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9 votes
Accepted

Prove that for every Nash equilibrium $\sigma^*$, the probability distribution $p_{\sigma^*}$ is a correlated equilibrium

A strategy profile $\sigma^*=(\sigma_i^*,\sigma_{-i}^*)$ is a Nash equilibrium if for all player $i$, \begin{equation} u_i(s_i,\sigma_{-i}^*)\ge u_i(s_i',\sigma_{-i}^*), \quad \forall s_i\in\mathrm{...
Herr K.'s user avatar
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5 votes

Indifference Curve Analysis

Indifference curves represent preferences. Preferences are usually assumed to be stable, i.e. they do not change. So no, indifference curves don't shift.
VARulle's user avatar
  • 7,044
4 votes

Concavity of Cobb-Douglass Utility Function on Non-Open set

I assume the notation $\mathbb R^2_+$ refers to $[0,\infty)^2$. Note that the set on which a function is defined need not be the same set on which a function is differentiable. In particular, it's ...
Herr K.'s user avatar
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3 votes
Accepted

What is the meaning of the support set in game theory?

From Wikipedia: Suppose that $f : X \to \mathbb{R}$ is a real-valued function whose domain is an arbitrary set $X.$ The set-theoretic support of $f$, written $\operatorname{supp}(f),$ is the set of ...
Giskard'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
2 votes

When are marginal rates of substitution consistent with a utility function?

This is a rather indirect way. For $\omega, z \in \mathbb{R}_{++}$, define the (demand) correspondence: $$ D(\omega, z) = \left\{(x,y) \in \mathbb{R}^2_+| MRS(x,y) = \omega \text{ and } \omega x + y = ...
tdm's user avatar
  • 12.5k
2 votes

Proving an additive function is sigma additive (Measure Theory)

Sigma additivity requires that if you have a countable collection $(B_i)_{i \ge 0}$ of mutually disjoint sets ($B_i \cap B_j = \emptyset$ for $i \ne j$) then: $$ \sum_{i \ge 0} \mu(B_i) = \mu(\cup_{i \...
tdm's user avatar
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2 votes
Accepted

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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2 votes
Accepted

Budget Set- closed and boundedness

The budget set is always defined given a price vector $p=(p_i)_{i\leq l}$ (it seems like $l$ is the number of goods in your problem) and an income $w$. We usually implicitly assume that prices are ...
Bayesian's user avatar
  • 5,291
1 vote

Differentiability of value of convex optimization problem

A friend suggested a solution, which I will sketch below. It relies on some rank conditions that are hard to interpret, but it is better than nothing. (1) If either $f$ or $g$ is strictly convex, then ...
John Sturm's user avatar
1 vote

Calculating elasticity between terms in a regression equation

The formula of the OP is perfectly fine. Treating the derivative as a ratio of differential (long live Leibniz), with $x$ representing "age", \begin{align} \frac{d \ln w}{dx} &= b_2 + ...
Alecos Papadopoulos's user avatar
1 vote
Accepted

Supporting Hyperplane Theorem and quasiconcave utility function

This problem is quite specific to economics. The correct statement is: Proposition If $u(\cdot)$ is quasiconcave, strictly increasing, and continuous, then $\forall x$, there exists $p \gg 0$ and $w ...
Michael's user avatar
  • 2,619

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