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 — this is the Cournot convergence result.

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.