How to Calculate the Stackelberg Equilibrium
The Stackelberg leader produces (a − c) ÷ 2b and the follower half of that, (a − c) ÷ 4b, when inverse demand is P = a − bQ and both firms share marginal cost c.
The Stackelberg Model formula
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Enter the demand intercept, slope and marginal cost to get leader and follower output, the market price and both profits.
The price at which buyers would take nothing at all.
How much the price falls for each extra unit the industry sells.
Constant cost per unit, the same for the leader and the follower.
Moving first, the leader commits to 30 units, the quantity a monopolist would choose on this demand curve.
- Follower's output
- 15
- Total market output
- 45
- Market price
- $50
- Leader's profit
- $900
- Follower's profit
- $450
Facing that commitment, the best the follower can do is 15 units, half the leader's quantity.
The two together supply 45 units, more than a Cournot duopoly would and less than perfect competition.
Reading that total off the demand curve puts the price at $50.
The leader keeps $900, which is the payoff to being able to commit first.
The follower earns $450, half the leader's profit, on half the leader's output at the same margin.
How to calculate Stackelberg Model, step by step
- 1Put demand in inverse form. Write price as P = a − bQ, where Q is the sum of both firms' output, and give each firm the same constant marginal cost c.
- 2Solve the follower's problem first. The follower treats the leader's output as already fixed and sets its own marginal revenue equal to marginal cost, giving the reaction function q_follower = (a − c − b × q_leader) ÷ (2b).
- 3Substitute that reaction into the leader's profit. The leader knows how the follower will answer, so it replaces q_follower with the reaction function before maximizing. Working back from the last move is what backward induction means.
- 4Maximize the leader's profit. Setting the leader's marginal profit to zero gives q_leader = (a − c) ÷ (2b), the same quantity a monopolist facing this demand curve would pick.
- 5Work out price and profits. Feed the leader's output back into the reaction function for q_follower, add the quantities, read price off the demand curve, then take (P − c) × q for each firm.
Worked example: Stackelberg Model
Take P = 140 − 2Q with marginal cost of $20 at both firms, so a − c = 120. The leader produces 120 ÷ (2 × 2) = 30 units and the follower 120 ÷ (4 × 2) = 15 units, a total of 45. Price = 140 − 2(45) = $50, so the leader earns (50 − 20) × 30 = $900 and the follower earns (50 − 20) × 15 = $450, exactly half. The same market under Cournot would give 20 units each, a total of 40 and a higher price of $60.
Stackelberg Model questions
How is the Stackelberg model different from Cournot?
Cournot firms choose quantities at the same moment, so neither can commit first. In Stackelberg one firm moves first and the other responds, which lets the leader pick the best point on the follower's reaction function instead of meeting it halfway.
Why does the leader produce more than it would under Cournot?
Every extra unit the leader makes causes the follower to cut back by half a unit, so expanding costs the leader less in lost price than it would against a rival choosing at the same time.
Is the first-mover advantage always real?
Only when the commitment is credible and visible. If the follower cannot observe the leader's output before choosing its own, the game collapses back to Cournot and the advantage disappears.
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