Question

Suppose a firm has the following production function: Q = 2K L. The marginal product of capital is 2L, and the marginal product of labor is 2K. Assume that capital rents for $100 per unit per day, that labor can be hired for $200 per unit per day, and that the firm is minimizing costs. a. The firm will hire units of labor and - units of capital. b. What is the total cost of producing 60 units of output? TC60 units: $ c. Suppose you want to find the cost of producing a more general q. Using the same procedures you followed in parts a and b, but using the more general q instead of 60, find this firm's total cost function, which expresses total cost as a function of q. Begin by determining the quantities of labor and capital hired. The firm will hire units of labor and units of capital. d. What is the total cost of producing q units of output? TC = e. Divide the total cost function you found in part c by q to find this firm's average cost function. ATC = TE = 0

Answer 1

a) Equilibrium condition :

MPL/MPK = wage rate / capital rental rate

2L/2K = 200/100

L = 2K

Labor and capital will be hired in this ratio

If Q = 60

K = 3.87

L = 7.74

b ) Q = 60

Q = 2LK = 2 (2K) K = 4K2 = 60
K = (15)0.5

L = 2(15)0.5

Cost = 100K + 200L = 100(15)0.5​​​​​​​ + 200(2)(15)0.5​​​​​​​ = 500(15)0.5​​​​​​​ = 1936.5

c)  

Q = 4K2

K = (Q/4)0.5  = Q0.5/2 = 0.5Q0.5

L = 2K = 2 x 0.5Q0.5  = Q0.5

d) Total Cost = 100K + 200L = 100(0.5Q0.5) + 200(Q0.5) = 250Q0.5

e ) ATC = TC/Q = 250/Q0.5​​​​​​​