Question

Consider a monopolist who has a cost function of ?(?)=???. This monopolist faces two consumers, the first having demand ??(??)=??−?? and the second having demand ??(??)=??−??.
a) Calculate the profit-maximizing price and then the optimal quantity sold to each consumer under uniform pricing, i.e. the monopolist charges the same price for both consumers. What are the monopolist’s profits?
b) Suppose now that the monopolist can engage in third degree price discrimination. Find the monopolists profit maximizing prices for consumers 1 and 2 (they should be different), and calculate the monopolist’s profits. How do they compare to the profits in part (a)?

Answer 1

Answer :

a)

C(Q)=10Q

Marginal cost=MC=dC(Q)/dQ=10

Total demand=Q=q1+q2=60-P+50-P=110-2P

Combined demand is given by

Q=110-2P

or 2P=110-Q

P=55-0.5Q

Total Revenue=P*Q=(55-0.5Q)*Q=55Q-0.5Q^2

Marginal Revenue=MR=55-Q

Set MR=MC for profit maximization

55-Q=10

Q=45

Price=55-0.5*45=32.50

Quantity sold to customer 1=60-32.5=27.50

Quantity sold to customer 2=50-32.5=17.50

Total cost=TC=10*Q=10*45=450

Total Revenue=TR=P*Q=32.50*45=1462.50

Profit=TR-TC=1462.50-450=$1012.50

b)

In case monopolist charges different price to different customer than

Total output =Q=q1+q2

Marginal Cost=10 (we have calculated in first part)

Demand of first customer is

q1=60-P1 or P1=60-q1

Total Revenue from customer 1 =TR1=(60-q1)*q1=60q1-q12

Marginal revenue=MR1=dTR1/dq1=60-2q1

Set MR1=MC

60-2q1=10

q1=25

P1=60-q1=60-25=35

Demand of second customer is

q2=50-P2 or P2=50-q2

Total Revenue from customer 2 =TR2=(50-q2)*q2=50q2-q22

Marginal revenue=MR2=dTR2/dq2=50-2q2

Set MR2=MC

50-2q2=10

q2=20

P2=50-q2=50-20=30

Total Revenue=TR=p1*q1+p2*q2=35*25+30*20=1475

Total Cost=TC=10*Q=10*(q1+q2)=10*(25+20)=450

Price for customer 1=35

Price for customer 2=30

Profit=TR-TC=1475-450=1075

Profit has increased in case of price discrimination.