Question

3. Suppose a monopolist has a production function given by Q = L12K12. Therefore, L2 MPL K2 2/12 , and MPK 2K? The monopolist can purchase labor, L at a price w=16, and capital, K at a price ofr=9. The demand curve facing the monopolist is P = 360 -20. a) What is the monopolist's total cost function?

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

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

d) How Much Capital should the firm hire?  

e) What price should the monopolist charge?

f) What is the deadweight loss?

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

Answer 1

3.

Q= L^1/2 K^1/2

MP_L = K^1/2 / 2L^1/2

MP_K = L^1/2 /2K^1/2

W= 16

r =9

P= 360-2Q

Total revenue= TR= P x Q= 360Q-2Q²

Optimal condition:

MP_L /MP_K = K^1/2 / 2L^1/2/L^1/2 /2K^1/2

MP_L /MP_K = K/L

MP_L /MP_K = W/r

K/L= 16/9

9K = 16L

K= 16L/9 Equation 1

Use this equation in production function:

Q= L^1/2 K^1/2 = L^1/2 (16L/9)^1/2

Q= L^1/2 L^1/2 (4/3)

Q= 4L/3

L= 3Q/4 Demand for labor

Use this in equation 1:

K= 16(3Q/4)/9= 4Q/3 Demand for capital

a)

Total cost(TC) = LW+Kr

TC= 16L+9r

From demand functions for L and K:

TC= 16(3Q/4)+9(4Q/3)= 12Q+12Q= 24Q Total cost function.

b)

Monopoly maximizes profit where : MC=MR

MC= Differentiation of TC wrt Q= 24

MR= Differentiation of TR wrt Q= 360-4Q

MC=MR

24= 360-4Q

4Q= 336

Q= 336/4= 84 Profit maximizing quantity by monopoly

c)

Labor hire by monopoly can be calculated by labor demand function:

L= 3Q/4= 3(84)/4= 3 x 21= 63 Number of labor hired by the firm for Q=84

d)

Capital hire by monopoly can be calculated by Capital demand function:

K= 4Q/3= 4(84)/3= 4 x 28= 112 Amount of capital hired by the firm for Q=84

e)

Price charged by the firm:

P=360-2Q= 360-2(84)= 360-168= 192