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204PS62009

# 204PS62009 - Econ 204 Problem Set 6 Due Monday August 17th...

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Econ 204 Problem Set 6 Due Monday, August 17th Exercise 1 Consider f : R 2 ! R 2 such that f 2 C 3 ( R 2 ) . Now let F ( x; y; w; z ) = f ( x; y ) ° ( w; z ) and suppose that F ( x; y; w; z ) = 0 has solutions in R 4 : Let S ± R 4 be the set of solutions to this system. Show that there exists a set B such that B c has measure zero and for ( x; y; w; z ) 2 S where ( w; z ) 2 B , there is a local implicit function h : W ± R 2 ! R 2 ( W open) such that F ( h ( w; z ) ; w; z ) = 0 for all ( w; z ) 2 W and h 2 C 3 ( R 2 ) : Exercise 2 Let f : [0 ; 1] ! [0 ; 1] be a correspondence de°ned as f ( x ) = f 0 ; 1 = ( x + 1) g for x 6 = 0 and f (0) = f 1 = 2 g : Does f have a °xed point? If yes, °nd the point(s). Does any of the °xed point theorems you have learned apply here? Explain. Answer the same questions for f ( x ) = [ °; 1 = ( x +1)] for all x 2 [0 ; 1] where 0 < ° < 1 = 2 . Exercise 3 We say that a relation R on X is convex if whenever xRy and zRy then ( ±x + (1 ° ± ) z ) Ry for all ± 2 (0 ; 1) . (if x and y are in R , ² is an example of such relation). Let R i be a convex relation on R n for i = 1 ; 2 ; :::m , °x x 2 R n and let B i = f y ° x : yR i x; y 2 R n g .

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204PS62009 - Econ 204 Problem Set 6 Due Monday August 17th...

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