276 PART • Producers, Consumers, and Competitive Markets This is the same result as (A7.5)—that is, the necessary condition for cost minimization The Cobb-Douglas Cost and Production Functions • Cobb-Douglas production function Production function of the form q ϭ AKa Lb, where q is the rate of output, K is the quantity of capital, and L is the quantity of labor, and where A, a, and b are positive constants Given a specific production function F(K, L), conditions (A7.13) and (A7.14) can be used to derive the cost function C(q) To understand this principle, let’s work through the example of a Cobb-Douglas production function This production function is F(K,L) = AK aLb where A, a, and b are positive constants We assume that a and b 1, so that the firm has decreasing marginal products of labor and capital.2 If a + b = 1, the firm has constant returns to scale, because doubling K and L doubles F If a + b 1, the firm has increasing returns to scale, and if a + b 1, it has decreasing returns to scale As an application, consider the carpet industry described in Example 6.4 (page 221) The production of both small and large firms can be described by CobbDouglas production functions For small firms, a = 77 and b = 23 Because a + b = 1, there are constant returns to scale For larger firms, however, a = 83 and b = 22 Thus a + b = 1.05, and there are increasing returns to scale The Cobb-Douglas production function is frequently encountered in economics and can be used to model many kinds of production We have already seen how it can accommodate differences in returns to scale It can also account for changes in technology or productivity through changes in the value of A: The larger the value of A, more can be produced for a given level of K and L To find the amounts of capital and labor that the firm should utilize to minimize the cost of producing an output q0, we first write the Lagrangian ⌽ = wL + rK - l(AK aLb - q0) (A7.15) Differentiating with respect to L, K, and l, and setting those derivatives equal to 0, we obtain 0⌽/0L = w - l(bAK aLb - 1) = (A7.16) a-1 b (A7.17) 0⌽/0l = AK L - q0 = (A7.18) 0⌽/0K = r - l(aAK L) = a b From equation (A7.16) we have l = w/AbK aLb - (A7.19) Substituting this formula into equation (A7.17) gives us rbAK aLb - = waAK a - 1Lb (A7.20) or L = br K aw (A7.21) For example, the marginal product of labor is given by MPL = 0[F(K,L)]/0L = bAK aLb - Thus, MPL falls as L increases