Question

The productivity of a South American country is given by the function when x units of labor and y units of capital are used. (a) What are the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 25 units and 10 units, respectively. (b) Should the government encourage capital investment rather than increased expenditure on labor at this time to increase the country's productivity?

Answer 1

Given f (x, y) = 20 x^3/4 y^1/4 labour = x units capital = y units f (x, y) = 20 middot 20 x^3/4 y^1/4 f_x = 20 (3/4) x^(3/4-1) y^1/4 = 15 x^-1/4 y^1/4 f_x = 15 (y/x)^1/4 f_y = 20 middot (1/4) x^3/4 y^(1/4-1) = 5 middot x^3/4 middot y^-3/4 f_y = 5 middot (x/y)^3/4 \r\n The required marginal productivity of labour (MPL) f_x (25, 10) = 15 (10/25)^1/4 = 0.795 times 15 = 11.929 11.929 units per unit increase in labour expenditure. The required marginal productivity of capital (MPC) f_y (2, 10) = 5 (25/10)^3/4 = 1.988 times 5 = 9.94 9.94 units per unit increase in capital expenditure No, (MPL > MPC). The govt. should not encorage capital investment rather than increased expenditure on labour at this time since we see that a unit increase in labour expenditure result in a much faster increase in Productivity that a unit increase in capital expenditure.