Rubinstein2005-page71

Rubinstein2005-page71 - Inferior x 1 Inferior x n that...

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October 21, 2005 12:18 master Sheet number 69 Page number 53 Demand: Consumer Choice 53 fnding an % optimal bundle is equivalent to solving the problem max x B ( p , w ) u ( x ) . Since the budget set is compact and u is continuous, the problem has a solution. To emphasize that a utility representation is not necessary For the current analysis, let us study a direct prooF oF the previous claim, avoiding the notion oF utility. Direct Proof: ±or any x B ( p , w ) defne the set Inferior ( x ) ={ y B ( p , w ) | x  y } .By the continuity oF the preFerences, every such set is open. Assume there is no solution to the consumer’s problem oF maximizing % on B ( p , w ) . Then, every z B ( p , w ) is a member oF some set Inferior ( x ) , that is, the collection oF sets { Inferior ( x ) | x X } covers B ( p , w ) . A col- lection oF open sets that covers a compact set has a fnite subset oF sets that covers it. Thus, there is a fnite collection
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Unformatted text preview: Inferior ( x 1 ) , . . . , Inferior ( x n ) that covers B ( p , w ) . Letting x j be the optimal bundle within the fnite set { x 1 , . . . , x n } , we obtain that x j is an optimal bun-dle in B ( p , w ) , a contradiction. Claim: IF % is convex, then the set oF solutions For a choice From B ( p , w ) (or any other convex set) is convex. Proof: IF both x and y maximize % given B ( p , w ) , then α x + ( 1 − α) y ∈ B ( p , w ) and, by the convexity oF the preFerences, α x + ( 1 − α) y % x % z For all z ∈ B ( p , w ) . Thus, α x + ( 1 − α) y is also a solution to the con-sumer’s problem. Claim: IF % is strictly convex, then every consumer’s problem has at most one solution....
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