Question

Given the Cobb-Douglas production function q = 2K 1 4 L 3 4 , the marginal product of labor is: 3 2K 1 4 L 1 4 and the marginal product of capital is: 1 2K 3 4 L 3 4 .

A) What is the marginal rate of technical substitution (RTS)?

B) If the rental rate of capital (v) is $10 and the wage rate (w) is $30 what is the necessary condition for cost-minimization? (Your answer should be in the form of an equation in K and L)

C) What is the cost-minimizing input combination for producing 100 units of output?

D) What is the total cost of producing 100 units of output? What portion of total cost is spent on labor and what portion is spent on capital?

Answer 1

A) MRS = MP of labor / MP of capital = 3/2K^1/4L^-1/4 / 1/2K^-3/4L^3/4

MRS = 3K/L

B) Condition for cost minimization -

MRS = w/v

3K/L = 30/10

K = L

The above is the condition of cost minimization

C) For producing 100 units-

100 = 2K^1/4 L^3/4

50/K^1/4 = L^3/4

and from part B we know that K =L

thus

50/L^1/4 = L^3/4

50 = L

and hence K = 50

D) Total Cost of producing 100 units of output -

TC = wL +vK

TC = 30(50) + 10(50) = 2000

Portion spent on labor = 1500/2000 = 75%

and thus portion spent on capital = 25%