10.10 Input demand elasticities The own-price elasticities of contingent input demand for labor and capital are defined as el c , w ¼ @l c

@w (

w l

c , ekc

, v ¼ @kc

@v (

v kc :

a. Calculate el c,w and ekc, v for each of the cost functions shown in Example 10.2.

b. Show that, in general, el

c, w þ ekc, v ¼ 0.

c. Show that the cross-price derivatives of contingent demand functions are equal—that is, show that @l c

/@v ¼ @kc

/@w. Use this fact to show that slel

c, v ¼ skekc,w where sl

, sk are, respectively, the share of labor in total cost (wl/C) and of capital in total cost (vk/C).

d. Use the results from parts

(b) and

(c) to show that slel c,w þ skekc,w ¼ 0.

e. Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input demand.