Question

1. [20] A monopolist with cost function c(Q) = 2 faces a consumer whose demand functions are given by. 01 20 P and Q2 40 2P. (a) 5] Suppose the monopolist cannot engage in any price discrimination. Find the firm's optimal pricing strategy. Calculate the firm's Lerner index. (b) 15) What is the deadweight loss (relative to the competitive market out- come) associated with this pricing strategy, if any? (c) 15) Now, suppose price discrimination is possible. Find the monopolist's optimal third-degree price-discrimination strategy. Which consumer is more inelastic at the optimal solution? (a) [5 What information is required for the monopolist to be able to use third-degree price discrimination? Give a real-life example where third- degree price discrimination is used.

Answer 1

Marginal cost (MC) = dc(Q)/dQ = Q

(a) With single pricing policy, Market output (Q) = Q1 + Q2 and profit is maximized when MR = MC.

Q = 20 - P + 40 - 2P = 60 - 3P

P = (60 - Q)/3

Total revenue (TR) = P x Q = (60Q - Q²)/3

MR = dTR/dQ = (60 - 2Q)/3

Equating with MC,

(60 - 2Q)/3 = Q

60 - 2Q = 3Q

5Q = 60

Q = 12

P = (60 - 12)/3 = 48/3 = 16

Lerner Index = (P - MC)/P = (16 - 12)/16 = 4/16 = 0.25 (= 25%)

(b) With competitive outcome, P = MC.

(60 - Q)/3 = Q

60 - Q = 3Q

4Q = 60

Q = 15

P = MC = Q = 15

Deadweight loss = (1/2) x Change in price x Change in output = (1/2) x (16 - 15) x (15 - 12) = (1/2) x 1 x 3 = 1.5

(c) With third-degree price discrimination, profit is maximized when MR1 = MC1 and MR2 = MC2.

Total cost = c(Q) = (Q1 + Q2)²

MC1 = $\partial$ c(Q)/dQ1 = 2(Q1 + Q2) = 2Q1 + 2Q2

MC2 = $\partial$ c(Q)/dQ2 = 2(Q1 + Q2) = 2Q1 + 2Q2

For market 1,

Q1 = 20 - P1

P1 = 20 - Q1

TR1 = P1 x Q1 = 20Q1 - Q1²

MR1 = dTR1/dQ1 = 20 - 2Q1

20 - 2Q1 = 2Q1 + 2Q2

4Q1 + 2Q2 = 20

2Q1 + Q2 = 10.........(1)

For market 2,

Q2 = 40 - 2P2

2P2 = 40 - Q2

P2 = 20 - 0.5Q2

TR2 = P2 x Q2 = 20Q2 - 0.5Q2²

MR2 = dTR2/dQ2 = 20 - Q2

20 - Q2 = 2Q1 + 2Q2

2Q1 + 3Q2 = 20...........(2)

(2) - (1) yields:

2Q2 = 10

Q2 = 5

P2 = 20 - (0.5 x 5) = 20 - 2.5 = 17.5

Elasticity in market 2 = (dQ2/dP2) x (P2/Q2) = -2 x (17.5/5) = -7

Q1 = (10 - Q2)/2 [From (1)] = (10 - 5)/2 = 15/2 = 7.5

P1 = 20 - 7.5 = 12.5

Elasticity in market 1 = (dQ1/dP1) x (P1/Q1) = -1 x (12.5/7.5) = -1.67

Since absolute value of elasticity is lower in market 1 (1.67 < 7), this market is more inelastic.

(d) The required condition for a firm to price discriminate is to know that there are at least two market segments with different price elasticity of demand, and which market has higher elasticity should be known so that higher price can be charged in the more inelastic segment and lower price can be charged in the more elastic segment.

A real-life example is the airline market where the airlines charge higher price from business travelers who has inelastic demand, and lower price from non-business travelers who have elastic demand.