lecture_05_handout

If the geometry still works we can nd a more general

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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 (Cobb-Douglas) 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 Cobb-Douglas as ! 1. [Hint: use L’ Hopital Rule] Compute the Walrasian Demand when < 1. Verify that these converge to the Walrasian Demands for Leontie¤ and Cobb-Douglas 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 Cobb-Douglas 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.

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