# Tag Info

6

So recently I have been thinking a lot about this fundamental question: Does the Sonnenschein-Mantel-Debreu theorem disprove the "Law of Demand"? It contradicts law of demand as a general law but it is worth noting that the law of demand is for over a century not considered actual general law but just a special case that is simply applicable to ...

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Would be strange to write it that way. If you had to define something like that, just do the following: Start with a type space $(T,\mu)$ with probability measure $\mu$. Let $\sigma: T \rightarrow \Delta(A)$. Then $\mathbb{P}(a) =\mu(\sigma^{-1}(a))$.

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First, the Sonnenschein-Mantel-Debreu theorem has nothing to do with demand functions. It is a result about excess-demand functions, which represent demand minus supply. They are formulated in the context of an exchange economy in which no production happens. The result says that if one only looks at prices that are not too close to zero, any continuous ...

4

Take a strictly increasing mapping $f:\mathbb{N} \to [0,1)$, such as $$f(y) = 1 - \frac{1}{y+1}.$$ Then $$U(x,y) = x + f(y)$$ represents the Lexicographic preference in the $\mathbb{N}^2$ choice space.

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You got to the quadratic equation $$\lambda^2 - (\rho - n)\lambda + \frac{c^\ast f''(k^\ast)}{\varepsilon}$$ The discriminant is given by: $$\Delta = (\rho - n)^2 - 4 \frac{c^\ast f''(k^\ast)}{\varepsilon}$$ So the two roots are: $$\lambda_1, \lambda_2 = \frac{(\rho - n) \pm \sqrt{(\rho - n)^2 - 4 \frac{c^\ast f''(k^\ast)}{2}}}{2}$$ As $f''(k) < 0$ ...

4

To answer the first part of your question, we do not need any more assumptions for the comparison of experiments (besides some measurability issues). Before going on, I'll fix some notations to ones that are standard in the game theory literature, and for the sake of my convenience. An experiment (or informtion structure) is defined as a tuple $(S,\pi)$ for ...

3

You take $p$ to be the corresponding correlated equilibrium with $A_k$ being the strategy space of player $k$ Conditions 1. and 2. mean that each player can compute the prescribed action they should play and that they do not know more than this prescription would give them. This is exactly what a correlated equilibrium requires (together with optimality, of ...

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The answer is no. If you want to have decreasing marginal rates of substitution then you need quasi-concavity. In order for the notion of decreasing marginal rates of substitution to make sense, you need a few assumptions. First you restrict yourself to a two good setting, say with goods $x$ and $y$ You assume that the utility function $u(x,y)$ satisfies ...

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Solving $Min(-F[x])$ s.t. $G[x]\leq 0$ is same as solving $Max(F[x])$ s.t. $G[x]\leq0$ So, the Lagrangian for the minimizing problem will be: $L = -F[x] - \lambda (0-G[x])$ For the maximization problem the Lagrangian will be: $L = F[x] + \lambda (0-G[x])$ In both cases, $\lambda \geq 0$

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Looking at the numbers again plan B is simply worse (assuming a non-negative interest rate). In any specific month you look at, you have to pay no less under plan B than under plan A.

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