More on the derived demand with two inputs The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry’s demand for inputs.

To do so, we assume that an industry produces a homogeneous good, Q, under constant returns to scale using only capital and labor. The demand function for Q is given by Q 5 D 1P2, where P is the market price of the good being produced. Because of the constant returns-to-scale assumption, P 5 MC 5 AC. Throughout this problem let C(v, w, 1) be the firm’s unit cost function.

a. Explain why the total industry demands for capital and labor are given by k 5 QCv and l 5 QCw.

b. Show that ∂k ∂v 5 QCvv 1 DrC2 v and ∂l ∂w 5 QCww 1 DrC2 w.

c. Prove that Cvv 5 2w v Cvw and Cww 5 2v w Cvw.

d. Use the results from parts

(b) and

(c) together with the elasticity of substitution defined σ 5 CCvw/CvCw to show that ∂k ∂v 5 wl Q # σk vC 1 Drk2 Q2 and ∂l ∂w 5 vk Q # σl wC 1 Drl 2 Q2 .

e. Convert the derivatives in part

(d) into elasticities to show that ek,v 5 2slσ 1 skeQ,P and el,w 5 2skσ 1 sleQ,P, where eQ,P is the price elasticity of demand for the product being produced.

f. Discuss the importance of the results in part

(e) using the notions of substitution and output effects from Chapter 11.
Note: The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was first suggested by Alfred Marshall.
The proof given here follows that in D. Hamermesh, Labor Demand (Princeton, NJ: Princeton University Press, 1993).