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3 votes

Intuition of two Measure theory statements

For the first statement, note that $\int_E f~\mathrm d\mu$ is by definition the same thing as $\int 1_E f~\mathrm d\mu$, and the integral of a function is defined as the limit of the integrals of ...
Michael Greinecker's user avatar
3 votes
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Prove: The law of demand holds if WA, Walras' law, homogeneity of degree 0, and homogeneity of degree 1 in wealth hold for Walrasian demand functions

We need to show that: $$ (p - p')\cdot(x(p,1) - x(p', 1)) \le 0. $$ Note that if $x(p,1) = x(p', 1)$ then this is obviously satisfied, so assume that $x(p,1) \ne x(p', 1)$. Note that the condition is ...
tdm's user avatar
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0 votes

Drawing a Probability simplex

For the strategy below we find the vector AC, scale it down to AD then set the vector's tail at A so that the tip is at D. Finally, adding the vector to A gives us point D. The vector from point A to ...
Goose's user avatar
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2 votes
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CES function: Lagrangian or Kuhn-Tucker

If $\rho < 1$ then the marginal utility (say with respect to $x_1$ equals): $$ \frac{\partial U}{\partial x_1}(x_1, x_2) = x_1^{\rho - 1} (x_1^\rho + x_2^\rho)^{\frac{1}{\rho}- 1},$$ This tends to $...
tdm's user avatar
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2 votes
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Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?

Note that: $$ f(Ax) = f(a_{11} x_1 + a_{12} x_2, a_{21} x_1 + a_{22} x_2). $$ So the derivative with respect to $x_1$ is given by: $$ f_1 a_{11} + f_{2} a_{21}. $$ Taking the derivative of this with ...
tdm's user avatar
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1 vote

Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)

Yes, I think for question (e), there is a bit of an "error" in the reasoning. The total demand change can always be decomposed in an income and substitution effect. If there are only two ...
tdm's user avatar
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3 votes

Continuous logit framework

This derivation follows the same procedure as the one used in discrete logit model. Let's denote the index of the maximum given $y$ is selected is $(y_i^m,\varepsilon_i^m)$, where $y_i^m = y$. We ...
Alalalalaki's user avatar
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2 votes

Continuous logit framework

As they state, $\pi(y|x)$ is the density of $\ln\left(\int_Y \exp(U(x,y) dy\right)$ (the density being the derivative of the cdf). If we take the derivative of $\ln\left(\int_Y \exp(U(x,y')) dy' \...
tdm's user avatar
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3 votes

Is this proof correct (measure theory)?

The first direction is fine, the second has some small problems and some scope for improvement. There is the notational issue that $\cup_{n=1}^\infty \in \mathcal{F}$ should be $\cup_{n=1}^\infty A_n \...
Michael Greinecker's user avatar
0 votes

Does the Sonnenschein-Mantel-Debreu theorem fundamentally undermine Mises' Economic Calculation Argument?

What the Economic Calculation Problem is about, in its essence, is judging the causal effect of capital goods on final consumer goods. Suppose you're a socialist economic planner and someone comes to ...
UtilityMaximiser's user avatar
1 vote

Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation

Let ${\cal B}$ be a collection of choice sets and let $C$ be a single valued choice function, picking one item $C(B)$ out of every set $B \in {\cal B}$. Let $x$ be revealed preferred to $y$ if there ...
tdm's user avatar
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2 votes

Resources for a Deep Dive into the New Keynesian / DSGE models

Wickens Macroeconomic Theory: A Dynamic General Equilibrium Approach, is good resource that has what you are looking for.
1muflon1's user avatar
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1 vote
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MWG Exercise 2.E.5

Answer to 1. $$ x_l(\mathbf{p},w) = \frac{\partial x_l(\mathbf{p},w)}{\partial w}p_l \cdot \frac{w}{p_l} $$ Let $\alpha_l = \frac{\partial x_l(\mathbf{p},w)}{\partial w}p_l$. This is nice, now use ...
Giskard's user avatar
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