $Q_{2}$ into $P_{1}$ to get:

$$P_{1} = 100 - 0.5Q_{1} - 0.4 (200 - 2P_{2} - 0.8Q_{1}) = 50 - 0.18Q_{1} + 0.8P_{2}$$

Player $1$'s maximization problem is:

$$\max_{Q_{1}} 50Q_{1} - 0.18Q_{1}^{2} + 0.8P_{2}Q_{1}$$

Which yields the first order conditions:

$$0.36Q_{1} = 50 + 0.8P_{2} \implies Q_{1}^{*} = \frac{1250}{9} + \frac{20}{9}P_{2}$$

Now we can substitute $Q_{1}^{*}$ into $Q_{2}$ to get:

$$Q_{2} = 200 - 2P_{2} - \frac{4}{5}(\frac{1250}{9} + \frac{20}{9}P_{2})$$

Now maximize $Q_{2}$ with respect to $P_{2}$. Can you take it from here?


Why does Samuelson say common stocks are better than bonds and preferred stocks during inflation?

This was most likely reference to the Fisher hypothesis. As argued by Engsted & Tanggaard (2002):

classical economic theory, especially the Fisher hypothesis, according to which expected nominal asset returns move one-for-one with expected inflation such that expected real returns are independent of expected inflation. A related implication is that assets which represent claims to real payments, such as common stocks, should offer a hedge against unexpected inflation, while assets which represent claims to nominal payments, such as bonds, should not be expected to offer such hedge possibilities.

Essentially this is because payment on both bonds and preferred stocks are usually fixed nominal terms. For example, bonds pay fixed coupon payment that is predetermined at a time you buy bond.  Preferred stocks also usually stipulate some dividend payment that is fixed. This means that inflation makes the value of the principal as well as the fixed stream of coupon payments/dividends lower in relative terms.
Moreover, as explained here bond prices fall when nominal interest rises and vice versa.
$$  P = \sum^T_{t=1} \frac{C_t}{(1+i)^t}+\frac{M}{(1+i)^T}$$
Where P is the price of the bond, C is the coupon i the interest rate, and M the value at maturity (i.e. the face value). By Fisher equation $i \approx \pi +r$ where $\pi$ is the inflation rate so as inflation rate increases the market value of the bond will be lower. Furthermore, as explained in this investopedia article proffered stocks are very similar to bonds except for legal treatment that is unrelated to this particular discussion. This being said bonds should according to theory still offer hedge against expected inflation because rational person would take any inflation expectation into account before investing into bond (although of course in real world people's expectations might not be completely rational).
A common stocks essentially represent claims to real payments as mentioned above and hence they should offer better hedge against inflation in general as they should also cover the unexpected inflation.
However, few caveats:

The empirical literature actually does not confirm the above result. See for example, . Fama and Schwert, 1977, Gultekin, 1983 or Barnes et al. (1999). The Samuelson's textbook was first published in late 40's even though it is being regularly updated with new edition it might be that this statement was not updated or perhaps Samuelson wanted to teach people about the Fisher hypothesis - that is up to speculation.

Inflation adjusted bonds and floating dividend preferred stock exist, but I think that Samuelson was implicitly talking about the  'standard' bonds and preferred stocks. Although of course we can only speculate about what Samuelson was thinking the passage would not make sense otherwise.


Has there been an applications of search and matching theory in the Neoclassical Growth Model?

Search and matching models are fairly standard in macroeconomics. I guess this is fair to say since Peter Diamond, Dale Mortensen, and Christopher Pissarides won the Nobel prize in 2010. These are the names behind the canonical DMP model that most graduate students have to study in their first year. Although I am not a macroeconomist, I am confident to say that search models recognized as a key ingredient to explain unemployment in equilibrium. I suggest to watch the interview with Per Krusell on the Nobel prize homepage and read the scientific background.


Why does average variable cost = marginal cost for this function?

So using your above statements in the comments it should be relatively clear how to find that $MC = AVC$.
$$MC = \frac{\partial TC}{\partial Q} = 12$$
and then using the fact that variable cost is $12Q$
$$AVC = \frac{VC}{Q} = 12$$


von-Neumann-Morgenstern v. Bernoulli Utility Function

Bernoulli utility represents preference over monetary outcomes. In a way, this is no different from the typical utility functions defined over consumption bundles.

vNM utility, in contrast, represents preference over lotteries of monetary outcomes. Thus, the argument of vNM utility is an object related to, but categorically distinct from, the object that is an argument of Bernoulli utility.

For example, Bernoulli utility function allows us to compare the utilities from having $\$5$ to having $\$7$, while vNM utility allows us to compare utilities from the lottery $(0.2\otimes\$5,0.8\otimes\$7)$ --- having $\$5$ with $20\%$ and $\$7$ with $80\%$ --- to the lottery $(0.6\otimes\$5,0.4\otimes\$7)$ --- having $\$5$ with $60\%$ and $\$7$ with $40\%$.

In this sense, the distinction between Bernoulli and vNM utility functions are necessary (as they are applied to different objects) rather than important (since they both represent some kind of preference in the end), as @denesp says in his comment.

Moreover, if we only consider degenerate lotteries, i.e. the probabilities are either $0$ or $1$, then vNM and Bernoulli utilities coincide.
That is, the Bernoulli utility of having $\$5$ is exactly the same as the vNM utility of having $\$5$ with $100\%$. Therefore, distinguishing Bernoulli from vNM utility functions enables us to examine the effects of uncertainty apart from the mere quantity of "stuff" (be it goods or money).

Lastly, defining utility over money also allows us to study people's attitudes towards risk.


what is the Intuition behind Cholesky identification in VAR/SVAR

The terms "ordered last" or "ordered first" do not necessarily relate to the place of the variable in the $y_t$ vector. Rather they describe the column (row) vector of the contemporaneous impact matrix pertaining to the variable. Once the restrictions have been set and the model is identified, where you exactly place the variables is an uninteresting computational issue that concerns only the implementation of the model in your software package. Here is what I mean in equations.
Let the structural model be: $y_t = c + Ay_{t-1} + Bu_t$
$y_t= \begin{bmatrix}
n_t\\
m_t\\
q_t\\
\end{bmatrix}$
and
$B = \begin{bmatrix}
b_{11} & b_{12} & b_{13}\\
b_{21} & b_{22} & b_{23}\\
b_{31} & b_{32} & b_{33}\\
\end{bmatrix}= \begin{bmatrix}
b_{11} & 0 & 0\\
b_{21} & b_{22} & 0\\
b_{31} & b_{32} & b_{33}\\
\end{bmatrix}$.
That is $B$ coincides with the lower triangular of a Choleski decomposition.
"Ordered first" means variables like $n_t$, that have only zeros in their row vectors where $i \neq j$. "Ordered last" are variables like $q_t$, that have only zeros in their column vectors where $i \neq j$ .
So "ordering a variable first" means assuming that it is not affected by the other variables contemporaneously and "ordering a variable last" means assuming that it does not affect the other variables contemporaneously.
It is not necessary that the model where we order some variable last actually has a lower triangular $B$. Perhaps we want to use partial identification (just like Cristiano, Eichenbaum, Evans (1999)) and we are not interested in the entire $B$.
In the specification you are showing, we wouldn't call any variable "ordered last" or "ordered first".
I once again encourage you to consult
Prof. Ramey's excellent exposition on identification
.
Consider especially Sections 2.3.1. and 2.3.2.
Presenting Blanchard Perotti (2002)  in section 2.3.2 on page 10 she writes:

Let ܻ$Y_1$ be net taxes, ܻ$Y_2$ be government spending, and ܻ$Y_3$ be GDP. They identify the shock  to  government  spending  using  a  Cholesky  decomposition  in  which  government  spending  is ordered first (i.e. ܾ$b_{21} = b_{23} = 0$). They identify exogenous shocks to net taxes by setting ܾ$b_{13}= 2.08$,  an  outside  estimate  of  the  cyclical  sensitivity  of  net  taxes.

The variable that is "ordered first" - government spending is actually the second component of $Y_t$:
$B= \begin{bmatrix}
b_{11} & b_{12} & 2.08\\
0 & b_{22} & 0\\
b_{31} & b_{32} & b_{33}\\
\end{bmatrix}$.


What is the classic paper that derives borrowing constraints from asymmetric information?

I don't know if you refer to the extensive margin (some borrowers not being able to get credit) or to the intensive margin (one borrower not being able to get as much credit as (s)he wants). If you are referring to the former, one of the theoretical papers for borrowing constraints on markets with asymmetric information is the following one:

Stiglitz and Weiss (1981) AER's paper on Credit Rationing in Markets with Imperfect Information.
http://www.jstor.org/stable/1802787?seq=1#page_scan_tab_contents

The argument is very interesting. If lenders cannot distinguish between high risk and low risk borrowers, then the ''market-clearing'' (loosely speaking) equilibrium interest rates will be such that identical (to the observer) consumers can be rationed out.

Other good papers are Bester's (1985) AER's paper on Screening vs. Rationing in Credit Markets with Imperfect Information. http://www.jstor.org/stable/1821362?seq=1#page_scan_tab_contents

Finally Chapter 16 on the Handbook of Monetary Economics by Jaffee and Stiglitz may help you.


If you have more detailed information about what kind of expenditures you have then you can use the CPI data. Current price data for government does not exist under the price program for BLS for you to use to convert the nominal to reals.

However, you can use the BEA industry data which does have a government sector price indexes for either valued added output, gross output concept of GDP or you can select the government sectors price indexes for intermediate inputs.  It depends on the topic you are analyzing.

https://apps.bea.gov/iTable/iTable.cfm?ReqID=51&step=1


How much did it 'cost' slave-owning nations to emancipate the slaves as a percentage of GDP?

Unfortunately, I'm not sure this will really answer the questions you raised to your satisfaction, but I can at least give some of them a try:

First Note: It wasn't 40% of the GDP of the Empire, but rather the 20 million GBP sum was equivalent to 40% of their annual governmental expenditures (or 5% of the Empire's contemporary GDP) source here.

Regarding your first couple of questions: unfortunately, there's no good, useful answer that I can find. If you were interested in calculating these numbers yourself, it might be worthwhile to look at this online database of payments made under the act. That said, it's important to remember that, as colonies of the British Empire, there might well be many reasons that could make the desired "contemporary GDP amount" difficult to calculate. For example, if there's an individual who owned plantations in Trinidad and the Bahamas, they themselves might live in London. Thus, for all practical matters, the compensation payment the plantation owner would receive doesn't flow to either colony, and shouldn't be considered in those colonies GDPs.

As for "how did they come up with 20 million GBPs?" While I haven't seen a specific answer, I'd imagine much the same way compensation pools are created today- by Parliamentary negotiation, estimation and extrapolation. The number itself is known exactly (that is, we know it's 20 million, and that's not an estimation on the part of the Guardian) since it's inscribed in paragraph XXIV of the act itself (have to scroll down a bit, or search on the page for "million"). That said, there's specific information in paragraph XLV of that act how the sum of 20 million GBP should be divvied up between the different colonies, and that's based on the local "value" of slaves in each colony for the previous eight years.

About additional compensation- no. Remember, there weren't really those "national governments" at the time. They all were, by definition of the act, only territories which were still completely ruled by the British crown. As far as I know, no regional governors supplemented that pool (I'm not even sure if they would have been allowed to, since the authorization for the payments in this case were from the British Parliament, and the vehicle to disperse the payments was a commission set up by the act).

Finally, I'm not exactly sure what you mean by "how would you cost the six years of unpaid labor." Who are you referring to "costing" this? The value of this labor wasn't included in the payments by Parliament. If you were thinking about what the effective cost of this provision was (that is, how much it cost the slaves whose freedom was delayed by those six years) then again, I think it might be an open question.

Again, I'll check back later to see if I can help by refining my answer. Let me know how this aligns with what you were looking for. Also, these additional resources might provide a bit of useful info!
NB: the total number of slaves freed uner this act is roughly 800,000


study question: change of Expenditure in terms of price and quantity

Adopting your notations, $E = P \times Q$. Taking natural log for both sides, we have $\log E = \log P + \log Q$.

When the percent change of some quantity, $\Delta x / x$, is small, we can approximate this percent change with $\Delta \log x$,  the change in $\log x$. This is because when $\Delta x / x$ is small,

$$
\log (x + \Delta x) - \log x = \log \frac{x + \Delta x}{x} = \log (1 + \frac{\Delta x}{x}) \approx \frac{\Delta x}{x}
$$

Now we come back to the equation $\log E = \log P + \log Q$. When the change is small, we can loosely say that the percent change in E is the sum of the percent changes in P and Q. But you are right, this relationship is not exact, and may not be appropriate when the percent change is large.


Why's it always better to choose a lump sum than periodic payments over time?

The answer depends on the assumed interest rate of the periodic payments, compared to the expected return of alternative investments.

To be viable as a 'good' choice, the period payments must total to an amount that is equal to or greater than what could be achieved by taking the lump sum and doing something with it. In practice, this means that a period payment must total to a larger amount than the lump sum payment, with the difference increasing as the total amount of time of payments does.

For instance, the very large lotteries in the United States offer a lump sum amount that is somewhere on the order of 60% of the 'advertised' amount, which is what would be earned over 30 years of periodic payments. This figure is calculated assuming a 5% year-over-year increase of the payment amount.

Whether one is better than the other depends on what would be done with the lump sum money in the meantime. If one could invest the money or start a business with an expected rate of return greater than the one used to generate the periodic payment amount, then it is a bad idea to take the periodic payment; conversely, if the rate used in the periodic payment calculation is greater than what one would expect to earn themselves, then it would be a bad idea to take the lump sum.

In the real world, since the periodic payment calculation is usually made with a conservative, guaranteed rate, the lump sum would be better. If the periodic payment calculation is made with an assumed rate of 0% (ie, the total of all payments is equal to the lump sum amount), the lump sum will always be superior, assuming it can be used to earn any money at all. It sounds like the author is implicitly assuming one of these two scenarios.


Long Term Economic Profit for Perfectly Competitive market

This is because we are talking about economic profit not accounting profit.
An economic profit takes into account opportunity cost. If you are skilled programmer that can earn $\\\$100000$ per year being employed at Google then doing something else like operating your own business incurs an opportunity cost of $\\\$100000$ per year.
Hence in this situation if the person would set up perfectly competitive business, the business would still record $\\\$100000$ accounting profit even if the economic profit would be 0.
Consequently, even in perfectly competitive business people have actual incentives to run those business.


Why would oil suppliers choose a lower priced contract?

This is because for the price comparison it is not relevant what the exchange rate is now but what it will be at the time of delivery. The difference between these corresponds to different expectations of what the USD-Yuan exchange rate will be in the future.

For example consider oil supplier that is incorporated in the US and eventially needs to convert the money to USD to pay its employees/dividends/taxes. Moreover, lets suppose that the future contract specifies that the 1 barrel of oil has to be delivered at the end of next month at the 205 yuan. Well if during the month the exchange rate between USD and yuan changes to 1\$ = 15 yuan the 205 yuan you are going to receive next month are now worth only 13.66\$. Hence in such scenario you would be worse off if you would make the future to be denominated in yuan as opposed to USD (assuming \$20 price). Due to the law of one price the two futures should be priced equivalently which implies that people think that USD will most likely become stronger at the time of delivery relative to yuan.


What justifies the authority of a bank to create money when they make loans?

This is due to Fractional Reserve System. In fractional reserve system when you make a deposit at a bank bank is required to keep a bit of it as a deposit and it is allowed to lend out the rest.

So for example if Central Bank makes an initial money injection into economy. Let’s say 100€ (and it does not really matter whether it’s by helicopter drop or directly putting them in banks) the money will eventually be put into deposit account sooner or later. Once money is deposited with fractional reserve being let’s say 10% the bank keeps 10€ and lends the rest 90€. The 90€ again sooner or later gets deposited once the lender spends it so it will end up at some bank which will keep 9€ as reserves and lends rest 81€. In the end using this system private banks create from original 100€ a 1000€ worth of money supply (the final effect can be calculated using the multiplier ($1/ra)$ where $ra$ is the reserve ratio so with 10% reserves original 100€ creates 10 times as much money. Although as pointed out in the article its not inevitable the money supply will increase to its maximum possible value given multiplier as its not guarantee that households will put later money into deposit accounts or bank could just decide too keep full amount as reserve because it does not find profitable to lend money or there can be lending restriction imposed by regulation so the 1000 should be viewed as an upper limit on the maximum new money that can be created.

You could also do this. For example, if you convince your friends to give you deposits and then you keep only a fraction of those deposits and lend the money out to some other friends you will create new money (although if you are hell bent on doing this first check local laws in some countries maybe you can’t just do this without license). But in principle anyone could do this. In past, like during the Renaissance or even in more ancient times even many private money lenders did this.