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Cournot Equilibrium, Monopoly, Duopoly

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Let there be 3 identical firms in an indusrty. The inverse demand curve is given by P=1-Q where Q= q1+q2+q3. The marginal cost for each is zero.

a) Compute the Cournot equilibrium.

b) Show that if two of the three firms merge (transforming the industry to a duopoly) the profit of these firms decrease.

c) What happens if all three firms merge?

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Solution Summary

Cournot Equilibrium, Monopoly, and Duopoly are emphasized.

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a) Compute the Cournot equilibrium.
For each firm, it decides output level by considering the other two firms' output. Then the total revenue for firm 1 is:
TR1 = P*q1 = (1-Q)q1 = (1-(q1+q2+q3)q1 = q1 - q1^2 - q1q2 - q1q3
Marginal revenue is then MR1 = dTR1 / dq1 = 1 - 2q1 - q2 - q3
For the firm to maximize its profit, the first order condition is MR1 = MC1 and we know that MC1 = 0. Then we have:
1 - 2q1 - q2 - q3 = 0
since the three firms are ...

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