# Cartel

Oligopolists can form a cartel: collude and act as a monopolist.

## Example

Zach and Yann, duopolist in the market for coffee, decide to collude.

The demand for coffee satisfies $$P = 1400 - 2Q$$ So their revenue is $$R = 1400Q - 2 Q^2$$ and their marginal revenue is $$MR = 1400 - 4Q$$

Their marginal cost is MC = 200

They maximize profit like a monopolist: when their marginal cost is equal to their marginal revenue \begin{align*} MC &= MR \\ 200 &= 1400 - 4Q \\ 4Q &= 1200 \\ Q &= 300 \end{align*}

If they share the production load, Zach's quantity will be Q_Z = 150, and Yann's quantity will be Q_Y = 150.

Note that such a cartel agreement can fail. Imagine that Yann commits to produce a quantity Q_Y=150 of coffee. From Zach's point of view, producing a quantity 150 is not optimal. Indeed, plug Yann's quantity Q_Y = 150 into Zach's best response function \begin{align*} Q_Z &= 300 - \frac{Q_Y}{2} \\ &= 300 - \frac{150}{2} \\ &= 300 - 75 \\ &= 225 \end{align*} If Yann produces a quantity Q_Y = 150, then Zach should produce a quantity Q_Z = 150 to maximize profit.

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