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 /α l for all k and l and thus x ∗ k = α k w / p k for all k . Example 4:
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