Question

I. Consider a three firm (n = 3) Cournot oligopoly. The market inverse demand function is P()-24 Q. Firm 1 has constant average and marginal costs of $12 per unit, while firms 2 and 3 have constant average and marginal costs of $15 per unit. p (Q) (a) Verify that the following are Nash equilibrium quantities for this market: q,-. and g2 = g3

Answer 1

Q = q1 + q2 + q3

Reaction curve firm 1:

MR1 = MC1 = 12

MR1 = dR1/dq1

R1 = pq1 = 24q1 - q1^{​​​​​2}​​​​ - q1q2 - q1q3

MR1 = 24 - 2q1 - q2 - q3 = 12

2q1 + q2 +q3 = 12

Firm 2:

Similarly,

2q2 + q1 + q3 = 24 - 15 = 9

Firm 3:

2q3 + q2 + q1 = 9

Solving simultaneously for q1, q2, q3:

q1 = 9/2 and q2 = q3 = 3/2

b)

Profit = Pq - AC X q

P = 24 - (9/2 + 3) = 16.5

Profit (firm 1) = (16.5 - 12) x 4.5 = 20.25

Profit (firm 2) = profit (firm 3) = (16.5 - 15) x 1.5 = 2.25