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Rubinstein2005-page72 - direction of reducing the...

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October 21, 2005 12:18 master Sheet number 70 Page number 54 54 Lecture Five Proof: Assume that both x and y (where x = y ) are solutions to the con- sumer’s problem B ( p , w ) . Then x y (both are solutions to the same maximization problem) and α x + ( 1 α) y B ( p , w ) (the budget set is convex). By the strict convexity of , α x + ( 1 α) y x , which is a contradiction of x being a maximal bundle in B ( p , w ) . The Consumer’s Problem with Differentiable Preferences When the preferences are differentiable, we are provided with a “use- ful” condition for characterizing the optimal solution. Claim: If x is an optimal bundle in the consumer problem and k is a con- sumed commodity (i.e., x k > 0), then it must be that v k ( x )/ p k v j ( x )/ p j for all other j , where v k ( x ) are the “subjective value num- bers” (see the definition of differentiable preferences in Lecture 4). Proof: Assume that x is a solution to the consumer’s problem B ( p , w ) and that x k > 0 and v k ( x )/ p k < v j ( x )/ p j (see fig. 5.1). A “move” in the
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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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