Stackelberg Leader

The game: two firms, one demand curve, two stages

Consider an industry with exactly two firms producing a homogeneous good. Demand is linear: the market price falls as total quantity rises,

P(Q) = a − b Q,   where Q = q_L + q_F.

Both firms have the same constant marginal cost c, with a > c > 0 and b > 0. What distinguishes Stackelberg from the simultaneous Cournot game is the timing. Stage 1: the leader chooses q_L. Stage 2: the follower observes q_L and then chooses q_F. The follower's decision is made with full knowledge of the leader's quantity; the leader's decision is made with full knowledge that the follower will optimise afterward. Both stages happen exactly once and both firms are profit-maximisers.

The model was introduced by the German economist Heinrich von Stackelberg in his 1934 monograph Marktform und Gleichgewicht ("Market Structure and Equilibrium"). Stackelberg's question was simple: what changes when one firm moves first? The answer turned out to be unintuitive — moving first can be a strict advantage, not a handicap — and the analysis became one of the foundational results of modern industrial organization.

Backward induction: solve the follower first

The standard procedure for sequential games is backward induction. Start with the last decision and work back. Given any q_L the leader has already committed to, the follower's profit is

π_F(q_F | q_L) = (P − c) q_F
              = (a − b(q_L + q_F) − c) q_F.

Differentiating with respect to q_F and setting the derivative to zero,

dπ_F / dq_F = a − c − b q_L − 2 b q_F = 0
           →  q_F*(q_L) = (a − c − b q_L) / (2b).

That is the follower's reaction function or best-response curve. It is decreasing in q_L: every extra unit the leader produces pushes the follower's optimum quantity down by half a unit. Intuitively, the leader steals some of the market, the residual demand shrinks, and the follower's profit-maximising response is to produce less.

The leader, anticipating this, treats q_F as the known function q_F*(q_L) rather than as a free choice. Its profit becomes

π_L(q_L) = (a − b(q_L + q_F*(q_L)) − c) q_L
        = (a − c − b q_L − b · (a − c − b q_L)/(2b)) q_L
        = ((a − c) / 2 − (b/2) q_L) q_L.

This is a quadratic in q_L, maximised at

q_L* = (a − c) / (2b).

Substituting back gives the follower's equilibrium output and the implied total,

q_F* = (a − c − b · (a − c)/(2b)) / (2b) = (a − c) / (4b)
Q*  = q_L* + q_F* = 3(a − c) / (4b)
P*  = a − b Q* = (a + 3c) / 4.

The leader produces exactly twice the follower's quantity. Equilibrium profits are

π_L* = (a − c)² / (8b)
π_F* = (a − c)² / (16b).

The leader earns precisely double the follower's profit. That ratio is a constant — it does not depend on a, b, or c, only on the structure of the game.

Stackelberg vs Cournot vs Bertrand: a complete ladder

The four canonical models of duopolistic interaction sort by how aggressively output is competed away. Working from least to most competitive:

Model	Timing	Total Q	Price P	π per firm	Consumer surplus
Monopoly	—	(a − c) / 2b	(a + c) / 2	(a − c)² / 4b	baseline
Cournot duopoly	Simultaneous quantities	2(a − c) / 3b	(a + 2c) / 3	(a − c)² / 9b (each)	higher
Stackelberg duopoly	Sequential quantities	3(a − c) / 4b	(a + 3c) / 4	L: (a − c)² / 8b · F: (a − c)² / 16b	higher still
Bertrand duopoly	Simultaneous prices	(a − c) / b	c	0	maximum

The pattern is striking. With the same demand curve and the same costs, just rearranging the rules of the game moves the outcome smoothly from joint-monopoly profit to zero profit. Stackelberg sits between Cournot and Bertrand on every welfare dimension. Total output is larger than under Cournot — so consumers are better off — but the leader's profit is larger than its Cournot profit, while the follower's profit is smaller. Total industry profit is lower under Stackelberg than Cournot, because the leader's output increase more than offsets the follower's contraction.

Why commitment is the real story

Stackelberg is often described as a "first-mover advantage" model, and the description is correct as far as it goes — but the deep reason the leader wins is commitment, not chronological priority. If the leader could re-optimise after seeing q_F, it would prefer to do so. Given the follower's actual quantity (a − c)/(4b), the leader's unconditional best response would be to produce its own Cournot best response to that quantity, which is smaller than (a − c)/(2b). The Stackelberg equilibrium is sustained precisely because the leader cannot reverse course: q_L is locked in before the follower moves.

If commitment fails, the equilibrium unravels. The follower, knowing the leader can re-optimise, treats the situation as effectively simultaneous and plays its Cournot quantity. The leader, also knowing this, plays its Cournot quantity in response. The system reverts to the Cournot equilibrium with each firm producing (a − c)/(3b) and earning (a − c)²/(9b) — a worse outcome for the leader than the Stackelberg payoff.

This is the central insight behind Avinash Dixit's 1980 entry-deterrence model. An incumbent firm builds productive capacity before a potential entrant decides whether to enter. The capacity investment is a sunk cost: it makes the threat to flood the market credible, because once the factory is built the marginal cost of producing the threatened output is low. The entrant, anticipating that flood, stays out. Capacity is the commitment device that turns a Cournot game into a Stackelberg game in the incumbent's favour.

A worked numerical example

Let a = 100, b = 1, c = 20. Demand is P = 100 − Q with constant marginal cost 20.

Quantity / outcome	Cournot	Stackelberg
Firm 1 (leader) output	26.67	40
Firm 2 (follower) output	26.67	20
Total output Q	53.33	60
Market price P	46.67	40
Firm 1 profit	711.11	800
Firm 2 profit	711.11	400
Total industry profit	1422.22	1200
Consumer surplus (½ b Q²)	1422.22	1800

Going from Cournot to Stackelberg, the leader gains 89 in profit while the follower loses 311, for a net industry loss of 222. Consumers gain 378, more than offsetting the producer loss, so total welfare rises. Stackelberg is a step toward the competitive outcome, but it is asymmetric: the leader is the one party that prefers it over Cournot.

Beyond linear demand

The clean closed-form results — q_L = 2 q_F, π_L = 2 π_F — are specific to linear demand and constant marginal cost. The qualitative results are not. For any downward-sloping demand and convex cost where the follower's best-response function is decreasing in q_L, the Stackelberg leader will produce more than its Cournot quantity, the follower will produce less than its Cournot quantity, and the leader's profit will strictly exceed its Cournot profit. The exact ratios depend on the demand and cost curves, but the ordering π_L,Stackelberg > π_L,Cournot is robust.

Two technical conditions are required for the leader's commitment to pay off. First, the follower's reaction function must be downward-sloping (quantities are strategic substitutes). If reaction functions sloped upward (strategic complements, as in Bertrand price competition with differentiated products), being a Stackelberg leader can actually hurt the leader compared with simultaneous play. Second, the follower must observe q_L. If q_L is private or noisy, the model collapses to a signalling game and the simple backward-induction logic no longer applies.

Variants and extensions

• Multiple followers. If there are n − 1 followers responding to a single leader, total follower output is (n − 1)(a − c)/(2bn) and the leader's output remains (a − c)/(2b). The leader's share of total output falls but its absolute output is unchanged.
• Price leadership. Stackelberg in prices rather than quantities. Used to model dominant-firm pricing and benchmark-setting industries. The dominant firm sets a price; smaller fringe firms take it as given and supply whatever they want at that price. The leader serves residual demand.
• Stackelberg with capacity constraints (Dixit 1980). The leader chooses capacity in stage 1; the follower observes capacity then decides whether to enter and at what scale. Used as the workhorse model of entry deterrence and limit pricing.
• Endogenous timing (Hamilton-Slutsky 1990). Allow firms to choose whether to move early or late. Both firms generally prefer to be leader, giving rise to a war of attrition over timing. With asymmetric costs, the low-cost firm typically emerges as the natural leader.
• Stackelberg with R&D or product quality. Stages 0–1 are R&D investment, stages 2–3 are quantity competition. Leadership in R&D translates to leadership in production. The model is the basis for much of the literature on technology races and innovation.
• Bayesian Stackelberg. The follower's type (its cost, demand, or capacity) is private information. The leader maximises expected profit over the type distribution. The basis for the Stackelberg-security-game literature in computer science (e.g. airport screening, wildlife protection).

Where Stackelberg shows up in real markets

• OPEC and the Saudi swing producer. The canonical real-world example. Saudi Arabia, controlling roughly 30% of OPEC capacity and historically the lowest-cost producer, has long played the role of "swing producer": it adjusts its output first to target a price, and the rest of OPEC plus the non-OPEC fringe adjust afterward. The post-2014 shale revolution and the 2020 price war complicated the picture but the underlying Stackelberg dynamic — one large player whose commitments are unusually credible because of sunk infrastructure costs — remains the textbook framing of the global oil market.
• Incumbent vs entrant. Established firms in capital-intensive industries (semiconductors, airliners, telecommunications) build capacity ahead of demand to deter entry. The sunk capacity acts as the leader's commitment device. The literature on limit pricing, entry deterrence, and predatory pricing is essentially applied Stackelberg.
• Dominant-firm pricing. A large firm sets a benchmark price; smaller fringe firms match it and supply residual demand. Found historically in steel (U.S. Steel as price leader in the early 20th century), tobacco, and ready-to-eat cereals. The leader's commitment device is its visibility and the cost to the fringe of starting a price war.
• Platform launches. The first credible commitment to a large user base — through network-effects investments, marketing, exclusive content — discourages later entrants from contesting the market. The mechanism is Stackelberg in quality or coverage rather than physical quantity, but the logic is identical.
• Stackelberg security games. A defender (police, customs, conservation agency) commits to a randomised patrol schedule that an attacker observes before choosing where to strike. The commitment is the publicly announced patrol policy. Real deployments include the Federal Air Marshal Service, LAX airport screening, and anti-poaching patrols in African wildlife reserves (PROTECT/ARMOR/IRIS systems).

Common pitfalls

• Confusing first-mover advantage with timing. Moving first by itself is not the source of the advantage; commitment is. A leader that can be observed reconsidering — for instance through quietly adjusted shipments or contractual escape clauses — receives no Stackelberg premium. The follower will rationally treat the situation as simultaneous Cournot.
• Applying it to price competition with substitutes. Stackelberg in quantities assumes strategic substitutes (reactions slope down). With differentiated products competing on price, reactions typically slope up (raising your price gives your rival room to raise theirs), and leadership becomes a handicap rather than an advantage.
• Forgetting that total profit falls. Industry profit is lower under Stackelberg than under Cournot. From an industry-wide perspective Stackelberg is a worse cartel — the leader's gain is smaller than the follower's loss. Firms with the ability to collude prefer Cournot or, better still, joint-monopoly outcomes.
• Ignoring the follower's exit option. If the follower's equilibrium profit falls below its outside option (or below zero with fixed entry costs), it exits and the leader becomes a monopolist. Models that treat the follower as captive miss this important boundary case — and it is exactly what makes Stackelberg useful for studying entry deterrence.
• Misreading "leader produces more, earns more" as a recipe. The leader produces more because committing to that level is the optimal exploitation of the follower's reaction curve. Doubling output for its own sake — without the structural ability to commit — just drives down price and profit for both firms.