British inflation up again, but crisis might come. What should the BoE do? According to the latest ONS report, inflation is rising as a consequence of the (pre/post) Brexit devaluation. This graph about the Input PPI is pretty revealing (taken from here):

Yet, with the risk of an oncoming economic crisis, and interest rates at record low (probably at the ZLB), what can the Central Bank do?

Some say inflation is a good thing, as it "greases the economy" by lowering real wages. But that surely will only hit the demand even more. I'm puzzled!

You could say what the chart represents.

It is in fact the changes in the input Producer Price Index http://www.ons.gov.uk/economy/inflationandpriceindices/bulletins/producerpriceinflation/july2016 with a headline statement of

  the total input price index rose 4.3%  in the year to July 2016, compared with a fall of 0.5% in the year to June 2016

But this effect is smaller as it affects the economy as a whole.  For the output Producer Price Index the headline statement is less dramatic:

  Factory gate prices (output prices) for goods produced by UK manufacturers rose 0.3% in the year to July 2016, compared with a fall of 0.2% in the year to June 2016.

This has not yet had a substantial effect on consumers, where for the Consumer Prices Index http://www.ons.gov.uk/economy/inflationandpriceindices/bulletins/consumerpriceinflation/july2016 the headline statement is:

  The Consumer Prices Index (CPI) rose by 0.6% in the year to July 2016, compared with a 0.5% rise in the year to June.

I would be surprised if any of today's data surprised the Bank of England:

Imported inputs for UK producers now cost more due to a weaker Pound sterling
This has a relatively smaller effect on output prices, meaning British exporters are potentially more competitive internationally
This has not yet had a substantial impact on consumers

What the central bank does not yet know includes:

how much of an effect this will have on consumer prices in the longer term
how negative this will allow real interest rates to be if nominal interest rates are kept close to zero
how much of an effect this will have on imports and exports
how much of an effect this will have on consumption
how much of an effect this will have on investment

though it might try informed guesses.  Investment, consumption and exports in particular will be affected by broader factors including predictions about the long-term impact of a future Brexit.


Capital accumulation I have following dynamic optimization problem
$$V(k_0)=\max \sum_{t=0}^{\infty} b^t((1-a)ln c_t+a ln(l_t))$$
Subject to $l_t+e_t=h$ and $y=Ak_t^pe_t^{1-p}$ and $k_{t+1}=i_t+(1-x)k_t$
Where c is consumption, i is inverse,ent, y is output, k is phisical capital per capita. $l$ is leisure time , e is working time. V is value function. x is depreciation rate. A is total factor productivity.

The question I asked to you is that when a is zero, the economy would accumulate more physical capital? Or not? If yes, Why?

I think that
Capital accumulates according to $k_{t+1}=i_t+(1-x)k_t$
Since investment $i_t=y_t-c_t$
So
$k_{t+1}=y_t+(1-x)k_t-c_t$
When a=0, the utility function becomes $u(c_t)=ln(c_t)$
That, househoulds get more utility from consumption with this new utility function in comparision to the previous utility function given. So consumption level increases, which decreases one period forward capital stock.
But the things that I said seems to be wrong to some extent. What do you think ? Do you have any other wise idea about this?

$a=0$ means implies the household does not enjoy leisure. In other words, the trade-off between consumption and leisure is switched off. It's not generally true that with $a=0$, utility is higher for a given value of $c_t$, as this depends on the previous value of $l$. Whether or not consumption rises as a consequence depends also on the parameter $b$, i.e. the utility discount factor.
If the household does not value leisure, he might work 80 hours per week and obtain a higher income. If he is very forward-looking however, he might also save a lot for later consumption.


Can the benefits provided by the Ex-Im Bank of the US be considered a Public Good? With the debate surrounding the upcoming re-authorization of the Ex-Im Bank of the US, I wanted to hear from someone who might know. What is the best case to be made for its continued operation? I understand the "loans as corporate welfare" case, but I think you could probably classify improved export conditions and the resulting job growth, tax revenue, etc. as a Public Good.

If this is true, do these benefits justify the bank's existence despite the corporate welfare?

Since you keep stressing the public good part of the argument..

From wikipedia:

  The Export-Import Bank of the United States (Ex-Im Bank) is the official export credit agency of the United States federal government.[1] Operating as a government corporation, the bank finances and insures foreign purchases of United States goods for customers unable or unwilling to accept credit risk. According to its charter, the Ex-Im Bank does not compete with private sector lenders, but rather provides financing for transactions that would otherwise not take place because commercial lenders are either unable or unwilling to accept the political or commercial risks inherent in the deal.

So the service at hand is lending to higher risk customers.

In economics, a public good is

  a good that is both non-excludable and non-rivalrous in that individuals cannot be effectively excluded from use and where use by one individual does not reduce availability to others.

Lending is very much excludable. To the extent that funds of the ex-im bank are limited, it is also rivalrous. Hence no, the services of the ex-im bank are not a public good.

Second order public good

To quote you:

  that might result in what you might call a Public Good  i.e. improved export conditions, job growth, tax revenue,

This part is irrelevant. Either you provide a public good, or you don't. Providing a private good, that may result in benefits for society doesn't make you a special provider that warrants special benefits.

Any producer of consumption goods creates "improved export conditions, job growth, tax revenue,". That doesn't warrant the government sponsoring them, because they create something "akin to a public good".


Verifying rationalizability of pure strategy Consider a static, complete information game with 2 players.
Strategy sets are $S_1=\{U,D\},S_2=\{l,m,r\}$.

Payoffs are irrelevant to this question as I am trying to get the concept of rationalizability correct.

Suppose I want to verify whether $m$ is a rationalizable strategy for player 2.

Then, I want to ask the following question:

$\exists \sigma_1=(q^*,1-q^*)\in\Delta(S_1)$ such that for $\sigma^*_2=(0,1,0),$ $u_2(\sigma_1,\sigma^*_2)\geq u_2(\sigma_1,\sigma_2)$ for all $\sigma_2\in\Delta(S_2)?$

Now, suppose I have payoff matrix such that I could find $(q^*,1-q^*)$ such that it satisfies both:

(1) $u_2(\sigma_1,\sigma^*_2)\geq u_2(\sigma_1,(1,0,0))$
(2) $u_2(\sigma_1,\sigma^*_2)\geq u_2(\sigma_1,(0,0,1))$.

This means, I could find a valid range of $q^*$ such that for player 2, choosing $m$ provides a weakly better payoff for her compared to the degenerate (i.e. pure) strategies of $l$ or $r$.

My question is:

If I could find such $q^*$ that satisfies both (1),(2), then I do not have to check for any other strategy profiles in $\Delta(S_2)$, that is any convex combo of $(1,0,0)$ and $(0,0,1)$? My intuition is that for any $\alpha\in[0,1]$, I could simple have:

(1)' $\alpha u_2(\sigma_1,\sigma^*_2)\geq \alpha u_2(\sigma_1,(1,0,0))$
(2)' $(1-\alpha)u_2(\sigma_1,\sigma^*_2)\geq (1-\alpha)u_2(\sigma_1,(0,0,1))$.

and show that (1)'+(2)'implies the degenerate strategy $m$ for player 2 is a best response to some belief, $\sigma_1\in\Delta(S_1)$. Hence, the bottom line is (1),(2) is sufficient, and I do not have to check the convex combo of the two other pure strategies.

Yes. Generally, the set of best replies is always a mixture over the set of pure best replies.  In particular, there is never a strict incentive to play a mixed strategy, a player playing a mixed best response is indifferent between all pure strategies in the support of the mixed strategy. All this follows from the linearity of expected utility.


Supply curve for a competitive firm with constant MC I know that for a perfectly competitive firm, the supply curve is given by $p=MC \ge AVC$, where $AVC$ is the average variable cost.
Now I get really confused when the $MC$ comes out to be a constant. How do I determine the supply curve in that scenario?

Consider a firm with constant marginal cost $c$ that chooses to supply a quantity $q \in [0, \bar{x}]$ when taking the price $p$ as given. The profits of such a firm can be written as

$$ \text{total revenue} - \text{total cost}  = q p - c q = q(p - c).$$

From this, we can immediately infer the optimal supply $q^*$:

If $p<c$, then $q^* = 0$.
If $p=c$, then any $q \in [0, x]$ is optimal.
If $p>c$, then $q^* = \bar{x}$ (the maximum possible quantity).

Notice we need a constraint like $q \in [0, \bar{x}]$ for the supply correspondence to be well-defined. If we had simply restricted $q$ to $\mathbb{R}^+$ (the non-negative real numbers), as is more usual, then there would be no solution to the profit maximisation problem in the case of $p>c$.


Cournot Competition game with 3 Firms Three firms are in Cournot competition. The inverse demand curve is denoted p(q) where p is the price if a total of q units are produced. Assumptions are: p(0)>0 and p'(q)<0 and p''(q) $\le 0$ whenever p(q) >0. The firms have identical and continuous cost functions, c($q_i$) >0 for all $q_i$>0. Assume c'($q_i$)$\ge 0$ , c''($q_i$) $\ge$ 0 for all $q_i$ $\ge 0$ and p(0) > c'(0). We consider pure strategy Nash equilibria. For interior quantities , derive firm i's first order condition.  Do you agree that all firms must supply the same quantity in any equilibrium? Do you agree that there is a unique equilibrium? Prove and explain your answers.

What I did is that

First I agree that all the 3 firms will supply the same quantity in the equilibrium and this equilibrium is unique. The Profit Maximization Problem for firm 1 is:

max  p($q_1$ + $q_2*$ + $q_3*$)$q_1$ -c($q_1$)

and the FOC is :

p'($q_1$ + $q_2*$ + $q_3*$)$q_1$ +p($q_1$ + $q_2*$ + $q_3*$) -c'($q_1$) = 0

If i had the inverse demand function and the cost function, I could have derive the best response function. But now the only that I can say is that $BR_1$($q_2$*, $q_3*$). Similarly for the other two firms : $BR_2$($q_1*$, $q_3*$) and $BR_2(q_1* , q_2*$). But I stuck to this point and I cannot show how this quantities are equal.
Any help will be appreciated. Thank you.

Firm 1's profit maximization problem is :
\begin{eqnarray*} \max_{q_1} & \ p(q_1 + q_2 + q_3)q_1 - c(q_1) \end{eqnarray*}
Firm 1's response to firm 2's and 3's quantity choices will satisfy the following:
\begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_1 - c'(q_1) = 0 \end{eqnarray*}
Likewise, we'll get the implicit best response functions for the other firms as well.
\begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_i - c'(q_i) = 0 \end{eqnarray*}
for each $i \in \{1,2 ,3\}$.

There is a symmetric equilibrium of the above game if there exist $q^*$ such that
\begin{eqnarray*} p(3q^*) + p'(3q^*)q^* - c'(q^*) = 0 \end{eqnarray*}
To show that there exist one, consider the following function
\begin{eqnarray*} g(q) = p(3q) + p'(3q)q - c'(q)\end{eqnarray*}
Observe that $g(0) = p(0) - c'(0) > 0$. Also, given that $p(q)$ is decreasing and concave, there exist $\overline{q} > 0$ such that $p(\overline{q}) \leq 0$. Consequently, $g(\overline{q}) \leq 0$. Therefore, by intermediate value theorem, there exist $q^* \in [0,\overline{q}] $ such that $g(q^*) = 0$ i.e.
\begin{eqnarray*} p(3q^*) + p'(3q^*)q^* - c'(q^*) = 0 \end{eqnarray*}
Also, $g'(q) = 4p'(3q) + 3p''(3q)q - c''(q) < 0$. Since $g(q)$ is decreasing in $q$, $q^*$ is unique.
So there is a unique symmetric equilibrium. To show that it is the only equilibrium, we'll show that there is no equilibrium where at least two of the firms choose different quantities. To do so, let us subtract the implicit best response function of firm 1 from that of firm 2,
\begin{eqnarray*} p'(q_1 + q_2 + q_3)(q_2 - q_1) = c'(q_2) - c'(q_1) \end{eqnarray*}
Notice that the above equality only holds for $q_2 = q_1$. This is because

when $q_2 > q_1$, $p'(q_1 + q_2 + q_3)(q_2 - q_1) < 0$ and $c'(q_2) - c'(q_1) > 0$,
when $q_2 < q_1$, $p'(q_1 + q_2 + q_3)(q_2 - q_1) > 0$ and $c'(q_2) - c'(q_1) < 0$.

Therefore, there is no equilibrium satisfying $q_2 \neq q_1$. Likewise, there is none satisfying $q_2 \neq q_3$ or $q_1 \neq q_3$.


Solvency and liquidity. How to understand them? Should Central Banks only lend to solvent but illiquid banks? Solvency: Long-term liabilities are covered by assets(with value measured at maturity, not at market-to-market)

Liquidity: short-term liabilities are covered by assets which can be sold in the short-term.

Is this it?

Should CB lend to insolvent banks? or only to solvent but illiquid banks? This related to the problem of moral hazard, where if banks are being bailed out, more can think that their liability side is being covered by the CB/Government...

Yes, your thinking about solvency and liquidity is in the right direction. An easier way to think about them is: Liquidity means that a bank has enough resources to meet its obligations and Solvency means the bank owns more than it owes.

Lending to solvent but illiquid banks is a less risky thing to do for the central bank because the bank needs more liquidity at the moment but it owns enough long-term assets to repay the debt. Bagehot was the first to publicise this insight. It is called Bagehot's rule.

Lending to insolvent banks (banks that owe more than they own) may be risky because it's long-term assets are significantly limited for the bank to repay this debt. This has the potential to cause a moral hazard because the bank is likely to come back to borrow more from the central bank in the future.


Simultaneous vs Sequential Games Is there a way to characterize the distinction between simultaneous vs sequential games? I'm trying to describe a situation where players can only take actions without knowledge of other players' actions in a given stage of the game (i.e. where the actions taken in any stage of the game are simultaneous), but where repeated games are permitted. I've read Gibbons (1992) but am nowhere wiser.

In the standard theory of games, simultaneous and sequential games are distinguished by the means of something called "Information Sets". Alluding to the origins of Game Theory (von Neumann and Morgenstern 1944), simultaneous move games can be thought of as special cases of sequential move games. In their landmark work, vN-M give an explicit method to construct simultaneous move version a sequential game. For them, every game is essentially a sequential move game. But that's just history.
From a theoretical standpoint, all that matters is what does a player "know" about the previously played moves while making a decision in a game. This notion of the "information that a player has before making a move" is characterized in sequential move games (and their game trees) via Information sets.
Consider, for example, prisoner's dilemma. But suppose that it is played sequentially, in two different ways:

Player 1 moves. Player 2 learns Player 1's move. Player 2 moves.
Player 1 moves. Player 2 does not know about Player 1's move, but knows that Player 1 has moved. Player 2 moves.
In most cases, 2. is essentially a simultaneous move game.

Here's how we conventionally represent a simultaneous move game in extensive form. In the above image, the dotted line joining the two decision nodes of Player 2 signify that while making her decision, Player 2 does not know at which node she actually is. If there were no dotted line, it would mean that Player 2 knows at which node she is, and hence has learned about Player 1's choice. In the pictured version, all that Player 2 knows that if she plays C, she would get either 3 or 1, but that's it. Note that this is exactly similar to the situation we have in a simultaneous move game. A repeated version of this one shot simultaneous move game is then easy to imagine.
In GT jargon, we say that these two nodes which are joined are contained in the same Information set. So, an Information set is the collection of nodes among which a player cannot differentiate.


Elasticity of subsitution: a redundant calculation? Elasticity of subsitution is a measure of the curvature of an isoquant1 and can provide us information regarding the nature of the production functions we are dealing with. this is visualized in Advanced Microeconomic Theory
by Geoffrey A. Jehle (Author), Philip J. Reny  (page 130 in the book) as:

However I wonder, isn't this information regarding the curvature of a isoquant given by the production function? what novel information does the calculation of elasticity of substitution provide us with?

1. I'm aware that elasticity of substitution can also be applied to consumer theory, just that its dominant in producer theory.

Why would one need to graph the function? Isn't any information in the graph already given in the function?

In  a certain sense all calculations are redundant as they are mathematical tautologies. Yet these indices make some things more clear, which is obviously useful.


How to find the mean of a two-part Poisson-based hurdle model If I have a hurdle model in which the distribution is one Poisson distribution for observations $=0$ and another distribution for observations not equal to zero, how do I go about finding the mean of such a distribution?

EDIT: this is the distribution I am given

$$
\Pr[y_i=j] = \begin{cases} e^{-\theta} & j=0, \\[10pt]
\dfrac{(1-e^{-\theta})e^{-\lambda} \lambda^j}{j!(1-e^{-\lambda})} & j = 1,2,\ldots \end{cases}
$$

Maybe I'm not understanding your question. Poisson distributions with expected values $\lambda$ and $\mu$ are given by
$$
\Pr(X=x) = \frac{\lambda^x e^{-\lambda}}{x!}\quad\text{ and }\quad \Pr(X=x) = \frac{\mu^x e^{-\mu}}{x!} \quad \text{ for } x=0,1,2,3,\ldots.
$$
Suppose the second one holds for $x>0$. It follows that
$$
\Pr(X=0) = 1 - \underbrace{\sum_{x=1}^\infty \frac{\mu^x e^{-\mu}}{x!}}_\text{starting at 1, not 0} = e^{-\mu}.
$$
Since the sum of the probabilities must be $1$, the probability that $X=0$ can only be $1$ minus the sum of the other probabilities, and that is $\dfrac{e^{-\mu} \mu^0}{0!} = e^{-\mu}.$

Postscript: In comments I am asked how to find the mean of this distribution.  Remember that
$$
\sum_{x=0}^\infty \frac{\mu^x}{x!} = e^\mu, \tag 1
$$
and from that we get the sum of the probabilities,
$$
\sum_{x=0}^\infty \frac{\mu^x e^{-\mu}}{x!} = 1 \tag 2
$$
We will use the sum $(2)$ in finding the mean.

The mean is
\begin{align}
\operatorname{E}(X) & = \sum_{x=0}^\infty x\Pr(X=x) \\
& = \sum_{x=0}^\infty x\cdot\frac{\mu^x e^{-\mu}}{x!} \\
& = \sum_{x=1}^\infty x\cdot\frac{\mu^x e^{-\mu}}{x!} & & \text{(starting with $1$ rather than $0$} \\
& & & \phantom{(}\text{since the $0$ term vanishes because} \\
& & & \phantom{(}\text{of the multiplication by $x$)} \\
& =\sum_{x=1}^\infty \frac{\mu^x e^{-\mu}}{(x-1)!} & & \text{(since $x/(x!) = 1/(x-1)!$ if $x\ge1$)} \\[8pt]
& = \mu \sum_{x=1}^\infty \frac{\mu^{x-1} e^{-\mu}}{(x-1)!} & & \text{(factors not depending on the index $x$)} \\
& & & \phantom{(} \text{can be pulled out)} \\
& = \mu \sum_{y=0}^\infty \frac{\mu^y e^{-\mu}}{y!} & & \text{where $y=x-1$} \\
& & & \text{so $y$ goes from $0$ to $\infty$ as $x$ goes from $1$ to $\infty$} \\
& = \mu \cdot 1 & & \text{by line $(2)$, stated above} \\
& = \mu.
\end{align}
So the mean is $\mu$.

Postscript:

Your new revision of the question isn't quite consistent with your statement that "the distribution is one Poisson distribution for observations $=0$ and another distribution for observations not equal to zero". You have
$$
\Pr(X=x) = \begin{cases} e^{-\theta} & \text{if }x=0, \\
(1-e^{-\theta}) \dfrac{\lambda^x e^{-\lambda}}{x!(1-e^{-\lambda})} & \text{if } x\ge 1. \end{cases}
$$
Thus you have a weighted average of

The distribution of a random variable constantly equal to $1$, and
The conditional distribution of a Poisson-distributed random variable given that it is $\ge 1.$

The first has expected value $1$.

The second has expected value
$$
\frac 1 {1- e^{-\lambda}} \sum_{x=1}^\infty \cdots = \frac 1 {1-e^{-\lambda}} \left( (-e^{-\lambda}\cdot0) + \sum_{x=0}^\infty \cdots \right) = \frac 1 {1-e^{-\lambda}} \cdot ( \lambda ).
$$

After that, use the fact that the expected value of the weighted average of two distributions is the weight average of their two expected values.


Models with Learning by Doing and Knowledge Spillovers - Barro, Sala-i-Martin (2003) Consider the model of endogenous growth with learning by doing and knowledge spillovers presented in Barro & Sala-i-Martin (2003), chapter 4, section 4.3, starting in page 212. In equilibrium, the growth rate of consumption in the decentralized economy is given by:
$$ \frac{\dot{c}}{c} = \left(\frac{1}{\theta} \right) \cdot \left( \underbrace{f(L) - L \cdot f'(L)}_{\phi_0} - \delta - \rho\right) $$
where $c$ is per capita consumption, $\theta$ is the inverse of intertemporal elasticity of substitution, $L$ is the size of the labor force ($\frac{\dot{L}}{L} = n$ is assumed to be equal to $0$), $\delta$ is the depreciation rate, $\rho$ is the subjective discount rate and $\phi_0$ is the marginal product of the capital.
If we now consider a central planner, the following expression from the growth rate of $c$ can be obtained:
$$ \frac{\dot{c}}{c}\bigg\vert_{\text{planner}} = \left( \frac{1}{\theta} \right) \cdot (\underbrace{f(L)}_{\phi_1} - \delta - \rho) $$
where $\phi_1 = f(L)$ is the average product of the capital.
The authors then go on to present an example using a Cobb-Douglas production given by:
$$ Y_i = A \cdot (K_i)^\alpha \cdot (KL_i)^{1-\alpha}, \quad 0 < \alpha < 1 $$
where the index $i$ corresponds to the firm $i$, i.e., $Y_i$ is the output of firm $i$.
Under the assumptions of the model, if we substitute $y_i = \frac{Y_i}{L_i}, k_i = \frac{K_i}{L_i}$ and $k = \frac{K}{L}$, and then set $y_i = y$ and $k_i = k$, the average product of the capital is:
$$ \frac{y}{k} = f(L) = A \cdot L^{1-\alpha} $$
and the marginal product of capital is
$$ f(L) - L \cdot f'(L) = A \cdot \alpha \cdot L^{1-\alpha} $$
and, therefore,
\begin{align*} \frac{\dot{c}}{c} &= \left(\frac{1}{\theta} \right) \cdot \left(A \cdot \alpha \cdot L^{1-\alpha} - \delta - \rho\right) \\ \frac{\dot{c}}{c}\bigg\vert_{\text{planner}} &= \left( \frac{1}{\theta} \right) \cdot (A \cdot L^{1-\alpha} - \delta - \rho) \end{align*}
Since $0 < \alpha < 1$, the growth rate of the decentralized economy is lower than that of the economy with a central planner.
Continuing, the authors write the following and I quote:

The social optimum can be attained in the decentralized economy by introducing an
investment-tax credit at the rate $(1-\alpha)$ and financing it with a lump-sum tax. If buyers
of capital pay only the fraction $\alpha$ of the cost, the private return on capital corresponds
to the social return. We can then show that the decentralized choices coincide with those
of the social planner. Alternatively, the government could generate the same outcome by
subsidizing production at the rate $ \frac{(1-\alpha)}{\alpha} $

I can't understand how the tax and subsidy could make the decentralized economy achieve a socially optimum level of growth rate. Intuitively, I can see how that might work but I cannot see how the math behind it would work in this case.
If anyone could explain that to me or point me in the right direction I'd appreciate it. Let me know if I need to add more details to the question.

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


Logit/Probit with numbers different than 0 or 1? I am trying to find an econometric model that helps me to answer the following. I would like to measure how much the increase in the cost of a production factor (my X) shifts the supply curve of a market "A" (my Y). That is, I want to regress Y on X. I have daily data of the supply curve (the pairs Price, Quantity) and the daily cost of the factor. As a first step, I calculated a dummy variable that takes the value 1 if in day t there was a shift to the left with respect to t-1. Then I calculated the variable that measures the displacement if there was a shift (my Y). Which econometric model could I use since Logit and Probit only apply when the variables are 0 or 1, but in this case, I want to know how much the curve moves? Should I take a subsample of all the dummies that are equal to 1 and work with that? I don't think this is the best answer.
Thanks in advance!

I don't see how Logit/Probit models are relevant here. They are designed to estimate conditional probabilities, not what you describe.
It appears you postulate a relation
$$S = f(P, C)$$
namely supply being a function of price and cost, and where you treat price as a "control" This could be written as a linear regression model, as a first-order approximation. Why don't you think estimating that model?


Why are elasticities defined as logarithmic derivatives? In my economics class, we often compute the elasticity of $Y$ with respect to $X$,
$$\eta = \frac{\partial \log Y}{\partial \log X}.$$
You can compute this from the slope of a line fit to a log-log plot.

Why is it more natural to consider this quantity than the much simpler quantity
$$\eta' = \frac{\partial Y}{\partial X}$$
which is just as easy to measure? There seems to be some implicit assumption that $\eta$ is perhaps 'more applicable', maybe that it's less sensitive to the value of $X$ than $\eta'$ is, so it's more useful for extrapolation. For example, I've seen a demand curve extrapolated by assuming that $\eta$ is constant, yielding a power law. But why not assume $\eta'$ is constant, yielding a line? Neither seems particularly more natural to me.

Why is $\eta$ a more useful quantity than $\eta'$ in economics?

If I understand your question, first the elasticity haven't units. The problem with $\partial Y/\partial X$ is that if you change the measure units the result is different. Is less problematic to express it in percentages:
$$ El_X(Y)=\frac{\Delta Y/Y}{\Delta X/X}$$
if $\Delta Y \to 0$ when $\Delta X \to 0$ then you obtain $$El_X(Y)=\frac{dY}{dX}\frac{X}{Y}$$ if $Y=Y(X,W,...,Z)$ you change $d$ for $\partial$.

Second, like you know $d \log(x)=\frac{dx}{x}$ thus $$ El_X(Y)=\frac{d Y}{d X}\frac{X}{Y}=\frac{d\log(Y)}{d\log(X)}.$$


Why can't price level be negatively correlated with money demand? Source: p 585, Economics, 3 Ed, 2014, by NG Mankiw, MP Taylor

  Although many variables affect the demand for money, one variable stands out in importance: the
  average level of prices in the economy. People hold money because it is the medium of exchange. Unlike
  other assets, such as bonds or stocks, people can use money to buy the goods and services on their
  shopping lists. How much money they choose to hold for this purpose depends on the prices of those
  goods and services. The higher prices are,
  the more money the typical transaction requires,
  [1.] and the more
  money people will choose to hold in their pockets and bank accounts.
  [2.] That is, a higher price level (a lower
  value of money) increases the quantity of money demanded.

I doubt 1 and 2 that feels too absolutely certain. (They're also restated, but rephrased, on pp 683 and 705.) Why must 1 and 2 be true?

Abbreviate Prior Money as PM, i.e. consumers' money held 'in their pockets and bank accounts' before the price rise.

Suppose that higher prices $\implies$ more money needed for each typical transaction. But even facing  a higher price level, consumers may already possess enough PM to transact, and so needn't more money.

Moreover, even with enough PM for their regular purchases, the price rise may cause consumers to spend less. Then money demand would DEcrease.

You're supposing the amount of money held for transactions by players is arbitrary. This may be true if you look at an individual - I typically withdraw fixed amounts of money when my purse starts feeling empty - but is not true on average and in the long run.

But cash (in the wider sense, money on regular bank accounts count toward this) is also held by companies and foreign countries.

Even if you feel like discussing the rationality of the average person, you cannot discuss the rationality of the average money holder: the opportunity cost of holding too much money may be negligible for an employee with low savings. But it it isn't for a big multinational or a multi-millionaire that wants to stay richer than his neighbours in Beverly Hills.

Consumers are also retailers, at some point, so you cannot suppose the price rise affects them in the way they buy less goods. A potato producer still produces 1kg of potatoes after a price rise, only they are worth more and he need more money to buy carrots. The money demand increases because there is less money available for the same goods. This is a very different situation than a productivity drop, in which case the money demand would decrease.


Am I correct that cyclical unemployment can be negative? Let's start with stating and emphasizing some truths that we will need later.
Overall unemployment rate=Frictional unemployment + Structural unemployment + Cyclical unemployment. Also, Natural rate of unemployment = Frictional unemployment + Structural unemployment. From this we can see that Cyclical unemployment would need to be equal to zero in order to our economy to have natural rate of unemployment.

But suppose we have overall level of unemployment that in LOWER than the natural rate of unemployment. Now things get interesting. We can't get overall level unemployment become lower than the natural rate of unemployment by decreasing frictional and/or stuctural unemployment. Fortunately, there is the third component of overall unemployment, namely cyclical unemployment. If we agree that cyclical unemployment can be negative, then this way we can allow overall unemployment to be under the natural rate of unemployment.

It looks reasonable mathematically, but I have some doubts because I have never heard that unemployment can be negative. Is my reasoning correct, can we have negative cyclical unemployment?

And if I'm NOT correct, if cyclical unemployment can NOT be negative, then HOW do we get overall unemployment below the natural rate of unemployment?

Cyclical unemployment $U^C$ is by definition difference between actual observed rate of employment $U$ and the natural rate of employment $U^N$ so we have: $U^C_t \equiv U_t-U_t^N$, see for example this paper.

Hence cyclical unemployment can be negative if actual observed rate of unemployment is lower than natural rate, because it is defined as a difference between the two.


Does unit elasticity has to be at exactly the middle of the demand curve? Can unit elasticity be anywhere else on the demand curve other than the midpoint?

No, that is only true in the linear case. For a simple counterexample consider
$$
D(p) = 1 - \sqrt{p}.
$$
$$
\epsilon(p) = \frac{d D(p)}{dp} \cdot \frac{p}{D(p)} = -\frac{1}{2  \cdot \sqrt{p}} \cdot \frac{p}{1 - \sqrt{p}} = \frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}}
$$
\begin{eqnarray*}
\epsilon(p) & = & 1 \\
\\
\frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}} & = & 1 \\
\\
\sqrt{p} & = & 2 - 2 \cdot \sqrt{p}  \\
\\
\sqrt{p} & = & \frac{2}{3} \\
\\
p & = & \frac{4}{9}.
\end{eqnarray*}
This is not in the middle of the demand curve in any sense.


Stone-Geary preferences and competitive equilibrium Does anybody know if a competitive equilibrium obtains under Stone-Geary preferences; are there multiple equilibria problems; do such preferences admit an analysis with more than one type of representative agents; any good+relevant resources on the subject?

This is a paper using a competitive monopolistic framework (in GE). It is an old paper by Stratz (1989). Recent work with Stone-Geary preferences relates to the issue of structural transformation following the work of Herrendorf, Rogerson and Valentinyi (2013), although there is considerable debate on whether these preferences appropriately capture the data. All these papers have a representative consumer, with expenditure shares changing over time due to growth.


What is the Thomson Reuters tool that is used in universities? I'm not an economics major but when I go to the library of the economics department I see that computers have access to something that is related to Thomson Reuters. given that I'm interested in finance, can somebody explain what is that tool, what can be done with it? and just give some info to somebody like me who doesn't know much. thank you so very much in advance

It is a huge financial database which consents you to export data easily via the datastream tool for excel. There you can find common data (such as those available via online databases, e.g. yahoo finance or investing) but also intra-day data, indexes, analysis and the platform consent you to apply directly some statistical techniques. It is a great source, useful especially when you cannot get data from free sources.


Is fiat currency better than a backed-currency? My question is would it be better if we went back to something like a gold standard but backed by a finite virtual material instead? It avoids the problem of fluctuations in gold supply. The money-supply is also fixed, so governments can't devalue currency with respect to the commodity, and cannot make more of it. If the economy grows, each unit of currency has more purchasing power, but you cannot print more money. Would this stop overprinting of money, help prevent hyperinflation?

My knowledge of economics is very poor so apologies in advance.
(Sorry I have edited this question so it makes more sense)

Yes it is. Lets first unpack some myths about about gold standard or commodity money in general.

They dont necessary prevent inflation. Spain managed to have under gold standard a very large inflation compared to other European nations (Ferguson, 2008), and this happened even without debasement.

Furthermore as you yourself mention debasement can occur as well. Commodity money or gold standard dont mean the money have to be solely made of that commodity. In case of the commodity money you can easily have gold standard where you debase the % of metal every year even to the point of having 0.0001% of gold and it would still be gold standard. The only difference between debasement and printing money is that debasement is little bit more expensive as it involves smelting - but in modern industrial world the cost is not prohibitive so it could be done almost as easily as just printing money.

Backing paper money with gold does not prevent inflation by itself either. If you for example set the exchange rate to 1 euro = 1 troy ounce of gold, there is literary nothing preventing government from deciding one day that suddenly 1 euro = 0.001 troy ounce just because they decide so. Again thats still gold standard. Even if 1 euro would be set to be equal to be equal to 0.0000001 troy ounce it would still be gold standard. So whats the point?

Is there any way that gold standard may prevent inflation? From economic side not really (aside its more expensive to remint coins in case of using direct commodity money), but from political side it might be more difficult to justify debasement or change in the convertibility as non-economists might perceive that as some sort of cheating, but aside from politics gold standard offers zero protections against inflation by itself.

Next, its important to understand that deflation is also a form of price instability as inflation and it has negative effects on economy in similar ways as inflation (even if we exclude sticky wages). Deflation still creates unnecessary menu costs for example, same as inflation.

In addition to that deflation has extra negative effects because of sticky wages. Sticky wages does not mean that when everything gets cheaper your wages do too. It means that your wage remains the same (at least for certain period depending on stickiness) and due to the fact it cannot be adjusted quickly enough you would get unemployed. Economy with deflation is economy with unnecessary high unemployment which is additional cost of deflation. For this reason wast majority of economists thinks government should on purpose target yearly inflation of about 1-2% (Blanchard et al. 2013).

Furthermore, the modern evidence shows that independent central banks, if they choose so, are extremely efficient in maintaining price stability. In recent period Euro-Zone had average inflation well below 1% which is lower than historically many countries had under Gold/Silver standard. This just further illustrates that it literary does not matter what the money are made of as long as the stuff they are made of is not inconvenient to use (i.e. food that quickly spoils). How big or little inflation country has is ultimately a policy choice no matter what the money are made of. Even in case of cryptocurrency, if its controlled by government, the government can always decide to change the number of money by just changing the source code.

So the question is why bother having gold standard if its literary just using a useful commodity that could be otherwise used in industry for jewelry/electronics etc. improving peoples material standards of living and it does not hold any real advantage over fiat currency except

So to sum up:

its a myth that gold standard by itself prevents inflation. In medieval era inflation was kept low mainly due to technical difficulties and due to general weak governments that could not just impose any debased coinage on its populus. Modern day government deciding to be on gold standard with coins with 0.000001% gold can do so as easily as imposing fiat money on its population just by demanding taxes being paid using said coins. There might be some purely political reason why it might be harder for government to increase inflation under gold standard but no direct economic reasons (aside that smelting might be bit more expensive than printing in case you want to have actual coinage and not just backing)
Deflation is not desirable. As inflation, deflation is a form of price instability. In fact in presence of sticky wages deflation has additional costs making it worse than inflation but even without sticky wages you would not want deflation but maintain price level stable.
Modern independent central banks can easily deliver price stability. In fact most mainstream economists nowadays are accusing central banks such as ECB to not doing enough to raise inflation to at least 1-2%. Euro is actually being routinely compared to gold standard - see for example Eichengreen & Temin (2010).
There is no reason to use a commodity which has valuable industrial uses as money when paper does the job in principle as well. Its just a waste of scarce resources that could otherwise be employed in productive uses.

References:

Blanchard, O., Amighini, A., & Giavazzi, F. (2013). Macroeconomics: a European perspective.

Eichengreen, B., & Temin, P. (2010). Fetters of gold and paper. oxford review of Economic Policy, 26(3), 370-384.

Ferguson, N. (2008). The ascent of money: A financial history of the world. Penguin.


negative interest rate lending ripple effects I was attempt to describe the ramifications of negative interest rates across benchmarks.

I can foresee the business practice changes amongst holders of government bonds, savings accounts, certificates of deposit and mortgages. My conclusion was a possible stall of lending in those markets, but thats a red herring to my real question. When it comes to corporate debt I wasn't so sure what ramification even lower interest rates would have on the attractiveness of investment grade corporate debt.

Is there anywhere I can read prevailing or new theories about that?

Decreasing the nominal interest rate to a negative value will cause businesses to have to raise their prices in order to maintain 0 economic profit. This is because wages are "sticky," and cannot change in a general quarterly or annual period.  People who live in the country have not been given any higher incentive to spend, but since prices are increased - aggregate demand will decrease.  Additionally, in order to survive the cut in demand and rise in prices, firms will look to lay off workers - raising unemployment.  Consumption should decrease as a result of a negative interest rate charged by the Central Bank.  In what was an attempt to increase GDP - output will decrease.  --> Stagflation is likely to occur


Nash equilibrium two players in a joint effortful project I have a problem where there are two agents in a joint project,
Each agent $i$ puts in effort $x_i$ $(0 \leq x_i \leq 1)$ which cost each $c(x_i)= x_i^2$. The outcome of the project is
$$
f(x_1, x_2)= 3 x_1 x_2
$$

which is split equally between both, regardless of their effort levels.
I am asked to formulate de situation as a normal form game and find the NE.

What I have done is that I have assumed some values for the efforts to evaluate the game and I have also maximized each player's payoff individually to get best responses and make them intersect to get to the NE.

In this illustration, NE are $(0,0)$; $(0.25, 0.25)$ and $(0.5, 0.5)$

$\max f(x_1, x_2)-c_1$ with respect to $x_1$ ; yields : $x_1 = (3/4)x_2$ and $\max f(x_1, x_2)-c_2$ with respect to $x_2$ ; yields : $x2 = (3/4)x_1$
Both maximization problems intersect when $x_1 = 0 = x_2$

Anyhow, I don't think this is the right way to go...

You are actually almost correct. Your first approach shows the normal form of the game, but in an incomplete way, which is why you find "extra" Nash equilibria. The possible strategies of the players include all positive effort levels not just the ones you show.

The NE (0.25;0.25) is not actually an equilibrium because either player can reduce their effort a tiny amount (but less than the amount in your game) increasing their payoff. For example the optimum amount of effort for player 1 if player 2 uses effort level 0.25 is 0.1875 which is not in your game. If player 1 opts for an effort level of 0.1875 player 2's optimal effort is 3/4*0.1875=0.140625.
Continuing this reasoning you'll find eventually there is only one NE: (0;0) which is what you had correctly identified using your second approach.


Smaller variance of estimators in matrix sense When comparing asymptotic variances, you usually conclude that a covariance matrix A is more efficient whenever the difference B - A ≥ 0 (where ≥ 0 means that the difference is a positive semidefinite matrix).

However, this comparison involves all elements in the matrices - both the diagonal elements (asymptotic variances of each beta estimator) and the off-diagonal ones (covariances between estimators) -, while we are usually interested in estimators for a particular beta coefficient having small standard errors.

Could anyone explain why is this method preferred to a comparison that entails only the diagonal elements? Could it be the case that some diagonal element is smaller in B while B - A ≥ 0?

Thank you!

Relative efficiency between two unbiased estimators $\widehat{\theta }_{A}$
and $\widehat{\theta }_{B}$ of an unknown parameter vector $\theta _{0}\in
\mathbb{R}^{K}$ is usually defined as follow (see for instance Ruud, 2001).
The estimator $\widehat{\theta }_{A}$ is said to be efficient relative to $
\widehat{\theta }_{B}$ if we have:
$$ \mathrm{V}\left[ \widehat{\theta }_{A}\right] \equiv \Omega _{A}<<\Omega
_{B}\equiv \mathrm{V}\left[ \widehat{\theta }_{B}\right] .$$

If $\Omega _{B}-\Omega _{A}$ is positive definite, then the diagonal terms
of $\Omega _{B}$ and $\Omega _{A}$ are necessarily such that $\sigma
_{Bii}^{2}>\sigma _{Aii}^{2}.$ Indeed if $v^{T}\left( \Omega _{B}-\Omega
_{A}\right) v>0$ for any $v\neq 0$ then for $v=e_{i}$ we catch the diagonal
terms out of $\Omega _{B}-\Omega _{A}$ and find that $\sigma
_{Bii}^{2}>\sigma _{Aii}^{2}.$ The converse, however, it is not true: the
mere conditions $\sigma _{Bii}^{2}>\sigma _{Aii}^{2}$ do not garantee that $
\Omega _{B}-\Omega _{A}$ be positive definite. Covariances matter.

The answer to your second question is: no it never happens that $\sigma _{Bii}^{2}<\sigma _{Aii}^{2}$ when $\Omega _{B}>>\Omega _{A}$.
The answer to your first question is: it is important to consider all covariance terms. If we want that the confidence ellipse (at any threshold) of $\widehat{\theta }_{A}$ be nested within the confidence ellipse of $\widehat{\theta }_{B}$ then we need $\Omega _{A}<<\Omega _{B}$ and not just $\sigma _{Aii}^{2}<\sigma _{Bii}^{2}$.

See Ruud, (2001, chapter 9) for a proof and detailed explanations. An example is provided here, illustrating that the confidence ellipses are nested when $\Omega _{B}-\Omega _{A}$ positive definite, and not nested if $\Omega _{B}-\Omega _{A}$ is not positive definite.

Example:
\begin{eqnarray*}
\widehat{\theta }_{A}\left( a\right) &\sim &\mathcal{N}\left( 0,\Omega
_{A}\left( a\right) \right) ,\qquad \widehat{\theta }_{B}\sim \mathcal{N}%
\left( 0,\Omega _{B}\right) \\
\Omega _{A}\left( a\right) &=&\left(
\begin{array}{cc}
4 & a \\
a & 9%
\end{array}%
\right) ,\qquad \Omega _{B}=\left(
\begin{array}{cc}
5 & 0 \\
0 & 10%
\end{array}%
\right).
\end{eqnarray*}

For $a=0$ the matrix $\Omega _{B}-\Omega _{A}\left( 0\right) $ is positive
definite and the $95\%$ threshold confidence ellipses are nested. The Figure
below (left panel) represents the iso-curves centered on $\theta _{0}=0$ and
whose equations are given by $v^{T}\left( \Omega _{A}\left( 0\right) \right)
^{-1}v=5.99$ and $x^{T}\Omega _{B}^{-1}x=5.99$ and illustrates that for $a=0$
the later is nested within the former. This is no longer true for $a=-5$
(right panel) in which case $\Omega _{B}-\Omega _{A}\left( -5\right) $ is no
longer positive definite. In this case the probability that $\widehat{\theta
}_{A}$ is further away than $\widehat{\theta }_{B}$ from the true value $%
\theta _{0}=0$ is positive, and $\widehat{\theta }_{A}$ is no longer
efficient (case with $a=-5$) relatively to $\widehat{\theta }_{B}$. Note
that the variances always satisfy $\sigma _{Bii}^{2}>\sigma _{Aii}^{2}$ but
this is not sufficient for the ellipses to be nested, the covariances must
also satisfy $\left( \sigma _{B11}^{2}-\sigma _{A11}^{2}\right) \left(
\sigma _{B22}^{2}-\sigma _{A22}^{2}\right) -\left( \sigma _{B12}-\sigma
_{A12}\right) ^{2}>0$, which is not the case in this example for $a=-5$.


Why has China's integration into the global economy caused it to slow it's output? In the following passage, what does it mean to be integrated into the economy and why does this mean that now China and India are integrated into the economy that their output has slowed down?

In the 1990s and early 200s, global commerce was growing at twice the
rate of output because big economies such as China, India and in
eastern Europe were being integrated into the global economy. Now they
have been more or less absorbed, it is only natural that things slow
down, but this does not mean we have reached the high-water mark, says
Okonjo-Iweala.

Ref: https://www.google.com/amp/s/amp.ft.com/content/87cb4674-6db7-41cf-a82e-44b83eaa436c

Being integrated into the global economy means that country increases its trade openness and lifts restrictions on capital flows (you can look at this paper).
The article does not say that China's output slowed down. It says its rate of growth slowed down. Usually, a country that used to be closed and did not trade much before experiences a boost in the rate of economic growth. This is because trading allows the country to specialize in a good where the country has a comparative advantage. This boosts growth by a lot at the beginning but the effect eventually wears down as the country gets fully specialized and all low hanging fruit was already picked.


How do you calculate the intrinsic value or fair value of real estate? How do you calculate the intrinsic value or fair value of real estate?
I mean how do you determine that the house/commercial estate is undervalued or overvalued?

same as for a company, if you assume that you are buying it for the rent it will get you then the value of the asset is the sum of the discounted future cash flows (called FFO in RE)
you can also apply the comparables methods, in this field we usually apply it with a slight twist we read it in terms of yield (inverse of the multiple) which is called a cap rate and compare it to similar projects in similar areas for similar purposes.
this pdf is a good primer on all of this :
https://www.merrilledge.com/publish/content/application/pdf/gwmol/bofaml-reit-primer.pdf


Finding the demand function Here is the problem:

Suppose that an oligopolist charges a price of 500 and sells 200 units of output per day. If the oligopolist increases its price about 500, quantity demanded will decline by 4 units for every 1 increase in price. On the other hand, if the oligopolist lowers price below 500, quantity demanded will increase by only 1 unit for every 1 decrease in price.

Here is what I did:
point 1: (200, 500)
point 2: (196, 501)

Then find slope using m=(y2-y1)/(x2-x1) and then use point-slope form for the line. I think I am interpreting the demand increase/decrease incorrectly, but am not sure?

Thanks!

I don't have a question to "correctly interpret" , but I think the way you are calculating the demand function is correct. The demand curve in an oligopoly can be kinked (bowed out), as this one is. I think it is mathematically accurate to write the function split across the domains, as such:

$Q = \begin{cases}
700 - P, & 0  \leq P\leq 500 \\
2200 - 4P, & 500< P< 550
\end{cases}$

where I have written quantity as a function of price, and assumed that the quantity is strictly positive.


Do general equilibrium models include money? An economics professor told me that General Equilibrium models are a sophistication of barter. Does that imply that they don't include money? And if that is the case, are there any attempts to include it?

Moreover, does this have any implications on the way we understand macroeconomics?

Thanks.

While many general equilibrium models do not need to model money to approach the questions that they would like to answer, there are many models that do include money to address questions that need money to be a relevant feature of the model. These models do it in a variety of ways -- some might be more relevant than others. I will try and describe two of the approaches briefly.

Cash-in-advance

When I think of a cash-in-advance (CIA) model, the first models that come to mind are Lucas (1982) and Lucas Stokey (1987). Models that take this approach have agents saving in assets and money, but they must finance any consumption solely through their money holdings (basically assets are less liquid than money which isn't the worst approach). There are many critiques of the CIA of which I find Wallace (1998) to be one of the more thoroughly presented.

New Monetarist

More recently there has been a push to use models of money that are less "ad-hoc" about how money works. Some foundational papers of this field are the Kiyotaki Wright papers (1989, 1991, 1993). These papers build a very basic model of money and show what role it plays in economics -- There is a good discussion on Randall Wright's Wikipedia page under the sub-title "Research contribution." While these models confirmed some of the reasons for money to exist in a model, they were quite simplistic and lacked many other features of the real economy. A monetary course I took labeled these types of models as "Generation 1 models of money." There was a group of papers that were known as "Generation 2 models of money, but these shared many of the simplifications made in Generation 1 (so I won't discuss them here).

Since this time Lagos Wright (2004) introduced a more complete framework (we referred to these as "Generation 3 models of money." One of the issues faced by these new monetarist models is that in order to solve the model, one must keep track of the entire distribution of money -- This is computationally infeasible unless the authors introduce a simplification in a different area of the model (in the Lagos Wright model, they simplify the form of utility in order to be able to compute the distribution of money next period). There is on-going research in this area and, in my opinion, is an open area for improvement in the literature.

References

Nobuhiro Kiyotaki and Randall Wright. "On Money as a Medium of Exchange." Journal of Political Economics. 1989.
Nobuhiro Kiyotaki and Randall Wright. "A Contribution to the Pure Theory of Money." Journal of Economic Theory. 1991.
Nobuhiro Kiyotaki and Randall Wright. "A search-theoretic approach to monetary economics." American Economic Review. 1993.
Ricardo Lagos and Randall Wright. "A Unified Framework for Monetary Theory and Policy Analysis." Journal of Political Economy. 2005.
Robert E. Lucas Jr. "Interest Rates and Currency Prices in a Two Country Model." Journal of Monetary Economics. 1982.
Robert E. Lucas Jr and Nancy L. Stokey. "Money and Interest in a Cash-In-Advance Economy." Econometrica. 1987.
Neil Wallace. "A Dictum for Monetary Theory." Federal Reserve Bank of Minneapolis Quarterly Review. Winter 1998.


Minimum wage divided by cost of living? Is there some commonly known indicator that is defined as:  Minimum wage divided by cost of living? I mean, I could call it that way, but perhaps people already have this indicator with a certain specific name or they have a better one for the purpose I'm looking for.

There is the Kaitz-Index, which relates the minimum wage to the average hourly wage, thus facilitating cross-country comparisons on the minimum wage level. The OECD has compiled it here. There you'll also find other metrics comparing minimum wage levels across countries.


Math needed for economic research I'm currently studying economics and am interested in getting into economic research. I'm aware of how math based economics research is. I have done some 1st math courses (probability, linear algebra, univariate calculus and discrete math), a 2nd year course in probability and have done/am going to do multiple courses in econometrics. However, I would like to learn the remaining math needed in my own time. When looking at advice pages on this topic people usually say some combination of needing: linear algebra, multivariable calculus, real analysis and probability. However, I'm hesitant to learn all of these topics from text books as I feel like I would learn a large number of topics that aren't used in economics. For example, in multivariable calculus divergence doesn't seem that applicable to economics. I feel that my time could be spend more productively going deep in the concepts that come up a lot in economics. I was wondering whether someone would be able to collate a list of all the useful concepts needed in each of these subjects in the format:
Multivariable calculus:

Langrangian
Hessian matrix
etc

Thanks in advance.

Given the level you say you are at I would recommend you to go through this book
Simon and Blume: Mathematics for Economists.
it will give you the core of math skills you need for undergraduate and graduate economics at most universities I would guess. In graduate courses you will probably have to learn more depending on the course and university but it is standard to learn some of the math on the fly when you do economics.
I could recommend other books but they would then be more specialized in certain topics and anyway Simon and Blume is the book that I have most often seen references to on various university courses.
Another virtue of the book is that the examples are to a large extent taken from economics. So you do not "waste your time" on nice mathematical examples that illustrate the mathematical methods but will never appear in economics.
For some graduate courses of course there will be special math skills that are required. Econ departments of some universities specialize in different areas so what you need can also vary from university to university. However, as an economist I would dare say you are very often learning the mathematics as you learn new areas of economics.
Choose a programming language
Although you did not as for a recommendation of this kind, I would strongly encourage you to start programming. Very often being able to simulate some problem in optimization, probability theory, dynamic programming etc. is an invaluable resource to get a deeper understanding of complicated math/economic models.
Currently Matlab is used a lot especially for structural economics. R is very versatile better than Matlab when it comes to econometrics. Python is strong for Machine Learning which is also finding its way into econometrics/economics these days.


Construction of a 3x3 simultaneous game How could I construct a 3x3 simultaneous game with 2 players with no equilibrium in pure strategies but 1 is mixed strategies .
I am able to write down one with no pure strategy Nash equilibrium but how would I then know if only one mixed strategy would show not more?

I was thinking if I could eliminate one row and one column ,I.e they are strictly dominated in mixed strategies then I would only get 1 MSNE otherwise I get a continuum of mixed strategy Nash  equilibria’s is that right?

Eliminating a column and a row strategy is a nice idea. If the remaining 2x2 game has no pure strategy Nash equilibria (PSNE) than Nash's theorem guarantues you exactly one MSNE.

However there are also very trivial 3x3 games with zero PSNE and exactly one MSNE, e.g. rock-paper-scissors.


Tests of rational inattention Can you provide a reference regarding a revealed preference test(s) of rational inattention models, as proposed by Sims, ergo, using a mutual information measure as the cost of putting attention. And if possible can you briefly explained it. A link that explains the model is given here http://sims.princeton.edu/yftp/Gerzensee/info.pdf

Empirical Evidence on rational inattention:

-Title: Attention Discrimination: Theory and Field Experiments with Monitoring Information Acquisition,
Authors: Vojtěch Bartoš, Michal Bauer, Julie Chytilová, and Filip Matějka.
Source: IZA Discussion Paper No. 8058.

Link: Attention Discrimination: Theory and Field Experiments with Monitoring Information Acquisition

-Title: What Can Survey Forecasts Tell Us about Information Rigidities?
Authors: Olivier Coibion and Yuriy Gorodnichenko.
Source: Journal of Political Economy, Vol. 120, No. 1 (February 2012), pp. 116-159

Link: What Can Survey Forecasts Tell Us about Information Rigidities?

-Title: A Rational Theory of Mutual Funds’ Attention Allocation,
Authors: Marcin Kacperczyk, Stijn Van Nieuwerburgh, Laura Veldkamp.
Source: NYU Working Paper No. 2451/28347 June 5, 2014

Link: A Rational Theory of Mutual Funds’ Attention Allocation


Does national debt and its repayment assume population increase? I assume that the repayment of an increasing national debt depends on an increasing GDP. And increasing GDP, in turn, depends mutatis mutandis  on increasing population. So an increase in national debt assumes increasing population and, likewise, a decreasing population would mean both a decrease in GDP per capita and an increase in national debt per capita. Is this, in general, correct?

tl;dr:  No, repaying national debt does not assume population increase although depending on the parameters of economy population increase can sometimes help with repaying the debt and also help with reducing the ratio of debt to GDP, but it is neither required nor assumed. Moreover, depending on parameters population growth can even be a hindrance as well.
Full Answer: No the statement in the question is not accurate for several reasons. It is true that GDP depends on population because if every person can produce $\\\$100$ worth of GDP then 10 people will be able to produce $\\\$1000$ of GDP. However, you are incorrect in saying:

decreasing population would mean both a decrease in GDP per capita

because GDP per capita is already calculated as GDP per person. If every person can produce $\\\$100$ then GDP per capita will be $\\\$100$ whether country's population would be ten people or thousands, because GDP per capita is by definition total GDP of a country divided by population.
Furthermore, you seem to be confusing concept of repaying debt with growing out of debt.
From economic perspective what matters the most is the size of debt relative to national income. If a poor person who earns $\\\$200$ a month has debt of $\\\$10000$ that is a problem, but if rich person who earns $\\\$2,000,000$ a month has $\\\$10000$ that is not a problem at all (at least not from economic perspective).
Hence there is a concept of growing out of the debt which says that provided that $GDP$ grows at faster rate than debt the debt to GDP ratio will become smaller and smaller over time. However, this has nothing to do with paying down that debt in principle as the debt is still there and just becomes less economically significant.
From this perspective what matters is what is total GDP and total debt not their per capita counterparts. GDP can grow thanks to population growth but population growth is only one of the factors that can cause GDP to grow. In fact most of GDP growth comes from advancement in technology which makes economies more productive (see Romer's Advanced Macroeconomics). However, at the same time debt can grow with population as well if government borrows to fund policies that depend on size of population (such as healthcare for example) rather ones that have mainly fixed cost such as many public goods like water filtration. Provided that total GDP increases at higher rate as a response to population ($Pop$) growth than total debt $D$ (that is $\partial GDP/ \partial Pop > \partial D /\partial Pop \quad |  \partial GDP/ \partial Pop > 0$) it can help to decrease the debt to GDP, but that is not the same as saying it helps to repay debt and even if it helps to reduce debt to GDP ratio it does not mean at all that this ratio cannot be reduced without any population increase. In fact empirically in recent years in developed countries most growth in total GDP comes just from technology/productivity increase as in many developed countries population more or less stagnates but most of them still experience some growth of GDP.  Furthermore,  note that if $\partial GDP/ \partial Pop > \partial D /\partial Pop \quad |  \partial GDP/ \partial Pop > 0$ does not hold population growth can even make the debt to GDP worse.
When we talk about repaying government debt we have to look at government budget constraint. The government budget constrain is given as:
$$G-T=\theta + \beta$$
where $G$ is government spending, $T$ is the net tax revenue after transfers and interest payments, $\theta$ is government financing by high powered money and $\beta$ is government financing by bonds (see for example Blinder & Solow (1973); Christ (1968); Tobin & Buiter (1976).
Government debt is accrued when government issues bonds $\beta>0$ and conversely government debt is repaid when  $\beta<0$. Hence in order to see what needs to happen to start repaying the debt we can just solve the budget constraint for $\beta$:
$$G - T- \theta = \beta$$
which tells us that $\beta<0$ when government spending $G$ will be smaller than the amount of taxes $T$ and high powered money $\theta$ government issues (so when $G<T+\theta$). Hence population growth is not necessary for repayment of debt or even assumed. Government will start repaying its debt if its spending is lower than net taxes plus high powered money it issues. Sure if population growth can increase net tax revenue $T$ while not causing government spending $G$ to increase it can help with repaying the debt but it is in no way necessary to assume it will happen for debt to be repaid (and it is also possible that increase in population could lead to more government spending than the increase in tax revenue where population increase would make situation worse).


Why have trickle-down economics failed to trickle down to the middle-class and poor in recent years? Trickle-down economics, also referred to as trickle-down theory, is an economic theory that advocates reducing taxes on businesses and the wealthy in society as a means to stimulate business investment in the short term and benefit society at large in the long term.

When it works, wealth created by the rich will trickle down to the middle-class and poor. Hence, the name trickle-down economics.

https://en.wikipedia.org/wiki/Trickle-down_economics

Strangely, one of the richest man in U.S, Warren Buffett, does not seem to think so. Why have trickle-down economics failed to trickle down to the middle-class and poor in recent years?

https://www.cnbc.com/2018/01/04/warren-buffett-on-the-failure-of-trickle-down-economics.html

  Buffett says there is a problem with that economic system, which made
  him a king: Many individuals suffer even as those at the top prosper
  wildly.

  He points to the Forbes 400, which lists the wealthiest Americans.
  "Between the first computation in 1982 and today, the wealth of the
  400 increased 29-fold — from $93 billion to $2.7 trillion — while many
  millions of hardworking citizens remained stuck on an economic
  treadmill. During this period, the tsunami of wealth didn't trickle
  down. It surged upward."

Trickle-down economics are widely criticized and many believe that it doesn't work. Asher Edelmanon explains brilliantly in the first 2min of this video (or in the written version here).

The basic logic is that, in practice, doing this only leads to a higher accumulation of wealth by the wealthy.


Global Education Indicators - dataset I've been looking for a dataset with a single education indicator by country so I can rank/compare them but I haven't found anything.

Already seen Unesco and Worldbank data but info is too much for what I really need at this point. Also had a look at the  Human Development Index (HDI)  but this is a composite statistic of life expectancy, education, and income per capita indicators so my analysi will be obscured by other elements.

I just need (for now) to be able to say for example Japan > Spain or Canada ranks 10th worldwide.

Does anyone have any idea as where I can download (if exists) this data set? API can work too. Ideally info comes with country iso code.

Thanks in advance.

You can use the PISA study from OECD. It evaluates the skills and knowledge in reading, mathematics and sciene of 15-year-old students. PISA does not combine these results, but commentators do so sometimes.

The most recent results are from 2012, you can find an overview of the rankings here. These rankings are composited from various tabels you can download using the links at page 297 from this document.
Results of 2009 are available in a xls-file. Scroll down this artcile to download.


Does recycling lead to fewer jobs? Say I buy a glass bottle, I can either put it in the landfill or recycle it.
My understanding is, glass will breakdown into sand. (Not sure how well though, since we sometimes find glass from Roman times.) Alternatively we can recycle it.

But I was wondering wouldn't this put some people out of work? Such as the people who have to mine all the silica for use in making the glass in the first place. And considering it is more economic to recycle glass, this seems like fewer humans would be needed.

Therefore could I make the case that not recycling my glass bottle is helping with employment. On the other hand, recycling might be helping the economy as a whole and there might be more money in the economy for public sector jobs. Which is right?

Any invention that replaces human labor puts an end to that specific task. Glass recycling eliminates (or decrease) the need for silica-gathering task. Typewriter eliminates the need for printing press typesetter. Etc.

Those people whose tasks are eliminated will get reallocated to their most productive use. This might be in the form of job change (silica miner move to coal miner), or might be in the form of task redefinition (the book Prediction Machines describe how self-driving school bus might shift the main task of a school bus driver to an adult who oversees and "teaches" the schoolchildren.)

Going back to your question, recycling also creates new jobs. They need people to sort bottles, maybe drive a recycling truck, etc. So whether an invention leads to more or less job is ambiguous.

Added: While the effect on a specific industry might be quantifiable, if you take into account job mobility, etc. then the effect on the whole labor market is ambiguous.

Regardless, an invention that increases productivity should increase the size of the pie. We can produce more from the same resources. How that bigger pie is divided, however, is another very important question altogether.


Is there a correlation between pharmaceutical prices and drug dealers? Question: Is there a quantitative or some kind of correlation between the number of drug dealers in a certain region and the price of pharmaceutical drugs?

This is for a case I'm working on, but I can't seem to find a direct correlation between the price of prescription drugs and drug dealers.
Right off the bat, it seems simple right? Lower prices force drug dealers out as consumers turn towards the brand name drugs which effectively lowers the amount of drug dealers in an area. However, I can't seem to find, anywhere, that says something along those lines.

Direct correlation seems unlikely. You would probably have to control for population or even better, population density. Leaving prices unchanged a city with ten times the population can support ten times as many drug dealers. Differences in average income levels would present a similar problem.

Finding reliable data, such as number of drug dealers and a good measure of price levels will be a challenge as well.


Growth Accounting with variable factor shares Under the assumption of competitive markets, the usual growth accounting equation using Cobb-Douglass function is as follows:

$\Delta Y/Y = \Delta A/A+w_l\Delta L/L+w_k\Delta K/K$

$Y$ is output
$K$ & $L$ are factors of production
$w_i$s are constant factor shares

Almost all empirical papers I've read so far assume constant factor share values $w_i$s regardless of the sample span. Usually the papers assume $w_l=0.7$ and $w_k=0.3$. I believe the results would be different if we let the factor share values vary over time in the growth accounting exercises.

Does anyone know of paper(s) that use growth accounting method by utilising actual data on factor shares?

Interesting question. In effect, while factor shares were thought to have remained fairly stable over a long time (the first of the Kaldor's facts), more recently they have varied, particularly in the direction of a fall in the labour share.

This short paper from (2012) shows that under such scenario, a growth accounting exercise which assumes constant factor shares leads to inconsistent results. The abstract reads:

  Recent evidence shows that factor shares are not constant. Consequently, growth accounting exercises rely on a false assumption and a measurement problem arises. We propose an empirical methodology to solve the measurement issue and estimate TFP growth.

A more complete and recent analysis from the same author (plus another co-author) is here (and a theoretical model with these insights is here).

As you can see, this is a very topical issue, and there seems to be very little empirical analysis so far. I only found this one.   As the authors in the second article linked conclude,

  any efforts to align empirical growth research with the reality of the factor share data cannot be carried out using standard techniques. New techniques need to be developed.

There might still be some time until a satisfactory empirical method is developed.


Physical goods in a typical household Estimating how much of physical goods a somehow typical person or household owns today (say in the Western world, compared to former times) is not so easy I guess, but there is one single number that might tell it: the typical weight/volume per person that is transported when moving a house. (This gives not the number, kind and economic value of physical goods, but only their cumulated weight/volume, but that would be OK for me for the moment.)

Removal companies know these numbers, and I wonder if they are available somewhere,  broken down by country, state or other region of the world, and of course broken down by time. (Of course the numbers from removal companies tell only about people who can afford to hire a removal company, but that would be OK for me for the moment, too.)

I would find it astonishing if these numbers had not grown significantly in the last decades, but I'm open to any answer. (For example, modern people might decide to dispose many of their physical goods before moving, and re-buy them at their new location. This would be interesting in its own right.)

If the approach via removal companies is not found sensible: Which statistics could you recommend alternatively? What about homeowner's insurances? Or are there official detailed statistics available, say for the U.S., Europe or e.g. Germany?

(If this forum is not the right one to ask such questions: Which one would you recommend?)

Those are some good starting points but I would refrain from using them as the end result. There are some fatal logical jumps in your example.

Moving house: A good one to use. If I were to guess, I think the ratio should have decreased due to tech. For example, we don't have those huge t.v. nowadays. No desktops for the most part. However, be careful about how wealth plays a role. Also, people moving into cities of California would be very different than of Colorado.

I think a better indicator would be the consumption category of GDP. Especially durable goods. You can find detailed numbers by year, state, category on the official website.

I hope that helps.

References:
https://www.bea.gov/data/by-place-us
https://de.wikipedia.org/wiki/Bruttoinlandsprodukt


Creation of Money - Hypothetical Situation II Continuation off this line of questioning. If I issue an IOU, it would need to be fungible, transferrable, etc in order to be considered "money." Understood - My follow up questions are:

Remaining agnostic as to whether or not my Note Payable is money: Say my friend takes out a cheaper loan because he uses my Note as collateral. So his interest rate 10% => 7%. While it may not technically be money (not all collateral is), its silhouette remains. Thus, has my IOU not increased the money supply? How is this conceptually treated?
Let's assume my IOU IS money because I'm heavily trustworthy, etc. For example, ABS, MBS, some CDOs (effectively) trade as information-insensitive debt. How much of the Monetary Supply is created Endogenously (in private situations like this), vs. Exogenously (by the Fed, etc) ?

No or not necessarily because you can't know if the actor will save income or pay down debt, or spend more. The interest rate is not a single factor that determines behaviour.
Nearly all of it is privately created debt money (97%), in circulation. Your 'money' is some bank's liability. You trade their IOUs and they are trusted/licensed to create them. (By all of it I mean in practical terms...there are nominally huge amounts of certain types of security but I don't think their relevance to mainstream social function is useful in the context of this kind of topic)

For more info: https://positivemoney.org/how-money-%20works/how-banks-%20create-money/

Video explanation:https://youtu.be/b6_SLwReMqo

Video from BoE: https://youtu.be/CvRAqR2pAgw

On the influence of the Fed, this is a huge topic. Basically since 2007/8 the Fed creates enough money to compensate for credit contraction in the private sector so that the appearance of growth is maintained. So if in the US nominal GDP growth is 2%, after removing fed stimulus you would see that actual growth is depression level negative. Also the fed sends signals. It could say it would start purchasing corporate bonds directly, which would trigger fund managers into a snowball doing the same. But basically the situation since 2008 is that the Fed is forced to keep monetary policy loose, stimulus ample, to keep lending up and credit growth going.


How can I build a fixed point theorem argument in pure strategies? To begin with, I am recalling the Banach Fixed Point Theorem.
Let $(X,d)$ be a non-empty complete metric space with a contraction mapping $T:X\to X$. Then $T$ admits a unique fixed-point $x^*$ in $X$ i.e. $T(x^*) = x^*$.
In order to prove that the combinantion of the best response strategies of $N \geqslant 2$ agents consitutes a Nash Equilibrium, we need to use a fixed point theorem argument. Is this necessary only in cases where the agents employ mixed strategies or this holds in the case where they also follow pure strategies? I am searcing for some guidance to use the Banch fixed point theorem in guadratic utility functions when agents act by submitting pure strategies. How can I build this argument? What confuses me the most is that the mapping $T:X\to X$ is about the strategies isn't it?
I can provide more details of my problem if you wish so. I would also be glad if you could answer with great details, since I do not have any idea about the topic

There is no apriori reason for the Best Response map to be a contraction in general. Here's a simple example (since Battle of Sexes has been my go-to for the past few days):
$$
\begin{array}{c|lcr}
\text{Player1/Player 2$\rightarrow$} & \text{F} & \text{T} \\
\hline
\text{F} & 3,1 & 0,0 \\
\text{T} & 0,0 & 1,3 \\
\end{array}
$$
Denote the strategy of player 1: $x = Pr(T)$ and that of player 2 by: $y=Pr(T)$. The best response of player 1  is a map $BR_1:[0,1]\rightarrow[0,1]$ defined by:
$
BR_1(y) = \begin{cases}
0 & \text{ if } y<\frac{3}{4}\\
[0,1] & \text{ if } y = \frac{3}{4}\\
1 & \text{ if } y>\frac{3}{4}\\
\end{cases}
$
Similarly, the best response of player 2 is a map $BR_2:[0,1]\rightarrow[0,1]$ defined by:
$
BR_2(x) = \begin{cases}
0 & \text{ if } x<\frac{1}{4}\\
[0,1] & \text{ if } x = \frac{1}{4}\\
1 & \text{ if } x>\frac{1}{4}\\
\end{cases}
$
Define the Best Response profile $BR:[0,1]^2\Rightarrow[0,1]^2$ as $BR(y,x) \equiv \big(BR_1(y), BR_2(x)\big)$. A Nash equilibrium of this game is a fixed point of $BR$.
This, however, need not be a contraction map. Take two points on the domain: $a = (0.8,0.5)$ and $b = (0.5,0.2)$. Its easy to see that $BR(0.8,0.5) =(1,1)$ and $BR(0.5,0.2)=(0,0)$. Thus the Euclidean distance $d\big(BR(a),BR(b)\big) = \sqrt{2}$ whereas $d\big(a,b\big) = 0.3\sqrt{2}$. It is clear that $\not\exists$ $k>0$ such that $d\big(BR(a), BR(b)\big)\leq k d\big(a,b\big)$.


Constant absolute risk aversion and certainty equivalent I need to prove that Constant Absolute Risk Aversion (CARA) is equivalent to
\begin{gather}
\int u'(x)dF(x) = u'(c(F,u))
\end{gather}

where $u(x)$ is a Bernoulli utility function, $F$ is the distribution of the lottery and $c(F,u)$ is the certainty equivalent.

I started from the fact that CARA is defined as $-\frac{u''(x)}{u'(x)}=a$, where $a$ is a constant and that the certainty equivalent is defined as $u(c(F,u))=\int u(x)dF(x)$.

I tried to mix up the two definitions but I am a bit lost with the interaction of integration and differentiation.

Do you have any hint?

This is perhaps not the best way to prove the statement. But notice that a CARA utility is equivalent to having the general functional form:
\begin{equation}
u(x)=-\alpha \mathrm e^{-ax-b}+\beta,\quad\text{where $a,\alpha>0$ and $b,\beta\in\mathbb R$}.
\end{equation}
Knowing this, the rest should be straightforward.


What is the difference between USD/EUR and EUR/USD when talking about Forex I was recently watching the Forex market and I want to know what does it means when you see something like

EUR/USD  -0.04% -1.1208

Does that mean the USD dropped with 0.04% against EUR or vice versa? Can you please also extend the explanation to stock indices?

When the forex market does a quote, it is done in currency pairs. When you sell a currency, you have to be buying something in return. That is why they are quoted in pairs.

For example when you buy an apple, you are selling your dollars to get it. Similarly in forex, when you sell EUR to get USD, you are using the EUR/USD currency pair, or USD/EUR depending on which way you want it, one is simply the inverse of the other.

However, these pairs are counter-intuitive in a such a way that they are not fractional terms. It is simply a quote. What I mean is that in the quote EUR/USD, it is expressing how much 1 EUR is worth in USD and in USD/EUR, how much 1 USD is worth in EUR. The first currency in a pair is called the "Base" currency and the second is called "Quote" or "term" currency. Think of it this way, the first currency in a pair is your "commodity" currency and the second currency is its price.

As for what "EUR/USD -0.04% -1.1208" means, I think I'll need more context to it. If by -0.04%, you mean that EUR/USD has dropped by 0.04%, in example, it simply means that if 1 EUR is 1.128 USD, it is now 1.1275 USD. Therefore, your EUR is now cheaper, USD has appreciated against the EUR or, EUR has depreciated against the USD, or a combination of both.

If you are confused, just remember the first currency in a pair is your "commodity" currency. Replace EUR with apples and you should see it clearer.

Read more about it here and here.


Has any person ever actually planned his or her consumption according to Modigliani’s LCH? https://en.wikipedia.org/wiki/Life-cycle_hypothesis?wprov=sfti1

It seems self-evident to me that nobody on Earth actually does this. Real people are short-sighted, undisciplined, and lack the computational ability to account for all the variables.

Some may attempt to, but will invariably fail, and most people don’t even try and instead live for the short-term. Richard Thaler covers this in his book “Misbehaving: The Making of Behavioral Economics”.

Has even one instance of LCH behavior ever been documented?

The life-cycle hypothesis as a model doesn't need to explain people's behavior perfectly to be of use in understanding the economy or the decisions that individuals make. Recall the old saying that "all models are wrong but some are useful." As a quick example, note that many people do in fact borrow when they are young (e.g., for college, save while their income is relatively high (e.g., a lot of people save for retirement), and dissave when they're older---just as would be predicted by this theory.


If billionaires stopped spending what would the economic effect be? For example they always invest every available wealth they have (or they burn money for cooking or dump money into the sea to make artificial islands)?

In some sense that money is not returned to the economic system.

In other words what are the effects besides the money supply decreasing?

I am asking this to satisfy my curiosity.

If one person decides to reduce his lifetime consumption by (say) \$100, then someone else (or many someone else's) is made richer by \$100.  For example, that $100 might be left to a single heir.  Or at the other extreme, the person might decide to burn a \$100 bill in which case the benefits are spread to everyone who holds money.

When people are richer, they can decide to consume more, or to work less, or some of each.  Because income elasticities of labor supply are empirically quite small, you'd expect most of this to come out in increased consumption.


How do we prevent big companies conspiracy? These days we can see a bunch of troubling trends.

Consumer goods are less quality.
Big companies merge.
Patent laws and strict requirements on appearance remove small entrepreneurs from the manufacturing business.

These problems had been subjects of dystopian sci-fi scenarios for many times. Even though it is profitable for a starting bussinessman to buy and sell as the supply and demand dictates, for a few big companies, ruling over the market it is profitable to keep the prices up.

What can lawmakers do about concluding conspiratorial agreements or merging big companies to keep the supply and demand principle?

The kinds of practices you describe are frequently controlled through competition/antitrust policy. For example:

In the US conspiratorial agreements (such as price-fixing cartels, bidding cartels, or other agreements that restrict competition) or conspiring to abuse a position of dominance in a manner that harms competition is illegal under the terms of the Sherman Act 1890. Likewise, these kinds of practices are prohibited within the EU by Article 101 of the Treaty on the Functioning of the European Union.
In the US, the Clayton Act 1914 provides for the prevention of mergers that are deemed anti-competitive. A merger between "big" companies is likely to be anticompetitive because it will result in a significant increase in market concentration. More formally, the DOJ writes, "Transactions that increase the HHI by more than 200 points in highly concentrated markets are presumed likely to enhance market power under the Horizontal Merger Guidelines issued by the Department of Justice and the Federal Trade Commission." Article 102 of the Treaty on the Functioning of the European Union serves the purpose of preventing anti-competitive mergers in the EU.

The laws mentioned above are regularly used to prevent conspiracies or mergers that harm competition, and firms are often subjected to significant penalties for violations of these laws. For example, here is a list of cases in which the European Commission has intervened to prevent cartels. Here is a similar list for merger cases.

The relevant EU laws are mirrored in national law within EU member states so that member state competition authorities can also intervene to prevent conspiratorial behaviour.

The issue of patents is a slightly more tricky issue. A patent is a state-granted right to behave anti-competitively. When a patent is awarded, the state is basically saying "we are going to let you behave like a monopoly over this technology for the next 20 years". This is harmful for all the usual reasons that monopoly is bad. But, there is an important trade-off. Firms invest a lot of resources into research and development. For example, developing a new drug is estimated to cost as much as $5,000,000,000. Companies would not be willing to make this investment if they thought that, as soon as the drug were invented, others would enter the market, copy the idea, and sell a cheaper competing generic alternative. If we want firms to innovate then there has to be some reward to innovation. Thus, the compromise we have is that firms are allowed to be a monopoly for a limited period of time in return for making their inventions public by writing them into a patent filing.


How can I get total home values on Zillow? I am interested in getting aggregate dollar values for listings/sales on Zillow (for the whole country summed together, for example). Unfortunately, Zillow's data page seems to only show the "ZHVI", an index value, not a total dollar value. Is there any way to get the total dollar value?

To answer your question of whether you can get a number for 'total home value,' the answer is no–at least not easily.

Zillow (somewhat) recently made available a data set similar to what you are searching for. I suspect, however, that they do not want the public to have access to aggregated value because it would showcase the volatility of Zillow home value estimates, i.e. their algorithm-based estimate fluctuates much more than the estimates coming out of research institutions, etc.

Sorry for the dissatisfying answer!


Hours worked increases with wage So I have been given a utility function = $ 48 R + Ry -R^2$ where $R$ represents leisure hours and $y$ represents labour income. $y=rl$, $r$ is wage rate and $l$ is labour hours. Find hours worked increases with wage.
So what I thought I can do was:

I equate MRS = Slope of Budget constraint and I got something like this
$(48 + y - 2R)/ R = r $
Then I equated in $R$ and got $R=48+(rl)/r+2$
Can I just now do differential of $R$ with respect to $r$

Can I continue with this method? Have I made some mistake so far? Or is this method completely wrong and is there some other method.

It's probably easiest to insert everything into Utility $U = U(R)$, such that it depends only on one choice variable:

$U(R) = 48R + Rr(L^{*}-R)-R^2$

Deriving w.r.t. $R$ yields the first order condition:

$48 - Rr + r(L^{*}-R)-2R \overset{!}{=} 0$

Rearranging yields leisure in terms of the wage rate:
$R(r) = \frac{48-L^{*}r}{3+r}$

Labor Supply $l(r)$ is hence given by:
$l(r) = \frac{3L^{*}+2rL^{*}-48}{3+r}$

Deriving this by $r$ should yield what you're looking for:

$\frac{d l}{d r} = \frac{3L^{*} + 16}{(r+3)^2}$


Cartel profit maximizing quantity question (Cournot game) Find the Nash equilibrium of Cournot’s game when there are two firms, the inverse demand function is P(Q) = α – Q when α ≥ Q and 0 otherwise, and the cost function of each firm I is Ci(qi) = qi2. If the firms collude in this situation to create a cartel to maximize their profits, how much would each firm produce?

I'm able to find the Nash equilibrium quantities when there is no collusion - however, I'm stuck on the cartel part.

We want to find joint profit maximizing quantities under collusion here. When the marginal cost to each firm is constant this is easy to do, since $C(Q) = C(q_1) + C(q_2)$. However, in this case, $C(Q) = q_1^2 + q_2^2$.

Thus, our joint profits are $\pi(Q) = Q(\alpha - Q) - (q_1^2 + q_2^2) \neq Q(\alpha - Q) - Q^2$. I'm not sure how to take this first order condition (with respect to $Q = (q_1 + q_2)$) given the non-linearity of the cost functions. Any ideas as to how to solve this would be greatly appreciated (not looking for a direct answer; simply a nudge in the right direction).

Hint

By colluding, presumably the two firms would like to figure out the least costly way to produce in order to maximize profit. Treat the two firms as two production plants of a single "parent company". The total cost of the cartel can then be found by solving
\begin{equation}
C_\text{cartel}(Q)=\min_{q_1,q_2}\;[C_1(q_1)+C_2(q_2)],\quad\text{s.t.}\quad q_1+q_2=Q.
\end{equation}
This will give you a the optimal way of distributing production across the two firms, $q_i^*(Q)$, as a function of aggregate output $Q$. Then the cartel's total cost is just the sum of the member firms' costs evaluated at the optimum: $C_\text{cartel}(Q)=\sum_i C_i(q_i^*(Q))$.


How is it not arbitrage that EURIBOR (EUR) is around -0.5% and LIBOR (USD) is around 0.5%? Why can I not borrow $1B of EUR at EURIBOR (pays me 50bps per year), convert it to USD (also pays me 50bps per year), lend it out at LIBOR, and collect a 1% spread? Am I missing somethign obvious?
EURIBOR
and LIBOR

Because of exchange rate and uncovered interest rate parity (UIP). By the UIP the following relationship should hold between nominal interest rates and exchange rates:
$$ (1+i_{t}^$) = \frac{E_t(S_{t+1})}{S_t}(1+i_{t}^€)$$
In short, the ratio of expected future nominal exchange rate ($E_t(S_{t+1})$) and current nominal exchange rate ($S_t$) should make the interest rates in US ($i^{\\\$}$) and Eurozone ($i_{t}^€$), equivalent eliminating any arbitrage opportunities.  If the equality above would not hold, then there indeed would be an opportunity for an arbitrage, but as soon as people would take advantage of the arbitrage opportunity trading on forex markets would move exchange rate to the proper position. You can learn more details on intuition behind UIP in Copeland Exchange Rates and International Finance.
In real life the above equation will sometimes be violated but empirical evidence shows trading typically eliminates arbitrage opportunities very fast. Usually within hours or days, or few weeks at the most save for some special circumstances (see discussion in Nelson International Macroeconomics and Finance in first chapter).
In addition, to the above mentioned UIP which may not always hold there is also related concept of covered interest rate parity (CIP) where expected future exchange rate is replaced with value of futures for the exchange rate and thus given by:
$$ (1+i_{t}^$) = \frac{F_{t}}{S_t}(1+i_{t}^€)$$
When it comes to intuition and mechanism of the model there is not much difference between UIP and CIP but evidence shows CIP is being violated less frequently (again see the Nelson reference above). This is precisely because value of future contracts is easier to measure and thus any possible arbitrage will closed much sooner or even prevented altogether by movements in price of futures.


Extensive form: backward induction & subgame perfect nash equilibria? I've been given the SPNE through backward induction, I just want to understand how to interpret the equilibria properly.

The SPNE: $(agi,de)$ and $(bgi,df)$

For instance, how would I interpret $(agi,de)$?

Please let me know if the following would be an accurate interpretation of the above:

If A chooses a first, then B will play d.
In the event that we end up on the right hand side of the extensive form, B will select e and A will subsequently select g. However, if given the opportunity, A will select i.

This is supposed to be a complete contingent plan of action for both players (i.e. if it were in matrix form then we would be looking at all possible strategies for all players).

I find the first part quite easy, i.e. If A selects a, then B will select d which can be represented by $(a,d)$. However, I am getting confused when I have to account for all possible strategies that could be played $\rightarrow$ $(a \color{blue}{gi},d \color{blue}{e})$.
I would appreciate it if you could offer some advice.

Until the last paragraph your interpretation is correct.
$(a,d)$ is not a strategy profile, it is merely a 'history', a way that the game can play out.

A strategy of a player is a function that chooses an action in each of her decision nodes. As you can see in your graph Player A has three decision nodes while B has two. Hence any strategy of player A will consist of three elements, while any strategy of B will consist of two elements. So $(agi,de)$ is indeed a strategy profile. The corresponding history, what will happen if the players play these strategies, is $(a,d)$. The corresponding payoff is (1,2).

I am unsure what you mean by having to account for all possible strategies.
Do you want to see what it means when an equilibrium is not subgame perfect?
That would be $(ahj,de)$. No player can gain by unilaterally switching to another strategy, but if we were to reach player A's second or third decision node his action would not be optimal. But $(ahj,de)$ can be an equilibrium because the history given by this strategy profile never reaches these nodes.

Do you want to find all the equilibria that are not subgame perfect? In that case you should write down all possible strategies:
There are 2^3 strategies for A, 2^2 strategies for B. Make a matrix using these as row and column labels. (Not sure if you really want to do this, as it is rather large.) Then you calculate the history and payoff for each strategy profile, creating a payoff matrix. In this you will find many equilibria. The ones that are not $(agi,de)$ and $(bgi,df)$ are not subgame perfect.