11.13 Cross-price effects in input demand With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.12.

a. Use steps (b), (d), and

(e) from Problem 11.12 to show that eK, w ¼ sLðr þ eQ, PÞ and eL, v ¼ sKðr þ eQ, PÞ:

b. Describe intuitively why input shares appear somewhat differently in the demand elasticities in part

(e) of Problem 11.12 than they do in part

(a) of this problem.

c. The expression computed in part

(a) can be easily generalized to the many-input case as exi, wj ¼ sjðAij þ eQ, PÞ, where Aij is the Allen elasticity of substitution defined in Problem 10.12. For reasons described in Problems 10.11 and 10.12, this approach to input demand in the multi-input case is generally inferior to using Morishima elasticities. One oddity might be mentioned, however. For the case i ¼ j this expression seems to say that eL, w ¼ sL (ALL þ eQ, P), and if we jumped to the conclusion that ALL ¼ s in the two-input case, then this would contradict the result from Problem 11.12. You can resolve this paradox by using the definitions from Problem 10.12 to show that, with two inputs, ALL ¼ (%sK/sL) Æ AKL ¼ (%sK/sL) Æ s and so there is no disagreement.