Question

Show that increasing returns to scale can co-exist with diminishing marginal productivity. To do so, provide an example of a production function with increasing returns to scale and diminishing marginal returns.

Answer 1

Let us consider the production function

Q = L^0.5K^0.6

When both inputs are doubled, new production function is

Q* = (2L)^0.5(2K)^0.6 = 2^0.5 x 2^0.6 x L^0.5K^0.6 = 2^1.1 x Q = 2.14

Q*/Q = 2.14 > 2

Since doubling both inputs more than doubles output, there is increasing return to scale.

Marginal product of labor (MPL) = Q/L = 0.5 x (K^0.6 / L^0.5)

As L increases, L^0.5 increases, so MPL decreases, reflecting diminishing marginal return of labor.

Marginal product of capital (MPK) = Q/L = 0.6 x (L^0.5 / K^0.4)

As K increases, K^0.4 increases, so MPK decreases, reflecting diminishing marginal return of capital.