Question

Show that in a Cournot duopoly with constant marginal costs (c1; c2) and

linear demand function, equilibrium quantities and profits of each firm are

decreasing functions of the own marginal cost and increasing functions on

the marginal cost of the rival firm.

Answer 1

Let there be two firms in a Cournot duopoly with marginal cost as c1 and c2

Assume inverse demand function as: p = a - bQ, where Q = q1+q2

Firm 1:

Total Revenue = p*q1 = (a-b(q1+q2))*q1 = aq1 - bq1² - bq1q2

Total Cost = c1q1

Profit 1 1= aq1 - bq1² - bq1q2 - c1q1

Similarly for firm 2 profit will be

Profit 2 2= aq2 - bq2² - bq1q2 - c2q2

Solving the first order condition for profit maximization and setting equal to zero we get,

From first equation we get, q1 = (a - bq2 - c1)/2b

From second equation we have, q2 = (a-bq1 - c2)/2b

Solving both the equations we get,

q1 = (a- 2c1 + c2)/3b

q2 = (a- 2c2 + c1)/3b

Price p = (a + c1 + c2)/3

1 TR - TC = (a-2c1+c2)²/9b

2 TR-TC = (a-2c2+c1)²/9b

Thus we can see that quantities and profits of each firm are decreasing functions of the own marginal cost [this is given by the negative sign with the own marginal cost, thus an increase in own MC decreases quantity and profit] and increasing functions on the marginal cost of the rival firm[this is given by the positive sign with the rival firm's marginal cost, thus an increase in rival firm's MC increases quantity and profit].