Question

1. Suppose that an economy’s production function is given by Y=10K^(1/3)L^(2/3). The firms in the economy are competitive and factors are paid their marginal products.

a. Show that the real wage is proportional to average labor productivity.

b. Show that the production function satisfies the Euler’s theorem.

Answer 1

a.

The production function is: Y = 10K^(1/3)L^(2/3)

The marginal product of labor = MPL = (∂Y/∂L) = 10*(2/3)*K^(1/3)L^((2/3)-1)

This implies,

MPL = (20/3)K^(1/3)L^(-1/3)

The average productivity of labor = APL = Y/L = 10K^(1/3)L^(2/3)/L = 10K^(1/3)L^(-1/3)

Thus, MPL/APL = (20/3)K^(1/3)L^(-1/3)/10K^(1/3)L^(-1/3) = 2/3

MPL = (2/3)*APL

Since the firm is perfect competition in product market, the condition Price =Marginal Revenue = Average Revenue holds. This implies, P = MR = AR

The factors such as labor are paid as per their marginal productivity

The equilibrium in the labor market is:

RealWage = Value of Marginal Product of Labor = Marginal Revenue Product of Labor = MPL*P = MPL*MR

Thus, RealWage = MPL*P

RealWage = (2/3)*APL*P

Since P is constant, it is observed that RealWage is directly proportional to the Average Productivity of Labor

----------------------------------------------------------------------------------------------------------------------------------------------------

b.

When F(L, K) is a production function then Euler's Theorem says that if factors of production are paid according to their marginal productivities the total factor payment is equal to the degree of homogeneity of the production function times output.

The given production function is: Y = 10K^(1/3)L^(2/3)

Let K'=xK; L'=xL

Y' = 10K^(1/3)L^(2/3)

=10(xK)^(1/3)(xL)^(2/3)

=x^(1/3 + 2/3) 10K^(1/3)L^(2/3)

=xY

Thus, the given production function is of homogeneity of degree 1

As per Euler's theorem, (∂F/∂L)L + (∂F/∂K)K = nF(L,K), where n= degree of homogeneity of the production function

This implies, MPL*L + MPK*K = Y, in the given case is to be shown to hold true

MPL*L + MPK*K

=(20/3)K^(1/3)L^(-1/3)*L + (10/3)K^(-2/3)L^(2/3)*K

=(20/3)K^(1/3)L^(2/3) + (10/3)K^(1/3)L^(2/3)

=10K^(1/3)L^(2/3) [ 2/3 + 1/3]

=10K^(1/3)L^(2/3)

=Y

Thus, MPL*L + MPK*K = Y holds implying the production function Y = 10K^(1/3)L^(2/3) satisfies the Euler’s theorem.