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Unformatted text preview: ive function is not quasi concave: must check second order
conditions.
these have to do with matrix of second derivatives;
look at Section M.K in GMW for details (boring and mechanical, but one needs
to know them);
see also Varian for a quick and dirty good guide. Summary If x solves max f (x ) subject to gi (x ) 0 with i = 1; ::; m then rL (x ; ) = rf (x ) m
X
i =1 i rgi (x ) = 0 with
i 0 and i gi (x ) = 0 for each i and
2
Dx L (x ; ) = D 2 f (x ) m
X
i =1 i rgi (x ) is negative semide…nite on the subspace fz 2 Rn : rgi (x ) z = 08i = 1; ::; mg...
...provided constraint quali…cation holds. Problem Set 3, (incomplete)
Due Tuesday, 18 September, at the beginning of class
1 Consider a setting with 2 goods. For each of the following utility functions, …nd the Walrasian demand correspondence. (Hint: draw pictures)
1
2
3
4 2 u (x ) = x1 x2 for ; > 0 (CobbDouglas)
u (x ) = minf x1 ; x2 g for ; > 0 (generalized Leontief)
u (x ) = x1 + x2 for ; > 0 (linear)
2
2
u (x ) = x 1 + x 2 The CES preferences over R2 are represented by the utility function (x1 ; x2 ) = [ (x1 ) +
1
2
3
4
5 x (p ;w ) 6 The elasticity of substitution between x1 and x2 is 1 ;2 = @ x1 (p ;w )
2
p @ p1
2 Prove that for CES preferences
3 1 (x2 ) ] Show that CES preferences are homothetic.
Show that these preferences become linear when = 1, and Leontie¤ as ! 1.
Assume strictly positive consumption and show that these preferences become CobbDouglas as ! 1. [Hint: use L’
Hopital Rule]
Compute the Walrasian Demand when < 1.
Verify that these converge to the Walrasian Demands for Leontie¤ and CobbDouglas utility functions as ! 1 and ! 1 repectively. 1 ;2 = 1
1 . What is 1 ;2 p1
p2
x (p ;w )
1
x (p ;w )
2 for linear, Leontie¤, and CobbDouglas preferences? (Properties of Di¤erentiable Walrasian Demand) Assume that the Walrasian demand x (p ; w ) is a di¤erentiable function.
1 2 Pn @ xk
@ xk
Show that for each (p ; w ),
for k = 1; : : : ; n
i =1 p i @ p i (p ; w ) + w @ w (p ; w ) = 0
Hint: x is homogeneous of degree 0 in (p ; w ). What does this imply?
(This is a special case of Euler’ formula for functions that are homogeneous of degree r . See GMW M.B for more.)
s
Suppose in addition that % is locally nonsatiated. Show that for each (p ; w ),
n
X
k =1 pk @ xk
(p ; w ) + x i (p ; w ) = 0
@ pi for i = 1; : : : ; n and n
X
k =1 pk @ xk
(p ; w ) = 1
@w Give a simple intuitive (short) version of these two results (sometimes called Cournot and Engel aggregation, respectively) in words....
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This note was uploaded on 05/13/2013 for the course ECON 2100 taught by Professor Board,o during the Fall '08 term at Pittsburgh.
 Fall '08
 Board,O

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