9.5 As we have seen in many places, the general Cobb–Douglas production function for two inputs is given by q ¼ fðk, lÞ ¼ Akal b, where 0 < a < 1 and 0 < b < 1. For this production function:

a. Show that fk > 0, f1 > 0, fkk < 0, fll < 0, and fkl ¼ flk > 0.

b. Show that eq, k ¼ a and eq, l ¼ b.

c. In footnote 5, we defined the scale elasticity as eq, t ¼ @fðtk, tlÞ

@t % t fðtk, tlÞ

, where the expression is to be evaluated at t ¼ 1. Show that, for this Cobb–Douglas function, eq,t ¼ a þ

b. Hence in this case the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9).

d. Show that this function is quasi-concave.

e. Show that the function is concave for a þ b ) 1 but not concave for a þ b > 1.