Consider the Production Function Y 25K13L23 a Calculate the

Consider the Production Function, Y = 25K1/3L2/3

(a) Calculate the marginal product of labor and capital

(b) Does this production function exhibit constant/increasing/decreasing returns to scale?

(c) If workers and the owners of capital are paid their marginal product, what are the equations for the real wage rate and the real rental rate of capital?

(d) What is the total income received by labor as a share of total output?

(e) What is the total income received by capital as a share of total output?

(f) What happens to your answers in part (d) and (e) if in the production function above 25 is replaced by 50?

Solution

Y = 25 K^1/3 L^2/3

We can write it as Y = 25K^1/3 L^2/3

(a) MP_K = dY/dK = 25(1/3)L^2/3K^-2/3 = 25/3L^2/3K^-2/3

      MP_K = dY/dL = 25(2/3)K^1/3L^-1/3 = 50/3K^1/3 L^-1/3

(b) It exhibits Constant Returns to Scale.

(c) The real wage and rental rates are equal to the marginal product of labor and capital, respectively. It is calculated in question (a)

(d) Total income received by labor = wL/Y = wL / 25 K^1/3 L^2/3

(e) Total income received by Capital = wL/Y = wK / 25 K^1/3 L^2/3

(f) Changing the productivity of technology has no impact on factor income shares.