## The duopoly cartel

September 29, 2011

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

The previous two models of competition, Cournot and Stackelberg, looked at different ways in which firms can compete with each other strategically by setting their quantity decisions.

Now we can look at the case where the firms decide to collude rather than compete, and form a cartel.

If the firms can trust each other, they can come together and behave like a joint monopoly firm, and just split the output.

When firm A had a monopoly, it produced output of 175 and sold at price of 325.

So if firms A and B come together, they can decide to each produce 87.5, so the market output is 175 and the price is 325. This means each firm gets profits of 325(87.5) – 150(87.5) = 15312.5.

However this arrangement relies on trust, because the cartel output is not on either firm’s best response curve. If firm A knows that firm B is going to produce 87.5, then given the best response function for firm A is $Q_A = 175-0.5Q_B$, the best response for firm A would be 131.25. As the firms are identical the same holds for firm B – if B knows A is producing 87.5, then the best response for B is to produce 131.25.

So what happens if one of them cheats and produces 131.25? Then the market output is 219 and the market price is 500-219 = 281. So the cheat will get profits of 281(131.25) – 150(131.25) = 17193.75, which is an improvement on the cartel equilibrium of 15312.5. The one that didn’t cheat will get profits of 281(87.5) – 150(87.5) = 11462.5, so they lose out.

But if they both cheat and produce 131.25, then the market output is 262.5 and the market price is 237.5, so they will both get profits of 237.5(131.25) – 150(131.25) = 11484.38. They would both have been better off by not cheating. So the cartel relies on trust!