Production in the country of Laboristan can be characterized

Production in the country of Laboristan can be characterized by the following function: y = 2K^02 L^08 and that Laboristan initially has a000 units of labour and 200 units of capital. Calculate the marginal product of labor and marginal product of capital for Laboristan. Show with a numerical example that Laboristan\'s production function exhibits diminishing marginal product of labor. Calculate the total payments of labor and capital in Laboristan. Suppose that in Laboristan there is a population boom so that the amount of labor increases. What happens to the real wage rate and real rental rate of capital in Laboristan? Briefly explain. Suppose that \"rich\" people in Laboristan own the capital (one unit of capital per \"rich\" person) and \"poor\" people in Laboristan are labor (one unit of labor per \"poor\" person). Briefly explain (no match necessary) what happens to inequality when the population boom occurs. In the production function for Laboristan briefly explain what the \"2\" represents.

Solution

Y = 2K0.2L0.8

(a)

MPL = dY / dL = 2 x 0.8 x (K / L)^0.2 = 1.6 x (K / L)^0.2

MPK = dY / dK = 2 x 0.2 x (L / K)^0.8 = 0.4 x (L / K)^0.8

(b)

Diminishing marginal product of labor states that as more of labor is used, MPL decreases.

Initial L = 1,000 & K = 200

Initial MPL = 1.6 x (K / L)^0.2 = 1.6 x (200 / 1,000)^0.2 = 1.6 x (0.2)^0.2 = 1.6 x 0.72 = 1.16

Let L increase by 1,000 and new level of L = 2,000

New MPL = 1.6 x (K / L)^0.2 = 1.6 x (200 / 2,000)^0.2 = 1.6 x (0.1)^0.2 = 1.6 x 0.63 = 1.01

So, as L has increased, MPL has declined.

(c)

Total cost, C = wL + rK = 1,000w + 200r

Also, in optimality, (MPL / MPK) = w/r

1.6 x (K / L)^0.2 / 0.4 x (L / K)^0.8 = w/r

4 x (K / L) = w/r

4 x 200 / 1,000 = w / r

w / r = 0.8

Total cost = 1,000w + 200r = 1,000 x 0.8r + 200r = 1,000r

[To find numerical value of total cost, we need value of either w or r].

(d)

As amount of labor increases, higher supply of labor lowers the wage rate. Amount of capital remaining unchanged, real rental rate increases relative to real wage rate.

NOTE: First 4 sub-questions are answered.