Two firms, one market — and everything turns on what they compete over, and when. Price-setting rivals land on the strangest result in oligopoly theory; a quantity-setting firm that moves first locks in a lasting advantage. Students searching for the best microeconomics tutors in London meet both models — Bertrand, then Stackelberg — usually in the same week.
1 · Bertrand: competing in prices
Two firms sell an identical product at the same constant marginal cost c. Each names a price; every consumer buys from the cheaper firm, and the market splits on a tie.
Now run the undercutting logic. At any common price above c, either firm can shave a penny off, steal the entire market, and multiply its profit — so no such price survives. The incentive to undercut only dies at P = c: raise your price and you sell nothing, cut it and you sell at a loss. Marginal-cost pricing is the unique pair of mutual best responses — the Nash equilibrium. Zero economic profit, from just two sellers: the competitive outcome without the crowd.
2 · The Bertrand paradox — and what softens it
That is the Bertrand paradox: one firm prices like a monopolist, yet a single rival supposedly collapses price to marginal cost. Real duopolies — aircraft makers, mobile networks — visibly earn margins, so something in the setup must bend. Three things do most of the work:
- Differentiation. A penny undercut no longer captures every customer, so prices settle above cost.
- Capacity limits. A firm that cannot serve the whole market gains little by stealing it; the price war loses its point.
- Repetition. Rivals who meet daily can hold prices up tacitly — undercut today and you trigger a war that costs far more tomorrow.
3 · Stackelberg: committing to quantity first
Now change both dials: firms choose quantities, in sequence. A leader commits to q₁; the follower observes it and picks q₂. With inverse demand P = a − b(q₁ + q₂) and marginal cost c, solve backwards.
The follower. Taking q₁ as fixed, it maximises (a − b(q₁ + q₂) − c)q₂; the first-order condition gives its reaction function, q₂ = (a − c − bq₁)/2b. Every extra leader unit shrinks the follower’s best response by half a unit. That half is the whole story.
The leader. It substitutes that schedule into its own profit before maximising, which yields q₁ = (a − c)/2b and q₂ = (a − c)/4b — the leader produces twice the follower’s output. The simultaneous-move (Cournot) benchmark — taught in full elsewhere on this site — sits at q = (a − c)/3b each; take it as given. Both outcomes sit on the follower’s reaction function — the diagram below draws exactly that.
4 · Why moving first pays
The leader’s advantage is commitment, not speed. Because its output is observable and irreversible — capacity built, aircraft ordered — the follower must treat q₁ as fact and rationally retreat. Relative to Cournot, the leader earns more and the follower less; total output rises and price falls, so consumers quietly gain. A reversible commitment would simply be ignored: credibility does all the work.
Worked example — a two-airline route
Two carriers fly the same short-haul route between Manchester and Dublin. Daily inverse demand for seats is P = 300 − Q (fare in £, Q total seats per day); marginal cost is £60 per seat for both. The incumbent publishes its schedule first.
Step 1 — Set up the game. The incumbent (leader) commits to q₁ seats; the entrant (follower) observes and chooses q₂. The fare is P = 300 − q₁ − q₂.
Step 2 — The entrant’s reaction function. The entrant maximises (300 − q₁ − q₂ − 60)q₂; the first-order condition solves to q₂ = (240 − q₁)/2 = 120 − q₁/2.
Step 3 — The leader’s problem, by backward induction. Substituting the reaction function into the incumbent’s profit gives π₁ = (120 − q₁/2)·q₁.
Step 4 — Solve. The first-order condition, 120 − q₁ = 0, gives q₁ = 120 seats; the entrant responds with q₂ = 120 − 60 = 60.
Step 5 — Price and profits. Output is 180 seats, so the fare is P = £120. The incumbent earns (120 − 60) × 120 = £7,200 per day; the entrant, 60 × 60 = £3,600.
Step 6 — Against the simultaneous benchmark. Scheduling at once, the Cournot outcome is 80 seats each, a £140 fare, £6,400 apiece. Commitment lifts the leader by £800 a day, costs the follower £2,800, adds 20 seats, cuts the fare by £20.
Step 7 — Interpret. The expansion pays only because the entrant sees it and retreats — the leader climbs to its favourite point along the entrant’s reaction function. The road not taken: had they fought over identical fares, undercutting would drive the fare to £60 — marginal cost, 240 seats, zero margin.
Bertrand or Stackelberg — could you set up the game before touching the algebra, and solve Stackelberg backwards? Naming the timing, deriving the follower’s reaction function, then maximising along it is the method these questions reward. Rehearsing it on past papers is exactly what a one-on-one microeconomics tutor does with you. Book a trial session.
Practice
Q1. Inverse demand is P = 140 − Q and both firms produce at marginal cost 20. Firm 1 moves first, in quantities. Find both outputs, the price, and both profits.
Q2. Two firms with marginal cost £25 sell an identical product to demand Q = 500 − 4P, naming prices simultaneously. Find the equilibrium price, total quantity, and each firm’s profit.
Q3. In the airline market above, the incumbent commits to q₁ = 100 instead of 120. Find the entrant’s response, the fare, and both profits. Does the incumbent regret it?
Answers. Q1: q₁ = 60, q₂ = 30; P = 140 − 90 = 50; π₁ = 30 × 60 = 1,800, π₂ = 30 × 30 = 900. Q2: undercutting gives P = £25 = marginal cost; Q = 500 − 100 = 400; each firm earns zero economic profit. Q3: q₂ = (240 − 100)/2 = 70; P = 300 − 170 = £130; π₁ = 70 × 100 = £7,000, π₂ = 70 × 70 = £4,900. Yes — £7,000 < £7,200: the fuller commitment was worth £200 a day.
Key takeaways
- Bertrand: identical products, identical costs, simultaneous prices — undercutting drives P to marginal cost and profit to zero.
- The paradox softens with differentiation, capacity limits, or repeated interaction — why real duopolies keep margins.
- Stackelberg is solved backwards: follower’s reaction function first, then the leader maximises along it. With linear demand the leader produces twice the follower’s output.
- First-mover advantage is commitment: against the Cournot benchmark the leader earns more, the follower less; output rises, price falls.
Why London students choose our microeconomics tutoring
- Intermediate micro specialists: tutors who teach oligopoly the way papers test it — state the timing, solve backwards, interpret the comparison.
- Your course’s notation: sessions work from your own slides and problem sets, so reaction functions look exactly as your examiner writes them.
- One-on-one and online: private sessions over a shared screen, built around past papers rather than generic notes.
FAQ
Q: Why does the Bertrand model predict zero profit with only two firms?
A: With identical products the cheaper firm takes the whole market, so any price above marginal cost invites a profitable undercut. The only equilibrium is both firms pricing at cost.
Q: What is a reaction function in oligopoly?
A: A firm’s best output as a function of its rival’s choice. In Stackelberg it is the follower’s whole strategy — the leader maximises profit along it.
Q: What exactly is the first-mover advantage in Stackelberg?
A: The leader commits to a large output and the follower rationally shrinks. With linear demand and equal costs it produces twice as much and out-earns its Cournot self.
Q: Is the Stackelberg outcome better than Cournot for consumers?
A: Yes — output is higher and price lower: 180 seats at £120 versus 160 at £140 above. The follower, not the consumer, pays for the leader’s gain.
Q: Does moving first always help?
A: No. It helps in quantity competition, and only if the commitment is observable and irreversible. In price-setting games moving first can even hurt — a posted price is easy to undercut.
Book a microeconomics tutor in London
Oligopoly questions reward students who set up the game before touching the algebra — timing, reaction function, backward induction. One-on-one sessions with a PhD micro tutor build that discipline on your module’s past papers. Tell us your course and exam date, and we’ll match you this week.
