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

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E C O N 4 0 0 H a r t m a n Winter 2011 Solutions for Problem Set VIII 1.a. The Lagrangean for the problem is ( ) ab x yM p x s y  L where is the Lagrange multiplier. The first order conditions are 1 0 xa xy p  L/ , 1 0 yb x y s   , and 0 Mp xs y  . The first two of these conditions imply that 1 1 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 p xM a     so that * (,, ) () aM xp sM abp . 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 yp s . 1.b. *0 * ( )() atM aM x tp ts tM t x p s M abt p   and * ( btM bM y tp ts tM t y p s M s s 1.c.(i) First find the elasticities: * * * 1 xp px  , * * * 0 xs sx  , * * * 0 yp py , * * * 1 ys ys sy , * * * 1 xM Mx , and * * * 1 yM My . Substitute to see that *** 1010 xp xs xM     and 0110 yp ys yM       . (ii) From part a above we see that the expenditure shares are p M and sy b M . Use these and the elasticities to see that ** (1 ) ( 0 ) 0 xp yp px sy px a b a MMM a b a b a b  , (0) ( 1) 0 xs ys px sy sy a b b M M M ab     , and (1) 1 xM yM px sy a b M M   . 1.d. Substitute * x and * y into the first of the first order conditions and solve to see that 1 *1* * 11 ()() a b ab a x y a aM bM ab a b p s M pp a b p a b s        .
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This note was uploaded on 04/04/2011 for the course ECON 400 taught by Professor Ellis,g during the Spring '08 term at University of Washington.

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

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