(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 302

CHAPTER 7 • The Cost of Production 277

(A7.21) is the expansion path. Now use Equation (A7.21) to substitute for L in
equation (A7.18):

AKa a

b
br
Kb - q0 = 0
aw

(A7.22)

We can rewrite the new equation as:
Ka + b = a

aw b q0
b
br A

(A7.23)

or
b

1

aw a + b q0 a + b
K = a
b a b

br
A

(A7.24)

(A7.24) is the factor demand for capital. We have now determined the cost-minimizing quantity of capital: Thus, if we wish to produce q0 units of output at
least cost, (A7.24) tells us how much capital we should employ as part of our
production plan. To determine the cost-minimizing quantity of labor, we simply
substitute equation (A7.24) into equation (A7.21):
b

1

br
br
aw a + b q0 a + b
L =
K =
£a
b a b §
aw
aw
br
A
a

L = a

(A7.25)

1

br a + b q0 a + b
b a b
aw
A

(A7.25) is the constrained factor demand for labor. Note that if the wage rate
w rises relative to the price of capital r, the firm will use more capital and less
labor. Suppose that, because of technological change, A increases (so the firm can
produce more output with the same inputs); in that case, both K and L will fall.
We have shown how cost-minimization subject to an output constraint can
be used to determine the firm’s optimal mix of capital and labor. Now we will
determine the firm’s cost function. The total cost of producing any output q can
be obtained by substituting equations (A7.24) for K and (A7.25) for L into the
equation C ϭ wL ϩ rK. After some algebraic manipulation we find that
a b/(a + b)
a -a/(a + b) q 1/(a + b)
§a b
C = w b/(a + b)r a/(a + b) £ a b
+ a b
(A7.26)
b
b
A
This cost function tells us (1) how the total cost of production increases as the
level of output q increases, and (2) how cost changes as input prices change.
When a + b equals 1, equation (A7.26) simplifies to
C = w br a[(a/b)b + (a/b)-a](1/A)q

(A7.27)