Unformatted text preview: direction of reducing the consumption of the k-th commodity by 1 and increasing the consumption of the j-th commodity by p k / p j is an improvement since v j ( x ∗ ) p k / p j − v k ( x ∗ ) > 0. As x ∗ k > 0, we can Fnd ε > 0 small enough such that decreasing k ’s quantity by ε and increasing j ’s quantity by ε p k / p j is feasible. This brings the consumer to a strictly better bundle, contradicting the assumption that x ∗ is a solution to the consumer’s problem. ±or the case in which the preferences are represented by a util-ity function u , we have v k ( x ∗ ) = ∂ u /∂ x k ( x ∗ ) . In other words, the “value per dollar” at the point x ∗ of the k-th commodity (which is consumed) must be as large as the “value per dollar” of any other commodity....
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- Fall '10
- Convex function, x∗, Differentiable preferences, pk /pj