Lecture7_Duality_CompStatics

Lecture7_Duality_CompStatics - Lecture 7: Indirect Utility,...

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Unformatted text preview: Lecture 7: Indirect Utility, Duality, and Comparative Statics Econ 101A: Microeconomic Theory UC Berkeley ring 2011 Spring 2011 Prof. Cristian Santesteban 1 U t i l i t y m a x im i z a t io n t r i c k y c a s e s F i r s t , r e- s o l v e C E S u t i l i t y f u n c t io n . m a x 1 ! 2 " # 1 + $ # 2 1 %# &' ' ) 1 1 + ) 2 2 * = S o lu t io n : 1 = * ) 1 1 + " $ 1 # 1 ) 2 ) 1 # # 1 ! 2 = * ) 2 1 + $ " 1 # 1 ) 1 ) 2 # # 1 ! S p e c ia l c a s e 1 : # 1 ( P e r f e c t S u b s t i t u t e s ) ,- # 1 1 # 1 = ,- # 1 # # 1 = ( h e r e n o t i c e t h e c o n v e r g e n c e f r o m t h e l e f t ) I f " $ ) 2 ) 1 . 1 ( o r ) 1 %) 2 / "%$ ) , " $ ! 1 # 1 ) 2 ) 1 ! # # 1 1 *%) 1 I f " $ ) 2 ) 1 / 1 ( o r ) 1 %) 2 . "%$ ) , " $ ! 1 # 1 ) 2 ) 1 ! # # 1 1 S o lu t io n f o r P e r f e c t S u b s t i t u t e s C a s e i s 1 = i f ) 1 %) 2 . "%$ *%) 1 i f ) 1 %) 2 / "%$ a n y 1 [ ! *%) 1 ] i f ) 1 %) 2 = "%$ 2 = *%) 2 i f ) 1 %) 2 . "%$ i f ) 1 %) 2 / "%$ 2 s u c h t h a t B .C . h o ld s i f ) 1 %) 2 = "%$ C a s e ) 1 %) 2 = "%$ h a s t o b e a n a l y z e d s e p a r a t e l y T h i s i s c a s e in w h i c h b u d g e t l in e a n d in d i e r e n c e c u r v e s a r e p a r a l l e l . A l l p o in t s o n b u d g e t l in e a r e t a n g e n t a n d h e n c e o p t im a l . S p e c ia l c a s e 1 : # = ( C o b b-D o u g la s ) 1 = " " + $ * ) 1 2 = $ " + $ * ) 2 S p e c ia l c a s e 1 : # 1 ( P e r f e c t S u b s t i t u t e s ) 1 = ( i f ) 1 %) 2 "%$ *%) 1 i f ) 1 %) 2 / "%$ 2 = ( *%) 2 i f ) 1 %) 2 "%$ i f ) 1 %) 2 / "%$ S p e c ia l c a s e 1 : # ( P e r f e c t C o m p l e m e n t s ) 1 = * ) 1 + ) 2 = 2 P a r a m e t e r # in d i c a t e s s u b s t i t io n p a t t e r n b e t w e e n g o o d s : # . . G o o d s a r e ( n e t ) s u b s t i t u t e s # / . G o o d s a r e ( n e t ) c o m p l e m e n t s U ( x 1 , x 2 ) = ( x 2 1 + x 2 2 ) 1 / 4 U x 1 ( x 1 , x 2 ) = ( 1 / 2 ) ( x 2 1 + x 2 2 ) 3 / 4 U x 1 x 1 ( x 1 , x 2 ) = ( 3 / 4 ) ( x 2 1 + x 2 2 ) 7 / 4 < d e c r e a s in g m a r g in a l u t i l i t y b u t t h e in d iff e r e n c e c u r v e s a r e c i r c l e s , w h i c h e x h ib i t in c r e a s in g M R S . U ( x 1 , x 2 ) = x 3 1 x 3 2 U x 1 ( x 1 , x 2 ) = 3 x 2 1 x 3 2 U x 1 x 1 ( x 1 , x 2 ) = 6 x 1 x 3 2 > in c r e a s in g m a r g in a l u t i l i t y b u t t h e in d iff e r e n c e c u r v e s a r e h y p e r b o l a s , w h i c h e x h ib i t D M R S . 2 . 3 C o n s u m e r s O p t im u m A n a l y t i c a l l y , t h e c o n s u m e r s p r o b l e m i s t o s o l v e m a x x 1 , x 2 U ( x 1 , x 2 ) s . t . p 1 x 1 + p 2 x 2 = I . H a v e a l o o k a t F i g u r e 1 5 . C l e a r l y , a b u n d l e ( x 1 , x 2 ) i s o p t im a l i f t w o t h in g s a r e t r u e : F i g u r e 1 5 : T h e c o n s u m e r c h o o s e s t h e b u n d l e t h a t l a n d s h e r o n...
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Lecture7_Duality_CompStatics - Lecture 7: Indirect Utility,...

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