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Consider the Cournot duopoly model where the inverse demand function is given by P(Q) = 100-Q...

Consider the Cournot duopoly model where the inverse demand function is given by P(Q) = 100-Q but the firms have asymmetric marginal costs: c1= 40 and c2= 60. What is the Nash equilibrium of this game?

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Answer #1

P = 100 - Q = 100 - Q1 - Q2 [Since Q = q1 + Q2]

c1 = 40, c2 = 60

For firm 1,

Total revenue (TR1) = P x Q1 = 100Q1 - Q12 - Q1Q2

Marginal revenue (MR1) = TR1/Q1 = 100 - 2Q1 - Q2

Equating MR1 & c1,

100 - 2Q1 - Q2 = 40

2Q1 + Q2 = 60..........(1) [Best response, firm 1]

For firm 2,

Total revenue (TR2) = P x Q2 = 100Q2 - Q1Q2 - Q22

Marginal revenue (MR2) = TR2/Q2 = 100 - Q1 - 2Q2

Equating MR2 & c2,

100 - Q1 - 2Q2 = 60

Q1 + 2Q2 = 40..........(2) [Best response, firm 2]

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

(2) x 2 yields:

2Q1 + 4Q2 = 80.......(3), and

2Q1 + Q2 = 60.........(1)

(3) - (1) yields: 3Q2 = 20

Q2 = 6.67

Q1 = 40 - 2Q2 [From (2)] = 40 - (2 x 6.67) = 40 - 13.34 = 26.66

P = 100 - 26.66 - 6.67 = 66.67

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