Depreciation of local currency and its effect on prices

I think you're getting tripped up by notation. Instead of LOC/\$, write \$/LOC, since international trade is most likely to take place using $USD as an intermediary currency.

Here's an example:

Suppose world prices for soybeans are \$150/ton USD.  Today, the Wakandan widget (W) trades at 75W/\$ - so, I suggest rewriting this as $1/75W. Then, absent tariffs, taxes, and other frictions, Wakandans can import US soybeans at 11,250W/ton.

Now suppose the widget depreciates - it now trades at 150W/\$.  Notice that the sign of this change is positive, because you need more LOC to buy a unit of USD.  But if you rewrite as $1/150W, the sign of this change is negative, fitting the intuition that the LOC has become less valuable.  And indeed, now Wakandans must pay 22,500W/ton to import US soybeans - a change that would be reflected by a positive sign.

The value change of the LOC is negative, leading to a positive change in the price for imports. Make sense?


What does Yanis Varoufakis mean by "surplus recycling mechanism"?

"Surplus recycling" is a term coined (to my knowledge) by Varoufakis to describe the fact that a country that enjoys a trade surplus should reinvest the surplus in the domestic economies of its trading partners. Such a policy was conduced with success by the United States in the years following WW2, where the Marshall plan and similar policies in Asia took place, mainly for political reasons.

As Varoufakis states it, there is no reason for the market to proceed to such a transfer, yet it does sound reasonable to keep trade partners in good shape.

In Europe, Germany's trade surplus is notable for being the only important European surplus. Surplus recycling is one way the Greek government is trying to get Germans to feel right about handing out money to European countries that are struggling with their economy, another infamous attempt being WW2 reparations.

I believe you're somewhat wrong in your interpretation because the question here is not whether to reinvest or not (this is a problem which is well treated by "classic" macroeconomics), but where to reinvest the money. The alternative is hence between domestic and foreign investments, and there is no known economic mechanism that supports foreign investments unconditionally. Varoufakis work aims at fostering foreign investment from Germany to southern Europe. I don't see anything controversial in his definition, but he is defining a policy, not a model.

The problem with solely-domestic investments is that, on the long term, it weakens trade partners, which weakens domestic economy in turn. I don't think there is much debate among the fact that trade balance is not a self-correcting problem. The real debate (and it's political for the moment, an open field for economic research) is on what end the policies to solve the problem has to be carried: the one for which it is easier to solve (Germany), or the one for which the problem is worse (Greece).

The amount of money we're talking about is sizable to say the least, and can be viewed here as suggested in a comment.


Difference between equilibrium and k-rationalizability

What is meant by fixed point reasoning here?

Specifically, Nash equilibria. One way to define a Nash equilibrium in words is "a strategy profile from which no player can be made better-off by unilaterally deviating to a different action".  In other words, a Nash equilibrium is a "fixed point" because if the game ends up there, play stops because there are no more optimal moves.

  Is there a difference between fixed point reasoning and iterative never-best-response elimination?

Yes, they are different solution concepts.  The key is the last paragraph on page 12 of the article - basically, think of k-rationalizability as half-assing the game while still keeping the assumptions of full info, rationality, belief in mutual rationality, and so on.  Instead of stopping play when there are no optimal moves, play stops when some kind of decision-making resource is exhausted. It is interesting precisely to the extent that it may or may not model how real people, who face real information costs and processing time limitations, might approach games.


It says

  If none of the companies with higher costs will gain positive profits

I assume this means they're making 0 profits, while strictly speaking it could mean that they make negative profits and leave the market.

So with 0 profits, this means that average costs = price for the high-cost firms.

First, you need to derive the equilibrium price, using that, for the equilibrium price, demand = supply.

$$ \text{total demand} = Q(p) = S(p) = \text{total supply}$$

A firm's supply is the solution to a maximization problem:

$$ q_h(p) = \arg\max_q (p-C_h(q))q \\
 q_l(p) = \arg\max_q (p-C_l(q))q$$

Then, total supply given price $p$ and amount of high-cost firms $n$ is

$$ S(p, n) = 50\cdot q_l(p) + n \cdot q_h(p)$$

Then, our equilibrium condition is simply

$$ S(p, n) = Q(p) $$

This is under defined, as we have one equation and two variables $(p, n)$ to solve for. But wait, we know more! 0 profits for the high-cost firms means

$$ p = C_h(q)$$

To get rid of the additional variable $q$, just use the solution to the firm's problem:

$$ p = C_h(q_h(p))$$

Now we have two equations and two variables, you are all set!

$$ S(p, n) = Q(p) \\
 p = C_h(q_h(p))$$


Is the BEA double counting benefits in Personal Income?

Statistical Agencies do not necessarily obey the habits of Economic Theory. There is no "mistake" here (certainly, there is no double-counting), but a different approach to what each concept means or "should mean" (which sometimes is also a consequence of historical precedence).

For the fun of it I post below how the same data could be arranged to conform with standard theoretical economics, where we make a point of distinguish between income earned, taxes in the broad sense, transfers, consumption, saving.

ALTERED/RE-ARRANGED BEA TABLE 2.1. "Personal Income and Its Distribution"


Intertemporal choice with possibility of death

It is safe to assume that if the consumer passes away, his creditors get his endowment. This actually happens in real life. $B$ should obviously be zero. As far as the second period is concerned you can represent it as a von Neuman-Morgenstern utility from the lottery:

$\beta u(c_2)$ with probability $p$ and $0$ with probability $1-p$.

I.e. $U(c_1, c_2) = log(c_1) + p\beta log(c_2).$


Does multicolinearity of variables imply complementary inputs?

A Leontief production does imply perfect multicollinearity between the inputs, since there is no substitutability between them. Thus, if a firm's production technology features non-substitutability between inputs, one would observe the firm choosing inputs in fixed proportions.

However, (perfect) multicollinearity does not necessarily imply non-substitutability. For instance, the Cobb-Douglas production technology of the form $y=x_1^{0.5}x_2^{0.5}$ is also consistent with a firm choosing inputs in fixed proportions. So the answer to your question is no (from a theoretical perspective).


Finding a nash equilibrium in pure or mixed strategies

Once again, thanks to denesp (and Herr K.) for the help. I thought that I'd try to answer the question just for reference purposes, and I guess that it's also a step toward improving the "answers per question" metric on Economics SE.

As aforementioned, the PSNE is $(u,r)=(3,3)$.

Player 2

  If $P1$ plays $u$, $P2$ can play either $l$ or $r$.
  $l$ would give $P2$ a payoff of $2$ whilst $r$ would give $P2$ a    payoff of $3$.
  Therefore, $P2$ will play $r$ with probability $\pi=1$

Player 1

  If $P2$ plays $r$, $P1$ can play either $u$ or $d$.
  $u$ would give $P1$ a payoff of $3$ whilst $d$ would give $P1$ a    payoff of $2$
  Therefore, $P1$ will play $u$ with probability $1-\rho=1$, $\rho=0$.

The result

$(\pi^*,\rho^*)=(1,0)$


Why didn't other oil pricing benchmarks such as Brent go negative?

The Brent price is for oil physically delivered in the UK. The price that went negative if for oil delivered in West Texas, US. The physical and economic conditions at the two places are different - volume available, flexibility of that volume, demand, storage, and so on. This leads to different prices.


Does the term 'marginal' refer to the last consumer?

Yes marginal refers to the last or one more unit or person. I don’t see there any contradiction if you have 2 people in room and third person joins it’s both the last person and one more person.

If you are looking for a precise definition then in economics the concept of margin is connected to the first derivative (instantaneous rate of change) of function at some point. For example, with utility function $U=q^2$ marginal benefit of consuming 10 units of $q$ would be $U’(10)=2(10)=20$.

Also, demand curve is an aggregate of demands of individuals who all might derive different marginal benefit of consuming the good, the point of the answer is that the marginal customer, last customer to participate in the market, will have marginal benefit of consuming good exactly equal to price (since if $MB<P$ the person would just not participate in market).


"The econony needs x workers of profession y". What does it mean and how is it derived?

Full-factor employment GDP (referring not only to labor but also to capital and any other factor), results from existing factor stock interacting with production technology. It is a (quasi) "engineering" relationship, not an economic one.

It says "if all labor and all capital were employed, given technology, output will reach its feasible maximum". It does not say that it is an optimal level in any sense. It does not allow for labor-leisure choice or lock-out (idle capital). It does not take into account whether current markets actually clear (or not).

Still we informally accept that "more is better" so we do examine what would it take to reach this feasible maximum, or how economic interactions, incentives, institutions etc may or may not lead us to such a full-factor employment GDP.

In the case examined by the question here, it appears that an expert body has identified an "engineering" mismatch in the composition of the labor force. Namely, this body of researchers has estimated the required compostion of the labor force that can lead the economy to "feasible maximum GDP"-and then found that the actual composition does not align with the former.

As to how one can indeed estimate such "required compositions" of the labor force, well, this is too technical to go into here, but it can be done.


What does the value of an industry mean?

If I'm understanding your question, this typically means the amount of revenue that an industry generates. Usually, the statement "this is a ____ industry" is labeled. For example, "this is a $1,000,000 a year industry." Although recognizing how much revenue an industry generates is nice and is helpful, it may not apply to a person looking to break into an industry. It's important to note that profit and revenue are not the same things.


Should the control variables in an econometric regression be correlated with both the dependent and the primary independent variables?

First of all, you mentioned that the disasters you are dealing with are "randomly" occurring.  So I want to provide the following as food for thought:

There is an important question here about what your "disaster" variable actually measures and at what level it was "randomized" (if at all).  If it is measured in spatial terms (was an individual at this particular place when disaster occurred), it is hard to imagine that this would be a binary variable (you experienced a disaster or not) and not something measured on a scale (how much damage you area experienced), unless treatment and control groups are far away from each other.  But in that case, it is hard to imagine that this variable would be random (both groups having equal probability of experiencing the same or similar type of disaster).  If it is measured in temporal terms (was there a disaster on day x) and there is no spatial variation in the data, then it is also not clear whether this variable is random because a lot of places experience the same disaster year after year and changes can be made over time in the same location based on previous experiences to better plan for future events.

But if the variable is, in fact, random, then by the beauty of randomized treatment you don't need to worry about control variables because the treatment variable is exogenous.

Question 1:

In answering your questions, I am going to assume that your "disaster" variable is not random, which seems like a more reasonable assumption.  I am also thinking of "experiencing a disaster" not just as whether you were in that place or time when the disaster occurred, but as something that had a physical/physiological/mental affect on you.  If it meant just being in that place or time then it is hard to think of anything affecting someone being in that place or time without specific context.

So we have to think about possible confounding variables.  Do any variables exist that affect whether someone "experiences" a disaster and that also independently affect someone's happiness level?  Yes, wealth fits that criteria, as a wealthier individual may have better safeguards in place (like earthquake or flood insurance) to allow them to deal with the impact of a disaster.  Other possible variables are age and health level.  Health affects happiness and during times of disasters those living in nursing homes or having chronic conditions like kidney failure or COPD are more affected by loss of power, for example.  Similarly, whether someone lives alone can be a confounder.  Occupation can be possibly, as emergency workers may be more impacted by disasters or some other professionals may be better trained to deal with the consequences of disasters.

Your thinking about not controlling for disaster-induced variables is correct.  You should not control for variables that are themselves not fixed at the time of "treatment".  They should be allowed to change when your "treatment" variable changes.

Question 2:

Like I said above, I would think of age as a confounding variable in the relationship between experiencing a disaster and happiness.  But if you have variables that you believe do not directly affect "experience" of a disaster but that do affect happiness, then it is still recommended to include them in the model because this reduces the error variance in the model (i.e., standard errors of coefficient estimates will be reduced).


Collapse of Lehman Brothers: What did they have in their books?

We do have information on this topic. I’d recommend starting with the Financial Crisis Inquiry Commission report chapter on it, and review the New York Fed article on its resolution. Detailed information on the assets should be available from bankruptcy filings, though those will involve quite a bit of work to obtain.


I believe that you may be looking at a spurious relationship. You are looking at the same data transformed by a scaling factor (the currency) and regressing it on itself. Of course they will be related. If they weren't scaled, they would be perfectly co-linear.

What you really want to know is if there is an effect of changes in the relative value of Yen to USD or Euros on the operating income of the company.

You should start by visualizing this relationship by graphing it. Graph the relative value of Yen in your time series, then graph operating income in your time series. Do they move together? Look for structural breaks. For instance, they may have only begun to move together in the last year, that can really mess up an econometric model if you don't account for it.

After you have finished analyzing it graphically, I suggest reading up on co-integration and the Engle-Granger method. Some literature that are good places to start are: Stock and Watson(1988), and Engle and Granger(1987).

This part of time series econometrics enables you to work with variables that are NOT STATIONARY. One problem with first-differencing such as you mentioned in your question, is that it throws away the very relationship you want to know about. The Engle Granger method enables you to analyze the relationships that matter economically, and is still statistically sound.

The world of time-series economics is a lot of fun, but make sure you stay grounded in the economics behind your data. Don't ignore the little things in your data, they may be the most meaningful. Don't be surprised if the relationship you thought was important actually is not important. That tells you just as much as if it is!

There's so much to talk about with the fascinating area you are delving into but I hope this can be of interest to you.


Write search model as a signal density inducing posterior belief

I am not entirely sure about what you mean with "Before choosing the DM can discover $z$ by searching sequentially and paying cost $c_y$ when uncovering $z_y$ for each $y\in Y$."

However, it sounds like you want your DM to choose both a stoping time, $t$, and an action, $y$. Since the DM is searching sequentially, the decision to continue searching or not will depend on how searching affects their beliefs and the cost of searching, which you are not fully specifying. Searching typically gives you access to some signal that will affect your beliefs, and the cost of searching is usually defined regardless of the result of the signal. You have not specified such searching signal, and it sounds like you only pay a cost for searching if you uncover state $z_y$. However, let me get to your question.

At a high level, you cannot re-write your problem through a signal density, $P_T(\cdot|z)$, because the re-write does not include the searching costs.

Argument for the statemet: Suppose there is a well-defined search problem and the decision-maker chooses $y\in Y$ only after the search phase has concluded (If these assumptions are not true, my argument is just strengthen). Given an optimal search strategy (that can only depend on the prior $F_Z$, the search costs $c_z$, and (perhaps) the interim beliefs during the search process), for each state of the world the DM will (probabilistically) end with some sequence of signals (one for each period they searched). Denote each possible sequence with a different element in $T$ and let $P_T(t|z)$ be the probability that the DM ends with sequence $t$ when the state is $z$. However, each sequence of signals will, in general, have a different cost. This cost cannot be, in general, captured by the probability density.

If searching was free, then sure, your problem is one of choosing $y$ to maximize the expected utility where expectations are taking with respect to some posterior beliefs that arise after searching, and the density $P_T(\cdot|z)$ is the object that summarizes the searching process that is assumed to be well-defined.


One important reason why it is better to have activity be done formally rather than informally (grey or black markets) is access to proper institutions.
Property rights, ability to enforce contractual obligations, above mentioned safety laws etc, are vital for economic activity (see Stiglitz Economics of Public Sector 3rd ed).
Some economic activity does not require institutions. You might hire someone for mowing your lawn, where you can easily observe if the job is done properly. At the same time the worker has little in stake since even if they can't force you to pay without contract they will not do business with you in future. But when it comes to more complex economic interactions which are often necessary as economic activity expands, such as establishing supply chains, having several workers at different locations etc it becomes more and more limiting if people do not have access to proper instiutions.
What even worse, in absence of government imposing institutions, gangs, mafia or local paramilitary might. For example, one reason why gangs exist in drug trade is that drugs are expensive and easily portable, so as all expensive things they attract thieves. If selling drugs was legal firms would be just able to either rely on police or in worst case scenario private security forces to defend their labs, storages or shipments. However, without access to this institutions they have rely on hiring thugs and gangsters. Moreover, in absence of access to court & legal system any disputes between drug producers about quality of drugs or payments between producers and dealers will be settled by gang conflict, intimidation and so on.
Second, you mention that formalizing sector brings in more government revenue, but then you just brush this point off as if it would be nothing even though this is crucial argument.
If millions of people work in informal economy (as you argue in your question), this seriously retards government ability to extract revenue. Even if you would happen to libertarian and reject any redistribution concerns government is still necessary to provide wide range of public goods (or in some cases quasi-public) goods such as national defense, infrastructure, law and order enforcement. If government can't efficiently raise enough revenue to properly fund these functions then whole economy will suffer. As a matter of fact provision of  public goods and access to well designed institutions,  economic development is thought to be virtually impossible  (see Acemoglu & Robinson. Why Nations Fail).
Consequently, it is vital for the country to have most of its economic activity be done in a formal way. That does not mean all grey economic activity should be completely eliminated, it would probably not be worth while to require even occasional babysitting should be only done by formally organized business, and have some space for economic activity between relatives. However, grey economic activity should not be encouraged and if as you put it millions of people are employed in informal economy then that would be a problem.
Lastly, you seem to assume that formalizing economic activity has to be costly but that is not true.  Formalizing economic activity does not mean that people there have to follow some draconian regulations. Maybe in your place government plans that, but many advanced countries will have lower regulation/administration/tax burden imposed on self-employed people or small businesses. Consequently, there is also no reason to think that formalizing informal sector should involve large costs a priori.


What are some sectors in which nations benefit from collaborating with other nations?

Nations can benefit from each other in all three of the broad sectors.

Collaboration in the primary sector (Involving raw materials, fishing, agriculture and mining) can lead to more sustainable extraction of resources. For example Nation A can choose to import Resource Y from Nation B rather than exhausting their own source. Nation A gains Resource Y and Nation B gets money.

Outsourcing is the most obvious example of collaboration in the secondary sector, Australian companies may outsource manufacturing work to Bangladesh, this is beneficial to both nations as money is going to Bangladesh and goods are going to Australia.

The tertiary sector involves providing services to consumers. One example of this could be tourism related services, if a tourist from overseas uses one of these tourist related services, the country they are visiting would receive money in exchange for the service, and the tourist would have their experience improved in some way. This example is small scale but when you consider how many people travel around the world it builds up.


Choice between dummy variables and Likert scale in Linear Regression

In your question you talk about coding so I will address primarily the question of coding.
You should definitely code it using suggested

0 for a, 1 for b, 2 for c and 3 for d

regardless of whether you plan to use it as ordinal variable or categorical variable.
Let me explain, nowadays virtually any language or program will have option to easily turn ordinal, or categorical variables into dummies with virtually zero effort.
In R you can simply use is.factor() function to turn any ordinal variable into separate dummies so you could use:
lm( dep_var ~ ind_var + is.factor(variable), data = XY)

In Stata you can use i.variable to turn any ordinal or categorical variable into dummies, again in Stata you would run something like:
reg depvar independentvar i.variable

Next, coding it as separate dummies is more time consuming (even if just marginally it takes more line of code) than just coding it into one variable.
Consequently, no matter whether in actual model you want to include it as oridinal variable or set of dummies, in most standard languages and programs it will be more efficient (i.e. take less time to code) if you just code it on Likert scale.
As for whether what you should do from scientific standpoint that depends on details of your research. I can imagine you would be being able to defend both including the variable as independent ordinal scale or as a separate dummies. Both of those would work in principle. My sense is that you would likely want to include it as dummies since that will have better interpretation,  but if you have little data and too many regressors then it might be worth while to just have it as ordinal variable.


What are the limitations that a country faces in increasing its Forex Reserve?

The foreign exchange reserves of a country are normally invested in short-term government bonds in countries with liquid currencies (or trade partners). It is a portfolio of assets, and so the reserves may be diversified. On average, it should earn the risk-free short-term rate of the reserve currencies.

The cost to the country holding reserves is that they have to issue currency/bank reserves as the flip side of holding the forex reserves. These excess reserves will need to be drained, and so in some cases needs to issue bonds or bills to do so. In other words, the country needs to pay the domestic interest rate in order to hold the forex reserves; you can think of them borrowing to finance the position. (They could raise reserve requirements, forcing local banks to effectively loan money to the central bank to finance the reserve. This pushes the interest cost onto the domestic banking industry.)

Since a central bank can create its own currency costlessly, it appears that this might be a good deal. But creating a very large reserve position is effectively a huge bet, and any losses are effectively a cost to the taxpayer. (The Swiss attempt to limit Franc strength versus the euro ultimately failed as the position was not politically viable.)

Furthermore, the countries where you hold the reserves may not be amused with the country holding its exchange rate at too low a level. The country may become the target for corrective trade policies if reserves are "too large."

Finally, the reserves are lent to another government. If relations with that country deteriorate, those reserves are at risk of being seized.


price space as a dual of commodity space

If you want to see a nice usage of duality in economics, in Mas-Colell, Whinston, and Green  there's a nice section on duality in the second or third chapter when talking about Hicksian demand. The treatment is pretty straightforward if the consumption space is real and has a finite dimension. I think understanding that basic usage of duality is illustrative of its general utility.

If the idea is to understand what duality is and how it operates, I think Conway is a great book.

(Just undergrad here, hope I'm not mixing up topics).


Precautionary Savings Two-Period model with a second period quadratic utility function

The two papers that explored first savings under uncertainty in a two-period setting are

Leland, H. E. (1968). Saving and uncertainty: The precautionary demand for saving. The Quarterly Journal of Economics, 82(3), 465-473.

and

Sandmo, A. (1970). The effect of uncertainty on saving decisions. The Review of Economic Studies, 37(3), 353-360.

They both deal with a general two period utility function $U(c_0,c_1)$ not necessarily additive. But both remark the following related to such a case:
The condition for precautionary savings depends on third cross partial derivatives, as well as on the third own derivative with respect to future consumption only.

If the utility function is additive $U=u(c_0)+Ev(c_1)$, then cross-partials are zero, so the result hinges only on the third own derivative for the second-period consumption.

It follows that with an additive over time utility function, it suffices that the second-period utility is quadratic (so third own derivative is zero), in order to not get precautionary savings, irrespective of the form of the first-period utility.


Calculating expected tenure of a customer

I am going to use a $t$-continuos scenario, and then discretization follows naturally.

Call $f(t)$ the distribution function of the variable $t$. The survival function is just

$$
S(t) = \int_t^{+\infty}f(u)~{\rm d}u \tag{1} = 1 - \int_0^t f(u)~{\rm d}u
$$

It is pretty clear from here that

$$
\frac{{\rm d}S}{{\rm d}t} = -f(t) \tag{2}
$$

and that $\lim_{t\to\infty}S(t) = 0$. Now consider the integral

\begin{eqnarray}
\require{cancel}
\mathbb{E}[t] &=& \int_0^{+\infty} t f(t) ~{\rm dt}
\stackrel{(2)}{=} -\int_0^{+\infty} t \frac{{\rm d}S}{{\rm d}t}~{\rm d}t  \\
&=& -\int_0^{+\infty} \left[\frac{{\rm d}(t S)}{{\rm d}t} - \cancelto{1}{\frac{{\rm d}t}{{\rm d}t}} S\right] {\rm d}t \\
&=& -\cancelto{0}{\strut{t S}\big\rvert_{0}^{+\infty}} +  \int_0^{+\infty}S(t)~{\rm d}t
\end{eqnarray}

So in summary

$$
\mathbb{E}[t] = \int_0^{+\infty}S(t)~{\rm d}t \tag{3}
$$

Now you can move to discrete time and represent this in the form

$$
\mathbb{E}[t] = \sum_{t = 0}^{+\infty} S(t) \delta
$$

for some small number $\delta$. In the paper you cite, they choose $\delta = 1$


Is price in this situation exogenous or endogenous?

It is endogenous, because a demand shock will affect prices.

To see this more clearly, add another regressor to the demand, for instance, income. It is clear that income is exogenous, as it is not affected by prices or quantities, or shocks. This cannot be said of prices.

This is also evident if you write the system as it should have been written in the first place:

$$ P = \alpha + \beta Q + \epsilon $$
$$ Q = c \quad \quad \ + \mu$$

This is a simultaneous equation system, where both $Q$ and $P$ are endogenous. Here you can see more clearly that a demand shock $\mu$ affects prices via $\beta$ (if $\beta=0$, the two equations are independent).

This means that the OLS estimation of $b$ is inconsistent. A consistent estimation can be obtained with 2SLS, using a proper instrument.


How is it possible that all currency exchange rates are fixed w.r.t. to each other?

International currency markets are highly liquid, with near-instant transfer of knowledge between trading centres.

This means that any arbitrage opportunities tend to get resolved within seconds.

To put it simply, the thing you tried to do, of trading two currencies via a third, is something that lots of real-world currency traders, and their automatic trading programs, are trying to do all the time; so any discrepancies are quickly mopped up by the market.


Structure of Non-profits and Moral Hazard

Although they are non-profit organizations, they still submit legal documents such as statement of cash flow, statement of financial position, etc.

These contain detailed information of the spending of such organizations.


The marginal cost function is meaningful only when the produced quantity is normalized

I am not sure I understand your question but if I do, then I do not agree.

Marginal cost is defined as the derivative of the cost function. If the producible units are too large then marginal cost is less likely be a good approximation of non-marginal cost changes. The importance of this is that the derivative of an estimated cost function will probably not correspond to the real cost increase when non-marginally increasing the output.


Using dependent variable's and exogenous variables' lags as instruments in 2SLS

Is this legit and what is the rational behind this?

Yes you will find this even as a recommendation in many textbooks (e.g. see Romer Advanced Macroeconomics pp 376)  so it is legit although with a caveat.
A good instrument should be correlated with the endogenous variable and be able to through it exert an effect on dependent variable. Well lags are more often than not highly correlated with the contemporaneous observation of the same variable.
Next, instruments should not be correlated with residuals and for lags this will often hold (but not always) in economics.  This is because in many cases when the residual reflects new information learned by people between $t$ and $t-1$, economic theory simply often tells us that any variable that is known as of time $t − 1$ is uncorrelated with the residual.
However, this being said lagged instruments can still turn out to be weak, or violate some of the other assumptions of IV (for overview of all assumptions required for IV see for example Verbeek, a Guide to Modern Econometrics or Angrist & Pischke Mostly Harmless Econometrics), and the above might not hold generally for any economic relationship. So these are no silver bullet, but they often do make sense in economics. Nonetheless, there is a criticism of their overuse in economics (see for example Wang & Bellemare 2019). However, this is to my best understanding not because they would not be legitimate instruments but rather many practitioners just  apply them without even bothering to check if other conditions that good instrument should satisfy are actually really satisfied.


What would happen if a country defaulted on its public debt?

A “full explanation” would be extremely long; depending on the situation, a great many things would happen.

For developing countries, defaults are typically on debt in foreign currency, issued under foreign law. They will end up in legal fights with bondholders, and/or end up dealing with the IMF (or other body) to restructure their debt. Economic problems will revolve around being cut off from the financial markets, such as the currency markets.

The scenario you are describing seems to more about countries that issue debt in their own currency, under their own law (for example, the United States, Japan, Canada, etc.)

For most of these countries, the government is a large borrower. If they refuse to make any payments on debts (which is rare; defaults are often partial), this would represent a loss that is comparable to one year’s production or income (GDP). This would represent severe losses for many savers.

Furthermore, banks hold government bonds to manage their liquidity. Meanwhile, they have very little equity when compared to the size of their balance sheets. Large losses on their bond portfolios would likely result in their bankruptcy. Their bankruptcy would then cause losses on their lenders - bank bond holders, depositors. Furthermore, a bankrupt bank cannot extend new loans, and so any businesses that survived would have a difficult time to borrow to fund expansion. As a result, there would be massive job losses.

In practice, defaults are rarely total (except after wars/revolutions),so we do not see the full potential effects. However, even partial defaults - such as in Greece after the Financial Crisis - typically result is a depression. (I suggest that you look up the Greek debt crisis for an example. The Greek situation is different than developed countries outside the euro, since it does not control the currency it is borrowing in, and so was forced to deal with supranational bodies.)


Is there a country in the world not in debt?

According to the World Economic Outlook database (April 2018), in 2018, the "General government net debt" of these countries was negative:

  Kazakhstan, Norway, Trinidad and Tobago, Luxembourg, Botswana, Oman, Estonia.

This database doesn't have this datum for Singapore, but I believe the Singapore government also has negative net debt.


Why did coronavirus cause a food shortage?

This question is too open-ended, as it does not even specify a country. There are a number of issues that have shown up. These comments are based on my reading of news stories, and arguably not authoritative.

Logistics have been disrupted, putting shipping, planting, and  harvesting at risk. This is a worry for countries that are dependent upon food imports.
Some countries have put into place export restrictions (as a results of the previous concern).
In the developed countries, a significant portion of eating is at restaurants. The supply chain for restaurants is different than grocery stores. There are two issues: people eat different foods at home than restaurants (The example I heard was that most fish is eaten at restaurants in the United States, so there is a surplus of fish.) People have more time, and are baking more now than even during the seasonal peak at Christmas. In addition to the supply chains being different, the sizing and labelling is different (e.g. nutritional information must be displayed on grocery store food, and so a waiver had to be given to allow food without such labels to be sold). Flour is sold in 80 pound bags to restaurants, and households cannot work with such bags. (At my store in Canada, flour is being sold in small generic plastic bags.)


When only half of the independent variables are non stationary, does it make sense to run a cointegration test?

For cointegration analysis all variables should be of the same order of integration. However this being said it depends on how much you trust your stationarity tests. Most unit root tests are riddled with low power problems especially in small samples (see some textbook on econometrics like Stock & Watson introductory econometrics for more details and sources).

Moreover, if the broader literature suggest that all the variables are let’s say I(1) I would even disregard the unit root tests results as they may be just false positives/negatives.

Also, if you want you can use the Pesaran bounds approach to cointegration which allows for both stationary and nonstationary variables being estimated within one model, although cointegration can only happen between stationary variables. This approach is usually not in textbooks yet so you might wanna check the original paper “Bounds testing approaches to the analysis of level relationships”


Price elasticity of demand always increases with price?

There are many functions where absolute value of elasticity is decreasing and which are continuous closed form.
One example of such function would be:
$$Q= a-\ln(p), p\geq1 \implies EL = - \frac{1}{p}\frac{p}{a-\ln p} = -\frac{1}{a - \ln p}, $$
which is continuous closed form  and decreasing in price in its absolute value.
Original Answer:
Originally I interpreted the request too mathematically I am leaving it here since other answer references it.
Yes for example consider the following demand function:
$$Q = e^{-p}$$
price elasticity of demand is given by:
$$EL = \frac{dQ}{dp}\frac{p}{Q} = -e^{-p} \frac{p}{e^{-p}} = -p$$
So the demand function has elasticity that is always decreasing in price $p$. The function has closed form, and it is continuous.


A classic example, related to the point made by @GuyLouzon, is access pricing in network industries.

A telephone network is a natural monopoly. The modern approach to regulating this market is to force the monopolist owner of the network infrastructure to allow competing telecomms services firms to buy access to the network. This allows those rivals to offer competing retail telephone services using the monopolist's infrastructure rather than having to build their own.

The price that the incumbent charges firms for network access is regulated. Ensuring that monopolist does not set the network access price too high means that downstream firms can profitably buy access and thereby enter the market for retail telecommunications services. While the fundamental natural monopoly problem has not gone away, regulating the access price at least helps to ensure the retail telehone service market is not also monopolised.

If you would like to read more about this example, I'd recommend "Competition in Telecommunications" by Laffont & Tirole.

One can easily build a model of this situation. Suppose that retail telephone firms earn profit $\pi_n$ when there are $n$ firms in the market. Make the standard assumption that $n\pi_n\leq m\pi_m$ whenever $n>m$ (industry profit is lower when there is more competition).

The wholesale firm sets a price, $p$, for access to its network. Firms will choose to buy access so long as $\pi_n>p$. The highest price the monopolist can charge if it wants to serve $n$ firms is therefore $\pi_n$. Thus, the monopolist can serve one firm and earn profit $\pi_1$, serve two firms and earn profit $2\pi_2$, serve 3 firms and earn $3\pi_3$, etc. Because $n\pi_n$ is decreasing in $n$, the monopolist optimally sets $p=\pi_1$ and grants access to a single firm, inducing a downstream monopoly.

A policy that constrains $p<\pi_2$ would, on the other hand, induce more than one firm to enter downstream and thus result in more downstream competition than when the price cap is not in place.


What does it mean by Portugal having a negative inflation?

Regarding whether negative inflation is a good sign: Deflation is generally speaking not a good sign for an economy. Most economist argue that it is good for an economy to have on average about $2\%$ inflation per year. There are many reason for this one example is that wages are sticky and hard to adjust but inflation can help to bring them down in real terms even if they dont change in nominal terms.  This is especially crucial during the times of economic crisis when it can help prevent unnecessarily high unemployment.

Regarding the potential causes: Inflation depends on many factors. Generally speaking the price level (changes of which are called inflation) is given by:

$$P=MV/Y$$

So the price level $P$ is increasing in money supply $M$ and velocity of money $V$ and decreasing in real output $Y$ (also note its driven not just by the variables itself but also by expectations of how they evolve). Moreover, Portugal is member of the euro-zone so its inflation rate depends on only situation in Portugal but across whole euro-zone.

Currently Portugal is experiencing recession, due to covid so one would expect that this would ceteris paribus lead to inflation as $Y$ drops. However, an important caveat is that during the recession velocity of money also drops as people are more reluctant to spend and prefer just saving their money 'under their mattresses' and that is what creates a lot of deflationary pressure. However, thats just an back of an envelope analysis of the situation.


You are right that for $|\epsilon|\leq1$, you would have issues with the wedge. However, this cannot occur in equilibrium, so it is not a problem.

Both your questions stem from the same issue. I will illustrate it for monopolists, but it is analogous for monopsonists.

First note, elasticity is not constant (except in the rare case of a constant elasticity of demand function). The elasticity of demand varies across the points on the demand function. In some regions it is $|\epsilon|>1$ and in others it is  $|\epsilon|\leq1$.

Second, note that the monopolist chooses the point on the demand curve, that she operates in, unlike under perfect competition.

It is a well known result that monopolists optimally only choose to operate in the elastic region of demand. This is because, if they are in an inelastic region, then in response to a price increase, demand is reduced proportionally less than the price increase. In such a case, the monopolist can increase its profits, so that cannot be an optimal price. The profit maximizing price chosen by the monopolist is always in the elastic region of the demand curve.

This means that $|\epsilon|>1$ always holds for a profit maximizing monopolist and the wedge is always positive.


Relationship of continuous and discrete time models

Poisson process can be interpreted as a continuous case of Bernoulli process.
Taking your example, consider that the buyer is consuming the good in batches in fixed intervals of time with probability $p$. So the consumption is allowed only after fixed intervals, and the r.v. $X_t \sim Bern(p)$, where $X_t=1$ indicates that the buyer consumes the good at time $t$. Further, how much she consumes in $n$ periods is distributed as $Bin(n,p)$. The expected level of consumptions in $n$ periods as we know is $np$.
Now consider that the buyer can consume at any instant and just not in intervals. Unlike in previous case where consumption could happen only at $t=1,2,...n$, now it can take place at any $t\in \mathbb{R^+}$. Naturally the probability that consumption happens at some given $t$ is $0$. So comes the rate $\lambda$, i.e., we say that the buyer on average consumes $\lambda$ amount in a given length $n$ of time. Now the total consumption between $t=0$ to $t=n$ becomes $Pois(\lambda)$.
So, $Pois(\lambda)$ is a limiting case of $Bin(n,p)$ as time interval $\to 0$.
Coming to your second question:
The buyer gets value $v$ when he uses the good at a given time, and the value gets depreciated after one time interval by $\delta$. The total value, as you mention, is: $v(1+\delta+\delta^2+..)$
However, if we assume the value is depreciating continuously at rate $r$ we can say:
\begin{align}
rv(t)&=\lim_{\Delta t \to 0} \frac{v(t+\Delta t) - v(t)}{\Delta t} \\
&=\frac{dv(t)}{dt} \\
\end{align}
From this you will get that the value from having the good over a time interval $[0,d]$ be $\int_0^d v e^{-rt} dt$
Lastly, to answer the last part: if you want to go to Bernoulli process from the described poisson process consider that arrivals can happen only in discrete time intervals such that on average in $d$ periods, $\lambda_d$ arrivals take place. So, this translates to the bernoulli process such that arrivals happens at a given time $t$ with probability $p=\lambda_d/d$.


What is the opportunity cost if I choose the option of highest value

The opportunity cost of any decision is the value of the next-best alternative forgone.

So in this case, if you choose \$40, your opportunity cost is \$30, because that's the value of the next-best alternative forgone.

(Implicit assumption: These are your only three alternatives and you must choose one of these three.)


Remember that increases in a standard IS-LM model an increase in government expenditure generates an increase in the real interest rate.
As such, private investment decreases. Traditional economic theory would suggest that what you recommend will crowd out private investment. The question can then be modified as follows-  does a 1 Dollar increase in government expenditure decrease private investment by less than 1 Dollar? In other words, is the government multiplier greater than 1? A lot of empirical work has been done on this.


"Why can't farmers simply sell their wheat ahead of time to buyers who are capable of storing this wheat..."

"...sell their wheat ahead of time..." is precisely what a futures contract does.

To imply that it is necessarily speculators on the other side is misleading.
On a futures exchange, trading is anonymous and identity of counterparty is not known. The counterparty can be a hedge fund taking a long speculative position or a cereal company who intends to take physical delivery.

For the short hedger (farmers in your example), it doesn't matter who is on the other side of the hedge. The futures position eliminates price risk regardless. It's a risk sharing arrangement. The party which has more incentive to shed risk must induce the other side to take the trade. If the short side is more risk averse, this leads to backwardation. Otherwise, contango.

(In particular, empirically expectation hypothesis does not hold for physical commodities. For financial products, no arbitrage would tell you EH holds not under the physical measure but the risk neutral measure.)


Why do we need at least as many instrumental variables as endogenous regressors to identify parameters in 2SLS?

In standard linear regression model
$$y = x^\top \beta + \epsilon$$
with exogeneity $\mathbb E[x\epsilon] = \mathbf 0$ you have $K$ parameters because $\beta$ is $K \times 1$ and you have $K$ equations
$$\mathbb E[x\epsilon] = \mathbb E[x(y - x^\top \beta)] = \mathbf 0 \Leftrightarrow \mathbb E[xy] = \mathbb E[xx^\top]\beta,$$
where $\mathbb E[xy] = \mathbb E[xx^\top]\beta$ is a system of linear equations. Only if the matrix $\mathbb E[xx^\top]$ is invertible it follows that $\beta := \mathbb E[xx^\top]^{-1}\mathbb E[xy]$.
For each endogenous variable in $x_k$ you lose one identifying equation because for the endogenous variable $x_k$ it does not hold that $\mathbb E[x_k\epsilon] = 0$. So for example with 2 endogenous variables you would be trying to identify $K$ parameters using only $K-2$ equations which amounts to trying to solve a system of $K-2$ linear equations for $K$ unknowns. So assuming that you know how systems of linear eqautions work it should come as no surprise that you need to find at least two instruments. This would allow using the true equation $\mathbb E[z_k (y - x^\top \beta)] =0$ instead of the untrue $\mathbb E[x_k (y - x^\top \beta)] =0$ for each $k$ (which amounts to assuming $z_k$ is exogenous while $x_k$ is not).


In this setting, both buyers and sellers are usually assumed to be risk neutral. So adding an additional layer of uncertainty over seller's valuation is not really going to change the model's prediction in any "qualitative" way, in the sense that market unraveling (perhaps only partial in this case) is still an inevitable outcome, even though the exact equilibrium may be different.

Suppose the buyer is unsure of the seller's valuation, but the buyer believes that the seller's valuation follows a distribution with a mean of $3000$. Further, let's assume that under this distribution, a fraction $p\in[0,1)$ of the plum seller's value is below $2750$, the buyer's average value of the two types.

So, by offering to buy a car of unknown quality at $2750$, the buyer gets an expected payoff of
\begin{equation}
\frac{p}{2}(4000-2750)+\frac12(1500-2750)=1250\left(\frac{p}2-\frac12\right)<0.
\end{equation}
This is worse than the zero payoff the buyer gets by offering $1500$ for a lemon.

Since the mean of seller's valuation is $3000$, there is always a positive fraction of the sellers with values greater than $2750$, i.e. $1-p>0$. These sellers will leave the market as they don't anticipate fetching a price higher than their own values for the car. The exit will then trigger the usual cycle of downward revision of buyer's conditional expected value and more exits. Depending on the distribution of seller value, full market unraveling may or may not occur.

Nevertheless, what underpins the unraveling of (necessarily a fraction of) the plum market is still the same: the buyer's anticipation of a positive fraction $(1-p)$ of the plum sellers not willing to sell at $2750$.


Amazon vs Netflix, closely resembles which type of competition?

It certainly isn't a Cournot scenario. That would require homogeneous products. In fact there's a ton of differentiation between the two products (Amazon prime and Netflix). They have different TV shows, different movies, different documentaries.

I would argue that it is this differentiation that allows them to charge different prices.

Hotelling's law is much more applicable. However I would argue that they do not maintain minimum differentiation. Yes they may share similarities in their web presence, and they may have adopted the strategy of producing content in house; but they also are so different that it is hard to say they are copying each other on almost everything.

Amazon prime offers reduced price shipping, music streaming, and reduced prices on some items on Amazon. Netflix has none of those things.

That said, Netflix includes hard copies of DVDs as rentals included in their subscription, whereas you would have to buy the DVDs on Amazon. So even though some of their business is similar or matching, they mostly represent differentiated products.


Is perfectly inelastic demand always coupled with inelastic supply?

I assume you are talking about Price elasticity of demand/supply and not cross elasticity or income elasticity of demand/supply.

Not necessarily.

Perfectly inelastic demand means that a change in price leads to no change in quantity demanded. Imagine a country where, say, tomato is super crucial to survival, any change in its price will not make people want to buy more/less of it.

Inelastic supply can exist separately from perfectly inelastic demand. This is just how responsive supply is to changes in price. Just because the price increases/decreases, doesnt mean producers can make more/less of it more easily, even if they wanted to, e.g. think about sudden weather changes for crop production as an example.

In conclusion, no perfectly inelastic demand is not always coupled with inelastic supply, but it can be.


Bayesian incentive compatibility for a general distribution

Your reformulation is certainly correct for independent values.

However, if value $v_i$ carries information about $v_{-i}$, you cannot write the incentive compatibility like that.
For independent value draws, the probability of winning and the expected payment in the direct mechanism depends only on your reported type $v'_i$, not on the true type $v_i$. That is the reason why you can write
$$u_i(v_i) \geq u_i(v_i^{'}) = v_iq(v_i^{'})-m_i(v_i^{'})$$
instead of $\widetilde u_i (v_i^{'},v_i) = v_iq(v_i^{'},v_i)-m_i(v_i^{'},v_i)$. That is, you cannot formulate your expected utility like this. If this was the expected utility, you could also formulate the expected transfer in a similar fashion and derive a revenue-equivalence result, but revenue equivalence requires independent values.


Why in maximization in OLG model with 1 generations we assume that both consumptions are positive?

I think all the textbook is trying to say is that there can be no corner solutions to your maximization problem.

Not refusing consumption in any period, I think translates to consumption in each period > 0.


Is there a study of world poverty rates over time pre 1800?

This is only for Europe unfortunately, but Bob Allen has a famous paper in which he presents reconstructed data for real wages in different European cities from 1350 all the way to 1799:

  Allen, Robert C. "The great divergence in European wages and prices from the Middle Ages to the First World War." Explorations in economic history 38.4 (2001): 411-447.

From real wages and price data, he then infers a measure of (extreme) poverty.

Look at Figure 5 in the paper. The scale of the y-axis is a measure of poverty in which 1 is the poverty threshold. Values above 1 on the y-axis correspond to real wages lower than survival rate (i.e. that do not allow a survival caloric intake), and values above 1 indicate a real wage above survival rate.


Money demand is fixed because it depends on the real economy....that's an assumption.
So when M changes only P can change. That's the theory. It's not really a theory. Just an accounting equality, since V is never defined exogenously. It's bad economics. :/


What is the graphic intuition behind Marginal rate of subsitution

This isn't a perfect answer to your question, but I tend to show all my students pages 1-4 of Dixit's Optimization book. It describes exactly what the MRS is doing and why it must be tangent to the budget constraint.

https://www.scribd.com/document/390573490/Dixit-Optimization-in-Economic-Theory-pdf (pages 1-3 here unless you sign in)


What are the books usually called for studying the computation of GDP?

The ultimate reference is the UN's SNA 2008 manual. SNA stands for System of National Accounts. The SNA 2008 is the international statistical standard for the national accounts


Dornbusch model for exchange rate undershooting

This will be long, but the subject is worth it.

The gist of the "exchange rate overshooting" model in Dornbusch, R. (1976). Expectations and exchange rate dynamics. The Journal of Political Economy, 1161-1176. can be given using two equations:

Uncovered Interest Rate Parity (UIRP)
$$i  = i^*+ \dot e\tag{1}$$

where $i$ is local interest rate, $i^*$ is international interest rate, and $e$ is the logarithm of the foreign exchange rate, i.e. local currency units per unit of international currency basket. So an increase in $e$ reflects depreciation of the local currency. A decrease in $e$ reflects appreciation of the local currency (I would prefer to use the local exchange rate, where conceptually we link "rise" with "rise", but it is unfriendly to the phase diagram that it will follow). Then $\dot e$ is the expected percentage change on the exchange rate.

Money-market Equilibrium (in logarithms)

$$m-p = -\lambda i +\phi y \tag{2}$$

where $m$ is money supply, $p$ is the local price level, and $y$ is local output.

The assumptions are: international interest rate is fixed. There is perfect capital mobility, so the local interest rate is a "jump" variable, being able to adjust instantaneously (and so eq. $(2)$ holds always). Local output is at full-employment level. Local prices are "sticky": this means that the local price-level is not a "jump" variable anymore, but its evolution is governed by a difference/differential equation that reflects gradual adjustment. Money supply is long-term neutral.

The famous "overshooting" effect from an unanticipated permanent expansion of $m$ can then be derived: an increase in $m$ raises the left-hand side of $(2)$. Since prices are sticky (don't jump) and output is fixed, the only way that equilibrium can continue to hold in the money market is by a decrease in $i$ (due to the minus sign), so that that eq. $(2)$ continues to hold.
Given the fall in $i$, and the assumption of constant $i^*$, the only way now that $(1)$ can hold with a lower $i$ is for $\dot e$ to be negative, i.e. if the local currency is expected to appreciate. But raising the long-term neutral local money supply can only create expectations for local currency depreciation, not appreciation.
"Overshooting" cuts the Gordian Knot: the immediate response of the exchange rate $e$ is to depreciate by more (overshooting) than the eventual depreciation that is consistent with the increase in the money supply, and then it starts to appreciate to reach its new level (and so we can have $\dot e<0$).

The question asks about "fiscal expansion". How is this to be represented in the model? The only possible way is to assume that output can rise due to the fiscal expansion.

A fiscal expansion increases $y$ and raises the right-hand-side of $(2)$. Assuming that money supply is unchanged (and prices of course sticky), to equilibrate the money market, the local interest rate must now rise (to reduce demand for money balances that tend to rise to facilitate transactions on the higher $y$). Turning to $(1)$, with a higher local interest rate, we need expectations for currency depreciation ($\dot e >0$) in order for UIRP to hold. But a fiscal expansion, be it temporary or permanent, given a full-employment output or not, is usually associated with local currency appreciation (higher $e$).
So it appears that here too the overshooting effect obtains: the local currency jumps and appreciates by more than it is consistent with the fiscal expansion, and then it starts to depreciate, and we obtain the needed $\dot e >0$.

Let's put this in a (semi) phase diagram. We need a bit more specification, how the exchange rate and the price level change. Denoting $\bar e$ and $\bar p$ their long-run equilibrium values, we specify

$$\dot e = \theta(\bar e -e)\; \;\;\;\text{eq.}(3a)\\ \dot p = -v(p-\bar p)\; \;\;\;\text{eq.}(3b)$$

Notes: Both parameters are positive. The above look like ad-hoc "adaptive" schemes, but: the $\theta$ parameter can be determined so as to be compatible with rational expectations, the $v$ parameter is essentially determined endogenously, while the sluggishness of the price level is an  a priori assumption. This means that while both are governed optimally by a differential equation, the exchange rate can jump, while the price level cannot -it is "slaved" to its differential equation, while the exchange rate is only optimally characterized by its own equation, not chained to it.

By setting $\dot e=0$ we can obtain the expression for the long-run price level:

$$\bar p = m+\lambda i^* - \phi y \Rightarrow m = \bar p-\lambda i^* + \phi y\tag{4}$$
Inserting equation $(3a)$ into $(1)$, then $(1)$ and $(4)$ into $(2)$, we have

$$\bar p-\lambda i^* + \phi y -p = -\lambda[i^*+ \theta(\bar e -e)] +\phi y$$

Simplifying and re-arranging we get

$$ e = \bar e -\frac {1}{\lambda \theta}(p-\bar p) \equiv QQ\tag{5}$$

Equation $(5)$ is the saddle-path of the economy in the $e-p$ space (although in Dornbusch's paper time it was rarely called that). And it alone can tell us about overshooting. Note that the current exchange rate level depends positively on both its long-run value but also on the long-run price level. Following Dornbusch, we normalize the initial relative price of local to foreign output to unity, and so we have that $\bar e = \bar p$. This means that in the $e-p$ space the long-run equilibrium will be on the $45^{\text{o}}$ line. Putting $p$ on the vertical axis, the $QQ$ schedule will be downward sloping. Then the phase-diagram (without fixed-point loci, we don't need them since we have the saddle-path) is, for the case of a fiscal expansion,

We assume that we start at the long-run equilibrium point $A$, (and we will discuss this point later). Then, fiscal expansion happens. If it is permanent, it is associated with long-run currency appreciation and a lower long-run price level. This means that the saddle-path of the economy moves permanently to the left and becomes the $Q'Q'$ schedule. How can the economy find itself on the new saddle-path? It must jump. But it cannot jump vertically because prices are sticky. It can only jump horizontally: so it goes from point $A$ to point $B$ which is one the new saddle-path, and then starts to travel towards the new equilibrium, that is point $C$. But at $B$ corresponds exchange rate $e_B<\bar e_2$, the latter being the new long-run equilibrium exchange rate. And lower $e$ means more appreciated currency. So the exchange rate "overshoots" the appreciation level, and then starts to depreciate as the economy moves towards $C$.

Assume that the fiscal expansion is considered "temporary" (for decades, this was a good laugh in real-world macroeconomics -not so in the last couple of years). Still, even though the long-run equilibrium point remains $A$, there is short-term and mid-term horizons that must adjust. Here the point $C$ is some sort of "temporary" equilibrium point, which subsequently will start again to approach $A$. But the initial adjustment will happen exactly in the same way, and we will have overshooting.

Imagining a monetary expansion, amounts to shift the saddle-path outwards -and obtain overshooting related to depreciation.

A final note: if one plays around with the diagram, he will realize that
if we do not start at the long run-equilibrium level, and if we are sufficiently far away from equilibrium on the $QQ$ schedule, we may obtain undershooting in either case of policy. For the case at hand (fiscal expansion), imagine that we are at point $D$ on $QQ$ that at the time of the expansion corresponds to a price level below the one that corresponds to new equilibrium point $C$, and we are moving upwards towards $A$. When the fiscal expansion happens and the new saddle-path is $Q'Q'$ we will jump horizontally alright -but we will jump on the upward moving part of the new schedule (draw an horizontal line from $D$ to see that), i.e. we will undershoot the needed appreciation, and then we will gradually continue to appreciate the currency.
But this case is not unique to the fiscal expansion scenario -it can also happen in the monetary expansion scenario.

But if in principle the model permits everything to happen, why everybody went bonkers over Dornbusch paper back then, and it continues to be considered one of the high peaks of economic theory? For the historical framework one can read this. The fact that the model offered a clear and optimizing reasoning to account for the continuous ups-and-downs of the exchange rates (a new "bizarre" phenomenon at the time) I guess was reason enough (and also perhaps because open economies sufficiently far away from the equilibrium point of their nominal variables are not considered very likely -prices do adjust, after all).


Where does the money required for reserves and loans come from?

Well I think that the key to understanding this is to realize that money supply as other macroeconomic aggregates must be defined over period of time.

If you deposit 100\$ bill at a time $t$ with  10% reserve system money supply can increase up to 1000\$ because first bank keeps 10\$ and lend outs 90\$ next keeps 9\$ and lends out 81\$ etc. and sum of this infinite sequence with 10% reserves is 1000.

You are completely correct to say that if people would start withdrawing money let’s say at time $t+1$ money supply would contract. So in this example at $t-1$ money supply would be 100\$, at time $t$ assuming no borrowing constraints and perfect markets money supply would be 1000\$ (because of the multiplier) and at time $t+1$ when you withdraw the money (so there is run on a banks with this one depositor) then money supply would contract back to its original 100\$.

People keep having this misconception about money supply, that in only increases if there is some permanent irreversible increase in quantity of money but that not correct. For example, consider another macroeconomic aggregate - unemployment, you would probably say unemployment increased if a person lost job even if in a year or so that same person gets their job back. The same applies to all aggregates.


What is the difference between Solow growth, classical growth, and neoclassical growth?

Classical growth theory: we can put many authors in it but I'd say Ricardo (if we put Malthus aside) is the first one to have describe a real growth theory. For their defenders, growth should somehow end at a certain point in time and is also very unstable for various reasons: Ricardo thinks growth would end because of diminishing returns (cf. Ricardo's On the Principles of Political Economy and Taxation (1817)). In the classical growth theory, you can also include the theory of Harrod-Domar which basically explains that growth is unstable: there is either too much investment in comparaison to savings, or there is not enough investment (cf. Harrod's "An Essay in Dynamic Theory", The Economic Journal (1939)).
Neoclassical growth: Solow's model is a neoclassical model of growth. It comes in response to the Harrod-Domar model and tries to explain that in the long run, growth is stable (no growth of capital per head).. The only extra growth per head can only come through technological progress (cf. Solosw' "A contribution to the theory of economic growth", Quarterly Journal of Economics (1956)), which is exogeneous.

I encourage you also to look at two other models such as: the AK (accumulated knowledge), which is based on the theory of Solow but tries to make technological changes a endogenous variable in the model (in the Solow model technological change is not the consequence of specific policies); Schumpeterian model created by Aghion and Howitt (1998) which also describes the policies that would be best to stimulate innovation and therefore growth...


How is the differential definition of money supply, velocity, and GDP defined?

That is not a differential form of the equation. $\Delta$ there stands just for changes not differential. You can derive the equation in such form by log-linearization.
Take the original equation:
$$MV=YP$$
Apply natural log to both sides:
$$ \ln M + \ln V = \ln Y + \ln P$$
And now you are done, since in an equation that is in log-log form you can interpret everything in terms of changes.
Hence:
$$ \ln M + \ln V = \ln Y + \ln P \implies \Delta M + \Delta V = \Delta Y + \Delta P$$


Why, in supply and demand curves, does price go on the y-axis?

As @KennyLJ points out I think the reason is partly historical. However it is a useful convention for several reasons.

Demand curves are frequently not functions but mappings, e.g. in the case of perfectly elastic demand. So you could not guarantee 'nice' functions anyway.
In this coordinate system the surpluses (producer's and consumer's) are areas under the curves. The explanation is more intuitive and it is somewhat easer to see which integral you have to take.


Why are simple returns additive over securities?

We have a portfolio of two assets, with prices $P_1(0), P_2(0)$ at $t=1$. In the next period, their prices are $P_1(1), P_2(1)$.

The portfolio market value at $t=0$ is $V(0) = W_1 P_1(0) + W_2 P_2(0)$.

The portfolio weights $\omega_i$ (for $i =1,2$) are:
$$
\omega_i = \frac{W_i P_i(0)}{W_1 P_1(0) + W_2 P_2(0)}.
$$

(We can easily see that $\omega_1 + \omega_2 = 1$.)

The return for an asset $R_i$ is given by:
$$
R_i = \frac{P_i(1)}{P_i(0)} - 1.
$$

The market value of the portfolio $V(1)$ is equal to:
$$
V(1) = W_1 P_1(0)(1+R_1) + W_2 P_2(0) (1+R_2).
$$
Then,
$$
V(1) = (W_1 P_1(0) + W_2P_2(0))(1 + \frac{W_1P_1(0)R_1}{W_1 P_1(0) + W_2P_2(0)} +
\frac{W_2P_2(0)R_2}{W_1 P_1(0) + W_2P_2(0)}),
$$
$$
V(1) = V(0) (1 + \omega_1 R_1 + \omega_2 R_2).
$$
If we define the portfolio return $R$ as $\omega_1 R_1 + \omega_2 R_2$, then
$$
V(1) = V(0) (1 + R).
$$

This is easier to see if we normalise prices.


Why's a penny stock basically a call option with a strike price of 0 that doesn't expire?

Your misunderstanding comes from the fact you are treating a penny stock as an option. They are not the same this is just a metaphor. The difference that makes the difference here is ownership.

The reason the strike price would theoretically be zero is, for a penny stock you own the underlying asset, so you would not pay to own something you already own. A strike price is how much you need to pay to purchase the asset, but you own the asset in this case.

The OP Reddit poster is dead wrong and is referencing a huge amount of underlying assumptions about penny stock trading and options trading. You can't treat them as the same.


How to deal with challenging exam questions and truly understand concepts?

I'm saying this from the perspective of a guy who was a student and a TA.

Understanding the material

Saying from the perspective of a TA, as you have hinted, the point of learning economics is not to memorize ways to drive equilibrium outcome for Cobb-Douglas utility function. The point, rather, is to understand how the demand function is derived (starting from the consumer's maximizing her utility). Often, in the process, simple utility functions have many simplifying assumptions involved (continuouity, differentiability, etc.) and recognizing this will help you become more aware of other things... a good example is when you have linear or quasilinear utility function so that you need to consider endpoints as well.

If you truly understand the material, then you have no need to memorize things. You can derive them all in the exam room.

Your question, of course, is how to do that. I feel that if you work on the example questions and focus not just on the mechanics (take derivative of this, solve for $p_x$, plug into demand, etc.) but rather truly think about what the economic agent (consumer, producer) is actually doing... what they want to achieve, and what their constraints are... what would the market demand look like... etc., that might help.

One more technique I use if I have some spare time is to tweak the questions myself. Instead of having this utility function, what would happen if the utility function looks like that instead? Sometimes you work it out and check with friends. Sometimes it doesn't work out (price becomes negative!) and you could figure out why the question you set up yourself doesn't make sense.

Exam format

This leads me to my second point, and where I might feel a bit different from you (and that you didn't ask).

As a student, I often find exams too long. There are too many questions and the time is too short. The only thing, I felt, that the exam is testing is whether the student could memorize that, when met with this kind of problem, the demand function is in the form of this and that. Many of my friends resort to memorizing shortcuts.

I would have much prefer exams you have... something that tests the understanding of the material, rather than your memory of the material. Of course, with this kind of problem, there needs to be enough time so that students could carefully go through and really think about the problems.


What's the difference of economic capital and liquidity buffer?

Different sides of the balance sheet.

Capital represents long-dated instruments (infinite lifetime in the case of common equity) issued by a bank that are subordinated in a way to make them comparable to equity. (Can include instruments that are technically liabilities.) The existence of capital means that assets are greater than non-capital liabilities, i.e., the bank remains solvent.
A liquidity buffer are assets that are deemed liquid, which means that they can be readily sold at prices near carrying value to meet payment requirements. Note that a solvent entity that only holds hard-to-liquidate assets may be unable to meet a sudden payment obligation. That is, it is solvent, but illiquid.


What is the effect of expansionary fiscal policy in case liquidity trap situation?

If I've understood correctly (it's not entirely clear from the question), you're asking about a policy of fiscal expansion in a depressionary environment, accompanied by a monetary expansion.

It is conceivable that an increase in money supply, coupled with an increase in money demand, would lead to unchanged interest rates, if those two opposing forces balanced out.


The paper computes the value of your coefficient $\gamma$ on its own with comparison to $\alpha +\gamma$1.

The interpretation in this context, would be that there is a limit of the returns to RightShare given the presence of observations existing contemporaneously with RightPM and InRange.

1. this of course assumes that $z=1$


How to calculate GDP increase using Cobb-Douglas production function?

I think this approach might be useful.The Cobb douglas production function with constant returns to scale is
\begin{equation}
Q=K^{\alpha}L^{1-\alpha}
\end{equation}
For simplicity, we use logarithms for the rates of change of its determinants
\begin{equation}
ln(Q)=\alpha ln(K)+(1-\alpha)ln(L)
\end{equation}

Differentiating with respect to time on both sides of the equation

\begin{equation}
\frac{1}{Q}\frac{dQ}{dt}=\alpha\frac{1}{K}\frac{dK}{dt}+(1-\alpha)\frac{1}{L}\frac{dL}{dt}
\end{equation}

\begin{equation}
\begin{array}{l}
 \text{Growth rate of GDP:}\frac{1}{Q}\frac{dQ}{dt}=G_{Q} \\
\text{Growth rate of Capital:}\frac{1}{K}\frac{dK}{dt}=G_{K} \\
\text{Growth rate of Labor:}\frac{1}{L}\frac{dL}{dt}=G_{L}
\end{array}
\end{equation}
\begin{equation}
G_{Q}=\alpha G_{K}+(1-\alpha)G_{L}
\end{equation}
Where $\alpha$ is the capital share in the gross domestic product. In your case $(1-\alpha)=70\%$, hence $\alpha=30\%$

Thus,
$G_{k}=0\%$ (The capital in this case does not change.), $G_{L}=1\%$
Substituting in the original equation
\begin{equation}
G_{Q}=0.7\cdot 1\%=0.7\%
\end{equation}
for further growth rate.

Adding to the previous growth rate ($3\%$), we obtain
$3\%+0.7\%=3.7\%$
To find the new GDP after 20 years.
\begin{equation}
 (1+0.037)^{20}=2.068117
\end{equation}


Interest rates in Quantitative Easing of 2008

Quantitative Easing is the purchase of bonds by the central bank. The purchases leave private banks with balances at the central bank (reserves). The private banks are “lending” the Fed money (not that it needs it). The rate if interest paid is set by the Federal Reserve.
The rate paid is shown at: Link to FRED seeies.
(Note that some balance sheet expansion was repurchase agreements - repos - which are a matched pair of transactions where the Fed buys a Treasurybond, then sells it back at a pre-determined price. This can be interpreted as a loan, with an implied interest rate - the repo rate.  These are “open market transactions,” and the exact details are not publicly disclosed. However, the repo market is a large market, and repo trades near the Fed Funds rate, as both represent lending backed by the Federal government.)


Inequality and Growth: the Role of Democracy

Alesina and Rodrik (1994) wanted to study whether the impact would be different for democracies vis-a-vis non-democracies because they show that

  in a democracy (where the ``median voter theorem'' applies) the rate
  of taxation is higher and the rate of growth lower, the more unequal
  is the distribution of wealth.

Their model implies that democracies with a more unequal distribution of capital ownership grow less rapidly than more egalitarian democracies. They argue this is because
the median voter has a relatively small endowment of capital when wealth is unequally distributed, and thus favors high taxes on capital which keep growth low.

They wanted to estimate this prediction by adding a dummy variable for democracies vis-a-vis non-democracies


You are correct that there have in fact been periods of time when growth was flat, or even negative. For example, if one could observe the economy of the Cahokia over time, one would expect to see a period of negative growth. Further, the rapid growth we're to which we are accustomed is a very recent phenomenon.

It's impossible to know whether growth will continue at such a pace or slow dramatically; prominent economists such as Robert Gordon have argued that future growth may be much slower (though few argue that it will be flat or negative), a set of hypotheses known as "secular stagnation".

So while your textbook, showing cyclical variation around a positive long-term trend, correctly depicts modern business cycles, there's no reason to believe that such a trend must appear in all societies at all times.


Interrelationship between inflation and foreign capital flow

I think the flaw in your reasoning is expecting that foreign capital inflow would lead to more ₹ in circulation.  When investors bring in capital, they are buying ₹ that were previously in circulation.  This additional demand without additional supply is exactly what drives the price of the ₹ up.  There is no increase in the quantity of ₹ circulating unless the government/central bank responds in some way.


Why can't a country print its own money to spend it only abroad?

This wouldn't work - for the money to have any value abroad, it has to find its way back to the source country.

Consider an example; the President of Ruritania wants a Swiss watch. He prints a thousand Ruritanian dingbats and sends them to his ambassador in Bern with the instruction to buy him a watch. However, the Swiss watch-maker doesn't want to be paid in dingbats - he wants Swiss Francs. So the ambassador goes to a Foreign Exchange (Forex) broker and asks to buy Swiss Francs using Ruritanian dingbats. What rate does the Forex broker give him? The broker has to consider what he could do with the dingbats - all he can do is sell them to some Swiss guy who needs dingbats for some reason. So it's the demand for dingbats in Switzerland that determines the rate.

Let's assume the Forex broker gives the ambassador some Swiss Francs and off he goes and gets the watch. What happens next? The broker sells the dingbats to a Swiss importer who is buying Ruritanian apples. The Swiss guy uses the dingbats to pay his supplier in Ruritania and now, the dingbats have ended up back in Ruritania!

So the Ruritanian money supply has increased and, despite the best efforts of the President, inflation is stoked.


Wealth Creation: Where does the money come from?

If we view "wealth" as the subjective experience of pleasure, then "wealth" can spring up ex nihilo: if people used to get two dollars' worth of pleasure from having a cup, but now they get ten dollars' worth, then in some sense eight dollars of wealth has come "out of nowhere".

However, it's more likely that at least one party has misvalued the cup. Perhaps the original seller was unaware that there was someone out there who would get ten dollars' worth of pleasure from the cup, or maybe the buyer has miscalculated how much pleasure they will get from the cup. In that case, wealth has not been created, it has simply been transferred to you.

A third possibility is that you have added value to the cup. Perhaps you own a retail store, and 2 dollars is the wholesale price while 10  dollars is the retail price. The person who paid 10  dollars for the cup doesn't want to buy in bulk like you do to get the wholesale price, and they don't want to deal with buying from the manufacturer. Maybe they bought a bunch of stuff from your store, and it would have been a lot of work to arrange purchases from all the different manufacturers. You've added to the end user's convenience, and therefore added wealth to the economy.

The issue of currency is a side matter. Currency is a measure of wealth; it isn't wealth in and of itself. If you had made the cup all by yourself, clearly you would have created wealth, even if no money changed hands. Money enables price signals to let other people know how much you value their activity, but that value exists independent of an explicit monetary figure been ascribed to it. If you create a cup and then sell it for 10 dollars, the wealth creation occurs when you make the cup. When you sell the cup, you are indeed not creating 10  dollars of wealth at that moment; it already has existed since you made the cup (however, depending on how one does the accounting, some of the wealth may be created when you transfer possession, if the person you sell it to has more of a use for it than you do, but it's the transferring possession, not the payment of currency, that causes the increase in wealth).


What does annualised GDP shrinkage of 75% and growth of 54.6% mean?

Growth of GDP is generally reported quarterly on YoY basis in real terms, i.e., the growth rate of 5% means increase of GDP in current quarter by 5% as compared to GDP in same quarter previous year in real terms.
As with most macroeconomic variables, GDP has seasonality and that is why the QoQ is not reported.
For example for india the FY Q1 (Calender Q2) has a less than 1 seasonal factor. So the QoQ of this quarter will always be low and therefore misleading. For Q1 FY 2020-21, India's QoQ growth rate is -27.1%.
Coming to annualized rate: it is the growth rate of GDP if the economy continued to grow (shrink, in this case) at the same pace as the current QoQ growth rate.
So if the QoQ is, say, 2% then annualized rate would be:
$$((1+0.02)^4)-1)*100=8.24$$
In the above example:
$$((1-0.2928)^4)-1)*100=74.98$$
Now in general in Central bank communications and research analysis work, the annualized rate is calculated on deseasonalized QoQ growth, otherwise you ate misleading the reader. The newspaper here seems to have ignored that, perhaps intentionally to magnify the contraction of economy and is most certainly misleading.
On the second part of your question: $75%$ economic of economic contraction does mean collapse. In worst of recessions this does not happen in a period of just one year. In theoretical terms, it is equivalent to everyone's income in the economy to decrease by $75%$.
You can rather think like this: Overall income last year was say $100$. Q1 of India's FY generates income of little less than $25$ (say, $22$ - seasonality effect). Given a negative growth rate of about $23.9%$, actual income generated is about $16.75(=(1-0.239)*22)$. Now as economy opens up, income generated in Q2-Q4 may not be $78$ but somewhere around may be $70$ (or $65$, who knows). So total income would be about $82-87$. Overall, therefore, the annual contraction may come-out to be of around $13-18%$. Certainly not $75%$!!


Why does the Fed feel the need to reduce its balance sheet?

This is veering into the direction of an opinion-based answer, but I would argue that there is no particular evidence that the Fed “needs” to reduce the size of its balance sheet.

If the Fed wanted to revert to its pre-2008 mode of operations - where interest is not paid on reserves - it would be required to reduce its balance sheet so that there are no excess reserves in the system, if it wanted the Fed Funds rate to be greater than zero. In the aftermath of the Financial Crisis, many commentators  assumed that this was the case. However, the pre-2008 Fed operating procedure was unusual relative to what other central banks do, and there is no “need” to revert to it. By paying interest on reserves, it can place the overnight rate wherever it wishes (with small spreads).
The Fed does not mark-to-market its portfolio. Market value losses on bonds does not present an issue, as they can hold to maturity. The only “risk” is that the cost of reserves is higher than coupons received. However, like all central banks, it can operate with negative equity. At most, negative equity is a political embarrassment.
Are there economic effects of a large balance sheet that need to be undone? This is a disputed area of research. My opinion is that the side arguing that the size of the balance sheet matters have not presented strong evidence of this claim. One would need to look at claims about the effects of the balance sheet growth, then see whether they hold up. From what I saw of the post-Crisis literature, the effects were not clear-cut.


What are main methods for econometrics of Macroeconomics?

There are many different econometric methods used in macroeconometrics, but the basic ones are advanced versions of least squares methods and Generalized Method of Moments (GMM). Structural VARs, ARCH/GARCH models and several others are widely used as well.

Since you're already familiar with econometric methods, you might try glancing through Fair's "Macroeconometric Modeling" (http://fairmodel.econ.yale.edu/mmm2/mm.pdf). You don't necessarily have to read it completely, but I found it a really useful document in understanding how to build and use macroeconometric models.

Based on your background and interest, I'd recommend Favero's "Applied Macroeconometrics" and De Jong's and Chetan's "Structural Macroeconometrics" as general books on the subject.


Which economic research articles or books use anthropological methods?

In economics, people usually use mathematics as an intermediary to understand some economic mechanisms that would be difficult to understand without maths.
Economists usually focus more on mechanisms in an analytic way than some historical facts in these mechanisms. That's a little bit what it is said "homo economicus"

Youtube Lecture Link

This link could be interesting for you but I don't know really (and don't think) some mainstream orthodox economists interested in the link between anthropology and economics.


Why do central banks increase their rate to fight inflation?

From Monetary Policy:

  There are several monetary policy tools available to achieve these
  ends: increasing interest rates by fiat; reducing the monetary base;
  and increasing reserve requirements. All have the effect of
  contracting the money supply; and, if reversed, expand the money
  supply.

Increasing rates makes it more expensive to borrow and thus slows economic growth where the US in the early 1980s would be a good example to see the recession that was had at the hand of high interest rates at the time.


Why have so many legitimate smaller biotechs sprung up and thrived?

I would prefer to take a broader tact with this: Why aren't all firms bought at a smaller size? Why are there any small R&D firms?
One answer is related to uncertainty.

Some firms look like they will be profitable, but they are not. Perhaps their research fails: they cannot get approval, the drug doesn't work, or it has severe undesirable side effects. This uncertainty is part of the business.
Firms produce signals of their quality. The signals reduce uncertainty and it becomes clearer which startups will be profitable, increasing their prices. Owners will be less willing to sell, and other buyers will make competing offers.
The increased price of the firm leads to a larger asking price, so any net profit of purchasing the new firm is reduced.
Since the profit margins are reduced, the remaining uncertainty leads to continued hesitation in purchasing.
This process continues until the small startup is no longer small enough to buy.

Sometimes, a profitable firm can be identified with sufficient certainty before the final step 5, and that is where you see purchases.


What is the difference between HM Treasury and HM Exchequer?

Collection of tax revenue in the UK is the responsibility of neither of those but of HM Revenue & Customs which has the status of a non-ministerial department.

HM Treasury is the UK's economic and finance ministry. Its responsibilities include advice to ministers on taxation policy and government expenditure. Confusingly, perhaps, the most senior minister at HM Treasury is known as the Chancellor of the Exchequer.

HM Exchequer is not a UK government department but a rather old-fashioned name for the Government's account into which is paid money from taxation and other sources and from which government expenditure is paid. The account is more commonly known as the Consolidated Fund.


Is there anything like ratio between number of manufacturing industries to number of retail shops?

Economists are generally more interested in the aggregate output of different sectors of the economy rather than the number of firms or outlets.  The difference between lots of small shops and a few large shops is quite different to the difference between the majority of GDP coming from industry or services.

If a country has a lot of small retail outlets and a few large manufacturing firms, that suggests that there are much larger economies of scale in manufacturing than retail.

The aggregate output of agriculture, industry and services are normally reported in GDP tables (here's an example for Rwanda).

A larger services sector is normally associated with richer, more advanced countries.  These images compare countries based on the relative size of industry, services and agriculture, and the proportion of the labour force working in each:

In poorer countries, agriculture tends to be lower productivity than other sectors, so those countries tend to show up as greener in the labour share than the GDP chart.


The two (pure) Nash equilibrium in this game is (Betray, Silent) and (Silent, Betray).
Let us see why (Betray, Silent) is an equilibrium.
Let us look at person A.
Person B is playing Silent.

If she plays Betray, she gets $2$.
If she deviates to Silent, she gets $1$. So she would not play Silent.

Now consider person B.
Person A is playing Betray.

If she continues playing Silent, she gets $0$.
If she deviates to Betray, she gets $-1$. So she would not play Betray.

No player has an incentive to switch strategies. So this is an equilibrium.
The same argument holds for the (Silent, Betray) equilibrium.


Deriving intertemporal budget constraint from flow constraint

Take your second equation, move it forward one period, and rearrange. You get:

$$ B_t = \frac{p_{t+1} s_{t+1} + \Delta M_{t+1}}{R_{t+1}}  + \frac{B_{t+1}}{R_{t+1}} $$

Then, define the nominal primary deficit as $D_{t+1} = -(p_{t+1} s_{t+1} + \Delta M_{t+1})$. The above transforms into:

$$ B_t = -\frac{D_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$

which has the format you are looking for. To find the infinite-period intertemporal budget constraint, simply proceed with forward substitution.

The notation I've used (regarding the deficit) is taken from Wickens, Chapter 5. The above is roughly equal to equation (5.11) in the book. There, the author has a more general expression, which includes inflation, GDP growth, and debt-to-GDP ratio. Your example is a simplification of his. I suggest you go through Section 5.4 of the book, which it gives a very detailed analysis of the Government budget constraint (it also gives the solution you are looking for, in case you want to compare).

To find the final expression you are looking for (after edit), take your second equation and replace $B_t$ with the infinite-period intertemporal budget constraint:

$$\frac{R_{t}B_{t-1} + M_{t-1}}{p_{t}} = \frac{B^*_{t} + M_{t}}{p_{t}} + s_{t}$$

where $B^*_t$ is the result of the forward substitution exercise.


While it's not explicitly mentioned in the question, it seems safe to assume that the manager gets to observe separately the outcome of each worker, i.e. the value of $v_i$ for $i=1,2$. If this is the case, the IR and IC conditions depend on the exact terms of the contract.
For example, suppose the manager conditions each worker $i$'s wage on both $v_i$ and $v_{-i}$. That is,
\begin{equation}
w_i(v_i,v_{-i})=
\begin{cases}
w^{11}&\text{if $v_i=10,v_{-i}=10$}\\
w^{10}&\text{if $v_i=10,v_{-i}=0$}\\
w^{01}&\text{if $v_i=0,v_{-i}=10$}\\
w^{00}&\text{if $v_i=0,v_{-i}=0$}
\end{cases}
\end{equation}
Further, let $p_h^{11}$ denote the probability that both workers produce the high-value outcome (indicated by the superscript $11$) when $i$ works hard (indicated by subscript $h$) and the fellow worker $-i$ also works hard (we suppress the second $h$ in the subscript as we always assume that $-i$ works hard). In more compact notations,
\begin{equation}
p_h^{11}=\Pr(v_i=10,v_{-i}=10 \mid a_i=a_h, a_{-i}=a_h).
\end{equation}
Probabilities $p_h^{10},p_h^{01},p_h^{00},p_l^{11},p_l^{10},p_l^{01},p_l^{00}$ are similarly defined.
Then, the IR constraint for worker $i$, assuming worker $-i$ is going to work hard, would be
\begin{equation}
p_h^{11}\sqrt{w^{11}}+p_h^{10}\sqrt{w^{10}}+p_h^{01}\sqrt{w^{01}}+p_h^{00}\sqrt{w^{00}}-a_h\ge 1.
\end{equation}
The IC constraint for worker $i$, again assuming worker $-i$ is going to work hard, would be
\begin{multline}
p_h^{11}\sqrt{w^{11}}+p_h^{10}\sqrt{w^{10}}+p_h^{01}\sqrt{w^{01}}+p_h^{00}\sqrt{w^{00}}-a_h\\ \ge p_l^{11}\sqrt{w^{11}}+p_l^{10}\sqrt{w^{10}}+p_l^{01}\sqrt{w^{01}}+p_l^{00}\sqrt{w^{00}}-a_l.\end{multline}
From the information given by the problem, you should be able to figure out the eight probabilities.
If the wage is conditioned simply on $v_i$, or
\begin{equation}
w_i(v_i)=
\begin{cases}
w^1&\text{if $v_i=10$}\\
w^0&\text{if $v_i=0$}
\end{cases}
\end{equation}
then the IR and IC for each worker would be
\begin{align}
0.7\sqrt{w^1}+0.3\sqrt{w^0}-a_h&\ge 1 \tag{IR}\\
0.7\sqrt{w^1}+0.3\sqrt{w^0}-a_h&\ge 0.3\sqrt{w^1}+0.7\sqrt{w^0}-a_l. \tag{IC}
\end{align}


Heuristics with time series measured in dollars

If your analysis is trying to provide predictions for the particular sector at hand in isolation, then one would ask why would you go for more than the index for that sector. However, one then notices that your index is the most comprehensive one possible for your research due to the isolation.

The last point brings one to ask the question of what happens whether the analysis will need a different index if the aforementioned isolation is relaxed. In that case, it is highly likely that you would need to change the benchmark index depending on the purpose. For instance, if a capital owner is trying to find a sector to invest in, it might be in his best interest to see how different sector are doing with respect to each other (or a benchmark) to come up with a better educated decision. An equivalent concern would be subsidies for instance.


Uniqueness of equilibria in first-price auction with discrete valuations

Ruling out (strictly) overlapping intervals should be possible using the fact that players must be indifferent among all actions in the support of the mixed strategy, or that these type of auctions is efficient.

Uniqueness is tricky, simply because there are many possible interval structures like that one.  But I guess you mean, equilibria are always of this form. In that case, uniqueness should be possible among symmetric equilibria, after ruling out the overlapping intervals, you need to show that the support of the mixed strategy is convex (which should be the case, otherwise money is left on the table), that the intervals are ordered (which comes from efficiency),  and put restrictions on the bidders with the lowest valuation.

Without symmetry, you can have situations where players with the same valuation use different mixed strategies  (even with different support), so the problem there seems untractable to me.


Is free-trade beneficial if living costs are unequal between countries?

The classic answers is transfers through welfare programs, re-training etc. See this question which is almost a duplicate. I'm re-quoting from there, just with a different emphasis:

  But just because the size of the pie expands [due to free trade], it doesn’t mean that everyone is better off. There are going to be some losers whose slice of the pie is so much smaller that they would have been better off with less trade. However, because the overall size of the pie has got bigger, the government can compensate the losers which can still make everyone better off. [...] But remember: thanks to free trade, you can afford that [better welfare programs], because the overall size of the pie is bigger.

Or

  Trade enables countries to experience economic growth and a rising standard of living by increasing access to physical capital and export markets. However, not everyone is better off as a result of international trade. Some domestic businesses and industries fail in the face of foreign competition, which results in job losses for workers. Economists suggest, however, that policy solutions that impose trade barriers are harmful to the economy. Rather, they propose policies such as those that provide job training programs to assist those displaced by trade.

(If you want to ask about the Eurozone crisis, please ask separately, I don't want to get into details on that here. Suffice to say that it wasn't simply a matter of free trade, but also of a currency union etc.)