(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 301

276 PART 2 • 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 6 1 and b 6 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 7 1, the firm has increasing returns
to scale, and if a + b 6 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) = 0

(A7.16)

a-1 b

(A7.17)

0⌽/0l = AK L - q0 = 0

(A7.18)

0⌽/0K = r - l(aAK

L) = 0

a b

From equation (A7.16) we have
l = w/AbK aLb - 1

(A7.19)

Substituting this formula into equation (A7.17) gives us
rbAK aLb - 1 = waAK a - 1Lb

(A7.20)

or
L =

br
K
aw

(A7.21)

2
For example, the marginal product of labor is given by MPL = 0[F(K,L)]/0L = bAK aLb - 1. Thus,
MPL falls as L increases.