9.5 The general CobbDouglas production function for two inputs is given by q = f (k, l ) = Ak I , where 0 <

9.5 The general Cobb–Douglas production function for two inputs is given by q = f (k, l ) = Ak

αI

β, where 0 < α < 1 and 0 < β < 1. For this production function:

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

flk > 0.

b. Show that eq, k = α and eq, l = β.

c. Scale elasticity can be deined as eq, t =

�f (tk, tl )

�t

·

t f (tk, tl )

, where the expression is to be evaluated at t = 1.

Show that, for this Cobb–Douglas function, eq, t =

α + β. Hence in this case the scale elasticity and the returns to scale of the production function agree.

d. Show that this function is quasi-concave.

e. Show that the function is concave for α + β ≤ 1 but not concave for α + β > 1.