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Question #5: Consider a Cournot duopoly, the firms face an (inverse) demand function: Pb = 110...

Question #5: Consider a Cournot duopoly, the firms face an (inverse) demand function: Pb = 110 - 7 Qb. The marginal cost for firm 1 is given by mc1 = 5 Q. The marginal cost for firm 2 is given by mc2 = 7 Q. (Assume firm 1 has a fixed cost of $ 112 and firm 2 has a fixed cost of $ 148 .) How much profit will firm 2 earn in the duopoly equilibrium ?

Expert Solution

Demand function: P = 110 - 7Q = 110 - 7Q1 - 7Q2 [Where Q1, Q2: Output by firm 1 and 2 and Q = Q1 + Q2]

MC1 = 5Q1

MC2 = 7Q2

For firm 1,

Total revenue (TR1) = P x Q1 = 110Q1 - 7Q1² - 7Q1Q2

Marginal revenue (MR1) = TR1/Q1 = 110 - 14Q1 - 7Q2

Equating MR1 and MC1,

110 - 14Q1 - 7Q2 = 5Q1

19Q1 + 7Q2 = 110........(1) [Reaction function, firm 1]

For firm 2,

TR2 = P x Q2 = 110Q2 - 7Q1Q2 - 7Q2²

MR2 = TR2/Q2 = 110 - 7Q1 - 14Q2

Equating MR2 and MC2,

110 - 7Q1 - 14Q2 = 7Q2

7Q1 + 21Q2 = 110........(2) [Reaction function, firm 2]

Cournot duopoly equilibrium is obtained by solving (1) and (2). Multiplying (1) by 3,

57Q1 + 21Q2 = 330.........(3)

7Q1 + 21Q2 = 110.........(2)

(3) - (2) gives:

50Q1 = 220

Q1 = 4.4

Q2 = (110 - 7Q1) / 21 [From (2)] = [110 - (7 x 4.4)] / 21 = (110 - 30.8) / 21 = 79.2 / 21 = 3.77

Q = 4.4 + 3.77 = 8.17

P = 110 - (7 x 8.17) = 110 - 57.19 = 52.81

MC2 = 7 x 3.77 = 26.39

Profit of firm 2 = Q2 x (P - MC2) - Fixed cost = 3.77 x (52.81 - 26.39) - 148 = 3.77 x 26.42 - 148 = 99.60 - 148 = -48.4

Rahul Sunny answered 1 year ago

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