Cournot Competition

Cournot Competition is the seminal mathematical model of strategic imperfect competition in microeconomics, industrial organization, and Business & Economics Theory, introduced by the French mathematician and economist Antoine Augustin Cournot in his 1838 treatise Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth). In a Cournot Oligopoly, a fixed number of N ≥ 2 firms produce a homogeneous (undifferentiated) product and compete by simultaneously and independently choosing their physical output quantities q₁, q₂, ..., q_N. Once all firms commit their output to the market, an auctioneer or market-clearing mechanism determines the single uniform market price P(Q) along the inverse market demand curve as a function of aggregate industry output Q = ∑_i=1^N q_i, consistent with the law of Supply and Demand.

Published more than a century before John Nash's 1950 formulation of non-cooperative Game Theory, Cournot's 1838 derivation of intersecting reaction curves (best-response functions) was the first formal solution of a Cournot–Nash equilibrium. The model provides a continuous bridge across the entire spectrum of market structures: it reduces to pure Monopoly when N = 1, generates intermediate markups and strategic output restriction for oligopolies (2 ≤ N < ∞), and converges smoothly to the zero-markup, Pareto Efficiency outcome of perfect competition (P = MC) as the number of competing firms N approaches infinity.

Best-Response (Reaction) Functions in a Linear Duopoly

Consider first a Cournot duopoly (N = 2) facing a linear inverse market demand curve:

where a > 0 is the choke price, b > 0 is the demand slope, and each firm i ∈ {1, 2} has a constant marginal and average production cost C_i(q_i) = c_iq_i with 0 ≤ c_i < a. Taking rival output q₂ as given, Firm 1 chooses q₁ ≥ 0 to maximize its profit based on its residual demand curve P₁^res(q₁) = (a − bq₂) − bq₁:

Setting the partial derivative ∂Π₁ ÷ ∂q₁ = 0 yields Firm 1's first-order condition (marginal residual revenue equals marginal cost):

Solving for q₁ as a function of q₂ gives Firm 1's Best-Response (Reaction) Function BR₁(q₂), and by symmetry Firm 2's reaction function BR₂(q₁):

Notice that the reaction slope is ∂BR_i ÷ ∂q_j = −0.5 < 0: in Jeremy Bulow, John Geanakoplos, and Paul Klemperer's (1985) taxonomy, Cournot quantities are strategic substitutes. When Firm 2 expands its output q₂, it depresses the market price and lowers Firm 1's marginal revenue on every unit, inducing Firm 1 to contract its optimal output q₁.

Solving for the Cournot–Nash Equilibrium

The Nash Equilibrium (q₁*, q₂*) occurs at the unique intersection of the two reaction functions where q₁* = BR₁(q₂*) and q₂* = BR₂(q₁*):

When both firms share identical marginal costs (c₁ = c₂ = c), the symmetric Cournot duopoly equilibrium simplifies to:

• Firm Output: q₁* = q₂* = (a − c) ÷ (3b) (each firm produces one-third of the competitive output Q_c = (a − c) ÷ b).
• Aggregate Industry Output: Q* = q₁* + q₂* = 2(a − c) ÷ (3b) (two-thirds of the competitive output, strictly exceeding the monopoly output Q_m = (a − c) ÷ (2b)).
• Equilibrium Price: P* = (a + 2c) ÷ 3.
• Firm Profit: Π₁* = Π₂* = (a − c)² ÷ (9b), so combined industry profit is 2(a − c)² ÷ (9b) ≈ 0.889 Π_m (strictly less than the collusive Prisoner's Dilemma cartel profit Π_m = (a − c)² ÷ (4b)).

Generalization to N Symmetric Firms

For N symmetric firms with constant marginal cost c, firm i's first-order condition is a − b(2q_i + ∑_j≠i q_j) − c = 0. Setting q_i = q* for all i yields a − b(N + 1)q* − c = 0:

(Note: Relative deadweight loss DWL ÷ W_c = 1 ÷ (N + 1)² declines quadratically with the number of competing firms.)

In an industry where N Cournot firms have heterogeneous marginal costs c₁, c₂, ..., c_N (for example, due to differing Economies of Scale or process patents) and face a general downward-sloping inverse demand curve P(Q) with Price Elasticity of Demand ε = −(dQ ÷ dP)(P ÷ Q), Firm i's first-order condition is:

Dividing by P(Q) and letting s_i = q_i ÷ Q ∈ [0, 1] denote Firm i's equilibrium output share yields the Cournot–Lerner Index for Firm i:

This equation shows that in a Cournot equilibrium, lower-cost firms (c_i < c_j) command strictly larger market shares (s_i > s_j) and higher profit margins. Taking the market-share-weighted average markup across all N firms in the industry yields a direct microeconomic link between industry profitability and the Herfindahl–Hirschman Index (H = ∑_i=1^N s_i², scaled on [0, 1]):

This fundamental identity provides the theoretical justification for why competition authorities use the Herfindahl–Hirschman Index to evaluate horizontal mergers and industry market power.

In which real-world industries does the Cournot model apply best?

Following the Kreps–Scheinkman (1983) theorem, Cournot competition applies to industries where firms must commit to physical production capacities, extraction volumes, or inventory runs well in advance of sales, and where expanding output beyond installed capacity is slow or prohibitively expensive. Classic empirical examples include crude oil extraction, bulk cement and aluminum smelting, DRAM semiconductor wafer fabrication, agricultural commodity harvests, and wholesale electricity generation auctions.

Why can a horizontal merger between two Cournot firms be unprofitable ("The Cournot Merger Paradox")?

Stephen Salant, Sheldon Switzer, and Robert Reynolds (1983) demonstrated that in a linear Cournot model with constant marginal costs c and N ≥ 3 firms, if two firms merge into a single entity (reducing the number of independent competitors from N to N − 1), the merged entity internalizes the price-depressing externality between its two plants and contracts its combined output. Because quantities are strategic substitutes, the N − 2 non-merging outsider firms respond along their best-response curves by expanding their output, capturing so much market share that the merged firm earns less profit than its two constituent firms earned prior to the merger unless the merger covers at least 80% of the industry or generates substantial marginal-cost synergies (Economies of Scale).

How does Stackelberg leadership alter the Cournot duopoly outcome?

In Heinrich von Stackelberg's (1934) sequential quantity game, Firm 1 (the Leader) commits irreversibly to its output q₁ before Firm 2 (the Follower) chooses q₂. Anticipating that Firm 2 will respond along BR₂(q₁) = (a − c) ÷ (2b) − 0.5q₁, Firm 1 substitutes BR₂(q₁) directly into its objective function and chooses the monopoly output q₁^S = (a − c) ÷ (2b), forcing Firm 2 back to q₂^S = (a − c) ÷ (4b). Total output rises to 3(a − c) ÷ (4b) (75% of competitive output), lowering price and deadweight loss relative to simultaneous Cournot competition.

1. Cournot, A. A. (1838). Recherches sur les principes mathématiques de la théorie des richesses. L. Hachette. (English translation by N. T. Bacon, 1897, Researches into the Mathematical Principles of the Theory of Wealth, Macmillan). https://oll.libertyfund.org/titles/cournot-researches-into-the-mathematical-principles-of-the-theory-of-wealth
2. Cowling, K., & Waterson, M. (1976). Price-cost margins and market structure. Economica, 43(171), 267–274. https://doi.org/10.2307/2553125
3. Salant, S. W., Switzer, S., & Reynolds, R. J. (1983). Losses from horizontal merger: The effects of an exogenous change in industry structure on Cournot-Nash equilibrium. The Quarterly Journal of Economics, 98(2), 185–199. https://doi.org/10.2307/1885620
4. Bulow, J. I., Geanakoplos, J. D., & Klemperer, P. D. (1985). Multimarket oligopoly: Strategic substitutes and complements. Journal of Political Economy, 93(3), 488–511. https://doi.org/10.1086/261312
5. Vives, X. (1999). Oligopoly Pricing: Old Ideas and New Tools. MIT Press. https://doi.org/10.7551/mitpress/5131.001.0001