400ps6-11-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 VI 1.a. Solve the constraint to get y mx  and substitute for y in the objective function to see that we want to choose x to maximize or minimize 2 () zxmx m xx  . The first order condition is /2 0 dz dx m x   which implies that xm . Substitute this into the constraint to get ym . Since 22 0 dz d x   , and / 2 gives a maximum. 1.b The Lagrangean for this problem is ( ) x x y    L . The first order conditions are 0 xy  L/ , 0 yx , and 0 mxy     . The first two imply that x y , and we can use this and the constraint to see that and / 2 . The bordered Hessian is 21 31 01 1 11 1 1 1 0 1 1 (1 ) 1 ) 11 2 0 10 0 1 0 xx xy x yx yy y    LLL where the second equality follows by evaluating the determinant down the first column. It follows that and / 2 gives a maximum.
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