3 (34 points) Suppose a monopolist has a production function given by Q- LinK2.

Question

3 (34 points) Suppose a monopolist has a production function given by Q- LinK2. Therefore, L12 K1/2 MPL=- K", , and Mh-2K2 16, and capital, K at a price ofr 9. The The monopolist can purchase labor, L at a price w demand curve facing the monopolist is P 360-2Q 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? o) (6 points) How much labor should the firm hire to produce this output? d) (4 points) How Much Capital should the firm hire? ) (4 points) What price should the monopolist charge? f) (4 points) What is the deadweight loss? R) (4 points) What is the Price Elasticity of Demand at the profit-maximizing price and quantity?

Explanation / Answer

a) Given MPK = L1/2 /2K1/2 and MPL = K1/2 /2L1/2

wage rate w= 16 and capital rate r = 9

Under equilibrium condition,

MPK / MPL = r/w

or, L1/2 /2K1/2 /K1/2 /2L1/2 = 9/16

or, L/K=9/16

or, L= 9/16K

Production function is given as Q= L1/2 K1/2 ....................(i)

Substituting the value of L in (i), we get

     Q= (9/16K)1/2 K1/2 or, Q= 3/4K or, K=4/3Q

Total cost(TC) = Variable cost +Fixed cost

or, TC = wL+rK= 16L+9K = 16.9/16K+9K= 18K= 18.4/3Q

or, TC = 24Q

b) The demand function is given as,

   P = 360-2Q

Total revenue is given by multiplying either side by Q,

PQ = 360Q-2Q2

Therefore takking partial derivative of either side with respect to Q,

d(PQ)/dQ= 360-4Q which is marginal revenue function

Under profit maximization condition of monopoly,

                   Marginal revenue = 0

           or, 360-4Q =0 or Q=90 which is the profit maximizing output of the firm

c) From (a) production function is given by Q= 3/4K

                                   or, Q= 3/4.16/9 L = 4/3L

Profit maximizing output of the firm is, Q= 90

   or, 4/3L=90 or, L=90*3/4=66 (omitting decimal figure)

d) Production function is,

                Q= 3/4K or, 90=3/4K or, K= 90*4/3=120 which is required capital the firm should invest

e) The profit maximizing price of the monopolist firm is

   P = 360-2Q = 360-2.90= 180

f) From a) the total cost function is

TC =24Q

Taking partial derivative of either side w.r.t. Q,

   d(TC)/dQ =24

which is marginal revenue function

Under equilibrium condition of a monopoly market,

                      Marginal revenue = Marginal cost

              or, 360-4Q = 24 or, 360-4Q=24

or, Q = 84

Equilibrium price is P = 360-4Q = 360-4*84= 24

Deadweight loss = 1/2*(P2-P1)*(Q1-Q2)    here P1= Equilibrium price =24, P2 = profit maximizing price = 180

     Q1= Equilibrium quantity = 84 and Q2= profit maximizing quantity = 90

Therefore, deadweight loss = 1/2*(180-24)*(84-90)= -468

g) Price elasticity of demand = (Q2-Q1)/(P2-P1) *P/Q = (180-24)/(90-84)*84/24 = 91

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