200a_f08_ps_4_ak

200a_f08_ps_4_ak - University of California, Davis ARE/ECN...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: University of California, Davis ARE/ECN 200A Fall 2008 Joaquim Silvestre & Shaofeng Xu PROBLEM SET 4 ANSWER KEY The Constant Elasticity of Substitution (CES) Utility Function 1.1 u is homogeneous of degree &: Preferences are strictly convex as the function is strictly quasiconcave, and actually strictly concave if a & 1 : Preferences are strictly monotone, as the function is strictly increasing, i.e., it has strictly positive rst order derivatives. (see AK to Mathcamp PS6 for details) 1.2 For L = 2 ; the UMAX [ p; w ] problem is Max x 2 R 2 ++ & & 2 j =1 j x & j =& s:t: p x & w; given that the problem parameters are ( p; w ) 2 R 3 ++ , ( 1 ; 2 ; ; & ) 2 R 3 ++ and 2 ( 1 ; 0) [ (0 ; 1) : The utility function is clearly locally nonsatiated, and therefore WL is satised. Since the consumption set is restricted to R 2 ++ ; we do not have to worry about boundary solutions. Therefore, with the other given conditions, we have only one constraint, the budget constraint, which we can treat as an equality constraint. Consequently, we have the FOCs: & & & 2 j =1 j x & j ( & & ) =& r i x & & 1 i = p i ; i = 1 ; 2 p 1 x 1 + p 2 x 2 = w We could solve the Walrasian demand functions: ~ x i ( p; w ) = ( p i = i ) 1 = ( & & 1) ( p 1 ) &= ( & & 1) ( 1 ) 1 = (1 & & ) + ( p 2 ) &= ( & & 1) ( 2 ) 1 = (1 & & ) w; i = 1 ; 2 : Then for xed prices, each demand function can be written in the form ~ x i ( w ) = wK 1 i ; for constants K 1 1 and K 1 2 so that all Engel curves are straight lines emanating from the origin. If instead, wealth and the otherprice is xed, then each demand can be written as ~ x i ( p i ) = w= p i + ( p i ) 1 = (1 & & ) K 2 i : With < 1 ; then 1 = (1 ) > ; so that ( p i ) 1 = (1 & & ) is an increasing function on p i : Then clearly, demand is falling with higher price, and the demand curve does not intercept either axis. Note that this curve is always bound in by (to the left of and below) the curve x i = w=p i ; and hence in the limit must also asymptotically approach both axes. 1 1.3 The general UMAX [ p; w ] problem is Max x 2 R L ++ & & L j =1 & j x & j =& s:t: p & x w; given that the problem parameters are ( p; w ) 2 R L +1 ++ , ( & 1 ; :::; & L ; ) 2 R L +1 ++ and 2 ( 1 ; 0) [ (0 ; 1) : We have the following FOCs: & & L j =1 & j x & j ( & & ) =& r i x & & 1 i = p i ; i = 1 ; :::; L p & x = w We could solve the Walrasian demand functions and value function: ~ x i ( p; w ) = ( p i =& i ) 1 = ( & & 1) & L j =1 ( p j ) &= ( & & 1) & & j 1 = (1 & & ) w; i = 1 ; :::; L: v ( p; w ) = u (~ x ( p; w )) = & L i =1 & i ( p i =& i ) 1 = ( & & 1) & L j =1 ( p j ) &= ( & & 1) & & j 1 = (1 & & ) w !...
View Full Document

This note was uploaded on 07/01/2009 for the course ARE 200a taught by Professor Silvestre,j during the Winter '08 term at UC Davis.

Page1 / 8

200a_f08_ps_4_ak - University of California, Davis ARE/ECN...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online