US GDP vs sub-$50/bbl oil How would we expect US GDP to be affected by oil prices dipping into the 30-50 USD/bbl range?  If you need to make additional assumptions to answer this question, please note these, but still answer the question.

I would expect that this would lower the cost of a major fundamental input and therefore increase the US GDP dramatically.  The US is both the largest producer and consumer of oil.  However, the US produces more oil these days from more expensive sources (shale) -- so perhaps this is a naive assumption?

This article suggests a slight increase -- but perhaps a drop larger than $10 would change this analysis?

Consider a basic case of linear supply and demand functions for oil.

Demand: $q_d = \alpha + \beta P + \gamma X + \epsilon$ Think of X as a demand shifter, so Z could move around if the Chinese economy goes into recession or changes growth speed, someone invents a cheap hybrid car, or Americans wake up one day and decide to drive less.

Supply: $q_s = \eta + \zeta P + \psi Z + \phi$ Think of Z as a supply shifter, so if Z could move around if we have a shock to oil production like an expansion of fracking, a return of Libyan oil production to market, or the Saudis decide to reduce production for strategic reasons.

Now in equilibrium supply equals demand ($q_d = q_s$):

$\alpha + \beta P + \gamma X + \epsilon = q_d = q_s = \eta + \zeta P + \psi Z + \phi$

Which we can solve for P:

$\Rightarrow P = \frac{(\alpha - \eta) \gamma X - \psi Z  + \epsilon - \phi}{\zeta - \beta}$

We can also solve for $\Delta P$:
$\Rightarrow \Delta P = \frac{ \gamma \Delta X - \psi \Delta Z  + \Delta (\epsilon - \phi)}{\zeta - \beta}$

When we observe that P has fallen, that may be because $\Delta Z>0$ or because $\Delta X<0$

But for the economy, we'd expect quite different results from $\Delta Z>0$ and $\Delta X <0$. A negative demand shock ($\Delta X <0$) for oil probably reflects people being poorer and the economy heading into recession. A positive supply shock ($\Delta Z>0$) probably reflects technological improvements that make oil extraction more cost effective. The former bodes poorly for the economy, the latter bodes well.

So it isn't enough to say that it is an exogenous change in oil prices, we have to know more about why oil prices are changing. Economics blogger Scott Sumner warns about this particular issue, advising his students and readers to never reason from a price change. However, if we can take a stand on where the shock is coming from we can get some place. For example, if we assume it is entirely a oil supply shock then because we have estimates of the price elasticity of demand for oil, we can ask how much the supply curve would have had to move to drop prices as observed. From a supply curve shift and a sense of the demand function we could then calculate the increase in social surplus(measured in dollars) from the movement in the supply curve. Or you could treat it as a pure positive income shock to american consumers, negative to American oil producers, and use estimates of fiscal multipliers and marginal propensity to consume to estimate the effects.  I suspect you could do something similar with demand shocks.

As a thought experiment, I tried to think of this as a global supply shock. In my scenario oil prices are lower because production is up but there are no additional knock on effects on demand for exports or the US dollar. We also have to take a stand on if this is temporary or permanent. In general you'll get much bigger effects from permenant shocks. I'm going to assume the price shock lasts a single year. I'm not positive that's a coherent scenario but it gives me something to plug into my calculator.

According to the US Energy Information Administration the US is a net importer of about 5,700,000 barrels of oil and oil products a day. If the price of oil falls from $\approx$\$102 to  $\approx$\$40 a barrel that's about \$127 billion in savings by the American public. That works out to about \$410 a person a year or 0.75 percent of GDP. What about the fiscal multiplier you ask? The marginal propensity to consume numbers are like 4% in representative agent models, which means that if everyone were the same and we have a multiplier of 2 we'd get something like 0.81 percent of GDP. Which suggests that the direct effect of oil price changes can't be very important. That is, if the distributional consequences are not important.

The aggregate net export numbers obscure that some people are huge net exporters (the people in oil business) but the vast majority of us are modest net importers. Aggregate production was something like 3.4 billion barrels produced by about 600,000 workers call it an even two million to include support staff and that's a massive \$100,000 less per worker (noting that capital will bear much of the adjustment here, but still, that's huge). With 9.1B barrels (5.7B imports  + 3.4B domestic production) in domestic consumption , that works out to more generous /$1,800 a year in savings for the 308 million or so of us who don't work in the oil business. But if everyone is the same, none of that really matters, income losses buy the oil guys are offset but income gains by consumers except to the extent that we are net importers. Do we think that's true here?

In general, we think that poorer people have a higher marginal propensity to consume than richer people, which generally means that shocks that affect poor people have stronger short-run economic effects than shocks of the same size affecting rich people. The people who work in the petroleum business are much higher paid than the average American (about 91k a year compared with say 43k for the typical worker and of course not everyone works).

Imagine America had one guy (some trillionare) who owned all domestic oil production and consumed so little of his money that oil price shocks didn't influence his consumption (MPC = 0). Everyone else was a consumer of oil. In that world the short run aggregate effects would be more like that of giving everyone the \$1,800 wealth effect from cheaper oil. So something like 3-4% of income is a non-trivial shocks, and by some accounts lower income people have MPC as high as 0.5, so that could be say 1.5 - 2 percent of GDP directly.  You'd also get a knock on effect of whatever you think the fiscal multiplier is. Some people would say about 2. So that tells you 3-4 percent of GDP stimulus effect should be the upper bound.

So in my thought experiment I'd say something between 0.75 - 4% of GDP effect from the oil price drop of \$62 a barrel from a pure supply effect.


Does inflation/expected inflation in a given country devalue its currency or increase its value? I've seen both arguments. From my understanding an increase in the rate of inflation should decrease the value of the currency since it's by definition a depreciation in its purchasing power. But I've heard people argue that an increase in inflation actually increases the value of a currency since that country's Central Bank is expected to hike its rates which makes that country's financial assets more attractive and thus increases that country's currency value through an increase in the demand for that currency.

Which of these two views make more sense theoretically and which is more supported empirically?

I've seen both arguments. [...]

You are comparing two arguments that are not incompatible. The first one scenarizes a situation within which the CB does nothing, while the second argument implies a CB that counteracts. Note that when the CB hikes its rates, it also has some forex-unrelated domestic consequences going in the same direction.

  Which of these two views make more sense theoretically [...] ?

As explained, both. It is all about politics actually. There is no deterministic math here. Put differently, human is an unpredictable variable in the model.

  Which [...] is more supported empirically ?

Once again, both. Notorious investments such as Soros's against the British pound in 1992 perfectly illustrates the second argument. But if Britain hadn't done anything to keep the pound's value high (in relation to the German mark), it would have decreased because of inflation.


Set of rationalizable strategies for this 4 x 4 matrix I would like to find the set of rationalizable strategies for this 4x4 game:

The first thing I did was try and find all PSNE. I found two, the ones I bolded.
Thus, my answer to this question is that the set of rationalizable strategies for Player 1 is {T, Y} and the set of rationalizable strategies for Player 2 is {B, C}.
Is this answer correct?

The set of rationalizable strategies is the set of strategies that survive the iterated elimination of strictly dominated strategies, i.e., strategies that are never a best response. It is a weaker concept than Nash equilibrium.
For player 1, you can eliminate strategy M, which is strictly dominated by T. You cannot eliminate any strategy for player 2 as there is only a weak dominance.
In a second step, you can eliminate A which is dominated by C.
As far as I can see, you cannot eliminate any other strategy in the next step and, hence, all remaining strategies are rationalizable.


Does investment directly influence demand? I quote from the first page of India's Economic Survey 2019 (an official document) -

  Investment, especially private investment, is the “key driver” that
  drives demand, creates capacity, increases labour productivity, introduces new
  technology, allows creative destruction, and generates jobs.

Does investment directly drive demand? If so, how? I can only see investment leading to higher productivity which increases supply and displaces jobs.If not, is the increase in demand a ripple effect of investment? If yes, what's the connection?

As mentioned by @Henry: investment is itself a demand for capital goods (physical, human, etc.).

If I decide to buy trucks this manifests itself as an increase in demand to the truck factory.


Why does profit maximization implies that both of the following are equal? Why Profit maximization implies that rate of return to capital equals the net marginal product of capital. Prove and give intuition why both are equal.

I also wonder that why it doesn't implies rate of return to capital equals the net marginal product of labor?

The highlight part and equation (6.32) is my concern

Please zoom in the web browser to view the image more clearly(press ctrl + turn the scroll wheel of the mouse simultaneously).

The production side is modeled as follows:
There are $m=1,...,j$ identical firms (the number of firms is not necessarily equal to the number of workers of course), that operate in a perfectly competitive environment. This means that firms are price takers, both in the goods market and in the production input market, i.e. they take prices as given when they seek to attain their objective. It also means that markets "clear": in particular, prices adjust without frictions/delay so that all capital and all labor are employed. The firms solve a static (not intertemporal) problem: maximize profits period-by-period separately. Labor is provided totally inelastically, no labor-leisure choice here from the part of workers. Moreover each firm has a constant-returns to scale production function in capital and labor, i.e. the function is homogeneous of degree one.

Omitting the time subscript, the typical firm's production function is

$$F(K_j, L_j),\;\; j=1,...,m$$

and the objective of the firms is to maximize profits which are defined as the surplus of production over payments to labor $wL_j$, net payments to rented capital $rK_j$, and depreciation $\delta K_j$:

$$\max_{K_j, L_j} \pi = F(K_j, L_j)-wL_j-rK_j-\delta K_j$$

Note that these are "real" magnitudes in the economics sense of the word, i.e. that we have divided throughout by the price of output (we do not show it usually, we just say, "expressed in real terms").

Denote $\kappa_j \equiv K_j/L_j$, the capital-labor ratio at firm level. Due to the homogeneity of degree one we can re-write the maximization problem of the firm as

$$\max_{\kappa_j} \pi = L_j\cdot \big[F(\kappa_j, 1)-w-r\kappa_j-\delta \kappa_j\big]$$

Note that $L_j$ has become a multiplicative factor, so we can maximize only the term in brackets, and so only with respect to the capital-labor ratio. We also set $F(\kappa_j, 1) \equiv f(\kappa_j)$ to arrive at

$$\max_{\kappa_j} \pi = \max_{\kappa_j} \big[f(\kappa_j)-w-r\kappa_j-\delta \kappa_j\big]$$

The first order condition for a maximum is for the first derivative to be set equal to zero so

$$f'(\kappa_j) - r - \delta = 0 \implies f'(\kappa_j) - \delta = r$$

This is not yet the funky equation $(6.32)$,although it looks a lot like it, because the latter is expressed in terms of "per capita" capital $k\equiv K/N_1$,  i.e. at the level of individuals/consumers/workers, not at firm level.

How do we arrive at $(6.32)$? Well, since we have assumed that all firms are identical, that labor is provided inellastically, and also that the markets for production inputs clear, we have that

$$mK_j = K \implies K_j = K/m,\;\;\; mL_j = N_1 \implies L_j = N_1/m $$

So

$$\kappa_j = \frac {K_j}{L_j} = \frac {K/m}{N_1/m} = K/N_1 \equiv k$$

and now we have obtained $(6.32)$.

Note how all the assumptions made have been used in order to arrive at this result.


Is the lack of information on customer behaviour good for banks? Discussing chapter 18 on Microeconomic Theory Basic Principles and Extensions - Nicholson and Snyder book - regarding Asymmetric information a student raise an interesting point of view regarding the available information on credit utilization, defaults and so on. According to him, in Brazil, the information from each customer (e.g.: if he has already defaulted anytime during his life) is not available and this is also because banks do not want. He mentioned that banks would have to provide better prices and fees for customers with a good historical behaviour on the use of credit lines and bank products. However, I was thinking and if we consider the capital ratio of a bank, provisioning and other charges banks has to deal with due to the lack of information of their clients, there would be a trade off. And banks wouldn't have great benefits from this lack of information on customers behavior. Additionally, banks could charge a reasonable interest and fees from all customers even if they know the probability of default of the customer is low.
How does this work in other countries? Is the student opinion reasonable?

Basel II opened the possibility towards "your individual interest rate" by allowing (and encouraging) banks to use more advanced methods of estimating Credit Risk, individualizing even, customer-per-customer, for example using Advanced-Internal ratings-based approach.

This means that the banks can lower interest rates towards clients that score better according to these presumably more sophisticated methods of credit risk assessment. They are not obliged to do so (to the degree that interest rates are determined by the interaction of market participants and not decided by the state). On the other hand, by making credit risk assessment more accurate, banks can select the customers with good credit ratings because this, in the end, lowers their overall credit risk exposure and ultimately, lowers also the obligatory capital that they must hold.

So indeed, individual credit rating and individual interest rates may increase the competition among firms and bring contractual interest rates down, reducing the banks' nominal profit rate -while it may keep intact the actual profit rate, since credit risk (and so, statistical losses) will also go down.
Moreover this may lessen the burden of capital reserves. In some cases (and I would say, in the current situation), banks may even care more about the latter than the former.


Repeated games without a beginning and an end I'm interested to know results on repeated games indexing each stage game by $\mathbb Z$ which contrast with those indexed by $\mathbb Z_+$.

It seems to me this could be quite different from repeated games with a starting stage, because there's no need to concern ourselves about whether or not a node could be obtained. Is it the case that we only need to consider stationary equilibrium?

What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z_+$ have the same cardinality. So it makes no substantive difference which index set you use.

Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) Markov strategy in a continuation game with infinitely long history.

If a history of infinite length is what you really care about, then may be looking into the literature of continuous-time repeated game will provide some insight.


Virtual valuation when the distribution is discrete The virtual valuation  of a buyer in an auctin is a function used to calculate the expeced revenue of a seller from that buyer. When the buyer's value comes from a continuous distribution with pdf $f$ and cdf $F$, the virtual valuation is:
$$
r(v) := v - \frac{1-F(v)}{f(v)}
$$
How exactly should the virtual valuation be calculated when the distribution is discrete?

For example, suppose the valuation of a buyer can take three values: 1 USD - with probability 0.3, 2  USD - with probability 0.3, 3  USD - with probability 0.4. What exactly are $F$ and $f$ in this case?

Virtual valuation is the derivative of the expected revenue function with respect to the quantile $q$ that is then evaluated at value $v$. The Revenue function is

$$R(q) = q\cdot v(q),\;\;\; q = 1-F(v),\;\;\; v(q) = F^{-1}(1-q) \tag{1}$$

and

$$r(v(q)): = \frac{dR(q)}{dq}=  \frac{d}{dq}\big[q\cdot F^{-1}(1-q)\big] \tag{2}$$

written, after the calculations, in terms of $v$, so

$$r(v)=v - \frac{1-F(v)}{f(v)} \tag{3}$$

If $v$ is discrete we have

$$q_j = 1-F(v\leq v_j),\;\;\; j=1,...,k \tag{4}$$

where now $F$ is a distribution function of a discrete rv, an where $j$ counts the ordered discrete values that $v$ may take, $v \in \{v_1,...,v_k\}$.

Discretizing the relationship we could define

$$r(v(q_j, q_{j+1})):= \frac{R(q_{j+1})-R(q_j)}{q_{j+1}-q_j} \tag{5}$$

which in terms of $v$ becomes

$$r(v(q_j, q_{j+1})):= \frac{v_{j+1}\cdot [1- F(v\leq v_{j+1})] -v_{j}\cdot [1- F(v\leq v_{j})]}{1- F(v\leq v_{j+1})-1+F(v\leq v_{j})} $$

$$=\frac{v_{j+1}\cdot Pr(v > v_{j+1}) -v_{j}\cdot Pr(v > v_{j})}{-Pr(v=v_{j+1})}$$

Using

$$Pr(v > v_{j}) = Pr(v=v_{j+1}) + Pr(v>v_{j+1})$$

we get

$$r(v_j,v_{j+1}) = v_{j} - (v_{j+1}-v_j)\cdot \frac{Pr(v>v_{j+1})}{Pr(v=v_{j+1})}$$

or

$$r(v_j,v_{j+1}) = v_{j} - (v_{j+1}-v_j)\cdot \frac{1-F(v_{j+1})}{Pr(v=v_{j+1})} \tag{6}$$

Since to a first approximation, if $v$ is continuous, and $v_j,v_{j+1}$ are "close enough" we have

$$Pr[v \in (v_j,v_{j+1})] \approx f(v_{j+1}) \cdot (v_{j+1}-v_{j})$$

It is clear how the obtained relation for the discrete case can be seen as the analogue of the continuous case.

For practical applications, one could think also of using a "continuity correction" for the probability in the denominator of $(6)$.


What's the special notation? I'm studying "ALGORITHMIC AND HIGH-FREQUENCY TRADING". Page 249 of it tries to find a formula for $h(t,q)$. First, it supposes $h(t,q) = \bar\kappa log  \space\omega(q,t)$ then let $\omega(t)=[\omega(t,\bar{q}),\omega(t,\bar{q}-1),...,\omega(t,q)]'$.

My question is: "What is the meaning of notation $'$, in $\omega(t)=[\omega(t,\bar{q}),\omega(t,\bar{q}-1),...,\omega(t,q)]'$?
Is it Lagrange's notation for differentiation?

Following, I added a photo of this page. I marked the notation which I mean.

It is the transpose, in this case of the vector.


Solution method in Smith (2006) A Closed Form Solution to the Ramsey Model I am trying to understand the way Smith demonstrates that the general solution to his equation (12) is (13) (see page 6).

  (12) \begin{eqnarray}
\dot{z} &=& (1- \alpha)\left[1-\left(\delta + \frac{\bar{x}}{1+\bar{x}Ae^{\bar{x}t}}\right)z\right]
\end{eqnarray}

  In the Appendix I demonstrate that the general solution to Equation (12) is:

  (13)\begin{eqnarray}
z &=& \frac{1}{\bar{x}+\delta} 2F1(1-\alpha,1,d;\omega)+B\bar{x}^{\alpha-1}e^{-(1-\alpha)(\bar{x}+\delta)t}(1+\bar{x}Ae^{\bar{x}t})^{1-\alpha}
\end{eqnarray}

  The Appendix:
  A.1
  \begin{eqnarray}
\dot{z} &=& -(1- \alpha)\left(\delta + \frac{\bar{x}}{1+\bar{x}Ae^{\bar{x}t}}\right)z
\end{eqnarray}
  This can be integrated to find the complementary solution:(A.2)
  \begin{eqnarray}
z_c &=& \bar{x}^{\alpha-1}e^{-(1-\alpha)(\bar{x}+\delta)t}(1+\bar{x}Ae^{\bar{x}t})^{1-\alpha}
\end{eqnarray}

  To find the particular solution to equation (12), I will use the method of variation of parameters. Conjecture that the particular solution is $z_p$ $=$ $z_c$$\Psi$, where $\Psi$ is an unknown function of time. Substituting this conjecture into equation (12), it follows that:
  (A.3)
  \begin{eqnarray}
\dot{\Psi} &=& \frac{1-\alpha}{z_c} &=& (1-\alpha)\bar{x}^{1-\alpha}e^{(1-\alpha)(\bar{x}+\delta)t}(1+\bar{x}Ae^{\bar{x}t})^{\alpha-1}
\end{eqnarray}

First, is (A.2) then just an integrated version of (A.1) or are there any other steps involved?

Second, I really do not see how he substituted the conjecture into 12 and how can he obtain (A.3) from this substitution? Especially here I am really lost and am lacking the imagination how he came up with $\dot{\Psi}$. Did he substitute $z_p$ for $\dot{z}$ or $z$ in (12)?

Citation: [Smith, William. (2006). A Closed Form Solution to the Ramsey Model. Contributions to Macroeconomics. 6.]

Let's solve the differential equation (12).

As a first step, we look for a "simpler" differential equation, namely (A.1).
(A.1) can be written
$$\frac{\dot{z}}{z}=-(1-\alpha)\left(\delta+\frac{\bar{x}e^{-\bar{x}t}}{e^{-\bar{x}t}+\bar{x}A}\right)$$
On the left-hand side, you have the derivative of $\ln(z)$. You can integrate to obtain the form of any solution $z_c$:
$$\ln(z_c)=-(1-\alpha)\left(\delta t-\ln(e^{-\bar{x}t}+\bar{x}A)\right)+ constant$$
You obtain $z_c=\kappa e^{-(1-\alpha)\delta t}\left(e^{-\bar{x}t}+\bar{x}A\right)^{1-\alpha}$, where $\kappa$ is the exponential of the constant in the previous equation.
This equation is equivalent to (A.2) for a particular constant, such that $\kappa=\bar{x}^{\alpha-1}$. This choice of the constant comes from some boundary conditions on $z$ (which should be stated in the paper).

As a second step, we use the method of variation of parameters, meaning we are looking for a solution of (12), $z_p$, that has a particular form, $z_p=z_c.\Psi$. If we find an expression for $\Psi$, then we have found a solution of (12).
We substitute $z_p$ in (12):
$$\dot{z_c}.\Psi+z_c.\dot{\Psi}=(1-\alpha)\left[1-\left(\delta+\frac{\bar{x}}{1+\bar{x}Ae^{\bar{x}t}}\right)z_c.\Psi\right]$$
This expression simplifies since $z_c$ satisfies (A.1):
$$z_c.\dot{\Psi}=(1-\alpha)$$
This is (A.3).
Then, I guess you we have to find a solution $\Psi$ of this equation, and we are done.


Do banks pay each other from their reserves? Suppose Alice wants to buy Bob's house but they use different banks. Alice gets a loan from bank A and needs to transfer her new deposits to Bob at bank B to pay for the house. Obviously Alice's deposits at bank A will decrease by the same amount that Bob's increases at bank B, but now bank B's liabilities have increased (more deposits) while bank A's have decreased (less deposits). How is this imbalance settled between the two banks?

I'm guessing bank A sends money to bank B through the central bank (since both banks have an account there), which means A is sending B some of its reserves. But it also seems possible to me that bank B could have an account at bank A that gets credited new deposits.

Similarly, when banks loan to each other to meet reserve requirements, do they loan money that the central bank credited to them or do they credit each other with the reserves they need?

The details of payments systems are complex, and vary by country. I am slightly familiar with how it works in Canada, and I believe that the principles would not be radically different in other developed countries. Here’s a primer on the Canadian payments systems: link.

The advantage of Canada is that there are no reserve requirements, so it’s easier to explain. If you look at the balance sheet data published by the Bank of Canada, the Canadian banks normally have a balance of \$0 at the central bank at the end of the day. In fact, that is what is expected. (In practice, there are small misses of the zero balance; a lot of money sloshes through the payments system.) In a country with reserve requirements (like the U.S.), banks would like to have a target balance equal to their required reserves. (At present, the Fed has forced excess reserves into the system.)

The amount of interbank transactions are very large. They send amounts back and forth on the payments system based on various transactions - their own, and client transactions. The job of a Canadian bank Treasurer is to make sure that the net balance of all payments ends up at \$0 at the end of the day, so that the net position at the Bank of Canada is unchanged. (In a country with reserve requirements, they will want to hit at least the required reserves level.) They will achieve this by any number of methods:

issue new securities;
borrow/lend in the interbank market;
buy or sell securities (mainly government bonds);
borrow/lend through repurchase agreements (repos);
if all else fails, run to the central bank and borrow against their assets. This option is greatly frowned upon, and will prompt an investigation from regulators as to why the bank cannot borrow in the markets. (This is termed a “lender-of-last resort” operation, and should only be necessary in a crisis).

(Since you asked about bank lending in particular, it is achieved by Bank A wiring money to Bank B through the payments system, in exchange for a contractual obligation to make a reverse future operation with interest.)

The multiplicity of options means that we cannot trace exactly what happens when customers transfer funds to someone at another bank; all we know is that at the end of the day, all of the bank’s transactions with the payments system should net to zero.


Is a Nash equilibrium anything more than what it is? (Sorry for the fuzzy title, could not think of something more informative. Feel free to suggest improvements)
This question is somewhat of a generalization of "Osborne, Nash equilibria and the correctness of beliefs".  Consider the normal form game

$G = \langle P, S, U \rangle$ with
$P = \{1,\dots, m\}$ the set of players,
$S =\{S_1,\dots,S_m\}$ the $m$-tuple of pure strategies for players in $P$
$U = \{u_1,\dots,u_m\}$ the $m$-tuple of payoff functions for players in $P$

The structure $G$ is rich enough to define the notion of a Nash equilibrium (NE).
However, authors sometimes describe NE based on richer models. For instance, the description by Osborne discussed in "Osborne, Nash equilibria and the correctness of beliefs" hinges on a structure

$\hat{G} = \langle P, S, B, U \rangle$ with
$P = \{1,\dots, m\}$ the set of players,
$S =\{S_1,\dots,S_m\}$ the $m$-tuple of pure strategies for players in $P$
$B = \{B_1,\dots,B_m\}$ the $m$-tuple of possible beliefs about each other's actions for players in $P$
$U = \{u_1,\dots,u_m\}$ the $m$-tuple of payoff functions for players in $P$

In this setting a NE is equivalent to

A strategy profile $s^*$ and a belief profile $b^*$ in which for all $i\in P$
$$ u_i ( s^*_i ~|~ s_{-i} = b^*_{i}) \geq  u_i ( s' ~|~ s_{-i} = b^*_{i}) \text{ for all } s' \in S_i$$
and
$$b^*_i = s^*_{-i}$$

Based on this equivalence, Osborne describes a NE as a situation in which

First, each player chooses her action according to the model of rational choice, given her beliefs about the other players' actions. Second, every player's belief about the other players' actions is correct.

Now in other settings, I have seen  justifications of NE based on other structures richer than $G$. For instance, one could have a game with incomplete information

$\tilde{G} = \langle P, \Theta, p , S, U \rangle$ with
$P = \{1,\dots, m\}$ the set of players,
$\Theta = \{\Theta_1,\dots,\Theta_m\}$ the $m$-tuple of possible types for players in $P$
$p$, a joint probability distribution over types
$S =\{S_1,\dots,S_m\}$ the $m$-tuple of pure strategies for players in $P$
$U = \{u_1,\dots,u_m\}$ the $m$-tuple of payoff functions for players in $P$

In $\tilde{G}$, a NE is equivalent to

A Bayesian-Nash equilibrium in which agent know each other types for sure, that is $p$ is degenerate.

In this case, one could therefore describe a NE as a situation in which

Players know each other's type for sure and play according to a Bayesian-Nash equilibrium

(See for instance "Featherstone, C., & Niederle, M. (2008). Ex ante efficiency in school choice mechanisms: an experimental investigation")
Now these two examples puzzle me. One seems to recommend to view NE as a situation in which beliefs about actions are correct, whereas the other suggests that NE be viewed as situations in which agent have perfect information about each other's preferences.
So my  question is :

Is any of these two argument better than the another in any meaningful way? Is any of the two generalization of $G$ (to $\tilde{G}$ or $\hat{G}$) any more helpful than the other in terms of understanding the necessary conditions for a NE to prevail?

My impression is that the concept of NE stands alone, independently of these generalizations and that there is no "correct" way of enriching the NE framework. As I understand it, "a NE is a situation in which agents are rational and beliefs are correct" is not more or less true than "a NE is a situation in which agents have perfect information on each other's payoffs". But as I see such somewhat conflicting claims showing up again and again, I am worried that I might be missing something.

The conventional definition of Bayesian Nash equilibrium (BNE) is as follows:

  A pure strategy BNE is a profile of
  type-contingent strategies $$(s_i(\theta_i),s_{-i}(\theta_{-i}))=(s_1(\theta_1),\dots,s_{i-1}(\theta_{i-1}),s_i(\theta_i),s_{i+1}(\theta_{i+1}),\dots,s_m(\theta_m))$$
  such that for each $i\in P$,  $$ s_i(\theta_i)\in\arg\max_{s_i'\in
 S_i} \sum_{\theta_{-i}}p(\theta_{-i}\mid \theta_i)
 u_i(s_i'(\theta_i),s_{-i}(\theta_{-i}),(\theta_i,\theta_{-i})). $$

If we want to define BNE with the language of beliefs about the other player's actions, we may propose the following definition:

  A pure strategy BNE is a profile of type contingent strategies
  $(s_i(\theta_i),s_{-i}(\theta_{-i}))$ and a profile of beliefs
  $b^*=\{b_i(\theta_{-i})\}_{i\in P}$, where $b_i(\cdot)$ is a
  distribution over $S_{-i}(\cdot)$ conditional on $\theta_{-i}$, such
  that

  $s_i(\theta_i)$ maximizes $i$'s expected utility (expectation over $\theta_{-i}$) given $b_i(\cdot)$ and a prior on types; and
  $b_i(\cdot)=s_{-i}(\cdot)$, i.e. beliefs in $b^*$ are correct.

Thus if the type distribution is degenerate, the definitions of BNE and NE coincide. Nevertheless, the correctness of beliefs (about the opponents' strategies) are present in both definitions.
The difference is that in BNE, players' beliefs are over a set of functions (due to the fact that in Bayesian games, strategies are type-contingent).


Why should the cost of home ownership and home renting be equal in the long run? According to the person in this video (at about 10:10) from the economist, the evidence shows that long run cost of owning and renting homes are approximately same.

The video itself tries to explain it in the minute before that person speaks about the evidence showing this, but I dont understand the explanation.

The video says that this is because home buyers have to make mortgage payments which include interest and maintenance costs such as insurance, repairs and so on.

But I still don't understand their explanation because when people rent they have to make rent payments forever and they never get the house. However, when people buy house they have to make mortgage payements maybe for 30-40 years and after that the home is theirs and now they only need to make the upkeep cost which I always thought are small (but I am not a home owner so I dont know that for sure). So how come the evidence shows that they are roughly equal? Or is the video incorrect?

Note that the "cost of home ownership" depends not only on the house and the market, but also the individual. For someone who moves every year, home ownership is generally not a good option - that person incurs closing costs every year, and most of their mortgage payments at the start of the loan go to interest, rather than building a meaningful amount of equity in the house.  It would be far better for that person to simply rent, rather than throwing away a few percent of their home's value every year in closing costs.

With this in mind, it's possible that the characteristics of the population make renting and home ownership roughly equal in cost, when averaged over a long period of time and the entire population. If you're not going to stay in a home for several years, it will generally be more favorable to rent, since you do not have enough time to recoup up-front costs associated with buying. I don't have hard numbers to show this equivalence, but it's worth pointing out that annual cost of home ownership vs. renting has a component that is unrelated to market value or interest rate - individual lifestyle choices play a role as well. Empirically, we might find that renting is the better financial option for about as many people as would be better off with home ownership.


Pareto efficient allocation and core allocation in 2 person exchange economy with discontinuous utility I understand that with continuous and strictly increasing utility function we can find Pareto efficient allocations by looking at the allocation that satisfies $MRS_1 = MRS_2$. However, I was not sure how we can find Pareto Efficient Allocations for discontinuous utilities (or one continuous and one discontinuous utility).

For instance, given exchange economy with one unit of initial endowment of the good $x$ and $y$, if we have person A's utility function as $$u_A(x,y) = x^\alpha y^{1-\alpha}$$
and person B's preference is represented by lexicographic: given $(x,y)$ and $(x',y')$, B prefers the former if
$$x>x'$$ or
$$x = x', y>y'$$
In this case, we can't calculate $MRS_B$. Does anyone have any suggestion how can we find the Pareto Efficient Allocation and Core Allocation? Thank you very much for your help in advance!

Here are two pertinent observations:

If A has a positive amount of the $x$-good, and $B$ a positive amount of the $y$-good, both can be made strictly better off (why?).
There is no efficient allocation in which $A$ has no amount of the $x$-good, but a positive amount of the $y$-good (why?).

Remaining are the allocations in which $B$ gets everything and allocations in which A has a positive amount of the $x$-good and all of the $y$-good. Verify that these are indeed Pareto efficient allocations.

Finding core allocations is not much different. There is a small difference if you define blocking so that all parties involved might be strictly better off. In that case, you can have allocations in which $A$ has none of the $x$-good but some of the $y$-good.


How much the wealth inequality increased in the USA in the last decades? Are there any studies trying to measure the increase of the wealth inequality in the USA?

Say for example how much it increased, comparing with 1990. Or compared with any other moment in time after 1970.

This statistic shows the Gini coefficient, an index for measuring
  income distribution, for U.S. households from 1990 to 2016. A Gini
  coefficient of zero expresses perfect equality, where all would have
  the same income,a Gini coefficient of one expresses maximal inequality
  among values. In 2016, the Gini coefficient for household income was
  0,48. This is an increase since 1990 of 0.5 points, indicating an
  increase of income inequality in the United States.

Source: Statista

In case you are interested in how data for such indeces are gathered, a recent study explaining its data process is Wealth Inequality in the United States since 1913.


Does neoclassical economics explain or assume private property rights? I understand that neoclassical economics is empirical, in contrast to the a priori base of Austrian economics. I therefore wonder if neoclassical economics also explains, rather than assumes, private property rights?
Intuitively private property rights offer a greater incentive to invest than mere use rights, such as leases.
Private property also reduces transaction costs by internalising, rather than contracting, activities (theory of the firm).
Is this how private property is explained in economics?

It is true that mainstream economics (which includes neoclasicall economics) is empirical but note that any explanation can  only arise from theory (whether theory is explicitly formulated or only implicit in the background), and any theory is based upon assumptions. Empiricism does not mean you don't build theoretical models with assumptions, it means you go and test those theoretical models rather than just rely on deductive reasoning alone.
For example, theory of relativity is explanation of gravitational attraction deduced from some assumed premises. This theory can be tested by observing whether light bends around massive object. However, note it does not work the other way around. Observing that light bends around massive objects does not provide any explanation for gravitational attraction - on its own it is just a empirical curiosity. This is not to say that one might not first observe something and then develop theory, but point is that explanations themselves can't be empirical.
That out of the way these are common explanations for property rights in mainstream economics with references that also contain empirical examples:

Private Property Solves the Tragedy of Commons: The tragedy of the commons is a situation that arises when goods/resources are commonly owned where individuals, acting independently behave contrary to the 'common good' by depleting the shared resource through their collective action. This happens because if everyone owns let's say fish in the lake, then everyone has incentive to extract those fish as soon as possible otherwise someone else will do so. This is an example of prisoner dilemma type situation. Hence, excessive pollution, overfishing, deforestation and other such examples are observed virtually  every time there is a lack of well defined property rights. Property rights (where possible to apply) solve this issue because if one person has right over some resource they have an incentive to exploit it sustainably  (see Mankiw Principles of Economics pp 218-223).  This is why ivory trade makes elephants almost extinct (as they are 'commons' in many places), while caws are not endangered even if they are much more intensely used for food or skin or why rainforests are disappearing while privately owned forests are managed sustainably.
Private Property Encourages Investment: This was already correctly mentioned in your answer. Some altruistic preferences/utility aside people will invest less in something when they will not reap all the benefits. This is why it is very uncommon to see people making large investment in leased car or property. If you rent a house why would you renovate it just so later on when your lease expires someone else is going to enjoy the investment.
This is even more important in context of investing in factors of
production like capital goods. This is crucial for economic development. Economic growth arises from investment (see Barro & Sala-i-Martin: Economic Growth 2nd ed). Empirically the above two are very important as there are virtually no societies that would achieve sustained economic development without property rights and other inclusive institutions, while a common factor among societies that remain  poor lack secure property rights (see Acemoglu & Robinson: Why Nations Fail? and sources cited therein).
Private Property Rights Encourage Productivity: Private property rights provide incentive for production and encourage productivity. For example, if you are a farmer but there are no property rights so anyone can just walk on your farm and take any produce they want this destroys any incentive for people to produce anything beyond immediate consumption (again see Acemoglu & Robinson: Why Nations Fail? and sources cited therein).
Private Property Rights are Prerequisites for Well Functioning Markets: This is connected to the two point above, but I list it as a separate reason as it goes little bit further. Lack of well defined and secured property rights is actually considered market failure and in fact government providing well defined and secure property right is considered to be necessary intervention to make markets work. As argued by Stiglitz (Economics of the Public Sector pp28):

An important activity of the government, but one that accounts for very little expenditure, is the establishment of the legal framework within which firms and individuals can engage in economic interactions. ... Without laws defining property rights, only the exercise of force would stop one individual from stealing from another. Washout the ability to protect property, individuals would have little incentive to accumulate assets. Needless to say, economic activities would be severely restricted.

Later on pp 77:

But even for markets to work, there needs to be a government to define property rights and enforce contract.

In fact Mueller in Public Choice III pp 10 argues enforcement of property rights is one of the reasons for people developing 'constitutional contracts' and moving beyond state of Hobbesian anarchy toward having some sort of organized state.

Private Property Reduces Transaction Costs: This was again already mentioned in your question and it is also one of the reasons you see in the literature. Often property rights can be used to provide better incentives for people to behave in certain way. For example, a firm my require employee to own certain asset (e.g. car) rather than provide it to ensure that its taken good care, which might be cheaper than having some monitoring in place  (see  Williamson The Economics of Transaction Costs and sources cited therein).

There might be other explanations,  that I have missed and hopefully someone other users can fill those gaps, but I think I listed all the major ones.


Why does China spend so much on agricultural/farming subsidies? I use the OECD's Total Support Estimates (TSE), which I think is a proper measure of total agricultural subsidies (but correct me if I'm wrong).

China's TSE is by far the largest at \$239B in 2017 and even its per-capita TSE (PPP*) is higher than the US's.

Here are some explanations I have come across for why agriculture is heavily subsidized in the rich democracies:

Rural voters wield disproportionate power.
The costs (of the subsidies) are diffuse and so no single voter has much incentive to organize/lobby against the subsidies.
Voter inattention: The voters who pay for the subsidies are simply unaware that they are doing so.

However, China is not a democracy and so it would seem that the above explanations do not apply. So what explains China's large agricultural subsidies?

*I think (but am not sure) the TSEs given by the OECD have not been corrected for PPP. And so, I have taken the liberty of converting them.

Google spreadsheet with data and sources.

The first thing that comes to mind is food security. China has a large population to feed and does not want to depend on foreign nations for its food.

A (speculative)  second reason may be prevention of domestic unrest. The largest share of China's population still lives in rural areas, and by supporting farmers they may silence protests.

A recent article analysing the drivers of China's agricultural policies by Lopez et al., including the fairly recent shift from taxes to subsidies can be found here:
https://www.sciencedirect.com/science/article/pii/S0305750X1630211X


When the global optimal is outside of the constraint set, what will be the demand? $u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex.
$a,b\in\mathbb R^n$ are two points. Our budget set is the (one-dimensional) segment $[a,b]$ that connects $a$ and $b$.
Given: $$x^*=\arg\max_{x\in[a,b]}u(x)$$
Let $b'$ be a point in the segment $[a,x^*]$. That is: $b'=\lambda a+(1-\lambda)x^*$ for any $\lambda\in[0,1]$.
Prove that:
$$b'=\arg\max_{x\in[a,b']}u(x)$$
Graphically this result is very straight forward but I don't know how to mathematically prove it.
I think we could start of proving that $u(\lambda a+(1-\lambda) x^*)$ is monotonically decreasing with $\lambda$.
Are there named theory related?

Argue that, given your assumptions on the utility function, $x^*$ is the essentially unique (and hence global) maximum. (You need this because there may be local maxima when the assumptions on the utility function is relaxed - this will violate the proposition you're trying to prove).

Now simply use the definition of global optima: for any $x\leq x^*$, $u(x)\leq u(x^*)$. This should be enough to give you the result.


Rothschild-Stiglitz (RS) Separating Equilibrium In RS model, where there are only two risk types (high and low risks), there is no pooling equilibrium but a separating one (when there are sufficiently large number of high risks.)

Additionally, they state that if there are continuum of risk types, there can never be an equilibrium.

I have 2 questions:

I am trying to understand the analytical proof of the continuum case. I have the technical report of their paper and will provide that section here. However, I think the proof is not well-organised and I appreciate a clear explanation to the proof.**
In the proof, there is a section (highlighted) stating "necessary condition for existence is that.." and giving an inequality afterwards. To my understanding, this inequality gives how large the high risk population should be in order to guarantee the separating equilibrium. (Please let me know if I am wrong at this point). However, they take the derivative of the incentive compatibility constraint to derive this inequality, i.e. they take the derivative of an inequality to reach another inequality. Don't you think this approach is wrong as taking the derivative of an inequality does not have to yield another inequality?

** In the continuum case, I understand their reasoning that it is enough to show the inexistence of an equilibrium by only showing the inexistence for 2 close risk types, and they start from the maximisation problem of low risks. However, rest is foggy.

I recommend reading 'Informational Equilibrium' by John Riley (1979). It discusses this setting more generally. How the "insurance model" of Rotschild and Stiglitz is a special case of this is presented on pg. 335, and then Theorem 3 on pg. 343 is what you need. The proof is not trivial, but - in my opinion - well presented.


World Stock Markets that went up in 2008 We all know that the US stock market(s) collapsed in price in 2008, see https://en.wikipedia.org/wiki/United_States_bear_market_of_2007%E2%80%9309.

I was wondering what countries' stock markets went up throughout this period.

Include things like frontier markets and not just developed countries.

I cannot find a data source online.

Google is your friend. E.g Australia did very well in the crisis of 2008

See: https://en.m.wikipedia.org/wiki/Financial_crisis_of_2007–2008#


Restrictions on goods and capital, but freedom of labour movement I recently had a discussion with a left-wing friend in the UK about about Brexit, and the possible repercussions of restriction of labour when the UK leaves Europe. He argued that as someone who is quite left-wing, he would be in favour of large tarrifs and other restrictions to prevent the flow of goods and capital, but little or no restrictions of the flow of labour between countries.

Can anyone outline the repercussions of such a policy?

The proposed policy you are describing is akin to a basic International Microeconomics model with free immigration but immovable capital/goods. Assume two economies, $A$ and $B$, where in the one wages are higher, say $w_A > w_B$.

The basic results will be:

1) Immigration flows from economy $B$ to economy $A$ will continue until the wage is equated in both economies.

2) This common wage $w^*$ will be in between the previous two: $w_A > w^* > w_B$. This is because, as labor is concentrated in host Economy $A$ its marginal product falls and so is the wage, while as labor becomes more scarce in economy $B$ its marginal product and so the wage there increases.

The basic distributional consequences will be

1) Workers from economy $B$, both those that emigrated to economy $A$ as well as those that "stayed behind", will have higher wages than before.

2) Workers in the host economy $A$ will see their wage get lower compared to previously.

3) Capital holders in the host economy $A$ will see their returns increase

4) Capital holders in the origin economy $B$ will see their returns decrease.

Introducing a dynamic aspect (capital accumulation, technological progress), would modify the above conclusions in the following way: immigration will partially "hold back" any increases in the host-economy wage, resulting in lower wage increases than what would happen without immigration.

So from the point of view of a host country like the UK is, under such a policy foreign workers (immigrants or not) and local capitalists will benefit, while local workers and foreign capitalists will get hurt.

In my eyes, the above constitute an interesting framework to contemplate the ideological principles alluded to by the OP, but this is a discourse that does not belong to this site.


"Normal" vs "actual" variables in empirical work In an early OECD study by Deppler and Ripley (1978), the authors distinguish between output per man hour, $OMH$, and what they call "Normal" output per man-hour, $NOMH$.

(table 2, p152)

At no point in the paper do they explain this distinction, which also occurs for the labour cost variables.

In the text (p154) they explain that:

  The cyclical component of labor costs is represented by the ratio of
  actual to "normal" output per man-hour.

This all seems very mysterious to me. Are they talking about a particular reference year value? Or is it a trend component?

"Normal" in this context means "trend" (usually a simple linear trend, but in principle could be a non-linear trend by using for example the Hodrick-Prescott filter device).

Decompose output $Y_t = T_t+Y_{c,t}$, where $T_t$ is the trend component and $Y_{c,t}$ is the cyclical component (note that the cyclical component can be positive or negative). Then

$$OMH_t = \frac {Y_t}{L_t} = \frac {T_t+Y_{c,t}}{L_t}$$

while

$$NOMH_t = \frac {T_t}{L_t}$$

Then the ratio of actual to normal is

$$\frac{OMH_t}{NOMH_t} = 1+ \frac{Y_{c,t}}{T_t}$$

This gives a "mark-up" (or "mark-down" if $Y_{c,t}<0$), on "normal costs" and it is in this sense that it "represents the cyclical component of costs". Namely, this ratio is the gross mark-up or mark-down we need in order to go from "normal" costs to actual costs.


Why is the ECB asking EU banks to hold off on buybacks? In the wake of the Covid-19 pandemic, the ECB requested EU banks not to pay dividends as well as refraining from buying back their own shares until the end of 2020. And that even if they had already set cash aside for future loan losses.
Banks however are lobbying for a resumption to shareholder payouts. Presumably because their stocks have been hit hard by the cancellation  of dividends (among other things).
Buybacks (allowed since 1982) can be  a hot political topic, scrutiny is imposed around them. They can affect a companies EPS and therefore executive rewards.
Why is the ECB asking EU banks to hold off on buybacks?  And why is it so important for banks to resume shareholder payouts ? What, in effect, is impact the payout (E.g. impact different if payout is 10% of banks profit VS 50% ) ?

The reason for the ECB to limit dividend payments is to preserve bank capital, to ensure safety and soundness of the banking system at a time of stress.  Remember that when banks pay dividends , the capital base is reduced.   If that happens at the same time that banks have to make provisions against bad loans , taxpayers could be at risk from bank bailouts.


Why does the government need to control interest rates? Context
As far as I understand, economic cycles are mostly dependent on the amount of demand (which is influenced by money supply, which is in turn influenced by the availability of credit). Since credit dwarfs cash as a % of total currency, credit is by far the most important driver of money supply.
In general, the government controls interest rates to slow down an overheating economy, or to expand a slowing economy.

Question
My question is, why do we need government to control interest rates? Assuming banks are rational and want to make money, wouldn't banks tighten credit rules in the midst of an overheating economy? If banks sense that debts are too high, wouldn't it make sense for them to increase interest, or stop lending to risky customers to avoid customers that default on their debts? I don't see why government is any more qualified at controlling interest rates than banks themselves. You could argue that banks are greedy, but at the same time, they're smart and have a long-term view. If a bank loaned money in an overheating market, they wouldn't be in business for too long.

Thanks!

why do we need government to control interest rates? Assuming banks
  are rational and want to make money, wouldn't banks tighten credit
  rules in the midst of an overheating economy?

I assume that by government you mean central bank. Otherwise, I agree with you by saying that governmental intrusion in monetary policy often creates inefficiencies and distortions.

The purpose of any [private] financial entity is to maximize its profits rather than the stability of the domestic currency. Thus, a central bank's primary goal is (or should be) to procure that currency stability, which it pursues by "printing money" and issuing bonds. The conventional term "control [of interest rates]", however, is a overstatement/misnomer from the standpoint that financial entities are not prohibited to transact with each other (or with the public) at lower or higher rates than the ones pursued by the central bank.

That being said, a central bank's pursuit of other goals in addition to --or in lieu of-- currency stability tends to impair its ability to achieve that primary goal (see also last paragraph of this answer). See, for instance, the Federal Reserve's assertions of its dual mandate of maximum employment and price stability. Notwithstanding Ben Bernanke's renown inflationary profile, his repeated calls in the midst of the Great Recession for Congress to enact reforms reflects his awareness that his monetary policy of essentially printing more money was improper, insufficient, and/or ineffective to address the core issues of the labor market. The vulnerability is more severe in those economies where a government exerts greater control on that nation's central bank.

  You could argue that banks are greedy, but at the same time, they're
  smart and have a long-term view. If a bank loaned money in an
  overheating market, they wouldn't be in business for too long.

Some financial institutions negligently or recklessly incur excessive risk, thereby causing severe repercussions when unexpected market conditions occur. Consider the Barings Bank. This entity was founded in the XVIII century, yet the lack of scrutiny of a strategy performed by one of its hitherto "star traders" caused its collapse in 1995.

My allusion to the Barings Bank diverts from your central inquiry, but it illustrates that an entity's longstanding prevalence in the economy does not disprove the necessity for a [purportedly disinterested] regulator or central bank to scrutinize anomalies that are potentially catastrophic.


Deriving demand function from utility function How do I derive the demand function from

$$U=y + 2\sqrt{x}$$

Currently I have $MRS = \frac{1}{\sqrt{x}} = \frac{P_x}{P_y}$, so, $P_y = P_x\sqrt{x}$

Using the budget line: $$I = xP_x + yP_y \implies I = xP_x +y(P_x\sqrt{x})$$

Which simplifies to $$x = \frac{2I + y^2 P_x \pm y\sqrt{Px(4I+y^2P_x)}}{2P_x}$$

I'm not sure if this is correct, should I have did $MRS = \frac{1}{\sqrt
{x}} = \frac{P_x}{P_y}$, so, $P_x = \frac{P_y}{\sqrt{x}}$ instead and substitute $\frac{P_y}{\sqrt{x}}$ into budget line? How do I determine what to substitute into the budget line?

This is an example of quasilinear utility. In general, a quasilinear utility function takes the form
$$u(x,y)=f(x)+y.$$
The perfect substitutes utility function, for example, is a special case of the quasilinear utility function. One feature of quasilinear utility is that the MRS is independent of at least one of the two goods:
$$\frac{\frac{\partial u(x,y)}{\partial x}}{\frac{\partial u(x,y)}{\partial y}}=f'(x),$$
meaning the slope of indifference curves is independent of the quantity of good $y$.

The Marshallian demand functions satisfy the equations:
$$ f'(x)=\frac{P_x}{P_y}$$
$$I=P_xx+P_yy,$$

which come from the first-order conditions of the constrained maximization problem. We can solve for the Marshallian demand function for $x$ directly from the first equation:
$$x^*=f'^{-1}(\frac{P_x}{P_y}).$$
Substituting this into your second equation gives
$$I=P_xf'^{-1}(\frac{P_x}{P_y})+P_yy$$
$$y^*=\frac{I-P_xf'^{-1}(\frac{P_x}{P_y})}{P_y}.$$

In your example, we get
$$x^*=(\frac{P_y}{P_x})^2.$$
$$y^*=\frac{I-P_x(\frac{P_y}{P_x})^2}{P_y}.$$

There is a fundamental problem with your solution. You write a demand function for $x$ that depends on the quantity of $y$. Demand functions depend either on $(P_x,P_y,I)$, if Marshallian, or $(P_x,P_y,U)$, if Hicksian.


What are alternative measures of risk? In finance, the variance of the returns of a security are used as a proxy for the associated risk of the security. I've seen some books include sentences like "if you take variance as a measure of risk...". In what way might variance be an incomplete measure of risk? What alternatives do we have?

A variance is an incomplete measure of risk in a sense, that it measures uncertainty in security payoffs, rather than uncertainty in holder's welfare. In the simplest way we can demonstrate this point as follows.

Suppose that agents want to marginally increase her holding of an asset by $\xi$ and a unit of asset provides a payoff of $x$, which is a random variable. Now we model agent as caring about volatility in her consumption $c$, which is more reasonable, than assumption that she cares about volatility in the payoff directly. Then $$Var(c+\xi x) = Var(c) + 2\xi\cdot cov(c,x) + \xi^2Var(x)$$ and the last term is negligible as $\xi$ is marginal change.

So, as marginal changes are considered in equilibrium, it is more reasonable to take covariance between consumption and payoff as measure of riskiness of the security, rather than just variance of the payoff. In practice the aggregate consumption is often used as a proxy for individual consumption.

Yet another more accurate measure of riskiness can be introduced if we were to presume not direct care for volatility in consumption, but consider an agent with a utility dependent upon his consumption stream. It is a rather lengthy technical exercise and those who are interested should look it up in an excellent textbook "Asset Pricing" by John Cochrane. I will limit myself with presenting the result for multiperiod discrete time model here.

If we were to defined a random variable $m_t = \beta\frac{u'(c_{t+1})}{u'(c_t)}$ called stochastic discount factor for agent with von Neumann-Morgenstern utility and discount factor $\beta$ then by the similar argument as above we may derive that $cov(m_t,x_t)$ should be considered to be even a better measure of riskiness of the security.

Obviously, any nonrandom transformation of the above may be considered to be measure of riskiness, namely, standard deviation and correlation instead of variance and covariance are quite popular. Also different nonrandom normalizations may be applied to $m_t$. For example $u'(c)$ may be used, as $\beta$ is nonrandom and $c_t$ is known at time $t$


Utility maximization in a 2-good scenario with an option to buy a combo of the two I am solving the following question:

Suppose that we live in a two good world, books (x) and movies (y), with utility function given by $u(x,y)=min(x+2y,2x+y)$. Prices of books and movies are 25 and 10 respectively. Income is 1000.
Consider the following two schemes:

The person gets one movie free on each book they purchase.
In addition to buying books and movies separately you also get an option to buy a book with four movies combo pack for 50 only.

Q1. What is the optimal consumption in scheme 1?

A. (40,40)
B. (0,100)
C. (10,75)
D. None of the above

Q2. What is the optimal consumption in scheme 2?
A. (0,100)
B. (10,75)
C. (10,80)
D. None of the above

My working:

For the first part I figured the following:
If I buy $x$ books, I only have to pay for $y-x$ movies, so therefore my budget constraint will look like this:

$$25x + 10(y-x) = 1000$$
$$\implies 15x + 10y = 1000$$

Now solving this budget constraint with the given utility function, I get the optimal consumption as $(40,40)$.

On to the next part, I apply the same logic as above. If I buy $z$ units of the combo for 50, I only have to pay for $x-z$ books and $y-4z$ movies. So therefore my budget constraint will look like this:

$$25(x-z) + 10(y-4z) + 50z = 1000$$
$$\implies 25x + 10y - 15z = 1000$$

My question here is two-fold:

Is my reasoning correct in the above two scenarios?
How do I solve for the optimal consumption in the second scheme, where I have the option to buy a combo?

Thanks!

Your answer to Q1 is correct.

For Q2, a little graphical intuition would help.

Suppose you spend all your income on the combo, you'd be able to purchase $80$ movies and $20$ books. This is the point $(20,80)$ on the $(x,y)$-plane.

If you buy $1$ less combo, you'll have $76$ movies, $19$ books, and $\$50$ cash. Spending the entire $\$50$ on movies would move you to $(19,81)$; spending all the cash on books would move you to $(21,76)$. (Of course you can split the cash between the two goods, but it's easy see on a graph that such a split is not efficient.)

Therefore, trading one combo for movies implies a relative price of 1 book for 1 movie, and trading one combo for books implies a relative price of 4 movies for 1 book. Using the information, we can plot the budget constraint as below:

The green segment, given by $y=100-x$ ($x\in[0,20]$), represents trade-off between the two goods implied by switching from combos to movies. The red segment, given by $y=160-4x$ ($x\in[20,40]$), represents the trade-off implied by switching from combos to books. It is clear from the graph that no other ways of using the income is more efficient. (As a reference, the budget line without the combo is plotted using the black dashed line.)

Given this budget constraint, it should be straightforward to solve for the optimal consumption bundle.


intuition about option values What is the relationship of the option value and the pay off to an option?
What happens if the option value > pay off? or if it is equal to the pay off?

I read that in a put option , the option value is a solution to $$\sup_\tau E[\max\left\{K-S,0\right \}]$$, where $K$ is the exercise price. Does this imply that the possible option values are just 0 or K-S. How important is the option value?

The put option allows you to decide to sell at the exercise price $K$ rather than the market price $S$.

So if you exercise the put option, in effect it is worth $K-S$ at that point as you could have sold at the market price.

You will only do that if it is positive. Hence the $\max\left\{K-S,0\right \}$ expression.

If you could only exercise at a particular future point in time, the option would now be worth the expected future benefit $\mathbb{E}[\max\left\{K-S,0\right \}]$

But with a so-called American put option you can exercise at any time up to a limit, making the value the highest of the future expectations, leading to an option value now of $$\sup_\tau \mathbb{E}[\max\left\{K-S,0\right \}]$$

Typically the option value now is higher than the payoff if exercised now of $K-S$.  This difference is called the time value of the option and reflects the benefit from the future payoff being asymmetrical since it cannot be worse than $0$


What is the proof for why border adjustments are exactly offset in theory by currency appreciation? Why is it true, in theory, that border adjustments are exactly offset by currency appreciation? I've read the colloquial explanations online, but am looking for something more rigorous.

For example, http://taxfoundation.org/blog/exchange-rates-and-border-adjustment says that

  Standard economic theory states that a border adjustment ends up in a wash for both exporters and importers because prices adjust to leave the trade balance unchanged. Specifically, the value of the domestic currency adjusts upward. Standard supply and demand shows why both the import tax and the export exemption work together to push the value of the domestic currency up.

Here is a reference by Feldstein, Krugman that is often quoted when discussing the FX neutrality of border tax adjustments:
http://www.nber.org/chapters/c7211.pdf

Another useful reference that takes a somewhat skeptical view and lists more references is
http://voxeu.org/article/exchange-rate-implications-border-tax-adjustment-neutrality. All models developed or mentioned in these references are non-monetary models, i.e. changes in the exchange rates

I have not digested these references completely yet. However, my understanding of the basic argument is as follows:

In present value terms exports must always equals imports. Changes to taxation may disturb this equilibrium which may then lead to an adjustment of the exchange rate. However, in a world with inelastic labor supply as assumed by Feldstein/Krugman a VAT with border tax adjustments does not have any distortive effects as it changes neither the consumption/savings decision, nor the present value of imports or exports it will not have any real effects.

In principle already from this result one can conclude that border tax adjustments that limit any tax impact on domestic economic agents cannot change any consumption or production decisions as long as the economy is small, labor supply is inelastic and tax revenue is redistributed in a lump sump fashion.

However, a more rigorous argument is as follows:

With inelastic labor supply, households have to decide on their consumption and savings. The first order condition resulting from a standard intertemporal optimization model is (following Feldstein, Krugman)

${U_C}_1/(\delta{U_C}_2)=(1+r)p_1/p_2$

This condition shows that the consumption/savings decision changes only when relative prices or interest rates change.

Consider now various types of firms underlying Auerbach's border tax adjustments, where we again model the firm as in Feldstein, Krugman:

Exporting firms

$maxV=p_X X^1+(1-\tau)(p_D^1 D^1-w^1l^1-p_D^1k^2)+[p_X X^2+(1-\tau)(p_D^2 D^2-w^2l^2)]/(1+r)$

subject to

$Q^1(l^1)=X^1+D^1$ and $Q^2(l^2;k^2)=X^2+D^2$

Where D denotes domestic and X denotes export sales.
Labor market equilibrium is given by

$\bar L=l^1$  and  $\bar L =l^2$

Equilibrium conditions are in the first period

$p_D^1=p_X/(1-\tau)$; $w^1=p_X/(1-\tau)Q_l^1(\bar L)$

From this follows that in the first period and increase in $\tau$ leads to a proportional increase in $p_D^1$ and $w$. This means essentially that the exchange rate will adjust to neutralize the impact of $\tau$.

These conclusions can be shown to hold also in period 2, where there is one additional equilibrium condition $1=Q_k^2(\bar L,k^2)$.

Exporting firms importing an intermediate good

$$maxV=p_X X^1+(1-\tau)(p_D^1 D^1-w^1l^1-p_D^1k^2)-p_m^1m^1+\frac{p_X X^2+(1-\tau)(p_D^2 D^2-w^2l^2)-p_m^2m}{1+r}$$
subject to
$Q^1(l^1;m^1)=X^1+D^1$ and $Q^2(l^2;m^2;k^2)=X^2+D^2.$

Compared to 1. there is one additional equilibrium condition in period 1: $p_m^1/p_X=Q_m^1(l^1;m^1)$ and one in period 2 $p_m^2/p_X=Q_m^2(l^2;m^2;k^2)$.  However, also these addtional equilibrium conditions do not change the overall conclusion that changes in $\tau$ will only impact $p_D$ and $w$.

These conclusions also hold in a more general context, for instance, if there are importing firms producing nontradeable goods.


Why do lump sum transfers affect prices? Let's suppose that I want to maximize total welfare (Social Planner Problem) using lump sum transfers to individuals.

It's known then that the price ratio changes relative to when there are no lump sum transfers. Total income and total endowments are fixed in the model, only distribution of initial endowments is changed between the individuals by planner.

My Question is :

Why do lump sum transfers affect the price ratio in the social planners problem?

Prices (and hence price ratios) carry information about what people want, and what is supplied.

You're only changing initial endowments, hence you're not affecting what is supplied. However, as you change the initial distribution, you change what each individual wants, and possibly (not necessarily) also what the economy as an aggregate wants. This change in wants and haves is reflected in prices.


Endogeneity and Partial effect interpretation of coefficients Is it possible to preserve the partial effect interpretation of the coefficients/parameters, when in the presence of endogeneity? I don't see how it's possible with the 2SLS... Wooldridge, in his graduate book speaks of a special case, in a random coefficient model, but where we can only talk of the Average Partial Effect (APE) interpretation.

Imagine we have the model: $y=\beta_0+\beta_1x_1+\beta_2x_2+u$. Where does the partial effect interpretation come from? Assuming $E(u|x_1,x_2)=0$, $E(y|x_1,x_2)=\beta_0+\beta_1x_1+\beta_2x_2$. Keeping everything known constant except for $x_2$(partial effect of $x_2$), i.e. $E(y|x_1,x_2=1)-E(y|x_1,x_2=0)=\beta_2$. Now if $x_2$ is endogenous, then $Cov(x_2,u)\neq 0 \Rightarrow E(u|x_1,x_2)\neq 0$

Then $E(y|x_1,x_2=1)-E(y|x_1,x_2=0)=\beta_2+E(u|x_1,x_2=1)-E(u|x_1,x_2=0)$.

Granted that IV estimate, when consistent, gives us $\beta_2$, but that's not the partial effect of $x_2$

Addendum:

Alecos,
So let's be specific about the interaction between $x_2$ and $u$. Assume that $u=\beta_{\nu} x_2\ \nu+v$, where $\nu$ is unobserved, and $v$ is an error term such that $Cov(x_2,v)= 0$

Then assuming that we can condition on the unobserved variable, we have
$\frac{\partial E(y|x_1,x_2,q)}{\partial x_2}= \beta_2+\beta_{\nu}q$.
There's no way to estimate this effect, or is there?

Any help would be appreciated.

I am not sure where the interpretation issue lies. If we look at a relation

$$y = b_0 + b_1x_1 + u \tag{1}$$

and we declare that we would like to estimate $b_0$ and $b_1$ but that $x_1$ is "endogenous", we are essentially revealing that we are interested in the partial effect of $x_1$ on $y$ in the relation

$$y = b_0 + b_1x_1 + \delta'\mathbf x{^*} + e \tag{2}$$

where $\mathbf x{^*}$ contains all the (known or unknown to us) variables that are correlated with $x_1$ (and have something to do with $y$ too), and therefore, $e$ is the conditional expectation function error (not just "any" error term), and as such, by construction uncorrelated with all the variables. Alas, $\mathbf x{^*}$ is unobservable or unavailable to us. Valid IV estimation will nevertheless allow us to estimate consistently $b_1$... which is the partial effect of $x_1$ on $y$ in the above relation $(2)$.

Even if we cannot account for endogeneity, the problem will be that our estimations will converge to someplace unknown. So while we will still be attempting to estimate the partial effect of $x_1$ in relation $(2)$, our numerical estimates will be unacceptable and suspect as regards their value (due to the deficiency of the estimation method) -and not because the desired (but unattainable) research target loses its fundamental interpretation. Our estimates will be "wrongly estimated partial effects". It is not the same as saying "they are not partial effects" (epistemologically speaking).

ADDENDUM
Responding on OP's comments and its enhancement of the question, nothing is stopping us from writing $\partial E(y \mid x, x^*)/\partial x$. At the same time our sample is realized, so is $x^*$ realized, irrespective of whether we know its realized value or not.
Now note that the coefficients/partial effects are assumed constant in this linear setting, irrespective of the variability in the regressors: the partial effect will be the same for all individuals in the sample, irrespective of their actual value for the observable regressor

$$\partial E(y_i \mid x_{1i}, x^*_{1i})/\partial x_{1i} = \partial E(y_j \mid x_{1j}, x^*_{1j})/\partial x_{1j} = \beta_1$$

per assumptions, where $i,j$ are two different observations/individuals in the sample.


Old Measure of Inflation Still Tracked? At this ZH link (paranoid blogger warning), the free documentary mentions that using an old measure of inflation, which was used during Paul Volcker's years at the Fed, inflation has been much higher than reported.  In fact, their chart, in some cases appears to be double what's reported.  The documentary implies that this measure of inflation actually included assets bubbles, like housing, which helped Volcker and company prevent too much market enthusiasm.  I also found this CNBC link, which shows the same.

Is this measure still tracked?  I've been Googling to find it and haven't had much luck (and Shadow Stats doesn't appear to be this calculation, but I might be wrong?), but I had no idea the measurements were changed and I'd be very interested in comparing the two.  Thanks!

There have been several changes to the way the CPI is calculated over the years. A significant change occurred with respect to weights and hedonic adjustments in 1998 (I believe the pre-1998 series is what ShadowStats, without having any of the underlying data, attempts to replicate), and one in 1983 that, among other things, changed the way that housing services were calculated from an "asset price" approach to a "rental equivalence" approach.

The 1983 change was made to more accurately reflect changes in consumer prices, rather than reflecting changes in asset prices, which is not the point of the CPI. The 1998 change was made in part to more accurately reflect changes in the quality of goods produced, because it had been apparent for some time, particularly in the tech sector, that a computer sold one year for $500 was much more "computer" than the one sold a year or so earlier for the same price, and was therefore actually cheaper in real terms once you accounted for the change in the quality of the goods.

None of the old measures are still tracked, for two reasons:

The old measures were targeted for these changes because they were not believed to accurately reflect economic realities.
Economic statistics agencies like the BLS have limited budgets, and calculating multiple versions of a series is more costly than calculating one version.

So the series you want doesn't exist basically because it'd be a waste of money to have the government calculating numbers that economists believe are wrong.

Old series are dropped all the time for these reasons; for example, around the time of the 1998 change to inflation calculations, the BEA adopted the NAICS industry classification and dropped the SIC system, because the NAICS system was believed to more accurately reflect the structure of industries at the time. For example, before NAICS, computers were in a category called "Industrial And Commercial Machinery And Computer Equipment" right between the pump/compressor/industrial oven category and the industrial refrigerator category. At the same time, guided missiles were lumped into the same category as space vehicles (under transportation), so that we didn't tell the Russians exactly how much we were spending on our intercontinental ballistic missiles (or so I've heard).

However, if you want an independent check on the BLS-calculated CPI numbers, you can check out MIT's Billion Prices Project (a similar project, the "Google Price Index", is no longer publicly available, to my knowledge). You will find that they generally come to the same conclusions as the CPI.


Why is Roy's Identity so important? I've been reviewing some of my microeconomics theory and have been reading up on Roy's Identity. Recall that Roy's identity is defined  as:
$$x^*_{i}(\text{p},m)=-\frac{\frac{\partial v}{\partial p_{i}}}{\frac{\partial v}{\partial m}}$$

being that the indirect utility function is just the utility function evaluated at the appropriate Marshallian demands, this result is not surprising.

What is an important application of roy's identity that I wouldn't know from just solving for marshallian demands?

When working with assumed functional forms, then you could be justified supposing that there is not much added value.

However, when you try to model behavior without functional forms, you will need all the help you can get in terms of results like Roy's identity to prove your results.

One important application of Roy's identity is in deriving the Ramsey Rule for optimal commodity taxation. A derivation can be found here on page 14.


Is belief in future dividends the only long term value of stock? Are future dividends the primary value backing  that a stock price will rise? (Let's assume that purchase by a competitor is an unlikely event and discount the value this might have).

It seems to me that many stocks are worth astonishingly many times more than their dividends pay, or are likely or even pay. In fact, many companies pay no dividends, and we have no expectation that that they will start paying dividends any time soon. Why buy those stocks?

If your expectation is that their value will go up, it must be because someone else will be will to pay more for it later. However let's imagine even if a company becomes phenomenally profitable, dominates over its competition to an extent that it cannot be acquired, and has lots of cash in the bank, if the company pays no dividends (let's assume its legally bound not to pay dividends for some reason, or has never paid dividends in the past or we have little reason to expect it will pay dividends in the future), where is the value in that stock to a future individual investor?

Disclaimer: There are a lot of interesting aspects to this question. Shareholder voting rights, control over the company, etc. are all interesting things to consider. Here, I only focus on a few parts of the question.
Excess Volatility
Claim: Are actual stock prices much too volatile to be explained by dividends? Is this evidence that the price of a stock depends on more than just future dividends?
It's perhaps not much of an exaggeration to state that the paper that won Robert Shiller his Nobel prize in economics is the paper "Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?" In this paper, Shiller looks at historical prices and dividend payments and asks the question, suppose market expectations of future dividends were on average correct? What should the prices of a stock been historically if those expectations were on average correct? He argues that we can do this by looking at the dividend payments and prices and constructing and "ex-post rational price" at each point in time, called $p^*$. He then compares that price to the actual historical prices, $p$. This is shown if Figure 1 below for the S&P Composite Stock Price index from 1871 to 1979, after detrending both series. The result is that $p^*$ appears to be far too smooth relative to $p$. He argues that this is a problem because it implies that actual stock prices are too volatile to be explained just by movements in future dividends.

References and other resources:

Shiller, Robert J. "Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends?" The American Economic Review 71, no. 3 (1981): 421-36. Accessed March 12, 2021. http://www.jstor.org/stable/1802789.
You can view the following online lectures here. This is from "Asset Pricing with Prof. John H. Cochrane. PART II. Module 3. Time Series Predictability, Volatility, and Bubbles" https://youtu.be/mhGQeJyKVvU?t=275

Important Note: In the current asset pricing literature, there is a strong belief that this "excess volatility" puzzle is not a puzzle at all. Empirical and theoretical investigation seems to indicate that this "excess volatility" arises from variation in discount rates. Shiller constructed this ex post ration prices by assuming that the discount rate was constant over time. Empirically, this seems to be far from the truth. For a good discussion of this, see the following:

Cochrane, John H. "Presidential address: Discount rates." The Journal of finance 66, no. 4 (2011): 1047-1108.

For a video of a presentation of this paper, see here: https://www.youtube.com/watch?v=ZDsOiftolUI

Class lectures of this can be seen here:

3.2 One-Period Identities
3.3 Campbell-Shiller Present Value Formula
3.4 Volatility

Rational Bubbles
Claim: Perhaps stock prices can be explained by a belief that prices will continue to grow in the future, even if dividends do not? That is, suppose the price of a stock is not derived from a belief in future dividends but in the belief that prices will continue to growth without bound.
There is a literature that investigates the possible effect of "rational bubbles" on the price of a stock. This is summarized nicely in the "Discount Rates" paper cited about. One way to investigate this question, as described in the Discount Rates paper is to use the Campbell-Shiller (1988) present value identity,
$$
dp_t \approx \sum_{j=1}^k \rho^{j-1} r_{t+j} - \sum_{j=1}^k \rho^{j-1} \Delta d_{t+j} + \rho^k dp_{t+k},
$$
where $\Delta d_{t+j} = d_{t+j} - d_{t+j-1}$, $dp_t := d_t - p_t = \log(D_t/P_t)$, $D_t$ is the dividend, $P_t$ is the price, $r_{t+1} = \log R_{t+1}$ is the log return of the asset, and $\rho \approx 0.96$ is a constant of approximation. This is an identity based on the definition of returns $R_{t+1} = (D_{t+1} + P_{t+1})/P_t$. This says that the normalized prices of an asset ($dp_t$) is derived from three possible terms:

Future returns on the asset (discount rates)
Future dividend growth paid by the asset (cash flows)
The future price of the asset (bubble term)

Investigation of the contribution of each of these terms to the volatility of normalized prices indicates the nearly all of the volatility comes from variation in the discount rates term and essentially none comes from the dividend growth and rational bubble terms. This also explains the above "excess volatility" puzzle. Note that this still means that dividend explain the value of a stock. It just rules out that changes in the price-dividend ratio of a stock are not explained by changes in future dividends nor by belief in rational bubble growth in prices.
You can view more of the details of the empirical investigation in the paper or the same videos linked in the previous two section.
Conclusion
The attack against the present discounted value model of stock prices embodied in the "excess volatility" puzzle seem to have an explanation in the fact that discount rates vary over time. An investigation of the contribution of future dividend growth, discount rates, and "rational bubbles" to dividend-price ratios indicates that "rational bubbles" do not seem to explain prices. Rather, dividend yields seem to be entirely explained by discount rate variation. Further investigation seems to indicate that prices and dividends seem to be cointegrated. All of this combined is consistent with the theory that prices reflect the present value of expected future dividends.
However, as noted in my disclaimer above, this leaves open many interesting questions. There is still much to do and learn in this literature.


how to find Cournot equilibrium for 2 firms having different MC? When MC function is different for both the firms, how will MR = MC work ?

Let the profit functions of the two firms be
$$
\begin{aligned}
\pi_1(q_1,q_2)&=p(q_1,q_2)q_1-C_1(q_1)\\
\pi_2(q_1,q_2)&=p(q_1,q_2)q_2-C_2(q_2)
\end{aligned}
$$
where $p(q_1,q_2)$ is the inverse demand depending on the total output of the two firms $q_1+q_2$, and $C_i$ is the total cost function of firm $i\in\{1,2\}$. Thus $C_i'$ is the marginal cost (MC) of firm $i$, and we can impose the condition that $C_1'\ne C_2'$ on the cost functions so that the firms have different MCs.

Now we can solve for the Nash equilibrium (NE). Profit maximization would yield the following first-order conditions for the two firms:
$$
\begin{aligned}
p'_1(q_1^*,q_2)q_1^*+p(q_1^*,q_2)-C_1'(q_1^*)&=0\qquad\qquad(1)\\
p'_2(q_1,q_2^*)q_2^*+p(q_1,q_2^*)-C_2'(q_2^*)&=0\qquad\qquad(2)
\end{aligned}
$$
These are the usual MR=MC condition (for each firm). Solving for $q_1^*(q_2)$ in $(1)$, we get the best response of Firm 1 as a function of Firm 2's output $q_2$. Likewise, from $(2)$ we get Firm 2's best response to Firm 1's output, $q_2^*(q_1)$.

Recall the a NE in a two-player game is a pair of mutual best responses. Therefore the NE of this two-firm Cournot game with different MCs is the pair $(q_1^*(q_2^*),q_2^*(q_1^*))$, where each firm is best responding to the other firm's best response (to the former firm). Practically, this means to plug $q_2^*$ into $q_1^*(q_2)$ and solve for $q_1^*$, and then plug $q_1^*$ back into $q_2^*(q_1)$ and solve for $q_2^*$.


Is there any nation which doesn't have any external debt? China and the US alone have a staggering 15 trillion dollars in debt. Is there any country which has no external debt and if so, then when did they pay back their debt?

The simplest answer is that yes, there are a few countries without external debt, all small.  Wikipedia has a list of the external debt of countries.  For instance Niue finished paying off its debt in October 2016.


Human capital for under developed countries I am currently undergoing a study on economic growth in 11 countries: Bangladesh, Egypt, Indonesia, Iran, Mexico, Philippines, Pakistan, South Korea, Turkey and Vietnam. I plan to use panel regression. I plan to follow an analysis such as Islam (1995) and Barro (1991) with an augmented Solow model with human capital.

As my human capital variable I am using the "Index of human capital per person" (based on years of schooling and returns to education), from the Penn-world tables. My question is, how would I go about finding the savings ratio $S_h$ i.e. the investment in human capital from the above index?

If you multiply the index by the population size (or the quantity used to compute the per person index), you get an index of the stock of human capital. For example, if for a given country the index is $h_t$ and population is $N_t$, then the stock is:

$$ H_t = h_t \times N_t $$

Now, by the definition a stock follows an accumulation rule, just as physical capital:

$$ H_{t+1} = H_t(1+d_t) + I_t $$

where $d_t$ is the "depreciation" of HK (e.g. because of death or skill obsolescence) and $I_t$ is gross investment in HK. Since you probably do not have data on depreciation (which I imagine is very difficult to compute), you can only calculate net investment, which is equivalent to net savings by definition:

$$ S^h_t = H_{t+1} - H_t $$

Finally, to compute the savings rate, you need to divide this by a measure of GDP. However, these need to be measured in the same units. Therefore, you need to transform $H_t$ or $S^h_t$ into monetary units. For this you need something like the average price of human capital for every period, which I have never seen, and seems not available in Penn-world tables.


How did investment go from exogenous to endogenous in the IS/LM model? Context: I am currently taking an introductory course to macroeconomics and we are on the topic of Keynesian Economic and the IS/LM model.
Initially we saw that the investment (I) component of aggergate/planned expenditure is considered autonomous expenditure, which I understand to mean it is not dependant on income.
Additionally when we were covering Withdrawals and Injections to the circular flow modelt. Here the textbook said investment(I) is a withdrawal from the circular flow model and withdawrals are exogenous. Exogenous was defined as not directly related to changes on income.
However when introducing the IS/LM model, a video tutorial said we are now going to make investment(I) endogenous to derive it IS curve by defining it as a function of income(Y) and interest rate(i).
My question is how did we get from investment (I) as exogenous investment (I) as to endogenous?I understand we used interest rate(i) to link our financial market with our goods market which makes sense but how does income factor into that after we have already said that investment (I) is autonomous spending/exogenous?
It seems there is something im missing here!

In the earlier model the fixed investment was a simplifying assumption made for didactic reasons (you dont want to overwhelm students), rather than because investment is actually fixed in general. In fact, it is not exogenous even in a full circular flow model (what you probably seen were simplified versions), but in a full circular flow model investment is actually dependent on income (investment is given by savings and savings are just unconsumed income) and endogenous as you can see below in a picture taken from Krugman Economics:

What you refer as the previous model was likely the basic goods market equilibrium for closed economy as in Blanchard et al Macroeconomics:
$$Y = C+I+G; \text{ with } C=b_0+b_1(Y-T)$$
The reason why investment in the above model is considered to be fixed/autonomous is not so much because it would actually be fixed, but because fixing it allows to make the model more simpler. However, this really models only small portion of the full economy as you can see from the picture above.  There are actually non-IS-LM extensions of the basic model where $I$ won't be fixed and you just assume there is some additional function $I(Y)= a_0 + a_1Y$ and you could solve the model for the multiplier and everything else.
However, most courses will just first cover goods market equilibrium and afterwards move onto IS-LM model due to lack of time.  IS-LM model will try to capture economy in its entirety so it will model the link between income and investment through savings in explicit fashion.
You will still have the goods market given by
$$Y = C(Y-T)+I(Y,i)+G$$
but now $I(Y,i)$ will depend on income and interest rate, which in turn also depend on equilibrium in the money (financial) markets described by LM equation ($M/P=L(i,Y)$). This really represents adding the arrow of private saving, borrowing and investments spending from the picture below into the model.
In summary, the main reason why it went from exogenous to endogenous is that in a sense that generally speaking it never was exogenous (a caveat is that more complex models can show that there are periods were it can be - but not generally and that whole another topic that is well beyond scope of this answer). The previous assumption was there so you can be first exposed to goods market equilibrium and multiplier idea without worrying about what happens to investment.


When does the "stationary" problem become and issue with panel data? I recently came by a post that states in panel data with small time dimensions the stationarity issue is not present, is this true? can anybody provide a source for this?

I'm working with a panel data set for 10 states over 11 years, with variables: unemployment, poverty rates, age controls in (%) and minimum wages.

Should I bother trying to test for unit roots? if so what tests would you recommend? I am using R and having difficulties trying to find and understand some tests used in panel data sets.

-thanks

The data sample is so small that formal testing for stationarity would be essentially worthless. Inspect visually your individual series for any obvious trend. This would be the case where even with a short sample non-stationarity would be a problem.


Up-to-date survey of panel data models I am looking for an up-to-date survey paper or a book chapter on panel data models.
I know there are excellent textbooks out there like

Wooldridge "Econometric analysis of cross section and panel data" (2nd edition, 2010) and
Hsiao "Analysis of Panel Data" (3rd edition, 2014)*,

but I am looking for a briefer introduction/overview. Among survey papers,

Baltagi & Raj "A Survey of Recent Theoretical Developments in the Econometrics of Panel Data" (1992) is no longer recent.
Chudik & Pesaran "Large Panel Data Models with Cross-Sectional Dependence: A Survey" (2013) is more recent but seemingly a bit narrow and rather technical.

*Chapter 13 looks pretty nice as it summarizes the entire book; it might be a good starting point.

As rightly pointed out by @1muflon1@ "Panel data is nowadays quite a big field - usually you will have separate chapters for panel IV, panel logit/probit, panel time series etc".
But if you are "looking for a briefer introduction/overview", I would recommend:

Econometrics Training: Module Three Panel Data (with William Greene and John Siegfried)

Dates back to 2010, but the authors propose a theoretical part on basic models and an applied part with data and programs.


Monetary Policy adjustments as a result of Brexit I would like to know the impact of interest rates within Monetary Policies in view of Brexit.

An increase in the Monetary stock would lead to a decline (at least in the short-term) in nominal interest rates. Conversely, decrease in money stock can lead to short-term increases in interest rates. What I would like to know is whether there might be significant increases in interest rates during Brexit?

I agree that Money supply/Money stock is negatively correlated with short-term interest rates.

Since Great Britain has been using the Sterling before and after Brexit, there is no fundamental change in Monetary Policy caused by Brexit. Your (implied) assumption that Brexit causes a change in interest rates is therefore highly speculative.

Rather, the Bank of England's Monetary Policy Committee decides on interest rates based on (forecasted) growth and inflation rates, and a big number of other factors.

The economic performance (and therefore, interest rates) of the UK in the aftermath of Brexit will depend heavily on the negotiated deals between the EU and the UK.

If the UK economy indeed slows down as a response to Brexit, as many assume, the BoE will most likely keep the interest rates down. However, this is not at all guaranteed to happen.


Duality of cost minimization and profit maximization The firm tries to maximize profits $\Pi$
\begin{align}
\max_{K,L}\{\Pi(K,L) = F(K,L) - RK - wL\}
\end{align}
where $F$ is the linear homogeneous production function, $R$ the rental rate of capital $K$ and $w$ the rental rate (wage) of labor $L$. FOCs are given by
\begin{align}
\Pi_K &= 0 \Leftrightarrow R = F_k\\
\Pi_L &= 0 \Leftrightarrow w = F_L.
\end{align}
Footnote 1 of page 33 in Acemoglu (2009) tells us that the FOCs can also derived by cost minimization.

With (2.6) and (2.7) being $w=F_L$ and $R=F_K$ respectively.
The firm tries to minize costs
\begin{align}
&\min_{K,L}\{RK + wL\}\\
\text{s.t.}~~& F(k,L) = Y
\end{align}
where $Y$ is some output level. Set up Lagrangian
\begin{align}
\mathcal{L} = RK + wL + \lambda(F(K,L) - Y)
\end{align}

FOCs are given by
\begin{align}
\mathcal{L}_K = 0& \Leftrightarrow R + \lambda F_K = 0\\
\mathcal{L}_L = 0& \Leftrightarrow w + \lambda F_L = 0\\
\mathcal{L}_\lambda = 0& \Leftrightarrow F(K,L) - Y = 0
\end{align}

I don't see, how we can conjecture $R = F_K$ and $w = F_L$ from those conditions?

If $F(K,L)$ is a homogeneous function of degree one then so is
$$
\Pi(K,L) = F(K,L) - R \cdot K - w \cdot L.
$$
This follows straight from the definition of homogeneity. (A definition of homogeneous function can be found here.) This means that if a maximal profit exists it is zero. Otherwise you could increase all inputs by say 100%, thereby increasing both revenues and costs and thus profits by 100%. So $\Pi(K^*,L^*) = 0$.

By Euler's Homogeneous Function Theorem we have $\forall K,L$:

\begin{align}
\Pi(K,L) &= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L  \\
\Pi(K,L) &= (F_K(K,L) - R) \cdot K + (F_L(K,L) - w) \cdot L.
\end{align}

Since $\Pi(K^*,L^*) = 0$, we have
$$
-(F_K(K^*,L^*) - R) \cdot K^* = (F_L(K^*,L^*) - w) \cdot L^*
$$
We know that $K^*,L^* \geq 0$, so if we can show that the signs of
$(F_K(K^*,L^*) - R)$ and $(F_L(K^*,L^*) - w)$ match we will have proven them to be equal to zero. Otherwise one side of the equation would be negative and the other positive. From cost minimization you have
\begin{align}
R + \lambda F_K & = 0\\
w + \lambda F_L & = 0.
\end{align}
If $\lambda >1$ then
\begin{align}
F_K(K^*,L^*) - R & < 0\\
F_L(K^*,L^*) - w & < 0,
\end{align}
if $\lambda =1$ then
\begin{align}
F_K(K^*,L^*) - R & = 0\\
F_L(K^*,L^*) - w & = 0.
\end{align}
and if $\lambda <1$ then
\begin{align}
F_K(K^*,L^*) - R & > 0\\
F_L(K^*,L^*) - w & > 0,
\end{align}

so the signs do indeed match, hence
\begin{align}
F_K(K^*,L^*) - R & = 0\\
F_L(K^*,L^*) - w & = 0.
\end{align}