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Consider a Cournot duopoly, the firms face an (inverse) demand function: Pb = 128 - 3...

Consider a Cournot duopoly, the firms face an (inverse) demand function: Pb = 128 - 3 Qb. The marginal cost for firm 1 is given by mc1 = 4 Q. The marginal cost for firm 2 is given by mc2 = 6 Q. (Assume firm 1 has a fixed cost of $ 65 and firm 2 has a fixed cost of $ 87 .) How much profit will firm 2 earn in the duopoly equilibrium ?

Expert Solution

Market demand: P = 128 - 3Q = 128 - 3Q1 - 3Q2 [Where Q1, Q2: Output by firm 1 and 2 and Q = Q1 + Q2]

MC1 = 4Q1

MC2 = 6Q2

For firm 1,

Total revenue (TR1) = P x Q1 = 128Q1 - 3Q1² - 3Q1Q2

Marginal revenue (MR1) = TR1/Q1 = 128 - 6Q1 - 3Q2

Equating MR1 and MC1,

128 - 6Q1 - 3Q2 = 4Q1

10Q1 + 3Q2 = 128........(1) [Best response, firm 1]

For firm 2,

TR2 = P x Q2 = 128Q2 - 3Q1Q2 - 3Q2²

MR2 = TR2/Q2 = 128 - 3Q1 - 6Q2

Equating MR2 and MC2,

128 - 3Q1 - 6Q2 = 6Q2

3Q1 + 12Q2 = 128........(2) [Best response, firm 2]

Nash equilibrium is obtained by solving (1) and (2). Multiplying (1) by 4,

40Q1 + 12Q2 = 512.........(3)

3Q1 + 12Q2 = 128.........(2)

(3) - (2) yields:

37Q1 = 384

Q1 = 10.38

Q2 = (128 - 10Q1) / 3 [From (1)] = [128 - (10 x 10.38)] / 3 = (128 - 103.8) / 3 = 24.2 / 3 = 8.07

Q = 10.38 + 8.07 = 18.45

P = 128 - (3 x 18.45) = 128 - 55.35 = 72.65

MC2 = 6 x 8.07 = 48.42

Profit by firm 2 = Q2 x (P - MC2) - Fixed cost = 8.07 x (72.65 - 48.42) - 87 = 8.07 x 24.23 - 87 = 195.54 - 87 = 108.54

Rahul Sunny answered 1 year ago

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