lecture_05_handout

Problem set 3 due next tuesday 1 ces preferences over

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Unformatted text preview: t 3, due next Tuesday. 1 CES preferences over R2 are represented byu (x1 ; x2 ) = [ (x1 ) + 1 Show that CES preferences are homothetic. 2 Show that these preferences become linear when ! 1. (x2 ) ] 3 4 5 6 = 1, and Leontie¤ as Assume strictly positive consumption and show that these preferences become Cobb-Douglas as ! 1. Compute Walrasian demand when < 1. Verify that it converges to the Walrasian demand for Leontie¤ and Cobb-Douglas utility functions as ! 1 and ! 1 repectively. The elasticity of substitution between x1 and x2 is Prove that for CES preferences and Cobb-Douglas preferences? 1 ;2 = 1 1 . What is @ 1 ;2 1 ;2 = x1 (p ;w ) x (p ; w ) 2 p @ p1 2 p1 p2 x (p ;w ) 1 x (p ;w ) 2 for linear, Leontie¤, An Optimization Recipe (Adapted from Kreps) max f (x ) subject to How to solve 1 Write the Langrange function L : Rn gi (x ) 0 with i = 1; ::; m Rm ! R as m X L (x ; ) = f (x ) i gi (x ) i =1 2 Write the First Order Conditions: n1 z }| { rL (x ; ) = rf (x ) | 3 m P @ f (x ) @ xj i =1 {z m X @ g i (x ) i @ x =0 j i =1 i rgi (x ) = 0 for all j =1 ;::;n Write constraints, inequalities for , and complementary slackness conditions: gi (x ) 0 with i = 1; ::; m i 0 with i = 1; ::; m (x ) = 0 with i = 1; ::; m i gi 4 } Find the x and i that satisfy all these and you are done...hopefully. The Recipe: An Example 1 Compute Walrasian demand x (p ; w ) when the utility function is u (x1 ; x2 ) = x1 x2 other words, solve 1 max x1 x2 . In x1 ;x2 2fp 1 x1 +p 2 x2 w ; x1 0 , x2 0 g The Langrange function is 1 L (x ; ) = x1 x2 w (p1 x1 + p2 x2 w) ( 1 x1 ) ( 2 x2 ) The First Order Conditions for x : rL (x ; ) = | {z } 11 x2 x1 (1 w p1 +1 w p1 + 1 ) x1 x2 = u (x1 ;x2 ) w p1 + 1 x1 1 ) u (xx2;x2 ) w p1 + (1 21 Write the constraints, inequalities for , and complementary slackness p1 x1 + p2 x2 w w ( p 1 x1 + p 2 x2 w 0, 0, 1 x1 0, and w ) = 0, 1 x1 Find a solution to the above (easy for me to say). 0, and 2 x2 0 0 = 0, and 2 x2 =0 1 ! = 0 0 The Recipe at Work: An Example 1 To …nd the...
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