What if one firm moves first? A leader that commits to its output, knowing the follower will react, plays a sequential version of Cournot Competition, solved by Sequential Games and Backward Induction.
Key idea
With $P = a - b(q_L + q_F)$ and common cost $c$, the follower plays its Cournot best response $q_F = \dfrac{a - c - bq_L}{2b}$. The leader, anticipating this, produces $$q_L = \frac{a - c}{2b}, \qquad q_F = \frac{a - c}{4b}.$$ The leader produces more and earns more than in Cournot; the follower produces less and earns less.
Why it works
Backward induction: the follower moves last, so its choice is its best response to whatever $q_L$ it sees. The leader substitutes that response into the price before maximizing: $$P = a - bq_L - b\cdot\frac{a - c - bq_L}{2b} = \frac{a + c - bq_L}{2}.$$ Its profit is $\left(\dfrac{a + c - bq_L}{2} - c\right)q_L = \dfrac{(a - c - bq_L)q_L}{2}$, maximized at $q_L = \dfrac{a - c}{2b}$. Moving first is valuable because the leader's big output is a commitment: the follower can only accept it and shrink.
Worked example: Leader and follower
$P = 100 - Q$, $c = 10$.
- Follower: $q_F = \dfrac{90 - q_L}{2}$.
- Leader: $q_L = \dfrac{90}{2} = 45$; then $q_F = 22.5$.
- $P = 100 - 67.5 = 32.5$. Leader's profit $(22.5)(45) = 1{,}012.5$; follower's $(22.5)(22.5) = 506.25$.
- Cournot gave each 900: the leader gains 112.5, the follower loses 393.75, and total output rises from 60 to 67.5.
Answer: $q_L = 45$, $q_F = 22.5$, $P = 32.5$.
Worked example: Why not just announce a big number?
In Cournot (moving at the same time), firm 1 says it will produce 45. Is that believable?
- If firm 2 believed it, firm 2 would produce 22.5; but firm 1's best response to 22.5 is $\dfrac{90 - 22.5}{2} = 33.75$, not 45.
- Without a way to commit first, the announcement is not credible. Stackelberg leadership requires that the leader's choice really is made, and seen, first.
Answer: No: only a real first move commits.
Common mistake
Giving the leader its Cournot best response, as if it ignored the follower's reaction. The leader's advantage comes from substituting $q_F(q_L)$ into its profit before maximizing. Check: compute the leader's profit at your answer and at the Cournot quantity; the Stackelberg choice must be at least as good for the leader.
More sequential games in Sequential Games and Backward Induction.