Question

4. Consider a Cournot duopoly with inverse demand function P = 100 – Q. Firm 1’s cost function is C1(q1) = 20q1, and firm 2’s cost function is C2(q2) = 30q2. Firms choose quantities once and simultaneously. (a) Write out each firm’s profit function. From these, derive the reaction functions of each firm, and solve for the Nash equilibrium quantities, price and profits. Illustrate your answer on a graph of the reaction functions (you do not need to draw isoprofit lines). (b) Now suppose firm 1’s marginal cost falls from 20 to 10. How do the reaction functions, Nash equilibrium quantities, price and profits differ from (b)? Illustrate on a graph how the reaction functions and quantities change. (c) Suppose that firm 2’s cost function is instead written C2(q2) = 30q2 + 225 if q2 > 0 C2(q2 ) = 0 if q2 = 0. That is, there is a fixed cost of 225 that can be avoided if the firm does not produce. How low must firm 1’s marginal cost be for firm 2 to be better off not producing?

Answer 1

(a)

P = 120 - Q = 120 - q1 - q2

MC1 = MC2 = 60

For Firm 1, Total revenue (TR1) = P x q1 = 120q1 - q1² - q1q2

Marginal revenue (MR1) = TR1 / q1 = 120 - 2q1 - q2

Equating MR1 and MC1,

120 - 2q1 - q2 = 60

2q1 + q2 = 60 ............(1) (Best response, Firm 1)

For Firm 2, Total revenue (TR2) = P x q2 = 120q2 - q1q2 - q2²

Marginal revenue (MR2) = TR2 / Q2 = 120 - q1 - 2q2

Equating MR2 and MC2,

120 - q1 - 2q2 = 60

q1 + 2Q2 = 60 ............(2) (Best response, Firm 2)

Cournot equilibrium is obtained by solving (1) and (2)

2q1 + q2 = 60 ..............(1)

(2) x 2 results in:

2q1 + 4q2 = 120.............(3)

(3) - (1) results in: 3q2 = 60

q2 = 20

q1 = 60 - 2q2 [From (2)] = 60 - (2 x 20) = 60 - 40 = 20

Q = 20 + 20 = 40

P = 120 - 40 = 80

Market share, firm 1 = q1 / Q = 20 / 40 = 0.5 = 50%

Market share, firm 2 = q2 / Q = 20 / 40 = 0.5 = 50%

(b) HHI Index = (50)² + (50)² = 2,500 + 2,500 = 5,000

(c) A monopolist maximizes profit by equating MR with MC.

P = 120 - Q

TR = P x Q = 120Q - Q²

MR = dTR / dQ = 120 - 2Q

Equating MR & MC,

120 - 2Q = 60

2Q = 60

Q = 30

P = 120 - 30 = 90

In a monopoly, HHI = 10,000

Change in HHI = 10,000 - 5,000 = 5,000 (Increase)