Question

. Discuss about returns to scale of following production function. (a) f(x1, x2) = x a 1 +x a 2 b , where a and b are positive constant. (Hint: ab < 1, ab = 1 and ab > 1.) (b) f(x1, x2) = √ x1 + x 2 2 . (Hint: Does it satisfy the definition of increasing return to scale, constant returns to scale, or decreasing returns to scale. How can this be?)

Answer 1

1. F(x1,x2) = x1^ab + x2^ab

F(kx1,kx2) = k^ab(x1^ab + x2^ab) = k^ab(F(x1,x2))

For ab = 1, the production function has Constant Returns to Scale.

For ab < 1, the production function has Decreasing Returns to Scale.

For ab > 1, the production function has Increasing Returns to Scale.

This production function has Constant Returns to Scale.

2. F(x1,x2) = $\sqrt{x1}$ + x2

F(kx1,kx2) = $\sqrt{k}$ *$\sqrt{x1}$ + kx2 < k($\sqrt{x1}$ + x2) = F(x1,x2) for k > 1. Hence, for k > 1, this production function has decreasing returns to scale.