Question

Two firms sell identical products and compete as Cournot (price-setting) competitors in a market with a demand of p = 150 - Q. Each firm has a constant marginal and average cost of $3 per unit of output. Find the quantity each firm will produce and the price in equilibrium.

Answer 1

p = 150 - Q1 - Q2 [Since in Cournot model, market quantity (Q) = Q1 + Q2]

For firm 1,

Total revenue (TR1) = p x Q1 = 150Q1 - Q1² - Q1Q2

Marginal revenue (MR1) = TR1/Q1 = 150 - 2Q1 - Q2

Equating MR1 and MC,

150 - 2Q1 - Q2 = 3

2Q1 + Q2 = 147........(1) [Best response, firm 1]

For firm 2,

TR2 = p x Q2 = 150Q2 - Q1Q2 - Q2²

MR2 = TR2/Q2 = 150 - Q1 - 2Q2

Equating MR2 and MC,

150 - Q1 - 2Q2 = 3

Q1 + 2Q2 = 147........(2) [Best response, firm 2]

Cournot equilibrium is obtained by solving (1) and (2). Multiplying (2) by 2,

2Q1 + 4Q2 = 294........(3)

2Q1 + Q2 = 147........(1)

(3) - (1) yields: 3Q2 = 147

Q2 = 49

Q1 = 147 - 2Q2 [From (2)] = 147 - (2 x 49) = 147 - 98 = 49

Q = 49 + 49 = 98

P = 150 - 98 = 52