Stackelberg Competition¶

Definition¶

Stackelberg competition is a quantity-setting duopoly with sequential moves. Firm 1 (the leader) chooses $q_1\ge0$ $0$ ; firm 2 (the follower) observes $q_1$ and then chooses $q_2\ge0$ $0$ . With linear inverse demand $p=a-b(q_1+q_2)$ $q_{2})$ , constant marginal cost $c$ and $a>c\ge0$ $0$ , $b>0$ $0$ , the profits are, for $i=1,2$ $1, 2$ ,

v_i(q_1,q_2)=\big(a-b(q_1+q_2)-c\big)\,q_i.

The leader's strategy is a quantity $q_1$ ; the follower's strategy is a function $s_2\colon[0,\infty)\to[0,\infty)$ giving her quantity after every possible $q_1$ .

Interpretation¶

The game is the sequential version of the Cournot duopoly and is solved by backward induction. The follower's best response after observing $q_1$ follows from the first-order condition $a-c-bq_1-2bq_2=0$ $0$ :

BR_2(q_1)=\max\Big\{0,\ \frac{a-c-bq_1}{2b}\Big\}.

The leader substitutes this into her own profit. For $q_1\le(a-c)/b$ this gives $v_1=\tfrac{1}{2}(a-c-bq_1)\,q_1$ $b q_{1}) q_{1}$ , a concave function maximized where $\tfrac{1}{2}(a-c)-bq_1=0$ $0$ .

Quantity plane with the follower's best response q2 = max((90 − q1)/2, 0), including its zero-output branch, the Cournot point (30, 30) on it, and the leader's iso-profit curve v1 = 1012.5 touching it at (45, 22.5).

The figure uses the numbers of the Examples ( $a =$ $a=100$ $100$ , $b=1$ $1$ , $c=10$ $10$ ). Because the follower plays $BR_2$ , the leader in effect chooses a point on that response graph, whose zero-output branch begins at $q_1=90$ $90$ . Her profit is highest where one of her iso-profit curves just touches it: the curve $v_1=1012.5$ touches at $(q_1,q_2)=(45,22.5)$ $(45, 22.5)$ . The Cournot point $(30,30)$ also lies on $BR_2$ but gives the leader only $900$ ; the difference is the value of commitment.

The leader gains because her quantity is a commitment: it is observed and cannot be revised, so she can push the follower to produce less. If the follower could not observe $q_1$ , the game would be strategically simultaneous and the Cournot outcome would result. A frequent error is to report the follower's equilibrium strategy as the number $(a-c)/(4b)$ $c) / (4 b)$ ; the strategy is the whole function $BR_2$ , and the number is only its value on the equilibrium path.

Key Results¶

Subgame-perfect Nash equilibrium (SPNE): $q_1^*=\dfrac{a-c}{2b}$ and $s_2^*=BR_2$ $B R_{2}$ , so on the path $q_2^*=\dfrac{a-c}{4b}$ $\frac{a - c}{4 b}$ .
Total output $Q=\dfrac{3(a-c)}{4b}$ and price $p=\dfrac{a+3c}{4}$ $\frac{a + 3 c}{4}$ .
Profits $v_1=\dfrac{(a-c)^2}{8b}$ and $v_2=\dfrac{(a-c)^2}{16b}$ $\frac{( a - c ) ^{2}}{16 b}$ .
Compared with Cournot ( $q_{i} =$ $q_i=\tfrac{a-c}{3b}$ $\frac{a - c}{3 b}$ , $v_i=\tfrac{(a-c)^2}{9b}$ $\frac{( a - c ) ^{2}}{9 b}$ ): the leader produces and earns more, the follower produces and earns less, total output is higher and the price lower.
There are Nash equilibria that are not subgame perfect, sustained by a follower strategy that does not best respond off the path.

Examples¶

Numbers. Let $a=100$ $100$ , $b=1$ $1$ , $c=10$ $10$ . The follower plays $BR_2(q_1)=\max\{0,(90-q_1)/2\}$ $q_{1}) /2}$ . The leader maximizes $\tfrac{1}{2}(90-q_1)q_1$ $q_{1}) q_{1}$ , so $q_1=45$ $45$ . Then $q_2=22.5$ $22.5$ , $Q=67.5$ $67.5$ , $p=32.5$ $32.5$ , $v_1=22.5\cdot45=1012.5=90^2/8$ and $v_2=22.5\cdot22.5=506.25=90^2/16$ $9 0^{2} /16$ . In the Cournot game each firm produces $30$ at price $40$ and earns $900=90^2/9$ $9 0^{2} /9$ .

A Nash equilibrium that is not subgame perfect. Let the follower use the constant strategy $s_2(q_1)=30$ for every $q_1$ , and the leader play $q_1=30$ $30$ . The leader's best response to a fixed $q_2=30$ is $(90-30)/2=30$ $30$ , and on the path the follower's $30=BR_2(30)$ is optimal, so this is a Nash equilibrium with the Cournot outcome. It is not subgame perfect: after, say, $q_1=45$ the follower should produce $22.5$ , not $30$ . The threat to ignore the leader's quantity is not credible.

Prerequisites¶

Sources¶

Game Theory · Week 3

Aliases: Stackelberg duopoly, Stackelberg model, leader–follower model, leader-follower model, Stackelberg game, Stackelberg leadership model Keywords: Stackelberg leader, Stackelberg follower, first-mover advantage, sequential quantity competition, q1 = (a−c)/(2b), q2 = (a−c)/(4b) Symbols: infinity

Last updated: 27 October 2026 11:52