Cournot Competition

The model in one breath

Two (or N) firms produce a homogeneous good. Each firm simultaneously and independently chooses how much to make, q_i. The market then clears: the price P is whatever the inverse demand curve says it must be given total industry quantity Q = Σ q_i. The simplest version uses linear inverse demand

P(Q) = a − b Q

and a constant marginal cost c that is the same for everyone. The intercept a is the choke price (the demand curve hits zero quantity), b is its slope, and c is what one unit costs to produce. We assume a > c so production is profitable at all and that the firms are not capacity-constrained at any of the relevant quantities. The single strategic choice each period is q_i ≥ 0. There is no hidden information, no advertising, no signalling — just a number per firm.

Augustin Cournot wrote this down in 1838, four years before the marginal-revolution debates that produced modern microeconomics. He was studying duopoly in spring water from two adjacent fountains. His insight — that each firm picks its quantity treating the rival's quantity as given, and that we should look for a fixed point — was the first explicit use of what we now call a Nash equilibrium, more than a century before Nash formalised it.

Best-response (reaction) functions

Firm i's profit is

π_i(q_i, q_{−i}) = ( P(Q) − c ) q_i
                = ( a − b(q_i + q_{−i}) − c ) q_i

where q_{−i} is the sum of all rivals' quantities. Treating q_{−i} as fixed (this is the key move), we differentiate and set to zero:

∂π_i / ∂q_i = a − 2 b q_i − b q_{−i} − c = 0
        ⇒  q_i*(q_{−i}) = ( a − c − b q_{−i} ) / ( 2 b )

This is firm i's best-response function, sometimes called its reaction function. Three things are worth pausing on:

• The intercept (a − c)/(2b) is what firm i would produce as a monopolist. Specifically, if q_{−i} = 0 (no rivals), firm i restricts output to half the competitive quantity — the textbook monopoly result.
• The slope is −1/2. Every extra unit a rival produces depresses firm i's optimal output by half a unit — but not by a full unit. Quantities are strategic substitutes: more aggression by one side pulls the other back, but only partly.
• The function is independent of the rest of the demand schedule. Only the local slope b and the cost c matter for the marginal calculation. This is why Cournot generalises so cleanly to non-linear demand: replace the parameters with local elasticities.

Nash equilibrium — the intersection

A Cournot–Nash equilibrium is a profile (q_1*, …, q_N*) such that each firm's quantity is a best response to all the others. With two symmetric firms (N = 2), the simultaneous solution of the two best-response equations gives

q_1* = q_2* = ( a − c ) / ( 3 b )
Q*   = 2 ( a − c ) / ( 3 b )
P*   = ( a + 2 c ) / 3

The equilibrium price (a + 2c)/3 sits strictly above marginal cost c (markup) and strictly below the monopoly price (a + c)/2. Industry quantity 2(a − c)/(3b) is two-thirds of the competitive quantity (a − c)/b. The duopoly is more efficient than monopoly but less than perfect competition — three quantities (monopoly, duopoly, competitive) line up like a thermometer of market power.

Why is this the right concept of "solution"? Because at any (q_1*, q_2*), neither firm has a profitable unilateral deviation. Move up — and your marginal revenue (which already accounts for your own price impact) falls below your marginal cost. Move down — and you leave units on the table whose revenue exceeds their cost. The Cournot intersection is the only profile from which no one wants to move.

N firms — the master formula

With N symmetric firms, impose q_i* = q* in every best-response equation. The total rivals' output for any single firm is (N − 1) q*, so

q*  = ( a − c − b (N − 1) q* ) / ( 2 b )
    ⇒ ( N + 1 ) q* = ( a − c ) / b
    ⇒ q* = ( a − c ) / ( b ( N + 1 ) )

Q*  = N q* = N ( a − c ) / ( b ( N + 1 ) )
P*  = a − b Q* = ( a + N c ) / ( N + 1 )
π*  = ( P* − c ) q* = ( a − c )² / ( b ( N + 1 )² )

These four equations are the most useful identities in oligopoly theory. They let you read off, with no further calculation:

• How total quantity Q* approaches the competitive (a − c)/b as N → ∞, with finite-N deficit (a − c)/(b(N+1)).
• How price P* approaches c as 1/(N+1).
• How firm profits shrink as 1/(N+1)² — entry is twice as painful as the price drop alone would suggest, because it both lowers price and shrinks your share.
• How the markup (P − c)/P, the Lerner index, equals 1/(N · η_D) where η_D is the demand elasticity at the equilibrium quantity. This is the prediction that empirical IO economists test in industry-level data.

Worked example: linear demand, two firms

Suppose inverse demand is P = 120 − 2Q (so a = 120, b = 2) and marginal cost is c = 30. Then:

a − c = 90
N = 2:
  q* = 90 / ( 2 · 3 ) = 15
  Q* = 30
  P* = 120 − 60 = 60       (compare MC = 30, monopoly P = 75)
  π* = ( 60 − 30 ) · 15 = 450  per firm

N = 5:
  q* = 90 / ( 2 · 6 ) = 7.5
  Q* = 37.5
  P* = 120 − 75 = 45
  π* = ( 45 − 30 ) · 7.5 = 112.5  per firm

N → ∞: P → 30, Q → 45, π → 0

Going from a 2-firm duopoly to a 5-firm market drops the price by 25 % and slices each firm's profit by three-quarters. Notice how disproportionate the profit drop is — three additional entrants are vastly more painful than the modest 25 % price decline would suggest. That asymmetry is why incumbents fight tooth and nail to block entry; they are correctly anticipating the (N+1)² profit shrinkage.

Cournot vs Bertrand vs Stackelberg

The three canonical oligopoly models differ in what firms choose and when.

Model	Strategic variable	Timing	Duopoly price	Markup at N = 2
Monopoly	Quantity	One mover	(a + c)/2	(a − c)/2
Cournot	Quantity	Simultaneous	(a + 2c)/3	(a − c)/3
Stackelberg	Quantity	Sequential (leader, follower)	(a + 3c)/4	(a − c)/4
Bertrand	Price	Simultaneous	c	0
Perfect competition	Quantity, price taker	—	c	0

The Bertrand result is the famous "Bertrand paradox": even with only two firms, undercutting drives the equilibrium price to marginal cost. The paradox dissolves once you add capacity constraints — Kreps and Scheinkman (1983) showed that a two-stage game in which firms first pick capacities and then prices reproduces the Cournot outcome. That is the standard rationalisation for treating Cournot as the right reduced-form model whenever capacity is a meaningful strategic commitment, as in airlines, electricity generation, or cement.

Stackelberg makes the leader strictly better off — and total industry quantity higher — by exploiting commitment. The leader produces the monopoly quantity (a − c)/(2b); the follower best-responds with (a − c)/(4b). Compare to the symmetric Cournot equilibrium where each produces (a − c)/(3b). The leader produces 50 % more than its Cournot counterpart by moving first, and pockets higher profit even though prices end up lower.

Comparative statics — what moves the equilibrium

• Demand shifts (a ↑). An outward demand shift raises q*, Q*, P*, and π* one-for-one in (a − c). The number of firms unchanged, every quantity scales.
• Cost shocks (c ↑). A symmetric MC increase passes through to price by N/(N+1) — incompletely. Higher N means more pass-through, with the limit reaching 100 % at perfect competition. With asymmetric cost shocks, the lower-cost firm gains share.
• Entry (N ↑). Quantities up, price down, individual profits down, total industry profit down (firms cannibalise each other). Consumer surplus rises by more than industry profit falls — so total welfare rises with entry, the standard textbook justification for antitrust skepticism of entry barriers.
• Free entry equilibrium. If entry is costless aside from a fixed cost F, firms enter until π* = F. With our parametrisation, (a − c)²/(b(N+1)²) = F gives N* = (a − c)/√(bF) − 1. Welfare-maximising N is generally smaller than the free-entry N (the "business-stealing" externality of Mankiw and Whinston 1986) — entrants ignore the loss they impose on incumbents.

Extensions and refinements

• Asymmetric costs. With firm-specific c_i, each firm's best response is q_i* = (a − c_i − b q_{−i})/(2b). Lower-cost firms produce more and earn higher margins; the model still has a unique linear equilibrium and is the workhorse of merger simulation in industrial organisation.
• Product differentiation. Replace the single inverse demand with q_i = f(p_i, p_{−i}) — Bowley demand, logit, etc. Firms can choose either quantities or prices; Cournot-with-differentiation predicts substantial markups even at N = 2 because of horizontal differentiation, the structure underlying every empirical demand model (BLP, Berry 1994) used today.
• Capacity constraints. With firm capacity k_i and unit production cost c only below k_i, the Kreps–Scheinkman game collapses to Cournot in capacities. This is the canonical micro-foundation for using Cournot in the field.
• Repeated games and collusion. If firms interact repeatedly, the folk theorem says they can sustain monopoly quantities through trigger strategies — punish defection by reverting to Cournot forever. The Cournot outcome thus serves as the off-equilibrium "punishment payoff" that disciplines collusive deviations.
• Bayesian Cournot. If firms have private information about costs, the equilibrium is a Bayesian Nash equilibrium in linear strategies — Vives (1984) showed how affiliated signals affect aggressive vs cautious play.
• Cournot with externalities. If firm i imposes a per-unit externality e on the rest of society, the social planner adds e to the firm's marginal cost in the welfare calculation but not in its private optimisation. The Cournot equilibrium ignores the externality; a Pigouvian tax τ = e restores the first-best output.

Empirical performance

Cournot does well in three classes of markets.

• Airlines. Berry, Carnall and Spiller (1996), Borenstein (1989) and many others find that entry on a route lowers fares by an amount roughly consistent with a Cournot model where each carrier's capacity is the number of seats it offers. A monopolised route's average fare exceeds a duopolised route's by ~20–30 %; adding a third carrier shaves another ~10–15 %.
• Electricity wholesale. Generators bid capacity into a clearing-price auction, the bid that meets the residual demand sets the price for everyone — almost a textbook Cournot game in capacity. Borenstein, Bushnell and Wolak (2002) used a counterfactual Cournot benchmark to estimate that California generators exercised market power worth ~$8 billion during the 2000–2001 crisis.
• Cement, semiconductors, fertilisers. Highly capacity-constrained homogeneous-product industries where firms cannot easily expand output in the short run. Cournot benchmarks are the standard tool in merger review.

It does poorly in retail goods, fashion, and any industry where prices are posted and inventories are easily replenished from elastic upstream supply. There, Bertrand-with-differentiation is the right benchmark.

Common pitfalls

• Confusing Cournot's "simultaneous" with "real-time". The model says firms commit without observing each other; it does not require literal simultaneity. Annual capacity decisions in cement plants are made over months but are strategically simultaneous: each firm chooses without seeing the rivals' choice.
• Assuming quantities are observable. The equilibrium derivation assumes each firm knows the rivals' marginal cost and demand. In practice firms see only realised market prices — which is why Bayesian-Cournot and learning dynamics matter for actual data.
• Treating Cournot and Bertrand as competing. They model different strategic variables. The right question is "which constraint binds first?" Capacity binds in airlines and power; price binds in supermarkets. Use Cournot for the former, Bertrand for the latter.
• Forgetting the homogeneous-product assumption. With differentiated products, the equilibrium price stays well above MC even with hundreds of firms — convergence to perfect competition relies on consumers regarding the output as a perfect substitute across firms.
• Reading the duopoly price (a + 2c)/3 as a universal constant. It is the duopoly price under linear demand and symmetric constant marginal cost. Curve the demand or asymmetrise the costs and the closed-form vanishes; the qualitative ranking (P_M > P_Cournot > P_PC) survives.