Question

Is there any relationship between returns to scale and economies of scale? Assume a production function q = 100(K^0.7*L^0.3), where K is capital and L is labor. Derive the marginal product of labor and the marginal product of capital. Show that the marginal product of labor is decreasing (hint: beginning with K = 2, and L = 50)

Answer 1

Yes, there exists a relationship between increasing returns to scale and economies of scale. Economies of scale for a firm indicate output produced at a larger scale at a lower per unit cost of production. Increasing returns to scale means when input involved in production process increase by some percentage then output rises at more than that percentage increase in inputs of production. When more output is produced with fewer units of inputs involved in production process, cost of input is also smaller which imply increasing returns to scale is same as economies of scale.

Given : production function

Marginal product of labor is given by  

Marginal product of capital is given by  

In order to show marginal product of labor is decreasing take marginal product of labor---------------

When K=2 and L=50

marginal product of labor when K=2 and L=50 is

When K remains same i.e. 2 and L increases by 1 unit i.e. L=51 then marginal product of labor is -----------------------

When K remains same i.e. 2 and L increases by 1 more unit i.e. L=52 then marginal product of labor is -----------------------

It is clear that when L increases then marginal product of labor is decreasing.

Other way to show marginal product of labor is decreasing is by taking partial derivative of marginal product of labor with respect to L i.e. . Since the partial derivative of marginal product of labor with respect to L is negative this implies marginal product of labor is decreasing.