In an oligopoly a few firms each know that their own output moves the price. Cournot's model treats quantities as the choice and looks for a Nash equilibrium. Each firm solves a small version of Monopoly with Calculus: a monopoly over the demand left over by its rival.
Key idea
With inverse demand $P = a - b(q_1 + q_2)$ and marginal costs $c_1, c_2$, firm 1's best response to $q_2$ is $$q_1 = \frac{a - c_1 - bq_2}{2b}.$$ The Cournot equilibrium solves both best responses at once: $q_1 = \dfrac{a - 2c_1 + c_2}{3b}$. With equal costs each firm makes $\dfrac{a - c}{3b}$, and total output lies between the monopoly and competitive levels.
Why it works
Firm 1 takes $q_2$ as fixed and maximizes $(a - bq_1 - bq_2)q_1 - c_1q_1$. Its marginal revenue is $a - 2bq_1 - bq_2$: like a monopoly facing the residual demand $a - bq_2 - bq_1$. Setting it equal to $c_1$ gives the best response. Each firm's best response falls as its rival produces more. The equilibrium is where the two best-response lines cross. Each firm ignores the fact that its extra output also lowers its rival's profit, so together they produce more than a monopoly (or a cartel) would.
Worked example: Symmetric duopoly
$P = 100 - Q$, $c = 10$ for both firms.
- Best response: $q_1 = \dfrac{90 - q_2}{2}$. By symmetry $q_1 = q_2 = q$, so $q = \dfrac{90 - q}{2}$ and $q = 30$.
- $Q = 60$, $P = 40$, each profit $(40 - 10)(30) = 900$.
- Compare: a monopoly makes 45 at \$55 (profit 2,025); competition ($P = MC$) makes 90 at \$10.
Answer: 30 each, $P = 40$.
Worked example: Different costs
$P = 100 - Q$, $c_1 = 10$, $c_2 = 16$.
- $q_1 = \dfrac{100 - 20 + 16}{3} = 32$ and $q_2 = \dfrac{100 - 32 + 10}{3} = 26$.
- $P = 100 - 58 = 42$.
- The low-cost firm produces more and earns more, but the high-cost firm still produces: unlike Bertrand competition, it isn't driven out.
Answer: $q_1 = 32$, $q_2 = 26$.
Common mistake
Maximizing each firm's profit as if it were a monopoly, leaving the rival's output out of the price. That gives each firm the monopoly quantity (45 above), a total of 90 and a price of 10: neither firm's choice is a best response to the other's. Check: plug your $q_2$ into firm 1's best response; you must get back your $q_1$.
Compare Cournot, Bertrand and Hotelling side by side in Competition: Cournot, Bertrand and Hotelling.