Lecture7_Duality_CompStatics

# Lecture7_Duality_CompStatics - Lecture 7 Indirect Utility...

This preview shows pages 1–8. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document

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.

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 ³ " # 1 + \$ # 2 ´ 1 %# &' ' ) 1 1 + ) 2 2 * = • S o lu t io n : 1 = * ) 1 Ã 1 + ³ " \$ ´ 1 # 1 ³ ) 2 ) 1 ´ # # 1 ! 2 = * ) 2 Ã 1 + ³ \$ " ´ 1 # 1 ³ ) 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 / "%\$ ) , Ã " \$ ! 1 # 1 Ã ) 2 ) 1 ! # # 1 1 *%) 1 — I f " \$ ) 2 ) 1 / 1 ( o r ) 1 %) 2 . "%\$ ) , Ã " \$ ! 1 # 1 Ã ) 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...
View Full Document

## This note was uploaded on 01/21/2012 for the course ECON 101a taught by Professor Staff during the Spring '08 term at Berkeley.

### Page1 / 21

Lecture7_Duality_CompStatics - Lecture 7 Indirect Utility...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online