ECON400HartmanWinter 2011 Solutions for Problem Set VI 1.a. Solve the constraint to get ymxand substitute for yin the objective function to see that we want to choose xto maximize or minimize 2()zxmx mxx. The first order condition is /20dz dxmx which implies that xm. Substitute this into the constraint to get ym. Since 220dz dx , and / 2gives a maximum. 1.b The Lagrangean for this problem is ()xxy L. The first order conditions are 0xyL/, 0yx, and 0mxy . The first two imply that xy, and we can use this and the constraint to see that and / 2. The bordered Hessian is 213101111111 01 1(1)1)11 2 0100 10xxxyxyxyyyLLLwhere the second equality follows by evaluating the determinant down the first column. It follows that and / 2gives a maximum.
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