The theory of the firm and industry equilibrium

Martin J. Osborne
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7.5 Collusion among oligopolists

We have seen that the total profit earned by the firms in a Nash equilibrium of Cournot's model of duopoly is in general less than the monopoly profit. That is, if the firms could agree to produce, between them, the output of a monopolist, they could both be better off. What stops them from doing so?

There may be laws that outlaw collusion. Such laws exist, for example, in Canada.

But more interestingly, there is a reason why, even in the absence of such laws, duopolists (and more generally oligopolists) may find it hard to collude. The reason is simply that a collusive outcome is not a Nash equilibrium! Given the output of one of the firms, the other firm can increase its profit by choosing an output different from its collusive output.

Specifically, suppose that the firms agree that each will produce half of the monopoly output. Then the total output is the monopoly output, so that the total profit earned by the firms is maximal. We know that in general this pair of outputs is not a Nash equilibrium, which means that at least one of the firms can increase its profit by choosing a different output, given the output of the other firm. When a firm deviates in this way, its rival's profit decreases by more than its own profit increases (the total profit cannot increase, since it is equal to the monopoly profit when the firms collude). That is, the deviating firm gains at the expense of the other firm. Certainly both firms would be better off if the agreement would stick, but the fact that each firm has a private incentive to “cheat” on any agreement means that a collusive agreement may not be stable.

Example
Each of two firms has the cost function TC(y) = 30y; the inverse demand function for the firms' output is p = 120 − Q, where Q is the total output.

We know that there is a unique Nash equilibrium in this case, in which each firm's output is 30, the price is 60, and each firm's profit is 900.

We know also that the monopoly output is 45, and the price is 75; the monopolist's profit is (45)(75) − (30)(45) = 2025.

Suppose that the firms agree to each produce half of the monopoly output, namely 22.5 units. Then each firm earns the profit 1012.5 (half of the monopoly profit).

Given that one firm is producing half the monopoly output, how much does the other firm want to produce? If it produces the output y then its profit is

y(120 − 22.5 − y) − 30y = y(67.5 − y).
The output that maximizes this profit is 33.75 (take the derivative and set it equal to zero). The firm's profit when it produces this output (and the other firm adheres to the agreement to produce half of the monopoly output) is
33.75(67.5 − 33.75) ≈ 1139.
Thus by violating the agreement and producing 33.75 units rather than 22.5 units of output, the firm can obtain the profit 1139, rather than the profit of 1012.5. When the firm "cheats" in this way, the other firm's profit decreases: it becomes
22.5(120 − 22.5 − 33.75) − 30(22.5) ≈ 759.
(Note that the total profit of the firms decreases, as it must: the total monopoly profit is 2025, while the total profit after the firm deviates is 1139 + 759 = 1898.)

In summary, if one firm adheres to the agreement to produce half of the monopoly output, then the optimal output for the other firm is 33.75, yielding it a profit of 1139, rather than the profit of 1012.5 that it gets in the agreement.