Therefore assuming that both constraints are binding

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Therefore, assuming that both constraints are binding, then Q 1 = Q 2 = Q and the Kuhn-Tucker condi- tions become λ 1 + λ 2 = 8 22 2 × 10 5 Q = 6 + λ 1 18 2 × 10 5 Q = 6 + λ 2 which yields the following solutions Q = K = 50000 λ 1 = 6 λ 2 = 2 P 1 = 17 P 2 = 13 Since the capacity constraint is binding in both markets, market one pays λ 1 = 6 of the capacity cost and market two pays λ 2 = 2 . 6
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Problems 1. Suppose in the above example a unit of capacity cost only 3 cents per day. (a) What would be the pro fi t maximizing peak and o ff -peak prices and quantitites? (b) What would be the values of the Lagrange multipliers? What interpretation do you put on their values? 2. Skippy lives on an island where she produces two goods, x and y, according the the production possi- bility frontier 200 x 2 + y 2 , and she consumes all the goods herself. Her utility function is u = x · y 3 Skippy also faces and environmental constraint on her total output of both goods. The environmental constraint is given by x + y 20 (a) Write down the Kuhn Tucker fi rst order conditions. (b) Find Skippy’s optimal x and y. Identify which constaints are binding. 3. An electric company is setting up a power plant in a foreign country and it has to plan its capacity. The peak period demand for power is given by p 1 = 400 q 1 and the o ff -peak is given by p 2 = 380 q 2 . The variable cost to is 20 per unit (paid in both markets) and capacity costs 10 per unit which is only paid once and is used in both periods. (a) write down the lagrangian and Kuhn-Tucker conditions for this problem (b) Find the optimal outputs and capacity for this problem. (c) How much of the capacity is paid for by each market (i.e. what are the values of λ 1 and λ 2 )? (d) Now suppose capacity cost is 30 per unit (paid only once). Find quantities, capacity and how much of the capacity is paid for by each market (i.e. λ 1 and λ 2 )? 7
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  • Fall '11
  • WendyWu
  • Economics, Utility, Constraint, lagrange multipliers, Ly, Zy, Kuhn-Tucker conditions

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