hw05a_2008 - UNC Wilmington Department of Economics and...

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UNC Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Homework 5 Solutions 1) max U = 50 + 30S – (1/4)S 2 + 300G – (1/10)G 2 S, G subject to: no constraints F.O.C.'s: (1) 0 S ) 4 / 1 ( 2 30 S U = - = (2) 0 G ) 10 / 1 ( 2 300 G U = - = Solve FOC (1) for S: S * = 60 Solve FOC (2) for G: G * = 1500 2) max U = 4D 1/4 ·J 1/3 D,J subject to: $0.20D + $4J ≤ $40 Converting the problem into an equivalent Lagrangian problem: max L = 4D 1/4 ·J 1/3 + λ (40 – 0.2D – 4J) D,J, λ subject to: nothing F.O.C.'s: (1) 0 2 . 0 J D 4 ) 4 / 1 ( D L 3 / 1 4 / 3 = λ - = - (2) 0 4 J D 4 ) 3 / 1 ( J L 3 / 2 4 / 1 = λ - = - (3) 0 J 4 D 2 . 0 40 L = - - = λ Combine FOC's (1) and (2) to eliminate λ : λ λ = - - 4 2 . 0 J D 4 ) 3 / 1 ( J D 4 ) 4 / 1 ( 3 / 2 4 / 1 3 / 1 4 / 3 4 2 . 0 D 4 J 3 = J = (0.067)D call this equation (4) Substitute equation (4) back into FOC (3) and solve for D: 40 - 0.2D - 4J = 0 40 - 0.2D - 4(0.067D) = 0 D * = 40/0.468= 85.47 1 The FOC equations provide us with three equations in three unknowns, D, J and λ . Solve these three equations for the solution values of D, S and λ . One way is illustrated below:
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UNC Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Substitute D * back into equation (4) to find J * : J = (0.067)D * J = (0.067)(85.47) J * = 5.73 Substitute D * and J *
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hw05a_2008 - UNC Wilmington Department of Economics and...

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