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

278 PART 2 • 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 =

1 2w 2 b ra £ a

a b
a -a
1
a b
a -a
1
b + a b § a b q0 = 2b wb ra £ 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 < 1 and ß < 1.

Therefore, C1 6 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
1. 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
2. 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?
3. 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?
4. 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?