Rubinstein2005-page78

Rubinstein2005-page78 - October 21, 2005 12:18 60 master...

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October 21, 2005 12:18 master Sheet number 76 Page number 60 60 Lecture Five Example 3: Consider a consumer who spends α k of his wealth on commodity k (where α k 0 and 6 K k = 1 α k = 1). This rule of behavior is not formu- lated as a maximization of some preference relation. It can how- ever be fully rationalized by the preference relation represented by the Cobb-Douglas utility function u ( x ) = 5 K k = 1 x α k k . A solution x to the consumer’s problem B ( p , w ) must satisfy x k > 0 for all k (notice that u ( x ) = 0 when x k = 0 for some k ). Given the differentiability of the preferences, a necessary condition for the optimality of x is that v k ( x )/ p k = v l ( x )/ p l for all k and l where v k ( x ) = du / dx k ( x ) = α k u ( x )/ x k for all k . It follows that p k x k / p l x l = α k l for all k and l and thus x k = α k w / p k for all k . Example 4:
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