400ps6-11-soln - ECON 400 Winter 2011 Solutions for Problem...

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ECON 400 Hartman Winter 2011 Solutions for Problem Set VI 1.a. Solve the constraint to get y m x and substitute for y in the objective function to see that we want to choose x to maximize or minimize 2 ( ) z x m x mx x . The first order condition is / 2 0 dz dx m x which implies that / 2 x m . Substitute this into the constraint to get / 2 y m . Since 2 2 / 2 0 d z dx   , / 2 x m and / 2 y m gives a maximum. 1.b The Lagrangean for this problem is ( ) xy m x y   L . The first order conditions are 0 x y    L / , 0 y x    L / , and 0 m x y   L / . The first two imply that x y , and we can use this and the constraint to see that / 2 x m and / 2 y m . The bordered Hessian is 2 1 3 1 0 1 1 1 1 1 1 1 0 1 1( 1) 1( 1) 1 1 2 0 1 0 0 1 1 1 0 xx xy x yx yy y x x     L L L L L L L L L where the second equality follows by evaluating the determinant down the first column.
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  • Fall '08
  • Ellis,G
  • Optimization, objective function, Fermat's theorem, 0 l

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