Suppose a monopolist has a production function given by Q = L 1/2 K 1/2 . Theref

Question

Suppose a monopolist has a production function given by Q = L1/2K1/2. Therefore,MPL= K^1/2 /2L^1/2 and MPk= L^1/2/2K^ 1/2.

The monopolist can purchase labor, L at a price w = 16, and capital, K at a price of r = 9. The demand curve facing the monopolist is P = 288 – 2Q.

a) (8 points) What is the monopolist’s total cost function?

b) (4 points) How much output should the monopolist produce in order to maximize profit?

c) (6 points) How much labor should the firm hire to produce this output?

d) (4 points) How Much Capital should the firm hire?

e) (4 points) What price should the monopolist charge?

f) (4 points) What is the deadweight loss?

g) (4 points) What is the Price Elasticity of Demand at the profit-maximizing price and quantity?

Explanation / Answer

(a) Profit is maximized when MPL/MPK = w/r = 16/9

MPL/MPK = K/L = 16/9

L = 9K/16 and K = 16L/9

Substituting in production function,

Q = L1/2(16L/9)1/2 = L1/2L1/2 (16/9)1/2 = L x (4/3) = 4L/3

L = 3Q/4

K = 16 x (3Q/4) / 9 = 48Q/36 = 4Q/3

Total cost (C) = wL + rK = 16 x (3Q/4) + 9 x (4Q/3) = 12Q + 12Q = 24Q

(b) Marginal cost (MC) = dC/dQ = 24

Monopolist will maximize profit by equating Marginal revenue (MR) with MC.

Total revenue (TR) = P x Q = 288Q - 2Q2

MR = dTR/dQ = 288 - 4Q

Equating with MC,

288 - 4Q = 24

4Q = 264

Q = 66 units

(c) When Q = 66, L = 3Q/4 = (3 x 66) / 4 = 49.5 units

(d) When Q = 66, K = 4Q/3 = (4 x 66) / 3 = 88 units

(e) Price = 288 - (2 x 66) = 288 - 132 = 156

NOTE: As per Chegg Answering Policy, first 5 parts are answered.

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