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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
CobbDouglas as ! 1.
Compute Walrasian demand when < 1. Verify that it converges to the Walrasian demand for Leontie¤ and
CobbDouglas utility functions as ! 1 and ! 1 repectively.
The elasticity of substitution between x1 and x2 is
Prove that for CES preferences
and CobbDouglas 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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 Fall '08
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