Cournot Duopoly Nash Equilibrium Calculator

In the Cournot duopoly model, two firms choose quantities simultaneously, each taking the other’s output as given. The inverse demand curve is linear:

P = a - b(Q1 + Q2)

where a is the demand intercept and b is the slope. Each firm i has constant marginal cost c_i.

Nash equilibrium quantities. Each firm maximizes its own profit given the other’s output. Solving the simultaneous best-response functions gives:

Q1* = (a - 2c1 + c2) / (3b) Q2* = (a - 2c2 + c1) / (3b)

Market price. P* = a - b(Q1* + Q2*)

Profits. pi_i = (P* - c_i) * Qi*

Comparing market structures:

• Perfect competition: P = min(c1, c2), zero profit
• Cournot duopoly: intermediate price and quantity, positive profit
• Monopoly (if c1 = c2 = c): Q_m = (a - c)/(2b), P_m = (a + c)/2

Cournot sits between these extremes. As the number of firms grows, Cournot equilibrium approaches the competitive outcome, which is the Cournot convergence result.

Market power in the standard antitrust measures

Two numbers make the Cournot outcome comparable with the tools regulators actually use, and both come straight out of the equilibrium quantities.

The Herfindahl-Hirschman Index is the sum of squared percentage market shares, so a duopoly always lands between 5,000 (an even split) and near 10,000 (one firm holding almost everything). Any duopoly is, by this measure, a highly concentrated market. The US antitrust threshold for that label is 2,500.

The Lerner Index is the markup as a share of price, (P - c) / P. In Cournot it has a tidy closed form: each firm’s Lerner Index equals its market share divided by the absolute price elasticity of demand at the equilibrium. That identity is not an approximation, it falls out of the first-order condition, and the calculator prints both sides so you can see them agree.

For linear demand P = a - bQ, elasticity at the equilibrium is P / (bQ). In the worked example below that comes to 0.765, and the low-cost firm’s 58.8% share divided by 0.765 gives a Lerner Index of 0.769, which is exactly (43.33 - 10) / 43.33.

Interpretation. If one firm has lower cost, it produces more and earns higher profit, but both firms earn positive economic profit in equilibrium. A firm with sufficiently high cost may find its Nash equilibrium quantity is negative. In that case it exits the market, and the survivor’s output has to be recomputed as a monopolist rather than left at the duopoly figure.

Worked example. Take demand P = 100 - Q, so a = 100 and b = 1, with Firm 1 at a marginal cost of 10 and Firm 2 at 20.

• Q1* = (100 - 20 + 20) / 3 = 33.33
• Q2* = (100 - 40 + 10) / 3 = 23.33
• Total output 56.67, so P* = 100 - 56.67 = 43.33
• Firm 1 profit = (43.33 - 10) x 33.33 = 1,111
• Firm 2 profit = (43.33 - 20) x 23.33 = 544

Note how a 10-unit cost disadvantage costs Firm 2 roughly half its profit. Cournot rewards cost advantage far more than proportionally, because the low-cost firm both sells more units and earns a wider margin on each one. Compare the outcome to the benchmarks: a monopolist at the lower cost would produce 45 units at a price of 55, while perfect competition would push output to 90 at a price of 10. Two firms already recover a large slice of the competitive gain, which is the practical lesson buried in the algebra.