The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially (hence, it is sometimes described as the leader-follower game). It is named after the German economist Heinrich Freiherr von Stackelberg who published Marktform und Gleichgewicht (Market Structure and Equilibrium) in 1934, which described the model. In game theory terms, the players of this game are a leader and a follower and they compete on quantity. The Stackelberg leader is sometimes referred to as the Market Leader. There are some further constraints upon the sustaining of a Stackelberg equilibrium. The leader must know ex ante that the follower observes its action. The follower must have no means of committing to a future non-Stackelberg leader's action and the leader must know this. Indeed, if the 'follower' could commit to a Stackelberg leader action and the 'leader' knew this, the leader's best response would be to play a Stackelberg follower action. Firms may engage in Stackelberg competition if one has some sort of advantage enabling it to move first. More generally, the leader must have commitment power. Moving observably first is the most obvious means of commitment: once the leader has made its move, it cannot undo it—it is committed to that action. Moving first may be possible if the leader was the incumbent monopoly of the industry and the follower is a new entrant. Holding excess capacity is another means of commitment.
Subgame perfect Nash equilibrium The Stackelberg model can be solved to find the subgame perfect Nash equilibrium or equilibria (SPNE), i.e. the strategy profile that serves best each player, given the strategies of the other player and that entails every player playing in a Nash equilibrium in every subgame. In very general terms, let the price function for the (duopoly) industry be P {\displaystyle P} . The price P ( q 1 + q 2 ) {\displaystyle P(q_{1}+q_{2})} is simply a function of total (industry) output q 1 + q 2 {\displaystyle q_{1}+q_{2}} where the subscript i = 1 {\displaystyle i=1} represents the leader, i = 2 {\displaystyle i=2} represents the follower, and q i {\displaystyle q_{i}} is the output of firm i {\displaystyle i} . Suppose firm i {\displaystyle i} has the cost structure C i ( q i ) {\displaystyle C_{i}(q_{i})} . The model is solved by backward induction. The leader considers what the best response of the follower is, i.e. how it will respond once it has observed the quantity of the leader. The leader then picks a quantity that maximises its payoff, anticipating the predicted response of the follower. The follower actually observes this and in equilibrium picks the expected quantity as a response. To calculate the SPNE, the best response functions of the follower must first be calculated (calculation moves 'backwards' because of backward induction). The profit of firm 2 {\displaystyle 2} (the follower) is revenue minus cost. Revenue is the product of price and quantity, and cost is given by the firm's cost structure, so profit is:
Π 2 = P ( q 1 + q 2 ) ⋅ q 2 − C 2 ( q 2 ) {\displaystyle \Pi _{2}=P(q_{1}+q_{2})\cdot q_{2}-C_{2}(q_{2})} . The best response is to find the value of q 2 {\displaystyle q_{2}} that maximises Π 2 {\displaystyle \Pi _{2}} given q 1 {\displaystyle q_{1}} , i.e. given the output of the leader (firm 1 {\displaystyle 1} ), the output that maximises the follower's profit is found. Hence, the maximum of Π 2 {\displaystyle \Pi _{2}} with respect to q 2 {\displaystyle q_{2}} is to be found. First differentiate Π 2 {\displaystyle \Pi _{2}} with respect to q 2 {\displaystyle q_{2}} and set to zero for maximisation:
∂ Π 2 ∂ q 2 = ∂ P ( q 1 + q 2 ) ∂ q 2 ⋅ q 2 + P ( q 1 + q 2 ) − ∂ C 2 ( q 2 ) ∂ q 2 = 0. {\displaystyle {\frac {\partial \Pi _{2}}{\partial q_{2}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{2}}}\cdot q_{2}+P(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0.}
The values of q 2 {\displaystyle q_{2}} that satisfy this equation are the best responses. Now the best response function of the leader is considered. This function is calculated by considering the follower's output as a function of the leader's output, as just computed. The profit of firm 1 {\displaystyle 1} (the leader) is
Π 1 = P ( q 1 + q 2 ( q 1 ) ) ⋅ q 1 − C 1 ( q 1 ) {\displaystyle \Pi _{1}=P(q_{1}+q_{2}(q_{1}))\cdot q_{1}-C_{1}(q_{1})} , where q 2 ( q 1 ) {\displaystyle q_{2}(q_{1})} is the follower's quantity as a function of the leader's quantity, namely the function calculated above. The best response is to find the value of q 1 {\displaystyle q_{1}} that maximises Π 1 {\displaystyle \Pi _{1}} given q 2 ( q 1 ) {\displaystyle q_{2}(q_{1})} , i.e. given the best response function of the follower (firm 2 {\displaystyle 2} ), the output that maximises the leader's profit is found. Hence, the maximum of Π 1 {\displaystyle \Pi _{1}} with respect to q 1 {\displaystyle q_{1}} is to be found. First, differentiate Π 1 {\displaystyle \Pi _{1}} with respect to q 1 {\displaystyle q_{1}} and set this to zero for maximisation:
∂ Π 1 ∂ q 1 = ∂ P ( q 1 + q 2 ) ∂ q 2 ⋅ ∂ q 2 ( q 1 ) ∂ q 1 ⋅ q 1 + ∂ P ( q 1 + q 2 ) ∂ q 1 ⋅ q 1 + P ( q 1 + q 2 ( q 1 ) ) − ∂ C 1 ( q 1 ) ∂ q 1 = 0. {\displaystyle {\frac {\partial \Pi _{1}}{\partial q_{1}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{2}}}\cdot {\frac {\partial q_{2}(q_{1})}{\partial q_{1}}}\cdot q_{1}+{\frac {\partial P(q_{1}+q_{2})}{\partial q_{1}}}\cdot q_{1}+P(q_{1}+q_{2}(q_{1}))-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0.}
Examples The following example assumes a generalised linear demand structure
p ( q 1 + q 2 ) = a − b ( q 1 + q 2 ) {\displaystyle p(q_{1}+q_{2})=a-b(q_{1}+q_{2})}
and imposes some restrictions on cost structures for simplicity's sake so the problem can be resolved.
∂ 2 C i ( q i ) ∂ q i ∂ q j = 0 , ∀ j {\displaystyle {\frac {\partial ^{2}C_{i}(q_{i})}{\partial q_{i}\partial q_{j}}}=0,\ \forall j} and ∂ C i ( q i ) ∂ q j = 0 , j ≠ i {\displaystyle {\frac {\partial C_{i}(q_{i})}{\partial q_{j}}}=0,j\neq \ i}
for ease of computation. The follower's profit is:
π 2 = ( a − b ( q 1 + q 2 ) ) ⋅ q 2 − C 2 ( q 2 ) . {\displaystyle \pi _{2}={\bigg (}a-b(q_{1}+q_{2}){\bigg )}\cdot q_{2}-C_{2}(q_{2}).}
The maximisation problem resolves to (from the general case):
∂ ( a − b ( q 1 + q 2 ) ) ∂ q 2 ⋅ q 2 + a − b ( q 1 + q 2 ) − ∂ C 2 ( q 2 ) ∂ q 2 = 0 , {\displaystyle {\frac {\partial {\bigg (}a-b(q_{1}+q_{2}){\bigg )}}{\partial q_{2}}}\cdot q_{2}+a-b(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0,}
⇒ − b q 2 + a − b ( q 1 + q 2 ) − ∂ C 2 ( q 2 ) ∂ q 2 = 0 , {\displaystyle \Rightarrow \ -bq_{2}+a-b(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0,}
⇒ q 2 = a − b q 1 − ∂ C 2 ( q 2 ) ∂ q 2 2 b . {\displaystyle \Rightarrow \ q_{2}={\frac {a-bq_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}}.}
Consider the leader's problem:
Π 1 = ( a − b ( q 1 + q 2 ( q 1 ) ) ) ⋅ q 1 − C 1 ( q 1 ) . {\displaystyle \Pi _{1}={\bigg (}a-b(q_{1}+q_{2}(q_{1})){\bigg )}\cdot q_{1}-C_{1}(q_{1}).}
Substituting for q 2 ( q 1 ) {\displaystyle q_{2}(q_{1})} from the follower's problem:
Π 1 = ( a − b ( q 1 + a − b q 1 − ∂ C 2 ( q 2 ) ∂ q 2 2 b ) ) ⋅ q 1 − C 1 ( q 1 ) , {\displaystyle \Pi _{1}={\bigg (}a-b{\bigg (}q_{1}+{\frac {a-bq_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}}{\bigg )}{\bigg )}\cdot q_{1}-C_{1}(q_{1}),}
⇒ Π 1 = ( a − b q 1 + ∂ C 2 ( q 2 ) ∂ q 2 2 ) ⋅ q 1 − C 1 ( q 1 ) . {\displaystyle \Rightarrow \Pi _{1}={\bigg (}{\frac {a-bq_{1}+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2}}{\bigg )}\cdot q_{1}-C_{1}(q_{1}).}
The maximisation problem resolves to (from the general case):
∂ π 1 ∂ q 1 = ( a − 2 b q 1 + ∂ C 2 ( q 2 ) ∂ q 2 2 ) − ∂ C 1 ( q 1 ) ∂ q 1 = 0. {\displaystyle {\frac {\partial \pi _{1}}{\partial q_{1}}}={\bigg (}{\frac {a-2bq_{1}+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2}}{\bigg )}-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0.}
Now solving for q 1 {\displaystyle q_{1}} yields q 1 ∗ {\displaystyle q_{1}^{*}} , the leader's optimal action:
q 1 ∗ = a + ∂ C 2 ( q 2 ) ∂ q 2 − 2 ⋅ ∂ C 1 ( q 1 ) ∂ q 1 2 b . {\displaystyle q_{1}^{*}={\frac {a+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}-2\cdot {\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2b}}.}
This is the leader's best response to the reaction of the follower in equilibrium. The follower's actual quantity can now be found by feeding this into its reaction function calculated earlier:
q 2 ∗ = a − b ⋅ a + ∂ C 2 ( q 2 ) ∂ q 2 − 2 ⋅ ∂ C 1 ( q 1 ) ∂ q 1 2 b − ∂ C 2 ( q 2 ) ∂ q 2 2 b , {\displaystyle q_{2}^{*}={\frac {a-b\cdot {\frac {a+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}-2\cdot {\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2b}}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}},}
⇒ q 2 ∗ = a − 3 ⋅ ∂ C 2 ( q 2 ) ∂ q 2 + 2 ⋅ ∂ C 1 ( q 1 ) ∂ q 1 4 b . {\displaystyle \Rightarrow q_{2}^{*}={\frac {a-3\cdot {\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}+2\cdot {\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{4b}}.}
The Nash equilibria are all ( q 1 ∗ , q 2 ∗ ) {\displaystyle (q_{1}^{*},q_{2}^{*})} . It is clear (if marginal costs are assumed to be zero – i.e. cost is essentially ignored) that the leader has a significant advantage. Intuitively, if the leader was no better off than the follower, it would simply adopt a Cournot competition strategy. Plugging the follower's quantity q 2 {\displaystyle q_{2}} , back into the leader's best response function will not yield q 1 {\displaystyle q_{1}} . This is because once the leader has committed to an output and observed the followers, it always wants to reduce its output ex-post. However its inability to do so is what allows it to receive higher profits than under Cournot.
Economic analysis An extensive-form representation is often used to analyze the Stackelberg leader-follower model. Also referred to as a "decision tree", the model shows the combination of outputs and payoffs both firms have in the Stackelberg game.
The image on the left depicts in extensive form a Stackelberg game. The payoffs are shown on the right. This example is fairly simple. There is a basic cost structure involving only marginal cost (there is no fixed cost). The demand function is linear and price elasticity of demand is 1. However, it illustrates the leader's advantage. The follower wants to choose q 2 {\displaystyle q_{2}} to maximise its payoff q 2 × ( 5000 − q 1 − q 2 − c 2 ) {\displaystyle q_{2}\times (5000-q_{1}-q_{2}-c_{2})} . Taking the first order derivative and equating it to zero (for maximisation) yields
q 2 = 5000 − q 1 − c 2 2 {\displaystyle q_{2}={\frac {5000-q_{1}-c_{2}}{2}}} as the maximum value of q 2 {\displaystyle q_{2}} . The leader wants to choose q 1 {\displaystyle q_{1}} to maximise its payoff q 1 × ( 5000 − q 1 − q 2 − c 1 ) {\displaystyle q_{1}\times (5000-q_{1}-q_{2}-c_{1})} . However, in equilibrium, it knows the follower will choose q 2 {\displaystyle q_{2}} as above. So in fact the leader wants to maximise its payoff q 1 × ( 5000 − q 1 − 5000 − q 1 − c 2 2 − c 1 ) {\displaystyle q_{1}\times (5000-q_{1}-{\frac {5000-q_{1}-c_{2}}{2}}-c_{1})} (by substituting q 2 {\displaystyle q_{2}} for the follower's best response function). By differentiation, the maximum payoff is given by q 1 = 5000 − 2 c 1 + c 2 2 {\displaystyle q_{1}={\frac {5000-2c_{1}+c_{2}}{2}}} . Feeding this into the follower's best response function yields q 2 = 5000 + 2 c 1