Question

Consider the Cobb-Douglas production function F (L, K) = (A)(L^α)(K^1/2) , where α > 0 and A > 0.
1. The Cobb-Douglas function can be either increasing, decreasing or constant returns to scale depending on the values of the exponents on L and K. Prove your answers to the following three cases.
(a) For what value(s) of α is F(L,K) decreasing returns to scale?

(b) For what value(s) of α is F(L,K) increasing returns to scale?

(c) For what value(s) of α is F(L,K) constant returns to scale?

Answer 1

the Cobb-Douglas production function F (L, K) = (A)(L^α)(K^1/2) , where α > 0 and A > 0.

1. The Cobb-Douglas function can be either increasing, decreasing or constant returns to scale depending on the values of the exponents on L and K.

(a)

It will show decreasing return to scale if the sum of (a+ b)<1, the b is 1/2 given, so the value of a must be less than 1/2, for making the Cobb Douglas production function decreasing.

(b)

It will show increasing return to scale if the sum of (a+ b)>1, the b is 1/2 given, so the value of a must be greater than 1/2, for making the Cobb Douglas production function increasing.

(c)

It will show constant return to scale if the sum of (a+ b)=1, the b is 1/2 given, so the value of a must be 1/2, for making the Cobb Douglas production function constant.