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
homogencous good, \(Q,\) under constant returns to scale using only capital and
labor. The demand function for \(Q\) is given by \(Q=D(P)\), where \(P\) is the
market price of the good being produced. Because of the constant returns-to-
scale assumption, \(P=M C=A C\). 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
\(K=Q C_{v}\) and \(L=Q C_{w}\)
b. Show that
\\[
\frac{\partial K}{\partial v}=Q C_{v v}+D^{\prime} C_{v}^{2} \quad \text { and
} \quad \frac{\partial L}{\partial w}=Q C_{w w}+D^{\prime} C_{w}^{2}
\\]
c. Prove that
\\[
C_{w v}=\frac{-w}{v} C_{v w} \quad \text { and } \quad C_{w w}=\frac{-v}{w}
C_{N w}
\\]
d. Use the results from parts (b) and (c) together with the elasticity of
substitution defined as \(\sigma=C C_{v n} / C_{\nu} C_{w}\) to show that
\\[
\frac{\partial K}{\partial v}=\frac{w L}{Q} \cdot \frac{\sigma K}{v
C}+\frac{D^{\prime} K^{2}}{Q^{2}} \text { and } \frac{\partial L}{\partial
w}=\frac{v K}{Q} \cdot \frac{\sigma L}{w C}+\frac{D^{\prime} L^{2}}{Q^{2}}
\\]
e. Convert the derivatives in part (d) into elasticities to show that
\\[
e_{K, v}=-s_{L} \sigma+s_{K} e_{Q, p} \quad \text { and } \quad e_{L,
w}=-s_{K} \sigma+s_{L} e_{Q, P}
\\]
where \(e_{Q, P}\) 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).
