Solution
Leader and Two Followers
Show the problem again
Three firms produce a homogeneous good with zero marginal cost and face inverse demand P = 120 − Q. Firm 1 first chooses its quantity q₁ publicly. Firms 2 and 3 then observe q₁ and choose q₂ and q₃ simultaneously. Everyone maximises profit and the leader anticipates the followers' behaviour (subgame-perfect equilibrium). What is the leader's profit?
Worked solution
Given q₁, the followers play a symmetric Cournot game on residual demand 120 − q₁ − q₂ − q₃, so each chooses (120 − q₁)/3. Together they produce 2(120 − q₁)/3, and the price becomes P = (120 − q₁)/3. The leader then maximises q₁(120 − q₁)/3, giving q₁ = 60. Followers each produce 20, total output is 100, P = 20, and the leader earns 60 × 20 = 1200 (each follower earns 400). Note the leader's output equals the monopoly quantity, since the followers' reaction cuts price by exactly the factor 1/3 regardless of q₁.
Source: Original problem written for AxiomIQ, extending the Stackelberg model of 1934.