A firm’s production function is given by the equation

Q = 12L.⁵K.⁵,

Where L, K, and Q are measured in thousands of units. Input prices are 36 per labor unit and 16 per capital unit.

a. Create a spreadsheet (based on the example shown) to model this production setting. (You may have already completed this step if you answered Problem S2 of Chapter 5.)

b. To explore the shape of short-run average cost, hold the amount of capital fixed at K = 9 thousand and vary the amount of labor from 1 thousand to 2.5 thousand to 4 thousand to 5.5 thousand to 7.5 thousand to 9 thousand units.

What is the resulting behavior of SAC? Use the spreadsheet optimizer to find the amount of labor corresponding to minimum SAC. What is the value of SAC_min?

c. In your spreadsheet, set L = 9 thousand (keeping K = 9 thousand) and note the resulting output and total cost. Now suppose that the firm is free to produce this same level of output by adjusting both labor and capital in the long run. Use the optimizer to determine the firm’s optimal inputs and LAC_min. (Remember to include an output constraint for cell I3.)

d. Confirm that the production function exhibits constant returns to scale and constant long-run average costs. For instance, recalculate the answer for part (c) after doubling both inputs.

e. Finally, suppose the firm’s inverse demand curve is given by

P = 9 − Q∕72.

With capital fixed at K = 9 (thousand) in the short run, use the optimizer to determine the firm’s optimal labor usage and maximum profit. Then find the optimal amounts of both inputs in the long run. Explain the large differences in inputs, output, and profit between the short run and the long run.

COST ANALYSIS Output36 Price8.50 8.00 Revenue306 Labor 1.00 MP 18.00 9.00 2.00 Capital MPk MR MRP MCK16.00 16.00 MRPL 144.00 MCL36.00 180 Ave Cos5.00 Cost 10 Profit 126 12 13