4Monotone Comparative Statics

# 4Monotone Comparative Statics - Monotone Comparative...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

4Monotone Comparative Statics - Monotone Comparative...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online