Examples and exercises on collusive oligopoly

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.