400ps8-11-soln - ECON 400 Winter 2011 Solutions for Problem...

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ECON 400 Hartman Winter 2011 Solutions for Problem Set VIII 1.a. The Lagrangean for the problem is ( ) a b x y M px sy   L where is the Lagrange multiplier. The first order conditions are 1 0 a b x ax y p   L / , 1 0 a b y bx y s    L / , and 0 M px sy   L / . The first two of these conditions imply that 1 1 a b a b ax y p bx y s which simplifies to bpx asy or bpx sy a . Substitute this into the budget constraint to see that 1 b px M a so that * ( , , ) ( ) aM x p s M a b p . Similarly, the first two first order conditions imply that asy px b which can be substituted into the budget constraint to give 1 a sy M b so that * ( , , ) ( ) bM y p s M a b s . 1.b. * 0 * ( ) ( , , ) ( , , ) ( )( ) ( ) a tM aM x tp ts tM t x p s M a b tp a b p and * 0 * ( ) ( , , ) ( , , ) ( )( ) ( ) b tM bM y tp ts tM t y p s M a b ts a b s 1.c.(i) First find the elasticities: * * * 1 xp x p p x   , * * * 0 xs x s s x , * * * 0 yp y p p y , * * * 1 ys y s s y   , * * * 1 xM x M M x , and * * * 1 yM y M M y . Substitute to see that * * * 1 0 1 0 xp xs xM        and * * * 0 1 1 0 yp ys yM       . (ii) From part a above we see that the expenditure shares are px a M a b and sy b M a b . Use these and the elasticities to see that * * ( 1) (0) 0 xp yp px sy px a b a M M M a b a b a b , * * (0) ( 1) 0 xs ys px sy sy a b b M M M a b a b a b , and * * (1) (1) 1 xM yM px sy a b M M a b a b . 1.d. Substitute * ( , , ) x x p s M and * ( , , ) y y p s M into the first of the first order conditions and solve to see that 1 * 1 * * 1 1 ( ) ( ) ( ) ( ) ( ) a b a b a b a b a b a b a x y a aM bM a b a b p s M p p a b p a b s      
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  • Fall '08
  • Ellis,G
  • Utility, Utility maximization problem, Hicksian demand function, Expenditure minimization problem, b

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