Question

# Suppose a firm’s production function is given by Q = L1/2*K1/2. The Marginal Product of Labor...

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?

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.

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