The most frequent causes of economic downfall?

Interestingly, Romer and Romer (2014) show that financial crises are typically not causing major downfalls in economic activity, if you restrict your sample onto the developed countries post WWII.

An advantage of their method is that they look at a real time narrative, so they don't try to judge ex-post what was a crises. So historians winner writes history criticism doesn't apply here.

It is certainly worse if the financial crises is global, but that's pretty infrequent.


Are government bonds internal or external debt?

Here’s a website from the IMF that has guides on their definition of external debt: link to IMF webpage.
In the 2013 guide, on page 5, the definition is debt owned by foreign entities. So foreign-owned bonds count as external debt.
From a practical perspective, this definition has problems. Tax avoidance schemes typically rely on using international transactions, so determining the true locale of the beneficial owner of securities is difficult. The other major issue is that for developed country government bonds, being a foreigner offers no special treatment - they have exactly the same economic rights with respect to the bond as locales. As such, some people do not consider domestic currency bonds as being “external debt”, but that is using a new definition.


Why would a digital Yuan (or some other national currency turned into a CBDC) "help fast-track international settlements" in that currency?

How China Influences the U.S. Dollar
https://www.thebalance.com/how-does-china-influence-the-u-s-dollar-3970466

For export transactions to United States companies/consumers, Chinese firms typically receive dollars as payment for their exports. These firms then deposit the dollars into local banks, in exchange for yuan, which can be used to pay their domestic workers and vendors. The local banks then send the dollars to China's central bank, the People's Bank of China (PBOC). The PBOC holds the dollars in its foreign exchange reserves and regularly adjusts these reserves by buying or selling dollars via foreign currency markets in exchange for yuan.
By stockpiling dollars, the PBOC reduces the supply of dollars available for trade, putting upward pressure on the dollar and downward pressure on the yuan. By selling dollars, the PBOC effects the opposite result.

It is important to note that the PBOC does not sit on cash reserves. It uses the dollars it accumulates to buy U.S. Treasuries, which are safe-haven assets that provide some incremental return over cash.

The Same Problems Plaguing the Yuan Will Plague China's Digital Currency
https://qz.com/1899402/why-chinas-digital-yuan-wont-internationalize-the-renminbi/

Because the digital currency is little different from the yuan itself, it will on its own “not be a game changer that elevates the renminbi’s role in international finance,” wrote Eswar Prasad, a professor of trade policy at Cornell University, using the currency’s official name. “After all, the Chinese government still restricts capital inflows and outflows, and the People’s Bank of China still manages the renminbi’s exchange rate. Neither policy is likely to change significantly anytime soon.”

What investors are looking for in a reserve currency isn’t the technology—it’s a currency that’s stable, underpinned by a strong economy, freely convertible, and able to be used widely.


Monopolistic and Bertrand (Nash) Competition

The following paper compares the efficiency of the Bertrand and Cournot game in the case of product differentiation. However, their utility function is more general than the Dixit/Stiglitz case. You also do not have an infinite number of firms competing in each differentiated product but just one firm per differentiated product (the number of differentiated products however may be infinite).

Vives, Xavier. "On the efficiency of Bertrand and Cournot equilibria with product differentation." Journal of Economic Theory 36.1 (1985): 166-175.


Why does borrowability of shares inversely correlate with probability of assignment or exercise?

I disagree with the author's premise. The primary reason that options are exercised early is because of discount arbitrage (an ITM option trades below its intrinsic value) or there is a dividend arbitrage in an ITM put (the time premium is less than the dividend, NOT the call).
No one with a lick of sense would exercise an option that has time premium remaining because that would be throwing away the time premium and incurring more transactional costs (B/A spread and commissions if you're still paying them).
I listened to another minute of the video and I think that the author really has no clue. He talks about the possibility of an OTM short option being assigned early because of a dividend. Now who wouldn't want that to happen to them? If the stock is \$102 and I'm short the \$105 call, please, make me sell the stock to you for \$105! It makes no sense at all.
He continues his errors with another example. A stock is \$101 and you are short a \$100 call that's worth \$2. The company announces a \$2 dividend and the call owner exercises and you are assigned. Why in the world would he do that? He could sell the call for \$2 and buy the stock for \$101. Exercising throws a dollar of time premium away.


What would happen if stock markets were open 24 hours?

You would have trading teams who work shifts like any other 24-hour business areas.

The power market is open 24-7; many companies have around-the-clock trading teams. A lot of analysis and other work that can be done in the daytime is still done in the daytime.

A major source of income during nighttime - in the power market - is just being there and executing trades - when other people are not there. I think that is the main change, that liquidity would shrink and give more arbitrage opportunities.


It isn't enough to have a large number of observations. You need multiple observations of each $i$ and $j$ for multiple time periods and doing trade with multiple countries. If, for example, if countries $k$ and $l$ only trade with each other, it wouldn't be possible to distinguish between $k$, $l$, and $lk$. In additions, you need multiple observations because you want to identify a time subscript on $i$ and $j$. If you have many countries, many periods, and many trade relationships, and the data are not sparse then this may not matter much.

You may find this paper of use:
Gravity or dummies? the limits of identification in gravity estimations (Hornok 2011)

  This paper argues that identification of trade policy effects with a
  gravity equation that includes country-time dummies to control for the
  theoretical Multilateral Trade Resistances (MTR) is severely limited.
  In most cases heterogeneous policy effects, i.e. more than one policy
  dummies, cannot be identified separately, because the policy dummies
  and the country-time dummies are perfectly collinear. Although a
  single policy dummy can b e identified, the estimate may not b e
  meaningful, because country-time dummies absorb to o much of the
  useful variation of the data. Standard estimation techniques often do
  not reveal these problems. The pap er demonstrates these arguments by
  taking four typical research questions on the effect of a trade
  policy, checking the
  identifiability of the corresponding policy effect and deriving the
  estimates. Empirical exercise on estimating the trade effects of EU
  enlargement complements the analytical findings.

And Estimating the gravity model without gravity using panel data
(Westerlund and  Wilhelmsson 2009)

  "...$\alpha_{ij}$ which is unidentified in the fixed effects
  formulation of the model. In order to identify $\alpha_{ij}$, a random
  effects assumption is needed. But such assumptions are generally not
  satisfied in practice..."

@ChinG, here is the basic gravity idea:
Newton's law of universal gravitation tells us that the force of gravity is
$$F = \frac{G M_1 M_2}{R^{2}_{1,2}}$$
which implies
$$f = g + m_1 + m_2 - 2 \cdot r_{1,2} $$
(where lowercase letters are the logs of upper case ones). In the trade equation, the force of gravity is replaced with the volume of trade. Usually, $M_1$ represents the economic size of economy one (say measured by $GDP_1$ and in your regression analogous to $I_t$), $M_2$ represents the economic size of economy two ($GDP_2$ and analogous to $J_t$), and $r_{1,2}$ is the geographic or effective economic distance between the two countries ($\alpha_{i,j}$). GDP isn't exactly the right measure because countries differ in fraction of the economy devoted to trad-able output. Think of it as analogous to the importance of that country in global trade (in an absolute and not relative sense).


Why is that when one person or group gets a good or service, someone else will have to live without it?

That statement is not generally true.

It is true of rivalrous goods (e.g. when I take a slice of a pizza, there's that much less left of the pizza for you).

It is false of non-rivalrous goods (e.g. when I step into the sunshine, there is usually* no less of the sunshine for you).

*Unless of course I somehow also block the sunshine for you.


State of general equilibrium theories Post Blanchard 2008

To understand why macroeconomists use DSGE as a tool, in general, it's a good idea to read up on the Lucas critique.

More colloquially, DSGE models provide macroeconomists with a laboratory that allows quantitative comparison (and ranking) of different economic policies.  Further, the process of writing a model disciplines thinking: if one cannot write down a minimally specified model that replicates (to varying degrees of success) some real world dynamics of interest, then do we really understand what is happening to begin with?  It's easy to play a casual story telling game that explains economic phenomena (with critical errors).  We unfortunately see this all the time in politics, or even on CNBC.

Also, DSGE models do not all share the same objective.  There are some medium and large scale models employed at central banks that are used to guide monetary policy.  Other models are much smaller and stylized, and seek to answer questions about how a government should optimally finance itself, or how to tax different factors of production, or how informational frictions affect prices.

Finally, to answer your question on the state of dynamic general equilibrium theory, researchers continue to apply this mode of analysis to deepen our understanding of economic dynamics.  This entails writing models of economic agents incentivized by objective functions, facing resource and informational constraints, interacting with one another and nature.

Every DSGE model has some fatal flaw which cripples it from perfectly explaining real world dynamics.  Perfection is not the point of DSGE models, so when macroeconomists say that DSGE models are "useful" or "good", it's important to ask questions about the context of those descriptive terms.  It's also important to remember that macroeconomists are, at the same time, some of the harshest critics of DSGE models.


I do not directly see what intuition you are supposed to get from this growth rate equation, and the resulting math exercise but here is a sketch of the proof.

First determine the steady state capital by setting $\Delta K_{t+1}=0$ and solving for $K^*$.

Substitute this value for the $(K^*)^{(2/3)}$ in the fraction. Replace the $ \frac{Y^*}{K^*}$ in the same expression with the value that you get capital-output steady state ratio, and work the $d$ into the parentheses.

The expression you have now can be shown to be the same as the growth rate of $y_t$ that can be found from the production function.


What was the cause of the decline in remittances sent to Pakistan between 1982 and 2000?

Perhaps more enlightening would be the reason why Remittances increased so much from 1976 to 1982, and then cooled off.
1977 was the year of a military coup in Pakistan (and the imposition of Martial Law), that was also, it appears, the beginning of deep structural socioeconomic changes. In such situation it is not uncommon to see a wave of emigration, among those that do not "fit well" in the new situation. But this rarely means that the emigrants severe all ties with the country back home, usually some family is left behind. Hence, increased Remittances.
In 1985 Martial Law was lifted and elections were held, and this could explain the downward trend.
So I would look into the emigration/repatrization flows in Pakistan during that era, coupled with the political situation and any economic reforms.


Rational preferences/individual decision-making theory

(i) Is not complete. For instance, (10,5) is not $\succeq$ (9,6), because $10>9$, but $5<6$. However, (9,6) is also not $\succeq$ (10,5) for the same reason. Hence, there exists a pair of bundles $A,B$ such that neither $A\succeq B$ nor $B \succeq A$. Thus, it is not rational.
(ii) and (iii) are both rational. You can either see this by showing transitivity and completeness directly or you exploit a central result in microeconomics: A preference $\succeq$ can be represented by a utility function $u$ only if it is rational.
For (ii), you can define $u(x_1,x_2)=x_1$ such that $x_2$ doesn't matter for comparisons as the preference states. For (iii), you can define Leontief preferences, $u(x_1,x_2)=\min(x_1,x_2)$.
(iv) is a funny case, because it shows why the result we exploited above is just "only if" and not "if and only if." Lexicographic preferences are rational, but they are not representable by a standard utility function.
For any two bundles either the first dimension differs, or the first one is equal and the second one differs or both are equal. So for any two bundles we have either $\succeq$ or $\preceq$ or both, i.e., the preferences are complete. They are also transitive because $X\succeq Y$ implies that either $x_1>y_1$ or $x_1=y_1$ and $x_2\geq y_2$
and $Y\succeq Z$ implies that either $y_1>z_1$ or $y_1=z_1$ and $y_2\geq z_2$. Therefore, either $x_1>z_1$ or $x_1=z_1$ and $x_2\geq z_2$, implying $X\succeq Z$.


As you pointed out, you need to distinguish between real and nominal GDP. The 3.7% growth from 1620 to 1680 is the nominal growth. To figure out real growth, you need to know the price index of the two years.


Are sales (discounts) an investment or a loss?

Suppose that the wholesale cost of the product is 40USD, and the maximum price that most customers are prepared to pay for the product is 50USD, but a few customers are prepared to pay 100USD for the product. Suppose also that the few customers who are prepared to pay more for the product will put less effort into waiting for discounts. Suppose also that some customers are more likely to buy a 50USD product if it previously costed 100USD than if it had always costed 50USD.

What would be the optimal pricing strategy in this circumstance?


Do the number of days in a month affect monthly return? (seasonality)

Let's make a fake example.

Compound interest of 0.1% over 30 days gives

$$1.001^{30} = 1.030439088$$

Compound interest of 0.1% over 31 days is

$$1.001^{31} = 1.031469527$$

And the percentage change is 1.000970497. That is, you would over estimate the growth rate of a stock by 0.009% if you would compare 30days and 31days without adjusting for it.


How does the economy "detect" if money is removed from circulation?

Money is only inflationary in so far as it's moving around. So for there to be an inflationary or deflationary signal, that means that either the amount of money moving around has changed, or its velocity has changed.

And what does it mean to have money "moving around"? It means that it's being used to buy goods or services. And that's how it affects prices - because it affects demand.

So, yes, to elucidate: it's as you say- "if the millionaire wouldn't have destroyed the money but spent it, he could have bought a lot of goods with it, and the sellers of those goods could have bought still more goods with that profit and so on. As now those goods are no longer being bought, the sellers have to slightly reduce prices, so their goods aren't wasted in their warehouses. This means prices will drop a little, and this is how the economy "feels" the difference"


Why don't central banks just donate money to their poor citizens, rather than QE?

Most federal banks don't have the legal authority to do this, but the notion is definitely sometimes discussed, often under the informal name "helicoptering money". Former Fed Chairman Ben Bernanke discusses his thinking on the subject here, and some economists have called for this response to the current economic crisis. Rep. Maxine Waters has proposed a bill that would authorize the Fed to do roughly this idea.


Can consumer surplus be negative if a consumer is forced to make a purchase?

Consumer surplus is their willingness to pay minus the price they pay,  and producer surplus is the price they receive minus their willingness to receive. So if you are assuming that consumers are forced to buy at a price of 100, yes the consumer surplus is negative. and according to your example, the producer surplus will be zero. You are right it does not make sense whatsoever to reflect it as producer surplus.


Linear Expenditure System of Demands, Derivation Help

I am not sure I understand your question. $\lambda$ is a Lagrange multiplicator which has a value in the optimum. It is not a parameter and hence you cannot leave it in your solution.
 When solving the Lagrangian the optimal solution has the form $(x,\lambda)$ and in addition to your conditions
$$x_1 = \frac{\alpha}{\lambda p_1} + b_1$$
$$x_2 = \frac{\beta}{\lambda p_2} + b_2$$
$$x_3 = \frac{\gamma}{\lambda p_3} + b_3$$
it also has to fulfill
$$
(p \cdot x - w) \lambda = 0.
$$
If $\lambda \neq 0$ this also means $p \cdot x = w$. $\lambda$ cannot be equal to zero because it is in the denominator of your optimal solution for $x$:
$$x(p,w) = (b_1, b_2, b_3) + \left(\frac{\alpha}{\lambda p_1},\frac{\beta}{\lambda p_2},\frac{\gamma}{\lambda p_3}\right)$$
and hence you were correct to use the Walras law or budget constraint.

For more on optimality conditions see the Karush-Kuhn-Tucker theorem.

About $b_1, b_2$ and $b_3$: I think this is supposed to be a bliss point.


Efficient Market Hypothesis and constant ex-ante real interest rates?

To test market efficiency, you always need to specify the market's model of price formation

Tests of the efficient markets hypothesis must always include a model of how the market forms prices. One of Fama's big contributions was that you cannot separate these two things in a test. Tests of efficient markets and models of price formations are inherently linked. So, in this case, the assumption of constant real interest rates is merely an assumption of how the market forms prices. Only after assuming this, can we test market efficiency. You might disagree with the model---more sophisticated models will repeat this experiment in that way---but the point is that you have to assume some model for price formation.

Further reading

For more information about this, check out Fama's website. You'll find some links to some books of his. It would be helpful to read at least the first two sections of chapter 5 of his book, "Foundations of Finance."

The argument that he makes in that chapter is approximately the following (borrowing his notation, but changing $\phi$ to $I$).
The point that he makes is that we want to test if the information sets are equal
$I_{t-1}^m = I_{t-1}$, where $I_{t-1}^m$ is the information that the market possesses. But because we can't test this directly, we would like to test
whether the distributions of prices are the same
$$
f_m(p_1,...,p_n \mid I_{t-1}^m) = f(p_1,..., p_n \mid I_{t-1}).
$$
However, this is impossible too. The equality possesses no testable content because we only observe $f(p_1,..., p_n \mid I_{t-1})$ and not $f_m(p_1,...,p_n \mid I_{t-1}^m)$ (see the top of page 137 of the linked chapter). I do not observe the latter because I do not know what $I_{t-1}^m$, except that $I_{t-1}^m \subseteq I_{t-1}$, and I do not know how the market uses this information. For this reason, we specify a model for how the market takes information and turns it into prices. Thus, we specify $f_m$ ourselves (in turn, also specifying what information $I_{t-1}^m$ the market uses). That is, we specify what data the market uses and the way in which it uses that data.

On page 134, Fama says

  the statement that prices in an efficient market "fully reflect" available information conveys
  the general idea of what is meant by market efficiency, but the statement is too general to be
  testable. Since the goal is to test the extent to which the market is efficient, the proposition
  must be restated in a testable form. ... this requires a more detailed specification of the process of price formation, one that gives testable content to the term "fully reflect."

Why assume constant rates in the example in Hayashi?

I think it's just an assumption made in that particular example. If you read a little further into the linked chapter, you'll see that Fama discusses 4 different models of market equilibrium. The first two are ridiculous, but he's doing it just to demonstrate the concept. (Part of the reason is that some of the previous ideas about market efficiency had some bizarre consequences, which he demonstrates through those examples.) The point is that any test of market efficiency is always tied to the model that is assumed. If the test fails, you know one of two things: either the market is inefficient or your model of the market is wrong. The unfortunate truth, however, is that you will never know which one it is.


Applications to Green's Theorem in Economics?

Here and here are two application of the theorem to finance.

Here is an application to game theory.

This is an application of the theorem to complex Bayesian stuff (potentially useful in econometrics).

Also, the Green's theorem is used in this proof, which relates to dynamics system, and therefore could be related to economic models (source of this here)

Last but surely not least, this book might have relevant stuff. At least Chapter 12 does use it.


Are there any other rational preference relations without utility function representations, besides Lexicographic?

Yes, there are many. Here are some examples :

Consider the weak preference relation $\succsim$ defined over $\mathbb{R}^2$ as:

Example 1

$(x_1, y_1) \succsim (x_2, y_2)$

if and only if

either ($x_1+y_1 > x_2 + y_2$) or ($x_1+y_1 = x_2 + y_2$ and $x_1 \geq x_2$)

Claim : $\succsim$ cannot be represented by a utility function.

Proof : Suppose by contradiction that there existed a utility function $u$ representing these preferences. For each $a > 0$, we have $(a, 0) \succ (0, a)$, and therefore, $u(a, 0) > u(0, a)$. We can therefore assign to $a$ a non-degenerate interval of values satisfying the above inequality
$I(a) = [u(0, a), u(a, 0)]$. For any $a > b > 0$, all commodity bundles generating utilities in the interval $I(a)$ are strictly preferred to those in the disjoint interval $I(b)$ and should therefore be assigned a greater utility level. Then in each of these intervals we can pick a distinct rational number in increasing order to represent preferences. Since $a \in \mathbb{R_{++}}$, there are uncountably many such intervals, but set of rational numbers are countable. This results in a contradiction.

Example 2

$(x_1, y_1) \succsim (x_2, y_2)$

if and only if

either ($\min(x_1,y_1) > \min(x_2,y_2)$) or ($\min(x_1,y_1) =  \min(x_2,y_2)$ and $x_1+y_1 \geq x_2 + y_2$)

Claim : $\succsim$ cannot be represented by a utility function.

Proof : Suppose by contradiction that there existed a utility function $u$ representing these preferences. For each $a > 0$, we have $(a+1, a) \succ (a, a)$, and therefore, $u(a+1, a) > u(a, a)$. We can therefore assign to $a$ a non-degenerate interval of values satisfying the above inequality
$I(a) = [u(a, a), u(a+1, a)]$. For any $a > b > 0$, all commodity bundles generating utilities in the interval $I(a)$ are strictly preferred to those in the disjoint interval $I(b)$ and should therefore be assigned a greater utility level. Then in each of these interval we can pick a distinct rational number in increasing order to represent preferences. Since $a \in \mathbb{R_{++}}$, there are uncountably many such intervals, but set of rational numbers are countable. This results in a contradiction.


Malaysia's national debt - who's correct?

From a quick glance, it looks like the difference is that the new finance minister is including promised payments that the government had previously made, but hasn't had to pay yet. For example, say I borrow \$10 from you. At the same time, say I had previously promised (in an unrelated interaction) to give you \$5 at the start of June. What's my current debt to you? Officially, it's \$10 (from the loan you had given me). But come the first of June, if I haven't paid you back yet, I'll find myself in debt to you for \$15: \$10 from the loan, and the additional \$5 from the promise. Effectively, the new finance minister is counting the debt to be the entire \$15 (that is, current debt obligations, as well as promised future payments to different projects in Malaysia), while the official figure only considers my debt to be \$10 (since I don't yet technically owe you the additional \$5).

Technically, the government official figures are correct, since the government can "get out" of the future payments by (I'd assume) changing the law to remove themselves from that responsibility. At the same time, since (it looks like) those payments are to Malaysian government/private company partnership projects, simply refusing to pay would ultimately undermine Malaysian projects, which would hurt citizens. Which is more important depends on the context, and the interests of the individual.


Why is the Marginal Cost (MC) of a monopoly horizontal

That is basically an assumption here. Often in monopoly problems we assume constant marginal costs (i.e. a linear cost function) to keep things simple. In that case the Marginal Cost Curve is horizontal in the graph.


Two  types  of  innovation,  namely  innovation  for  new  services  and  innovation  for  alternative  network  infrastructures,  underlie  competition  in  the  telecommunications  industry.  While  innovation  for  new  services  is  provided  mainly  by   telecommunications   operators,   equipment   suppliers   provide   most   of   the   innovation  for  new  network  technologies.  A  network  innovation  in  the  equipment  sector   is   followed   by   an   adoption   process   in   the   telecommunications   sector.   Operators have to decide whether and when to adopt the new technology. Indeed, an immediate   adoption   may   be   costly   and   risky.   Therefore,   adoption   resembles   innovation.   In   the   rest   of   the   paper,   innovation   will   refer   to   both   invention   (endogenous innovation) and adoption (exogenous innovation).

Generally,  regulation  can  affect  these  innovative  activities  via  two  different  channels.    First,    price    regulations    (or    more    specifically,    the    regulation    of    interconnection  charges  and  retail  prices)  alter  industry  profits,  hence  the  incentives  to innovate. Secondly, both price or entry regulations change the terms of entry, and hence innovation decisions regarding new entry.

Regulation and Innovation: M Bourreau and P Doğan 1 Regulation and Innovation in the Telecommunications Industry
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.477.412&rep=rep1&type=pdf
CLECs are given incentives to deploy capital investments that innovate in terms of services and network technology. ILECs are forced to offer competitive services and improve network technology to keep market share away from the CLECs. In the counterfactual world of natural monopoly the ILEC would not have incentives to innovate services or network technology. The ILEC would profit from investments in POTS - Plain Old Telephone Service - for a considerable period of time without any competitive forces making it adopt innovative services or network technology.


Substitution effect when price of both goods change by the same percentage

Consider the budget line with 2 goods:
\begin{equation}
p_xx+p_yy=m
\quad\text{or}\quad
y=\frac{m}{p_y}-\frac{p_x}{p_y}x,
\end{equation}
where the relative price is $\frac{p_x}{p_y}$.

Suppose now both goods become $10\%$ more expensive (e.g. due to a $10\%$ tax on them). Then the budget line becomes
\begin{equation}
1.1p_xx+1.1p_yy=m
\quad\Leftrightarrow\quad
p_xx+p_yy=\frac{m}{1.1}
\quad\text{or}\quad
y=\frac{m}{1.1p_y}-\frac{1.1p_x}{1.1p_y}x=\frac{m}{1.1p_y}-\frac{p_x}{p_y}x
\end{equation}
where the relative price is still $\frac{p_x}{p_y}$. From the equations, we see that a uniform percentage change in prices produces a pure income effect ($m$ drops to $m/1.1$), but no substitution effect, since the relative price remain the same after the change.


Won't countries be in trouble if all sovereign debtors demand payment?

The principal has to be repaid eventually, but with majority of formal debt instruments such as government bonds the creditor can’t just demand repayment whenever he  wants. Most debt instruments specify the date, and if not date, then conditions when and how the principal is being paid back.

For example, a standard 10 year bond will pay its principal in 10 years not a day earlier, unless you convince the debtor to voluntarily agree to pay it sooner).

The above works for sovereign countries too. A 10 year government bond will have on it printed that principal is paid back exactly after 10 years.

Now this is outside economics, but technically sovereign nations are in state of anarchy. So I can imagine one country could just decide to say to another — pay now or we will invade you/nuke you etc! But aside some military intervention, you can't legally force the country to pay.

In fact any government can, in principle, default on its debts anytime it wants. The only constraint on sovereign default is that if you constantly default on your debt, nobody will want to lend you money in the future, or if only at punitively high interest. Hence on sovereign level, governments as debtors have much better bargaining power than just individuals.

Government can be forced to pay its debt only through military action, or in some cases by diplomatic pressure (for example EU could threaten Greece by revoking its euro-zone or EU membership, US could threaten countries by economic sanctions etc.). Aside from that, there is nothing other country can do to get its money back if debtor nation says no.

Based on 1, and 2 you probably already know the answer. No that’s not possible. China could patiently wait for bonds to expire and then not buy new US bonds, but instead buy Australian or New Zealand bonds. Of course, since US government issues a lot of debt, bonds expire probably every month and are replaced by new ones, so China could start to refuse to buy new US bonds sooner than later.

Also, usually when people want to get rid of a debt, they won’t wait for maturity and for repayment of the principal. They will take the bond to bond market and sell it at market price to get money now. That creates a trouble for country because now it has to set lower price or higher interest on its newly issued bonds due to increased supply, but whether this becomes a problem depends on plethora of factors unrelated to the original debtor and creditor countries, such as how big is the demand for the debtors bonds etc.


Help with a choice under uncertainty exercise

Suppose $v=\alpha + \beta u$ for $\beta > 0$ i.e. $v$ is a positive affine transformation of $u$, then
$$ \frac{v(g^1) - v(g^2)}{v(g^2) - v(g^3)} = \frac{\alpha + \beta u(g^1) - \alpha - \beta u(g^2)}{\alpha + \beta u(g^2) - \alpha - \beta u(g^3)}  = \frac{u(g^1) - u(g^2)}{u(g^2) - u(g^3)}$$

Converse is not true because $v=-u$ also satisfy

$$ \frac{v(g^1) - v(g^2)}{v(g^2) - v(g^3)}   = \frac{u(g^1) - u(g^2)}{u(g^2) - u(g^3)}$$

for all $g_1, g_2, g_3\in\mathcal{G}$.

Added Later

By the way, converse is also true. We are given this information that $u$ and $v$ both represent $\succsim$. So, $v=-u$ is not a valid example as they cannot both represent $\succsim$.

Here is the proof of the converse:

Fix any two lotteries $b$ and $w$ such that $u(b) > u(w)$ and consequently, $v(b) > v(w)$. For any lottery $g\in\mathcal{G}$, we know that the following is true

$$ \frac{v(g) - v(w)}{v(w) - v(b)}   = \frac{u(g) - u(w)}{u(w) - u(b)}$$

So, $$v(g) = v(w) + (v(w) - v(b))\frac{u(g) - u(w)}{u(w) - u(b)} $$
or equivalently,

\begin{eqnarray*} v(g) & = & v(w)  - \frac{v(w) - v(b)}{u(w) - u(b)}u(w)+ \frac{v(w) - v(b)}{u(w) - u(b)}u(g) \\ & = &  \left(\frac{u(w)v(b) - v(w)u(b)}{u(w) - u(b)}\right)+ \left(\frac{v(w) - v(b)}{u(w) - u(b)}\right)u(g) \end{eqnarray*}

Therefore, $v = \alpha + \beta u$ where $\displaystyle\alpha = \frac{u(w)v(b) - v(w)u(b)}{u(w) - u(b)}$ and $\displaystyle\beta = \frac{v(w) - v(b)}{u(w) - u(b)} > 0$.


Has anyone ever done an analysis of the flexibility of the demand curve for summer camps?

It seems not, judging for the results of this and this search (but you might want to take a look yourself).

There are some more informal analysis of the market for summer camps (e.g. here) and some organisations recollecting some form of data (e.g. here). The former is interesting because it shows, unsurprisingly, that the services proved by a summer camp are very heterogeneous. They are also targeted to different audiences. This means that the good itself is heterogeneous, so thinking of a unified "demand for summer camps" might be problematic.

Still, there are studies estimating demand for such heterogeneous goods. Here (subscription-wall) is an example which perhaps might be related to your interest (not the least because it uses the word camp too, which is how I found it, btw). The study estimates the demand for camping park's amenities. The key is that these amenities differ across camping sites (e.g. electricity access, natural beauty, etc). The authors estimate, among other things, the "price and income elasticity" for 48 camping parks in a region of Canada. The main finding with respect to elasticities are:

  ... it is immediately evident that there exists no definite pattern with regard
  to the price elasticity; there appear to be almost as many cases of
  inelastic as there are instances of elastic demand. [...] Since our results also indicate a negative income elasticity, higher user fees would represent a regressive form of taxation. The principle of user pay may still be justified on efficiency, not equity, grounds.

If you want to do some form of econometric analysis of the market for summer camps, I think this method is a possible approach.


Is there an economic analysis of the rationality of buying lottery tickets?

There are definitely economic justifications for playing the lottery, even if all (I hope) players understand that it is unlikely to pay off.

One such justification is that what you actually buy when purchasing a lottery ticket is the fantasy of winning.

Here are a few sources. Lotterys are relatively well understood in economics.

The Economics of Lotteries: A Survey of the Literature(pdf)  - is an excellent article covering basically your entire question. It discusses the microeconomic forces at play, particularly income elasticity and risk aversion.
This source (note: this is now a dead link) has some excellent lecture notes on the market for risk. I recommend it highly. Some basic points are:

people don't want actuarialy fair games, often they would decline a coin flip with a payoff of \$1mil for a win and a cost of \$1m for a loss. Also, an actuarialy favourable game isn't necessarily desirable. If the payoff was \$1.1m people would still decline the game as the cost is just so high. In that sense, a lottery has an advantage. The payoff can be huge but the cost in the event of losing is almost negligible.

Lastly, this source is an article from the NY times. Definitely relevant, highlighting similar points to the first two articles but is a lot more accessible.


Does a price on the demand curve represent the highest or lowest price the consumer is willing to pay?

A distinction must be made between the individual demand curve and the aggregate demand curve.

An individual demand curve (or demand schedule) for a product/service, represents the pairs {price, quantity demanded} that co-optimize the consumer's utility maximization problem (together with all other quantities demanded of the other products that are present in the utility function).
It may appear that then this would be the "maximum" price, but it would be confusing to characterize it like that. Because, if the price changes, what would happen to individual quantity demanded depends on whether the good is normal or not, and in general whether there are "income-effects" or not (not all utility functions allow for income effects, by the way).
Moreover, at the individual level, the concept representing the consumer is quantity demanded, not price: the consumer observes prices and responds with quantity demanded.

Let's move now to the aggregate demand curve. It is simply the sum of individual demand curves for each price. Here we have two alternative approaches:

a) The more "traditional" one is that all consumers have a non-zero demand schedule for all price levels, each lowering its demand as price increases. Here the previous discussion applies.

b) The alternative but equally "standard" approach is to argue that for each consumer there exists a (possibly different) "threshold" price, above which the individual demands zero quantity. So as price decreases, we imagine that more consumers enter the market with non-zero demand, and this is why as price decreases aggregate quantity demanded increases.
This approach implies that the price level in each price/quantity combination is the minimum ("at least") price for some consumers, while others would be willing to pay more (and so they have contributed positive quantity demanded for higher price levels).

I believe this is the approach taken by the textbook.


Walrasian Equilibrium intuition given prices and some initial allocation

The "trick" of this question is that the fact that agents do not want to trade at the given prices does not mean the allocation is Pareto. The only thing you know is that if there is an allocation that is not a CE, there exists a vector of prices at which agents want to trade. Let me give an example:

You have 2 apples and I have 2 bananas. I'm willing to give you half a banana for an apple and you are willing to exchange a whole apple for a bite of a banana. Clearly, the initial allocation is not Pareto Efficient, nor a CE since it is profitable to trade and trade will occur if prices are somewhere between a bite of a banana and half a banana for an apple. The problem, however, assumes that prices are given. So in this toy example, if the price of an apple is set to one banana, there will be no trade (since I won't be willing to accept the prices).

From the lack of trades, it cannot be assumed that you are in a CE since the prices are not allowed to adjust to allow for mutually beneficial trading.


Fama Macbeth and double clustering presents inconsistent results

This is a large comment.

Fama-MacBeth only corrects for cross-sectional correlation in a panel and suffers from the error-in-variables problem (your $Z_t$, also check Chordia, Goyal and Shanken (2015)).

Depending on the exact nature of your dataset's variables the answer to your questions is "it depends"; please check Goyal (2012), Mitchell (2009) and Ibragimov and Mueller (2010).

Goyal (2012) accurately explains the F-M procedure (section 2.5). He states that "whether there is a bias in the traditional Fama–MacBeth approach if expected returns vary with time-varying characteristics is still unexplored.". Moreover, he says that "autocorrelation in returns (negligible at monthly frequency) leads to autocorrelation in risk premium estimates. This is easily accounted for by Newey–
West type corrections to variance formulas".
Mitchell (2009) discusses the SEs in financial panel data and clearly states that we should identify the presence of a firm or time effect to see whether the Fama-MacBeth standard errors are unbiased.
Ibragimov and Mueller (2010) "find that as long as year
coefficient estimators are approximately normal (or scale mixtures
of normals) and independent, the Fama–MacBeth method
results in valid inference even for a short panel that is heterogenous
over time."

You might also find Jagannathan, Skoulakis and Wang. (2014) enlightening to make proper arguments on the appropriateness of the methods if your study is asset pricing related; in corporate finance things are more "flexible".

Last, I am far from an expert in this area, but I think the "pre-made" Stata commands are not exhaustive in dealing with variables with different statistical characteristics (e.g. autocorrelated returns).


Screw foreign investment, lets just print money

"Foreign Direct Investment" is to be understood as bringing in an economy productive capital, and not just purchasing power.
When an economy is seriously below full employment (of capital and labor), then a case can be made that "printing money" (i.e. creating purchasing power out of thin air) may not result in just inflation, but it may indeed bring into production input factors (labor and capital) that are currently unemployed. Compared to Foreign Direct Investment, we have the following differences:
Creating purchasing power through printing money relies on an indirect effect: you strengthen demand and expect that this will lead to increased supply, which in turn translates into higher employment/output. This may mitigated by
a) increased imports rather than increased local factor employment to satisfy the demand
b) institutional issues that may obstruct bringing into production the idle local factors
c) imbalance in available unemployed factors - e.g. you may have too many unemployed people but not much unemployed infrastructure.
Foreign Direct Investment in almost all cases is partly translated into imports (the foreign investors bring in their funds which are then partly paid back abroad to import capital infrastructure that is not available in the local economy), but apart from that similarity it
a) increases supply and local employment directly and
b) by design it takes a priori into account the possible imbalance in the availability of local factors, and corrects from them, since it wants to arrive at a factor mix that will be able to produce.
In a more general perspective, we could say that by printing money, if the policy fails you will just burden the economy with inflation (which has transactional costs). But with Foreign Direct Investment, even if it fails eventually as a commercial enterprise, the local economy has nevertheless gained, even as a one-off, the part of the FDI that has been consumed locally: it is an inflow of real output from the outside world.


Can I use calculated data for regression

I'm afraid the accepted answer is not precise enough, and can be deeply misleading.

This issue is formally called the Generated Regression problem which arises when we use "generated" regressors in a regression. This is a very common feature of estimations like Heckman Selection Model, where we use the predicted Inverse Mills Ratio from the selection equation in the structural equation. We need to divide the issue in two:

Consistency: there is usually no problem of consistency when using generated regressors in an estimation. A formal proof can be found in Wooldridge (2010), page 123.
Statistical inference: here there might be problems. Under the null that generated regressors are zero, then standard errors and all tests are valid (you might still need to correct for heteroscedasticity and serial correlation as usual, using robust standard errors). However, if the generated regressor is statistically significant, then the standard errors and in consequence any test are invalid. What you need is a correct estimation of the asymptotic variance. The formulas depend on each case (more information in Newey and McFadden [1994]). However, a general solution to calculate the consistent standard errors is to use bootstrapping. As an aside, for example, if you use Stata, just run your normal command with the following code before:

bootstrap, reps(400) seed(10101): regres ...

reps is the number of repetitions and seed is for reproducibility.

TL;DR

If your generated regressor is statistically significant in your regression, then you need to estimate the standard errors using boostrapping because they are invalid. If they are insignificant, you are safe.


What is an unconditional model for a time series variable?

So the simplest response would be that an unconditional model is a model that does not include any other stochastic regressors. Lets say your variable of interest is $Y_t$ then a conditional model would be

$Y_t = \alpha + X_t\beta + v_t$

where you are implicitly conditioning on $X_t$ (i.e. treating it as fixed/exogenous). In time series $X_t$ can also include lags or transformations of $Y_t$. In the example that you gave, you could think of inflation (or lags of GDP) as potential conditioning variables.

So then, given this, what is an unconditional model? Well it is a model that does not include $X_t$ and instead only has deterministic terms (such as a constant or a trend). That is to say in the context of GDP, GDP growth doesn't depend on (i.e. isn't conditional upon) another variable but rather grows at a constant rate (or has a linear trend).


Intuition: Can someone explain me this contradiction please?

Figure 2 is a bit misleading. Both demand curves have an elastic and an inelastic region, just like the one in Figure 1. The meaning of "High elasticity " and "Low elasticity" in Figure 2 is that the steeper line has a much lower magnitude of elasticity than the flater line at the same quantity level.


How to choose instruments for GMM estimation?

As jmbejara says, better to be more specific, but let me guess.

You have a linear model $y = \beta_0 + \beta_1 x_1 + \cdots + \beta_{10} x_{10} + u$, where $x_1, x_2, \ldots, x_{10}$ are endogenous. Then, yes, you need at least ten instruments.

Can you use the same dependent variables as instruments for the regression? No, you can't. Instruments should be exogenous (uncorrelated with $u$) and relevant (strongly correlated with the endogenous regressors). But if some of the regressors are (believed to be) exogenous, then you use those exogenous ones as instruments. But you still need extra instruments for endogenous regressors. The rule is that you need at least $k$ extra instruments if there are $k$ endogenous regressors.

Finding good instruments (exogenous and relevant) is usually difficult. I suspect it would be really hard to find 10 instruments for a regression unless you have good economic justification.

That said, again it would be better to see your specific problem.


Study whether $\succsim$ represented by $u(x)=\lfloor x \rfloor$ is continuous

Are you sure that the definition you want?

With your definition then yes its true.

You should show that $x\succ y\succ z\implies x>y>z$. Hence, you can always find a $\alpha$ such that $\alpha x+(1-\alpha)y=z$, so $u(\alpha x+(1-\alpha)y)=u(z)$ which then implies $\alpha x+(1-\alpha)y\sim z$.

Perhaps the definition you want to use is for all $x^n$ and $y^n$ are two sequences with $x^n\to x$ and $y^n\to y$ and $x^n\succeq y^n$ for all $n$ then $x\succeq y$? But maybe not, I dont want to presume.


What happens if the "control variables" are also endogenous?

"But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?"

I don't want to emphasize this too much, but it's worth mentioning that this is not true in general. The following derivation will hopefully provide some understanding of the "contamination" you mention. As a simple counterexample, suppose that the data generating process is given by
$$
Y = X_1 \beta_1 + X_2 \beta_2 + Z \gamma + \varepsilon,
$$
where $Z$ is unobserved. Let $Cov(X_1,Z) = 0$, $Cov(X_2, Z) \neq 0$, and
$Cov(X_1,X_2) = 0$. Then, it
is clear that $X_2$ is "endogenous." But notice that because $Cov(X_1,Z) = 0$, our
estimate of $\beta_1$ will still be ok:
$$
\text{plim}\, \hat \beta_{1} = \beta_1 + \gamma \frac{Cov(X_1^*, Z)}{Var(X_1^*)} = \beta_1,
$$
where $X_1^* = M_2 X_1$ and $M_2 = [I - X_2(X_2'X_2)^{-1}X_2']$. Because $Cov(X_1,X_2) = 0$, $X_1^* = X_1$. So $Cov(X_1^*,Z)=0$.

"What can we do?"

One of the mains challenges of doing good econometrics is thinking of potential identification strategies. In the type of situation you describe, there is probably nothing you can do but to try to approach the problem a different way.


Method of Lagrange multipliers with random variables

If you treat $c_2$ as if it is a random variable, then the derivative of
  $E[u(c_2)]$ with respect to $c_2$ gives zero.

Why? Such an assertion doesn't follow from anywhere. The subtleties lie elsewhere. The problem with the Lagrangian considered by the OP

$$\mathcal{L} = u(c_1) + \mathbb{E}[u(c_2)] + \lambda(c_2 - f(c_1,Z))$$

is that it also is a random variable since now $Z$ appears outside the expected value (The direct substitution approach followed by maximization with respect to $c_1$ only, does not create any such issues).

Now, do we /  can we maximize a random variable? Well, no, because the essential characteristic of a random variable is that it is a function whose value cannot be set by command and control.

But one could say "ok, let's pretend that this Lagrangian is not a random variable, and just write down the conditions for maximization, even though we know that we can't force the solution".

But this won't work: if one attempts to do it one will eventually obtain

$$u'(c_1) + \mathbb{E}\left[u'(f(c_1,Z))\right] \cdot  \frac{\partial f(c_1,Z)}{\partial c_1} = 0 $$

which is not the same as the condition obtained through direct substitution, because here the partial derivative is outside the expected value.

(for those who may think "hey, then how do we apply maximization procedures in the maximum likelihood approach" the answer is that there, nothing is random anymore when we get to apply the maximization steps).


Willingness to accept is different from willingness to pay - is there a name for this

The discrepancy between WTP and WTA is known as the endowment effect, and this has been extensively studied in behavioral economics.


How to interpret coefficients in a dynamic OLS model?

yes the way how your model is set up $\beta_1$ would be immediate effect/multiplier and $\beta_1+\beta_2+\beta_3$ the long-run one.
However, an important caveat is that this is due to the way how you set up your model and not a general result. For example, in an ARDL model with stationary variables of the following form:
$$y_t = \alpha + \beta_1 y_{t-1} + \gamma_1 x_t + \gamma_2 x_{t-1}+ e_t$$
the long run multiplier would actually become: $ \frac{\gamma_1 + \gamma_2}{1 - \beta_1}$
or in more general case
$$y_t = \alpha + \sum_{p=1} \beta_p y_{t-p} + \sum_{q=1} \gamma_q x_{t-q+1} +e_t$$
the long run multiplier would be given by: $\frac{\gamma_1+\gamma_2+...+ \gamma_q}{1-\beta_1-\beta_2-...-\beta_p}$.
In your case you do not include any lags of dependent variable so you have a special case where the denominator is 1 and hence it is enough to add in the coefficients but I thought it might be good to mention that as long as you include lagged dependent variable the calculation of the long run multiplier changes (see Verbeek (2008) guide to modern econometrics for more details).


Simplifying a monoplist's objective function under incomplete information

I'll start from your last equation.

\begin{align*}
\int_{\underline{\theta}}^{\bar{\theta}}{u(\theta) dF(\theta)} & = \Big[ u(\theta)F(\theta) \Big]_{\underline{\theta}}^{\bar{\theta}} - \int_{\underline{\theta}}^{\bar{\theta}}{F(\theta) v_{\theta}(q(\theta),\theta) d\theta} \\
& = u(\bar{\theta})F(\bar{\theta})-u(\underline{\theta})F(\underline{\theta})-\int_{\underline{\theta}}^{\bar{\theta}}{F(\theta) v_{\theta}(q(\theta),\theta) d\theta} \\
& = u(\bar{\theta})-\int_{\underline{\theta}}^{\bar{\theta}}{F(\theta) v_{\theta}(q(\theta),\theta) d\theta} \text{ since } F(\bar{\theta})=1, F(\underline{\theta})=0\\
& = u(\underline{\theta})+(u(\bar{\theta})-u(\underline{\theta}))-\int_{\underline{\theta}}^{\bar{\theta}}{F(\theta) v_{\theta}(q(\theta),\theta) d\theta} \\
& = u(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}}{u'(\theta) d\theta}-\int_{\underline{\theta}}^{\bar{\theta}}{F(\theta) v_{\theta}(q(\theta),\theta) d\theta} \\
& = u(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}}{v_{\theta}(q(\theta),\theta) d\theta}-\int_{\underline{\theta}}^{\bar{\theta}}{F(\theta) v_{\theta}(q(\theta),\theta) d\theta} \text{ since } u'=v_{\theta}\\
& = u(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}}{(1-F(\theta)) v_{\theta}(q(\theta),\theta) d\theta} \\
& = u(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}}{\dfrac{1-F(\theta)}{f(\theta)} v_{\theta}(q(\theta),\theta) dF(\theta)} \\
& = \int_{\underline{\theta}}^{\bar{\theta}}{u(\underline{\theta})dF(\theta)}+\int_{\underline{\theta}}^{\bar{\theta}}{\dfrac{1-F(\theta)}{f(\theta)} v_{\theta}(q(\theta),\theta) dF(\theta)} \\
& = \int_{\underline{\theta}}^{\bar{\theta}}{\Big[u(\underline{\theta})+\dfrac{1-F(\theta)}{f(\theta)} v_{\theta}(q(\theta),\theta) \Big]dF(\theta)}
\end{align*}


What are the differences between hedging with swaps, options or futures?

The scope of this question is quite large. There are a lot of ways of hedging interest rate risk. Note that there is a quantitative finance stack exchange, and this question might fit better there.

All rates instruments have a sensitivity to interest rate changes. That is, their value goes up or down in response to interest rate movements.

If we assume that all interest rates move together in parallel (that is, spreads are unchanged), all you need to do to hedge interest rate risk is find instruments that have a value change that cancels the risk you want to hedge.

To a certain extent, most non-credit rates are correlated, and so one can use a wide variety of instruments to hedge.

However, rate movements are not perfectly correlated. As such, hedges may need to be structured to have a sensitivity closer to the risk to be hedged. How that is done is the full time job of interest rate traders. That is, too large a scope to be ansered in a single question here.


Is this Cost function concave or convex?

"In the context of economics", it is neither - because convexity in economics means the same it does in math. If you redefine its domain, such that it's only computed for any $t > 4.12$, this function's image is clearly convex: no line between two vectors inside it isn't entirely contained in the set.

P.S: Although convexity has the same definition in both fields, it has an important economic interpretation and is a very important hypothesis in many classical general equilibrium models.


Reading list: history of economic thought / political economy

My app is acting odd and won't allow me to comment, but here's my follow up comment.

Yeah, for sure! My list consists of this:

Wealth of Nations - Adam Smith
Capital - Karl Marx
Principles of Economics - Alfred Marshall
General Theory of Employment, Interest and Money - John Maynard Keynes.

Those are just the most definitive works in the field that I've found, so there are many more I've yet to discover. If I remember correctly I think Marshall put most of his analytic work in the appendices at the back of his Principles. Marshall had the body of his work written - not in math, but in words - to be concise, but yet understandable for most of the laymen of his day.

Anyways, I pulled out those four books because they establish a turning point in some way for the field (i.e. General Theory caused a bifurcation between macro and microeconomics where it left Marshall's Principles for micro and Keynes for macro.)

If you are pressed for time, then I'd suggest reading Backhouse's book along with the Teachings book by Heilbroner. Those will give you a very broad and quick view of where the fields come from and almost catch you up to modern day economics. And unfortunately a lot of the classics are lengthy and hard to decipher. Karl Marx writes his critique of capitalism in a full three or four volume set! (I can't remember the exact number.)

I think if you approach this adventure in the right manner, you may become hooked. That's how I got fascinated with the field, I read Worldly Philosophers when I was younger and just fell head over heels for the subject.

Hopefully this helps out, just let me know if you have any more questions or concerns.


There is quite a bit of work being done in that area. One very recent example is Straub and Werning's working paper "Positive Long Run Capital Taxation: Chamley-Juff Revisited."
The point seems to be that we need to consider the rate of convergence to the steady state.

Also, there is other literature that gives some competing results (e.g., "the new dynamic public finance" literature). For a more complete summary of mainstream objections, see Diamond and Saez, "The Case for a Progressive Tax: From Basic Research to Policy Recommendations," (JEP, 2011). They devote a whole section to why they think the Chamley-Judd result is not policy relevant.


Is there a proof for composite commodity theorem?

There is this nice expository note:
Carter M., 1995, "An expository note on the composite commodity theorem," Economic Theory, 5, 175-179
and an interesting generalization by
Lewbel, A., 1996, "Aggregation without Separability: A Generalized Composite Commodity Theorem," American Economic Review, 86, 524-543.

EDIT:
One reason why it is difficult to find a proof of the theorem, is that the so called "composite commodity theorem" does not really have the mathematical status of a theorem.
It is more a principle directly following from the reparameterization below.
Let $x\in \mathbb{R}^J_+$ denotes the disaggregate demand system, which depends on the price vector $p \in \mathbb{R}^J_+$, and the budget $b>0$. The relative prices are denoted by $\alpha = p/\textbf{p}$ where the aggregate price index is $\textbf{p} \in \mathbb{R_+}$. If $\alpha $ is a vector of constants, we can simplify the demand system such that it depends only upon the aggregate price $\bf{p}$:
$$ x(p,b) = x(\alpha \mathbf{p},b) \equiv \mathbf{x}(\mathbf{p},b).  $$
Note that now $\mathbf{x}:\mathbb{R^2_+} \rightarrow \mathbb{R}^J_+$.
If one is also interested in aggregating the elementary quantities, it is (for instance) possible to define the aggregate composite quantity as:
$$ \mathbf{X}(\mathbf{p},b) = \alpha^T\mathbf{x}(\mathbf{p},b),  $$ where $ \mathbf{X}:\mathbb{R^2_+} \rightarrow \mathbb{R}_+$. In this case, the elementary and aggregate expenditures are the same, in the sense that: $$\mathbf{p}\mathbf{X}(\mathbf{p},b)=p^Tx(p,b).$$


Suppose $(\sigma_1^*, \sigma_2^*)$ is the Nash Equilibrium of the game $G$ consisting of players $\{1,2\}$ having strategy sets $A_1$ and $A_2$ and the payoff functions $u_1:A_1\times A_2 \rightarrow \mathbb{R}$ and $u_2:A_1\times A_2 \rightarrow \mathbb{R}$. Consider the game $G'$ in which everything else is same as $G$ except that the payoff functions are $v_1:A_1\times A_2 \rightarrow \mathbb{R}$ and $v_2:A_1\times A_2 \rightarrow \mathbb{R}$ defined as: $v_1(a_1, a_2) = u_1(a_1, a_2) + d_1$ and $v_2(a_1, a_2) = u_2(a_1, a_2) + d_2$. We will now show that $(\sigma_1^*, \sigma_2^*)$ is also the Nash Equilibrium of $G'$.

Since $(\sigma_1^*, \sigma_2^*)$ is the Nash Equilibrium of $G$,
\begin{eqnarray*} \mathbb{E}(u_1(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(u_1(\sigma_1, \sigma_2^*)) & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(u_2(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(u_2(\sigma_1^*, \sigma_2)) & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

The above can also be rewritten as
\begin{eqnarray*} \mathbb{E}(u_1(\sigma_1^*, \sigma_2^*)) + d_1 \geq  \mathbb{E}(u_1(\sigma_1, \sigma_2^*)) + d_1 & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(u_2(\sigma_1^*, \sigma_2^*)) + d_2 \geq  \mathbb{E}(u_2(\sigma_1^*, \sigma_2)) + d_2 & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

Because $d_1$ and $d_2$ are constants, we can take them inside the expectation and rewrite the above inequalities as:
\begin{eqnarray*} \mathbb{E}(u_1(\sigma_1^*, \sigma_2^*) + d_1) \geq  \mathbb{E}(u_1(\sigma_1, \sigma_2^*) + d_1) & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(u_2(\sigma_1^*, \sigma_2^*) + d_2) \geq  \mathbb{E}(u_2(\sigma_1^*, \sigma_2) + d_2) & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

By definition of $v_1$ and $v_2$,

\begin{eqnarray*} \mathbb{E}(v_1(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(v_1(\sigma_1, \sigma_2^*)) & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(v_2(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(v_2(\sigma_1^*, \sigma_2)) & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

Therefore, $(\sigma_1^*, \sigma_2^*)$ is also the Nash Equilibrium of $G'$.


Where does investment originate from in the circular flow model?

Investment ultimately comes from households through savings. From a macroeconomic perspective savings is equal to to investment ($S=I$).
Investment comes from income because saving is portion of income that is not consumed. For example, if your income is $\\\$1000$ and you consume $\\\$700$ of it the $\\\$300$ you are left it is by definition saving and saving is ultimately equal to investment.
For example, when you put $\\\$300$ into your bank account you are actually inserting it into financial sector and bank will actually use the money on your deposit account to finance loans for investment even if you might not know about it.
This is why the arrows are as they are. Households supply saving to financial markets where firms who need money to make investments get it from.
The economy grows because even though everything is a circle the income increases each year as economy becomes more productive thanks to improvements in technology. For example, if today you can make 100 widgets then your income will be equal to the value of those 100 widgets. If thanks to increase in technology you can produce 200 widgets your new income is equal to the value of those 200 widgets.


Does the US stock market tend to rise after natural disasters?

This news article with some statements from financial workers makes a case that usually big storms don't impact the national economy that much, even despite the large localized damages. While insurance companies will suffer in the stock market because of all those payouts they'll have to give, oil prices as you mentioned will be impacted, but in this case, gas prices will rise because of the change in the supply curve, so they will actually benefit in the stock market. The various effects combined end up with the stock market not really going up or down. It's more or less a wash.

To think of it another way, a shock in the short term capital in the economy won't particularly change the steady state level of capital. If markets know this, there isn't a need for huge changes in prices. In the case of the insurance companies, their money isn't so much based on capital as it is based on states of the world, so they will end up needing to adjust prices.

There are other reasons why the stock market may not have moved up or down in particular, but for the most part it is speculation, and even my answer is just intuitive exposition.


Example of a game with no Nash equilibria but at least one correlated equilibrium

There are three players, 1,2,3, and the action spaces of players 1 and 2 are both $[0,1]\times\mathbb{N}$, the action space of player 3 is $[0,1]$.
The generic action of player 1 is written as $(x,m)$, the generic action of player 2 is written as $(y,n)$, and the generic action of player 3 is written as $(x')$.

The payoff of player 3 is simply $1$ if $x'=x$ and $0$ otherwise.

The payoffs of players 1 and 2 are more complicated. The payoff of player 1 and 2 is both $2$ if $y=x\neq x'$, and both $-2$ if $x=x'$. In the case that $y\neq x\neq x'$, the player with the highest number in the second coordinate gets a payoff of $1$ and the one with the lower number gets $-1$. If they pick the same number, they get both $0$.

This game has no Nash equilibrium. Suppose there were one. If player 1 would not randomize in equilibrium, player 3 can match them for a lousy payoff of $-2$. By randomizing the payoff will be higher, but it will also be the case that with positive probability $y\neq x\neq x'$, since the behavior of all players is stochastically independent. But then both 1 and 2 have to play the game of picking a higher number and that game has no equilibrium even in mixed strategies (if mixed strategies must be countably additive).

But there is a correlated equilibrium in which the behavior of 1 and 2 is given by a distribution on the diagonal $D=\{(x,y)\mid x=y\}$ without mass points, and player 3 plays anything stochastically independent. Player 1 and 2 can clearly do no better and there is no way for 3 to match 1 with positive probability.


No, there is not. Consider a game with two players, Ann and Bob. Both choose such vectors with entries $0$ or $1$ of the form $(a_1,a_2,\ldots,a_J)$ or $(b_1,b_2,\ldots,b_J)$, respectively. If $\sum_{i=1}^J a_i+b_i$ is odd, Ann wins and Bob loses. If the number is even, Ann loses and Bob wins. Clearly, one of them loses in every profile of pure strategies, and the loser can win by changing only one entry of their chosen vector. So there is no pure strategy profile at which the revealed preference inequalities hold.


Did real interest rates decrease during the Great Recession

The Great Recession precipitated in the financial sector, as a liquidity crisis, as I'm sure you know. It didn't precipitate in the real sector. I would represent that in the theoretical framework as a decrease in the stock of money:

That is because there was a lack of liquidity. The evidence is that interest rates soared. That is, the inter-bank lending rate soared, along with the increase in the perceived riskiness of financial instruments. The above shift correlates to a leftward shift of the LM curve.

The constraint on credit had real effects—a recession. The traditional way to smooth a recession is through monetary policy, but the Fed reached the lower limit when lowering the federal funds rate—a so-called "liquidity trap". So further, non-traditional methods were needed to address the recession (such as quantitative easing, where risky securities were purchased by the Fed in an attempt to lower those interest rates, and they also sought to lower forward interest rates).


Monetary Transmission in New Keynesian - Euler Equation

Real interest rate = Nominal interest rate - Expected inflation

Reducing the nominal interest rate by more than the decline in the expected inflation, with respect to the inflation target, leads the real interest rate to go down.
When the real interest rate goes down, the real exchange rate depreciates, both lead real monetary conditions to loosen. [Real monetary conditions = a1*RR_gap + (1*a1)*RER_gap].
Looser real monetary conditions lead to higher aggregate demand 'output gap' (higher domestic demand due to lower real interest rate and higher external demand due to the real exchange rate depreciation). [IS curve - Euler equation]
Higher aggregate demand puts more pressure on the real marginal costs, which in addition to higher inflation expectations that are directly affected by reducing the nominal interest rate, increase the inflation rate.


You have to use the midpoint method to resolve this (if I recall correctly).

https://quickonomics.com/how-to-calculate-price-elasticities-using-the-midpoint-formula/

The reason I may not recall correctly is because the second you introduce calculus into the study of economics, you discard these formulae entirely and (try to) forget they ever existed.  You seem to already know calculus, which is probably causing you to be even more confused rather than less, and understandably so.

The primary purpose of these non-calculus formulations of elasticity is to teach you the intuition around what elasticity is as a concept - I definitely have strong opinions about how well it accomplishes that. :)

EDIT: Just wanted to add that, if indeed the problem is as you're saying it (come hell or high water, Billy is going to spend \$10 and only \$10 on gas) then your intuition is right - Billy has unit price elasticity of demand for gas, although in a kludgy sort of way.  This is supported by your calculus-based analysis.


My Challenge in Econ 101 explanation of Marginal Benefit = Marginal Cost

So, you assume a finite stock of money units and also that the only thing any consumer desires is units of happiness. That is, consumers are utility maximizers and they have utilities that are functions of happiness only.

Now, if you impose some arbitrary value - say, one unit of money - for one unit of happiness you could construct any silly example. Say, a frat guy loves beautiful women. He can win the affection of a beautiful woman by taking her on a date that costs 10 units of money. The first day late with a beautiful girl gives him 20 units of happiness. Clearly, the date is worth it! This is beacause he is getting 2 units of happiness per unit of money which has, more or less, doubled the value of his money. Suppose he keeps dating beautiful women and that the 5th date - that costs 10 units - gives him exactly ten units of happiness. Now, suppose he knows date 6 will only give 8 units of happiness. The marginal benefit of the date is too low. He keeps his money and instead trades his units of money 1-1 for units of happiness until he depletes his remaining stock of money.

.

Something like that. Any arbitrary example where you can clearly demonstrate that if $MB<MC$, you're "losing" something by continuing to consume that good and $MB>MC$ means you're "gaining" something by continuing to consume that good.

Also - I have no graphing tools (typing from phone) but this example makes use of constant MC and so gives a good visual for what happens when MB is above and/or below MC by presenting two very easily identified regions where the MC curve partitions the graph. Above the curve is "good" and below is "bad".

This usually works well for me.


How to adjust salaries of more than two countries to a common year given that the data was collected in different years?

Your three stage procedure is not correct.

In effect, since the data is already in USD, to apply PPP you need to revert it back into LCU (local currency units). That is your step 1. Have in mind this might introduce errors if the one you use is not the same the authors used.

Then, since wages are not for the same year, ideally you do need a proxy for wage growth of each occupation, for each country and year, so that you can compute wages in LCU for the same year. You are attempting to do this in step 3 using GDP growth, which is not ideal (and the order is wrong too). I would rather use an inflation index. This is not ideal either, but the World Bank provides an homogenised one. Alternatively, you could get wage growth for occupational groups (or aggregate) from national statistical offices, but this is a time-consuming exercise.

After putting wages into the same year, you proceed with step 2, where you convert LCU to USD PPP, using a common year.


What are the concepts in Linear Algebra that model the idea of Identification Strategy in Econometrics?

In econometrics, identification can take several forms depending on the type of model you are working with (see this survey for a more comprehensive description). However, in general, identification pertains to the idea that either your econometric model or the exogenous variation in your data (or both) allows you to tease out the parameters that truly describe the data generating process.

To be more specific, suppose that you are studying outcome $y$ which depends (stochastically) on $x$ so that the data is generated according to $F(y|x)$ where $F$ is some distribution function. Then you have an econometric model that depends on a parameter, $\theta$ (this can be a vector of parameters), and for each parameter, it spells out a conditional distribution of $y$ in terms of $x$. That is your econometric model is a function $M(\theta)=G(y|x; \theta)$. We say that the model is identified if there exists a unique parameter, $\theta^*$, in the domain of $M$ such that $M(\theta^*)=F(y|x)$.

In reduced-form work, $M$ is assumed to be linear (or in general to be an injective function and have a relatively simple form), so ensuring identification has more to do with your data, $F$, having the right properties (mainly having enough exogenous variation).

In contrast, structural work typically deals with data that is not experimental or quasi-experimental so $M$ cannot credibly be assumed linear. Instead, you develop a rich enough model and identification requires $M$ to be invertible. If $M$ is invertible, then you have that $M^{-1}(F(y|x))=\theta^*$.

I think that in relation with identification, linear algebra tools are mainly useful to prove the invertibility of $M$. However, it can prove also useful when it comes to designing an estimation strategy, and when implementing the estimation strategy in the computer, etc.


How to prove convexity + quasilinear preferences imply concave utility?

Consider any $x_2'$ and $x_2''$ in $\mathbb{R}_+$. Without loss of generality, let $x_2'' > x_2'$. We can choose $x_1'=f(x_2'')-f(x_2') > 0$ so that $U(0,x_2'')=U(x_1',x_2')$. Let $\lambda(x_1',x_2')+(1-\lambda)(0,x_2'')$ be a convex combination of $(x_1',x_2')$, and $(0,x_2'')$. Since $\succsim$ is strictly convex and $U(0,x_2'')=U(x_1',x_2')$,
        \begin{eqnarray*}
 && U(\lambda(x_1',x_2')+(1-\lambda)(0,x_2''))  >  U(x_1',x_2') \\
 &\Rightarrow & U(\lambda x_1'+(1-\lambda)0,\lambda x_2'+(1-\lambda)x_2'')  >  \lambda U(x_1',x_2')+(1-\lambda)U(0,x_2'')\ldots(\because U(0,x_2'')=U(x_1',x_2')) \\
 &\Rightarrow &  \lambda x_1'+(1-\lambda)0 + f(\lambda x_2'+(1-\lambda)x_2'') >  \lambda (x_1'+f(x_2'))+(1-\lambda)(0 + f(x_2''))\\
  &\Rightarrow &  \lambda x_1'+(1-\lambda)0 + f(\lambda x_2'+(1-\lambda)x_2'') >  \lambda x_1'+(1-\lambda)0 + \lambda f(x_2')+(1-\lambda)f(x_2'') \\
    &\Rightarrow &  f(\lambda x_2'+(1-\lambda)x_2'') >   \lambda f(x_2')+(1-\lambda)f(x_2'')\end{eqnarray*}
Therefore, $f(\cdot)$ is strict concave.


Why should small open economies avoid interest rate policies?

The trilemma or impossible trinity is said to be that it is impossible to have all three of the following at the same time:

a fixed foreign exchange rate
free capital movement (absence of capital controls)
an independent monetary policy

The United States prioritises the second and third, and only worries slightly about the first and its impact on trade.  Being an extremely large economy, capital inflows can only make a small difference to domestic market conditions, though during the world financial crisis some moments of fear did tend to push the relative price of the dollar up as international money sought a safe haven (abbreviated to risk-off sentiments). Currently it seems that the US can make moderate increases to interest rates in contrast to Europe or Japan, without a major impact on US exchange rates

China prioritises the third and also the first in a managed exchange rate sense.  There are significant capital controls in China (more now than a few years ago), and they make it difficult for short-term capital flows to cross the border in either direction.  If somebody abroad wants to buy yuan-denominated shares or bonds, they will find it much easier in the rather peculiar markets in Hong Kong than in the domestic markets in Shanghai and the two often disconnect with significantly different prices for similar investments. This enables the Chinese government to pursue a domestic monetary policy without too many international concerns

A small open economy cannot follow the US example (being too small) or the Chinese example (being too open).  And in many cases smaller countries are more affected by international trade than similarly structured larger ones. So the exchange rate implications of changes in domestic monetary policy are more important to small open economies


Why, theoretically, is economic growth exponential?

There are two main theories of economic growth, the Solow-Swan growth model and Romer endogenous growth model. Both of these models allow for exponential growth.
The economic output (GDP) can be modeled through standard Cobb-Douglas production function. For example, a classical production function is given by:
$$ F(K,L) =  AK^{\alpha}L ^{(1-\alpha)},$$
where, $F(K,L)$ is the output (which we can measure by GDP), $A$ is a stock of technology, $K$ stock of capital and $L$ is a stock of labor.
In the standard Solow-Swan model we can show that in the steady state the GDP will grow at the combined rate of growth of technology and population $(g_A + g_L)$, and GDP per capita at the growth rate of technology $g_A$ (I omitted all derivations of this result for brevity but you can see it in Romer's Advanced Macroeconomics chapter 1). Hence Solow-Swan model offers solid theoretical reasons why growth should be exponential in approximately last 200 years.
According to Gordon's The Rise and Fall of American Growth the technology in last few centuries grew extremely rapidly and had more profound effect until the recent decade which he argues can explain both the unprecedented U.S. growth since industrialization and recent decline of U.S. growth. Gordon's work focuses on US but cited therein you will find sources that discuss other countries. Furthermore, population also grown rapidly in last few hundred years so if you care only about GDP not GDP per capita the growth of world's population is also part of the story.
Next we can also look at growth through the lenses of the endogenous growth theory. The main distinction between the above Solow-Swan exogenous growth model and endogenous growth model is that in the latter the rate at which technology itself grows is not just exogenously determined by outside forces (i.e. the inevitable march of human knowledge) but also endogenously determined within the model depending on how many resources are devoted to the discovery of new knowledge.
As with the Solow-Swan model I will skip any formal derivation, which you can review at your leisure in Romer's Advanced Macroeconomics chapter 3. In this model the relationship between technology and economic growth is more complex and depending on what parameters you assume for the model (especially how new technology is produced) the theoretical predictions can be virtually identical with Solow-Swan model or they can also suggest various positive feedback loops that can accelerate growth. For example, in one version of endogenous growth theory if world's population increases the number of people devoted to producing new knowledge (i.e. scientists) also increases further boosting growth. This offers some additional explanation in sense that it also tries to explain where the technologies come from instead of treating them as just exogenous shocks to the system.
The Solow-Swan model and its predictions have been rigorously tested in the past and especially the newer augmented versions of the theory (e.g. Mankiw, Romer and Weil (1992) augmented Solow model) are empirically quite successful (see the discussion of this in Romers's Advanced Macroeconomics). The endogenous growth theory is not well empirically tested, as it is difficult to do so, but it is becoming more popular. Paul Romer even recently earned Nobel prize for his contributions to endogenous growth theory.
Furthermore, aside from the above growth theory when we talk about economic development from historical perspective it is also important to acknowledge the role of institutions. One reason why prior to industrial revolution so little progress was done was that most countries had practiced extractive institutions. For example, countries without secure property rights where warlords can just take any possession of their subjects at will create environment in which people have no incentives to spend time creating new technologies, as doing so takes a lot of time and effort and without having secure property rights why would anyone bother if their inventions can easily be expropriated. Hence as argued by Acemoglu and Robinson (see their book Why Nations Fail), technological and economic growth could really start to grow rapidly only once proper inclusive institutions were adopted (i.e. secure property right, ban on slavery/serfdom/unfree labor, free entry into occupations, lift of arbitrary restrictions on commerce etc.).
Hence those historians are more or less correct that the growth can be attributed to technology, with a necessary caveat that having proper institutions is a pre-requisite for sustainable growth. The historians might explain the growth in post-hoc fashion because they might have no economic training but the above mentioned economic models offer both clear mechanisms of how growth depends on technology and offer testable predictions that can be applied equally to past, present and future. Also it is possible that historian's are actually aware of this (at least economic historians definitely are) and you just misread them - since you never provide any citation for your assertion it is hard to judge.
In addition the assertion in your title that growth is exponential is not generally valid. It has been so in recent history, but it wasn’t always exponential and it might not be exponential in the future if technological progress slows or countries start adopting worse institutions or if you buy into endogenous growth theory if people start investing less in new technologies and as population growth halts or even if it declines in the future.


How to tell the difference between consumer and capital goods

The only difference between consumer and capital goods is whether or not the good in question is purchased by a firm or a consumer.

The practical implications of this is whether or not this expenditure is included in $I$ or $C$ of the GDP equation.

Investopedia defines $I$ as: "the sum of all the country's investment, including businesses capital expenditures"

Going into detail on "capital expenditure":

  What is 'Capital Expenditure (CAPEX)'

  Capital expenditure, or CapEx, are funds used by a company to acquire or upgrade physical assets such as property, industrial buildings or equipment. It is often used to undertake new projects or investments by the firm. This type of outlay is also made by companies to maintain or increase the scope of their operations. These expenditures can include everything from repairing a roof to building, to purchasing a piece of equipment, or building a brand new factory.

It would appear that a staff fridge is a capital expenditure based on the above criteria because it contributes to maintaining the firms scope of operations.


In theory, it is bad because its inefficient. You protect your inefficient industries, I protect mine, we all end up consuming less. We would all be better off with totally free trade...

But of course, there are many complications to this. It might be that an industry needs time to become efficient and you want to protect it in the meanwhile. Or it might be that the other country is not more efficient, but is just dumping their products o your economy until your firms close, after which they will raise their prices and extract a handsome surplus...


why is market clearing price assumed in cournot model with homogenous products

[...]
   One way to interpret this result is to see our two-stage game as a mechanism to generate Cournot-like outcomes that dispenses with the mythical auctioneer.

A justification is given in Kreps-Scheinkmann (1983). They argue that this is the outcome if price competition follows production. For details see the linked article.


Would a Job Guarantee impact Cost of Living?

The answer depends on whether the provision of full employment is a public good that corrects for a negative externality (poverty). Suppose it is and we wish to create full employment.

First, we should consider what makes government provision of jobs different from policies like unemployment insurance? Instead of hedging against business cycles causing cyclical unemployment, we are now trying to effectively eliminate it. Then, we should consider if the government providing jobs as an employer of last resort would be any different than normal public sector jobs. Presumably, providing such undesirable jobs shouldn't crowd out the private labor market, but at the same time, what is to say that people wouldn't just reject these and continue to seek out jobs in the private sector?

The main route I see for making the government an employer of last resort would be to replace black markets, or to provide competition in inefficient markets. The former option has a prohibitively expensive political cost (and if allowed would be blockaded to the point of gross inefficiency). The latter option is harder to parse through. Most monopolies or inefficient markets we think of today are in technology or other places where monopolies benefit from large economies of scale. Treating the provision of goods in those markets as utilities and then subsidizing them is the go-to option. The other thing that could happen is that the government abuses its power to become a monopoly itself in the market it tries to provide employment for.

All that said, let's put those complications aside. Say the government tries to provide more public sector jobs (of some low perceived quality perhaps). The quantity is fixed to always provide jobs for every applicant at exactly the price where the worker would be indifferent between working and taking the public job. Thus, everyone searching can find employment, and in fact, everyone will be induced to join the labor market in this scenario. Note that the government has changed the number of people in the labor market.

You can form an economy with one representative private good, and one collective (public) good. The government levies taxes on consumers to freely to fund their jobs program if needed. The private sector produces the private good, while the government can produce the private or collective good. The government has the same technology as the private sector.

Thus, we have the different components of our economy:

$$
\begin{align}
\text{Utility:} & \qquad U(x_p, x_c, e) \\
\text{Prices:} & \qquad p(\sum x_p), p(\sum x_c) \\
\text{Production:} & \qquad f_p(\sum e), f_c(\sum e) \\
\text{Private Sector Profit:} & \qquad \pi(x_p) = f_p \cdot p_{x_p} - c(w, e)\\
\text{Government Profit:} & \qquad \pi(x_p, x_c, t) = f_p \cdot p_{x_p} + f_c \cdot p_{x_c} - c(w, e) + t\\
\end{align}
$$

You can add in the appropriate subscripts for the collective good to indicate what you desire about how the benefits or costs of the collective good are captured. We have $e$ as effort, $c(w, e)$ cost given wages and effort, $t$ as the tax, in this a case a simple lump sum. Solving for the equilibrium in the case with or without any government intervention here and then applying Shephard's Lemma would get you the consumer's expenditure function, which you can use to compare the cost of living in each case. If utility for each agent increases with the government provision or stays the same, then although the cost of living may increase, we would say in this scenario it is a Pareto improvement to have the government provision. Obvious extensions to the model could be changing the type of tax, making it so that you can levy a tax on production rather than consumption, or changing the competition so that you can use game theory to analyze price setting.

If this all seems rather theoretical (we have not specified the functional form of these equations), it's because your question is very broad. There is no way to succinctly analyze the effects of full employment on the cost of living. I gave my opinion above that I think it sounds infeasible, but if you wanted to set up a model and investigate for yourself, that might be a noble effort. My official answer thus is that it's unclear what effect a government provision of jobs would be on the cost of living.

(Boring safe answer, I know.)


Formula for the unconditional variance of the sum of observations from an autoregressive time series

For an AR(1) process (I omit any drift), the coefficient on the lag is the 1st-order correlation coefficient,

$$r_{t+1} = \rho r_t + u_{t+1}$$

So

$$r_{t,t+2} = \sum_{i=1}^2 r_{t+i} = r_{t+1} + r_{t+2} = \rho r_t + u_{t+1} + \rho r_{t+1} + u_{t+2}  $$

$$=\rho r_t + u_{t+1} + \rho \big(\rho r_t + u_{t+1}\big) + u_{t+2} = \rho(1+\rho)r_t+(1+\rho)u_{t+1} + u_{t+2}$$

and now the three components are independent. So

$${\rm Var}(r_{t,t+2}) = \rho^2(1+\rho)^2\frac {\sigma^2}{1-\rho^2}+ (1+\rho)^2\sigma^2 +\sigma^2$$

$$=\left( (1+\rho)^2\frac {\rho^2+1-\rho^2}{1-\rho^2}+1\right)\cdot \sigma^2$$

$$=\left( \frac {1+\rho}{1-\rho}+1\right)\cdot \sigma^2 = \frac {2}{1-\rho}\sigma^2$$

So we arrive at a different formula for ${\rm Var}(r_{t,t+2})$, compared to the one provided in these notes, and we will arrive also at a different expression for general $n$.

Note that my result is valid for $\rho =-1$, giving ${\rm Var}(r_{t,t+2}) = \sigma^2$ in this case since here

$$r_{t+1} = - r_t + u_{t+1}$$

and so

$$r_{t+1} + r_{t+2} = - r_t + u_{t+1}- r_{t+1} + u_{t+2} = - r_t + u_{t+1} - \big(- r_t + u_{t+1}\big) + u_{t+2} = u_{t+2}$$


Is that first claim right? I guess it depends what you mean by an income effect.

Say your quasi-linear utility form is:
$$U(x,v) = x + \ln(v)$$
Say $P_v = P_x=1$ and so for wealth $w = P_v\cdot v + P_x \cdot x = x + v$.
If $0<w\leq 1$ the household strictly prefers to buy $v$ to $x$ and buys no $x$ but $w>1$ is then exclusively $x$ is purchased beyond that. So demand for $x$ looks like this:
$$w<1: x^* = 0$$
$$w<1: v^* = w$$
$$w\geq1: x^* = w-1$$
$$w\geq1: v^* = 1$$

So the marginal demand for $x/v$ depends on $w$.


Is there any confliction in dataset and sample period description here?

Dasgupta in the study says he is using also other event windows in dynamic estimations:

To  explore the  dynamics  of  the issuance  activities  and leverage  change,  we  create  dummy variables corresponding to the following windows around the treatment  year:  from 1 to 4 years before the  treatment; the  treatment  year  and  the 2 years  after  treatment; the  next 3 years; and  the years beyond.

with window 1 to 4 years before treatment it makes sense.
Dasgupta also uses diff-in-diff and for that you need to test for pre-treatment trends so that also can be a reason.
Dasgupta also says that cartel data are from some 2013 publication so probably there was no point in getting more data from compustat. If they miss any data for their main variable of interest after 2012 then they can't run regression on longer than 2012.


First of all, the general form of the problem you got there is extremely demanding. In multidimensional screening problems, analytically often all hell breaks loose in a sense that it is just not tractable. One way out might be this recent approach by Gabriel Carrol: Robustness and Separation in Multidimensional Screening (also provides an introduction that hints at the intractibility of the problem).

Coming back to your question: $(2)$ does not imply $(1)$, because
$(2)$ only depends on $f$ and you can add a transfer rule that is not IC.
As a counterexample, consider a single good, $m=1$.
Take some strictly IC mechanism, $(f, p)$ and consider two types, $t$ and $t'$. So, $(1)$ is
$$U(t):= f(t)t - p(t) > f(t')t - p(t')\\
U(t'):= f(t')t' - p(t')> f(t)t' - p(t)$$
and by construction $(2)$ holds as well.
Let $t'>t$ and a lager allocation probability $f(t') > f(t)$ is accompanied by a larger expected transfer, $p(t') > p(t)$ - otherwise $t$ would love to fake being $t'$.

Now, construct $\widetilde{f}(s) = f(s)$ for all $s\in T$ and let $\widetilde p (t)= p(t')$ and $\widetilde p (t')= p(t)$. Since $(2)$ only depends on the allocation function $f$ and these functions are the same in both mechanisms, $(2)$ holds for $(\widetilde{f}, \widetilde{p})$.
Obviously, $(1)$ is violated for the reason named above:
$$\widetilde f (t') t - \widetilde p(t') > \widetilde f (t) t - \widetilde p(t).$$

To address your other problem (in your comment), use the integral formulation of expected utility.
Rewriting $(1)$ yields,
$$f( t)(t-t'))\geq  U(t) - U(t') \geq f (t' )(t-t')$$
implying that $U$ is Lipschitz continuous, implying $U$ is differentiable a.e., and equals the integral over its derivative:
$$U (t) = U(\underline t) + \int_{\underline t}^t f(s) ds$$
where $\underline t$ is the lowest possible type.
Then, you rewrite,
$$f (t) t - p(t) = U(\underline t) + \int_{\underline t}^t f(s) ds \\
f (t) t  - U(\underline t) - \int_{\underline t}^t f(s) ds = p(t).$$
Suppose $\underline t=0$, then
$$p(t) = p(0) + f(t)t - \int_{0}^t f(s) ds.$$
This implies the revenue equivalence theorem: If two auctions have the same allocation rule, payment functions (and thus revenue) can only differ by a constant. The same trick works for multi-unit auctions with multi-unit demand, see, e.g., Krishna, Ch 14.


Why aren't house prices included in CPI?

The CPI stands for a Consumer Price Index. As in the price of things that are consumed (at a particular moment in time). Real estate prices are not the price of something consumed because they contain the value of current housing consumption but also the capitalized value of future housing consumption. As such, including house prices would make the CPI a mixture of consumption at different times, and therefore unsuitable for comparing the price of consumption bundles at distinct times. Instead, they use a purer measure of the price of housing consumption: rents.

Rents reflect the price of consuming a flow of real estate services at a moment in time. Of course, many homes are owned by the occupants and not rented. Therefore, the calculators of national accounts generate something called "owner occupied rents", which is an attempt to calculate what the rents would be on homes that are occupied by their owners. This measure has problems, but for many purposes is quite adequate (Crone, Nakamura, Voith (2004)).


Why do Latin American countries fare among the worst at managing inflation

Perhaps you should look up the following collection of studies,

Dornbusch and Edwards (ed.), The Macroeconomics of Populism in Latin America(1991)

It is freely downloadable from its official site. Here is the opening paragraph of the Editors' piece

  "Latin America’s economic history seems to repeat itself endlessly, following irregular and dramatic cycles. This sense of circularity is
  particularly striking with respect to the use of populist
  macroeconomic policies for distributive purposes. Again and again, and
  in country after country, policymakers have embraced economic programs
  that rely heavily on the use of expansive fiscal and credit policies
  and overvalued currency to accelerate growth and redistribute income.
  In implementing these policies, there has usually been no concern for
  the existence of fiscal and foreign exchange constraints. After a
  short period of economic growth and recovery, bottlenecks develop
  provoking unsustainable macroeconomic pressures that, at the end,
  result in the plummeting of real wages and severe balance of payment
  difficulties. The final outcome of these experiments has generally
  been galloping inflation, crisis, and the collapse of the economic
  system. In the aftermath of these experiments there is no other
  alternative left but to implement, typically with the help of the
  International Monetary Fund (IMF), a drastically restrictive and
  costly stabilization program. The self-destructive feature of populism
  is particularly apparent from the stark decline in per capita income
  and real wages in the final days of these experiences."


Why do people think a stock should have a certain price based on the company's revenue?

If the price is too low (relative to any metric, be it earnings, growth, etc.), then anyone can profitably launch a takeover/acquisition bid.
Example. A company has 1B shares each priced at \$1 (so market cap is \$1B), pays no dividends, and earns \$100B (profits) per year. Then one could buy up all 1B shares for \$1B, own the company entirely, and enjoy \$99B in profits in just the first year alone.


Kuhn Tucker Conditions with fewer non-negativity constraints than number of variables

Non-negativity constraints have nothing special and are not critical for the general validity of the Karush-Kuhn-Tucker approach. First, realize that we could have $x \geq a >0$ and then we could write $x-a \geq 0$ and view this "non-negativity" constraint as just one more inequality constraint on the solution.

When a decision variable is weakly bounded, the new phenomenon that emerges is that we may have the partial derivative non-zero (strictly negative, for the maximization case), if at the solution this decision variable lies on the boundary.

If it is unbounded, than we must have the first-order derivative equal to zero. As another question pointed out, this may create issues in certain contexts, but it does not invalidate the approach.

As for your second question, indeed strict inequalities may create problems, but don't jump immediately to a "not well-defined" heavy statement. The problem will arise only if (stars denote the optimal solution)

$$\frac {\partial F(s^*,x^*,y^*,z^*)}{\partial s^*} <0$$

which would "push" the solution vector towards the unreachable zero-lower boundary for $s$. To avoid that, we require that at the solution the first-order derivative w.r.t to $s$ equals zero, and see whether we can indeed obtain a solution vector. If we can, it means that the $s>0$ constraint does not even "tend to" be binding, so you can ignore it.