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200a_f08_ps_4_ak

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

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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 satis¡ed. 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 ¢other¢price 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 !...
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