400ps6-11 - ECON 400 Winter 2011 Problem Set VI(for Monday...

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ECON 400 Hartman Winter 2011 Problem Set VI (for Monday, February 14) 1. Suppose the problem is to find the x and y that either maximize or minimize z xy subject to the constraint 0 m x y where m is a positive parameter. a. Solve this problem by the method of direct substitution. (The x and y you find will depend on m .) Use the second order condition to determine whether the x and y you have found give a maximum or a minimum. b. Solve this problem using a Lagrange multiplier. Use the second order condition to determine whether the x and y you have found give a maximum or a minimum. c. Let * ( ) x m and * ( ) y m denote the solution for x and y from part b, and let * ( ) m denote the Lagrange multiplier from part b. Substitute * ( ) x x m and * ( ) y y m into the expression to be maximized or minimized to get * * * ( ) ( ) ( ) z m x m y m . Show that * * ( ) dz m dm   . 2. Suppose the problem is to find the x and y that either maximize or minimize z x y
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