Short Run Costs
1. A firm’s short-run cost function is C(q)=200q-6q^2+0.3q^3+400. Determine the fixed cost, F; the variable cost function, AVC; the average cost, AC; the marginal cost, MC; and the average fixed-cost, AFC. (Mathematical problem)
2. A firm’s cost curve is C=F+10q-bq^2+q^3, where b > 0.
a. For what values of b are cost, average cost, and average variable cost positive? (From now on, assume that all these measures of cost are positive at every output level).
b. What is the shape of the AC curve? At what output level is the AC minimized?
c. At what output levels do the MC curve cross the AC and the AVC curves?
d. Use calculus to show that MC curve must cross the AVC at its minimum point. (mathematical problem).
3. A firm has two plants that produce identical output. The cost function are C1=10q-4q^2+q^3 and C2=10q-2q^2+q^3.
a. At what output level does the average cost curve of each plant reach its minimum?
b. If the firm wants to produce four units of output, how much should it produce in each plant? (mathematical problem).
Long Run Costs
4. Replace the production function in Solved Problem 7.4 with a Cobb-Douglas q=AL^aK^b, and use calculus to find the cost minimizing capital-labor ratio. (mathematical problem).
Challenge
5. A US apparel manufacturer is considering moving its production abroad. Its production function is q=L^0.7K^0.3 (based on Hsieh, 1995). In the United States, w=7 and r=3. At its Asian plant, the firm will pay a 50% lower wage and a 50% higher cost of capital: w=7/1.5 and r=3X1.5. What are L and K, and what is the cost of producing q=100 units in both countries? What would be the cost of production in Asia if the firm had to use the same factor quantities as in the United States? (mathematical problem).