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7 votes
Accepted

Optimal residential location and multivariate Frechet distribution

Consider a vector of stochastic variables $Z = (Z_1,...,Z_J)$. We assume each $Z_j$ is Frechet distributed $$Z_j \sim F(z_j) = \exp(-z_j^{-\theta}),$$ and that they are mutually independent such that $...
Jesper Hybel's user avatar
  • 3,531
5 votes

Proving that twice-differentiability implies supermodularity

Actually given the sign on the cross partial derivatives, the stronger inequality $$ \theta(x;q) - \theta(y;q) \ge \theta(x;r) - \theta(y;r), $$ holds for all $x \ge y$ and $q \ge r$. To see this, one ...
tdm's user avatar
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5 votes
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Why should $dp_2=dm =0$ in this problem?

The assumption $dm =0$ says that we examine the behavior of the consumer under a fixed nominal income, and this is something interesting to study, because it aligns to a large degree with the observed ...
Alecos Papadopoulos's user avatar
4 votes
Accepted

What are some helpful online tools (databases, etc..) for economic analysts?

St. Louis Fed: https://fred.stlouisfed.org/ Has very good US data. I am not aware of a database that systematically aggregates city-level data. I recommend state government sites and, for larger ...
123's user avatar
  • 2,911
4 votes

Total derivative evaluates to zero: problem while doing comparative statics

Consider the minimisation problem $$ OF(N,u,\eta,c) = \min_\phi C_1 S(\phi; N,u,\eta,c) $$ Here, I'm assuming that the minimisation is with respect to $\phi$. (don't know if this is correct as it is ...
tdm's user avatar
  • 12.5k
3 votes
Accepted

Proof of a comparative statics result in a maximization problem

The textbook way to derive this type of comparative statics is to use the implict function theorem. The first order condition gives: $$ \dfrac{\partial f}{\partial x}(x^\ast(\theta), \theta) - \dfrac{\...
tdm's user avatar
  • 12.5k
3 votes
Accepted

Comparative Statics: Income Effect

Taking the ratio of the rearranged second and third equations of The comparative static identities: $$ \begin{array}{r}B-x^{*} P_{x}-y^{*} P_{y} \equiv 0 \\ U_{x}\left(x^{*}, y^{*}\right)-\lambda^{*} ...
Giskard's user avatar
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3 votes
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Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32

If you assume $e_1=e_2=1$ as you have done (but not that $\alpha=1/2$), then the solution (as given in equation (48) in Simon and Blume) is: $$ \begin{align*} x_1=y_1&=\alpha\\ x_2=y_2&=1-\...
smcc's user avatar
  • 701
3 votes
Accepted

Brueckner's basic urban model, comparative statics

$\require{cancel}$ Start with the fact that $v = v(c, q)$ and use the triple product rule $$ \left(\frac{\partial v}{\partial c}\right)\left(\frac{\partial c}{\partial q}\right)\left(\frac{\partial q}...
caverac's user avatar
  • 1,216
3 votes
Accepted

Comparative statics question with an application

You can obtain a definite answer for $$\text{sign}\left\{\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }\right\}$$ given the assumptions. From $$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \...
Alecos Papadopoulos's user avatar
2 votes

Topkis' Theorem

I believe the analysis in E. Silberberg's "The Structure of Economics" (2n ed. 1990), is illuminating, chapter 7. The fundamental comparative-statics result (for constrained and unconstrained ...
Alecos Papadopoulos's user avatar
2 votes
Accepted

Comparative statics of a monopoly

The profit function is given by: $$ \pi = PQ(P) - C(Q(P), \mu) - tQ(P) $$ Assume demand is linear, s $$ p = \alpha - \beta Q \to Q = b(\alpha - P), $$ where $b = 1/\beta$. So: $$ Q_P = -b, $$ where I ...
tdm's user avatar
  • 12.5k
2 votes
Accepted

Muth-Mills Model

Define the marginal rate of subsitution $MRS = \frac{v_2}{v_1}$. This gives the slope of the indifference curve at the optimal choice. For fixed utility level $u$, this is a function of $q$ alone. The ...
tdm's user avatar
  • 12.5k
2 votes
Accepted

Comparative statics with multiple equations using calculus

Since this looks like a homework and there was no attempt made, in order to avoid giving full solution I will just give you conceptual explanation using equation 1 and 2 (you can later add equation 3 ...
1muflon1's user avatar
  • 57.2k
2 votes

Converse in supermodularity and single-crossing

I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function $h$, at the very least. It's pretty easy to show that $g(\...
Jay's user avatar
  • 71
2 votes
Accepted

How do I use total derivatives of an implicit function to solve this problem?

Given that $u_{1,2} = 0$, the utility function is additively separable. So we have the problem: $$ \begin{align*} \max_{x_1, x_2} u(x_1) + v(x_2) \text{ s.t. } &p_1 x_1 + p_2 x_2 = m \ell\\ & ...
tdm's user avatar
  • 12.5k
1 vote

Ratio of two Jensen inequality

A bit of a "non answer" but it appears that you cant say anything about how the a ratio of expectations is related to an expectation of ratios. To illustrate, let $f(X)$ and $g(X)$ be ...
EconJohn's user avatar
  • 8,407
1 vote
Accepted

Ratio of two Jensen inequality

If $X$ and $Y$ are independent positive random variables then the following are true by Jensen's Inequality: \begin{eqnarray*} \mathbb{E}\left(\dfrac{X}{Y}\right) \underbrace{=}_{\text{By Independence}...
Amit's user avatar
  • 9,106
1 vote

Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32

I just want to add an observation, too long for a comment, to the excellent explanations of smcc. You wrote [...]if my answer is correct; especially the step where I write "since the question ...
BakerStreet's user avatar
  • 4,047
1 vote

Income and Substition Effect: Assumptions: Normality vs. Inferiority of Goods

This income effect can only work in the opposite direction (more leisure, less labor) if we have the underlying assumption that leisure is a normal good, right? If you only talk about positive income ...
1muflon1's user avatar
  • 57.2k
1 vote

Why should $dp_2=dm =0$ in this problem?

While the x’s are functions of price, both P_2 and m are constants. Thus the zero-valued differential.
philE's user avatar
  • 1,040
1 vote

Why should $dp_2=dm =0$ in this problem?

This dx2=dm=0 assumption is necessary because it is the application of theory of partial derivative mathematics. If you don't follow this rule as you find dx1/dp1 you are not following partial ...
user676056's user avatar
1 vote

Comparative Statics on Balanced Growth Path

The approach is not correct, because $\mu^{BGP}$ is a result of underlying structural parameters. So to say "when $\mu^{BGP}$ increases..." immediately begs the question why it increases, which ...
Alecos Papadopoulos's user avatar
1 vote

What are some helpful online tools (databases, etc..) for economic analysts?

Also check out quandl.com. The data from the Fed and World bank type of organizations is usually free and updated regularly, as with most of the public economic data. There are other more curated ...
Jarrod's user avatar
  • 163

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