Joe and Rebecca are small-town ready-mix concrete duopolists. The market demand

Question

Joe and Rebecca are small-town ready-mix concrete duopolists. The market demand function is Qd= 10,000 –100P, where P is the price of a cubic yard of concrete and Qd is the number of cubic yards demanded per year. Marginal cost is $25 per cubic yard. Suppose that Joe and Rebecca compete in prices and competition in this market is described by Cournot model. Consumers perceive the ready-mix concrete produced by the two firms as identical products. Find the Nash equilibriumprices when the two firms set their prices simultaneously.

Explanation / Answer

(1) Cournot equilibrium:

Q = 10,000 - 100P

100P = 10,000 - Q

P = 100 - 0.01Q

P = 100 – 0.01Q where Q = q1 + q2

P = 100 – 0.01q1 – 0.01q2

So,

Total revenue of firm 1, TR1 = P x q1 = 100q1 – 0.01q12 – 0.01q1q2

Total revenue of firm 2, TR2 = P x q2 = 100q2 – 0.01q1q2 – 0.01q22

So,

Marginal revenue of firm 1, MR1 = dTR1 / dq1 = 100 - 0.02q1 - 0.01q2

Equating with MC1:

100 - 0.02q1 - 0.01q2 = 25

0.02q1 + 0.01q2 = 75

2q1 + q2 = 7,500 ......(1) [Reaction function, firm 1]

Marginal revenue of firm 2, MR2 = dTR2 / dq2 = 100 – 0.01q1 – 0.02q2

MC2 = 25

Equating MR2 = MC2,

100 – 0.01q1 – 0.02q2 = 25

Or,

0.01q1 + 0.02q2 = 75

q1 + 2q2 = 7,500 .....(2) [Reaction function, firm 2]

Equilibrium is obtained by solving (1) & (2).

2q1 + q2 = 7,500 ......(1)

(2) x 2:

2q1 + 4q2 = 15,000 .....(3)

(3) - (1): 3q2 = 7,500

q2 = 2,500

q1 = 7,500 - 2q2 = 7,500 - (2 x 2,500) = 7,500 - 5,000 = 2,500

Q = q1 + q2 = 2,500 + 2,500 = 5,000

P = 100 - 0.01Q = 100 - (0.01 x 5,000) = 100 - 50 = 50

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