Question

Suppose a firm’s production function is given by Q = L1/2*K1/2. The Marginal Product of Labor and the Marginal Product of Capital are given by:

MPL = (K^1/2)/2L^1/2 & MPK = (L^1/2)/2K^1/2)

a) (12 points) If the price of labor is w = 48, and the price of capital is r = 12, how much labor and capital should the firm hire in order to minimize the cost of production if the firm wants to produce output Q = 10?

b) (12 points) What is the firm’s Total Cost function TC(Q)?c) (6 points) What is the firm’s marginal cost of production?

c) (6 points) What is the firm's marginal cost of production?

Answer 1

We have MPL = (K^1/2)/2L^1/2 & MPK = (L^1/2)/2K^1/2). This gives MRTS = K/L. Wage rental ratio is given by w/r. At the optimum input mix, MRTS = w/r

K/L = w/r which then becomes K = wL/r

The production function is now Q = L^(1/2) (wL/r)^(1/2)

Q = L * (w/r)^1/2

Labor demand L* = Q*(r/w)^(1/2) and capital demand K* = Q*(w/r)^(1/2)

Cost function is C = wL + rK

C = wQ(r/w)^(1/2) + rQ(w/r)^(1/2)

C = Q(wr)^(1/2) + Q(wr)^(1/2)

C = 2Q(wr)^(1/2)

Marginal cost MC = 2(wr)^(1/2)

a) Labor demand = 10*(12/48)^(1/2) = 5 units and capital demand = 10*(48/12)^(1/2) = 20

b) Cost of production when Q = 10, w = 48 and r = 12 is C = 2*10(48*12)^(1/2) = 480

c) Marginal cost = 2*(48*12)^(1/2) = 48.