4Monotone Comparative Statics

4Monotone Comparative Statics - Monotone Comparative...

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Monotone Comparative Statics Let X and T be subsets of R and let f : X × T R . We consider how the set of maximizers of f ( x,t ) on X vary with the parameter t . Definition. f satisfies the strict single-crossing property (SSCP) in ( x,t ) if for every x 00 , x 0 in X and t 00 , t 0 in T , with x 00 > x 0 , and t 00 > t 0 f ( x 00 ,t 0 ) f ( x 0 ,t 0 ) implies f ( x 00 ,t 00 ) > f ( x 0 ,t 00 ) . The following simple Comparative Statics Theorem has many applications. Theorem (Easy Corollary to Shannon, 1995, Theorem 4) . Let f satisfy the strict single- crossing property (SSCP) in ( x,t ) and let t 1 , t 0 be any two points in T with t 1 > t 0 . If x i maximizes f ( x,t i ) on X for i = 0 , 1 , then x 1 x 0 . Note that the only assumption we made on f is the SSCP. X and T can be finite or infinite; if they are intervals, f need not be concave or differentiable or even continuous in x . Exercise 1. Prove the Comparative Statics Theorem. (Hint: just write down what it means for x i to solve the problem for
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This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.

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4Monotone Comparative Statics - Monotone Comparative...

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