(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 303

278 PART • Producers, Consumers, and Competitive Markets In this case, therefore, cost will increase proportionately with output As a result, the production process exhibits constant returns to scale Likewise, if a + b is greater than 1, there are increasing returns to scale; if a + b is less than 1, there are decreasing returns to scale The firm’s cost function contains many desirable features To appreciate this fact, consider the special constant returns to scale cost function (A7.27) Suppose that we wish to produce q0 in output but are faced with a doubling of the wage How should we expect our costs to change? New costs are given by C1 = 2w b £ a a b a -a a b a -a b + a b § a b q0 = 2b wb £ a b + a b § a b q0 = 2bC0 b b A b b A (+++1++)+++1++* C0 Recall that at the beginning of this section, we assumed that a < and ß < Therefore, C1 2C0 Even though wages doubled, the cost of producing q0 less than doubled This is the expected result If a firm suddenly had to pay more for labor, it would substitute away from labor and employ more of the relatively cheaper capital, thereby keeping the increase in total cost in check Now consider the dual problem of maximizing the output that can be produced with the expenditure of C0 dollars We leave it to you to work through this problem for the Cobb-Douglas production function You should be able to show that equations (A7.24) and (A7.25) describe the cost-minimizing input choices To get you started, note that the Lagrangian for this dual problem is ⌽ = AKaLb - o(wL + rK - C0) EXERCISES Of the following production functions, which exhibit increasing, constant, or decreasing returns to scale? a F(K, L) ϭ K2L b F(K, L) ϭ 10K ϩ 5L c F(K, L) ϭ (KL).5 The production function for a product is given by q ϭ 100KL If the price of capital is $120 per day and the price of labor $30 per day, what is the minimum cost of producing 1000 units of output? Suppose a production function is given by F(K, L) ϭ KL2; the price of capital is $10 and the price of labor $15 What combination of labor and capital minimizes the cost of producing any given output? Suppose the process of producing lightweight parkas by Polly’s Parkas is described by the function q = 10K 8(L - 40).2 where q is the number of parkas produced, K the number of computerized stitching-machine hours, and L the number of person-hours of labor In addition to capital and labor, $10 worth of raw materials is used in the production of each parka a By minimizing cost subject to the production function, derive the cost-minimizing demands for K and L as a function of output (q), wage rates (w), and rental rates on machines (r) Use these results to derive the total cost function: that is, costs as a function of q, r, w, and the constant $10 per unit materials cost b This process requires skilled workers, who earn $32 per hour The rental rate on the machines used in the process is $64 per hour At these factor prices, what are total costs as a function of q? Does this technology exhibit decreasing, constant, or increasing returns to scale? c Polly’s Parkas plans to produce 2000 parkas per week At the factor prices given above, how many workers should the firm hire (at 40 hours per week) and how many machines should it rent (at 40 machine-hours per week)? What are the marginal and average costs at this level of production?