Difficulty: Intermediate | Prerequisites: Profit maximisation (MR = MC), basic algebra, concept of Nash equilibrium.
Cournot competition is the workhorse model of oligopoly in intermediate micro. Two (or more) firms simultaneously choose how much to produce, taking the other firm's output as given. Each firm's optimal quantity depends on what the rival does, giving rise to reaction functions. Where those reaction functions intersect is the Cournot-Nash equilibrium. This framework lets you calculate market price, individual profits, and the deadweight loss that arises because oligopoly output falls short of the competitive level.
If you can set MR = MC and solve a two-equation, two-unknown system, you are ready.
In a Cournot duopoly, each firm picks its quantity to maximise profit given the other firm's output. The equilibrium is found by solving both firms' reaction functions simultaneously. The result sits between monopoly (less output, higher price) and perfect competition (more output, lower price), and produces a deadweight loss.
Cournot duopoly
A market structure with two firms that compete by simultaneously choosing quantities. Each firm treats the other's output as fixed when making its own decision. Think of it as: both firms guess what the other will produce, and each picks its best response to that guess.
Reaction function (best-response function)
An equation expressing one firm's profit-maximising quantity as a function of the other firm's quantity. Derived by setting MR = MC and solving for the firm's own q. In simple terms, it answers: "If my rival produces X, what should I produce?"
Cournot-Nash equilibrium
The pair of quantities where each firm is on its own reaction function simultaneously. Neither firm can improve its profit by unilaterally changing output. This is a Nash equilibrium in quantities.
Marginal revenue (MR)
The additional revenue a firm earns from selling one more unit. In a Cournot setting with a linear demand curve, each firm's MR depends on both its own output and the rival's output.
Deadweight loss (DWL)
The loss of total surplus that occurs because the duopoly produces less than the socially efficient (competitive) quantity. Graphically, it is the triangle between the demand curve and the marginal cost curve, from the duopoly quantity to the competitive quantity.
Each firm maximises profit by setting its own MR equal to MC.
Given: P = 100 - 4Q where Q = q1 + q2, and MC = 28 for both firms.
Firm 1's MR: MR1 = 100 - 8q1 - 4q2. Set MR1 = MC: 100 - 8q1 - 4q2 = 28.
Rearrange: 8q1 = 72 - 4q2, so q1 = 9 - 0.5q2. This is Firm 1's reaction function.
By symmetry (same cost structure, symmetric demand), Firm 2's reaction function is: q2 = 9 - 0.5q1.
Substitute Firm 2's reaction function into Firm 1's (or vice versa).
q1 = 9 - 0.5(9 - 0.5q1) = 9 - 4.5 + 0.25q1, so 0.75q1 = 4.5, giving q1 = 6.
By symmetry, q2 = 6.
Total output: Q = 6 + 6 = 12.
P = 100 - 4(12) = 100 - 48 = 52.
Profit = (P - MC) x q = (52 - 28) x 6 = 24 x 6 = 144.
Each firm earns a profit of 144. There are no fixed costs, so this is also economic profit.
The competitive output is found where P = MC: 100 - 4Q = 28, so Q_comp = 18.
The duopoly produces Q = 12, which is 6 units less.
DWL is the area of the triangle between the demand curve and MC, from Q = 12 to Q = 18.
At Q = 12, P = 52. At Q = 18, P = 28 (= MC). The base of the triangle is 18 - 12 = 6; the height is 52 - 28 = 24.
DWL = 0.5 x 6 x 24 = 72.
If Firm 2 produces q2 = 10, use Firm 1's reaction function: q1 = 9 - 0.5(10) = 9 - 5 = 4.
Firm 1 should produce 4 units to maximise its profit when Firm 2 produces 10.
Reaction function (general form for linear Cournot)
From MR_i = MC, with P = a - bQ and MC = c:
q_i = \frac{a - c}{2b} - \frac{1}{2}q_jFor this problem (a = 100, b = 4, c = 28):
q_i = 9 - 0.5\,q_jCournot equilibrium output (symmetric firms)
q_i^* = \frac{a - c}{3b}Here: q* = (100 - 28) / (3 x 4) = 72/12 = 6.
Market price
P = a - b(q_1 + q_2) = 100 - 4(12) = 52Profit per firm
\pi_i = (P - MC) \times q_i = (52 - 28)(6) = 144Deadweight loss
DWL = \frac{1}{2} \times (Q_{comp} - Q_{duopoly}) \times (P_{duopoly} - MC) = \frac{1}{2}(6)(24) = 72Students sometimes confuse the market demand slope with the individual firm's MR slope. In a two-firm Cournot model with P = a - bQ, each firm's MR has a coefficient of 2b on its own quantity, not b.
A frequent error is forgetting to account for the rival's output when computing MR. Each firm's MR includes a term in the other firm's quantity.
Some students think that because both firms have the same costs and face the same demand, one firm could profitably deviate from the Cournot quantity. It cannot: the Cournot equilibrium is a Nash equilibrium, meaning no unilateral deviation improves profit.
When computing deadweight loss, students sometimes use the monopoly quantity instead of the competitive quantity as the benchmark. The DWL triangle runs from the duopoly output to the competitive output (where P = MC).
Cournot is the default oligopoly model for most intermediate micro exams. Be prepared to derive reaction functions, solve the system, and compute price, profit, and DWL in a single question.
If the problem gives you MR directly (as this one does), use it. If it does not, derive MR from the firm's total revenue: TR_i = P x q_i = (a - b(q_i + q_j)) x q_i, then differentiate with respect to q_i.
"What if Firm 2 produces X units?" questions are testing whether you can use the reaction function for a specific value. Just plug in.
Symmetric firms always produce the same quantity in Cournot equilibrium. If costs differ, they will not.
True or False: In a symmetric Cournot duopoly, both firms produce the same quantity. (True.)
Fill in the blank: A reaction function is derived by setting ____ equal to ____. (MR; MC.)
True or False: The Cournot price is lower than the monopoly price but higher than the competitive price. (True.)
Fill in the blank: Deadweight loss from a duopoly is calculated as the triangle between the ____ curve and ____ from Q_duopoly to Q_competitive. (demand; MC.)
Q: Given P = 100 - 4Q, MC = 28, MR1 = 100 - 8q1 - 4q2. Derive Firm 1's reaction function.
A: Set MR1 = MC: 100 - 8q1 - 4q2 = 28. Rearrange: 8q1 = 72 - 4q2, so q1 = 9 - 0.5q2.
Q: What is the Cournot equilibrium output for each firm?
A: Substitute q2 = 9 - 0.5q1 into q1 = 9 - 0.5q2: q1 = 9 - 0.5(9 - 0.5q1) = 4.5 + 0.25q1. Then 0.75q1 = 4.5, q1 = 6. By symmetry, q2 = 6.
Q: What is the market price in the Cournot equilibrium?
A: Q = 12, so P = 100 - 4(12) = 52.
Q: What is the deadweight loss from this duopoly?
A: Competitive Q: 100 - 4Q = 28 gives Q_comp = 18. DWL = 0.5 x (18 - 12) x (52 - 28) = 0.5 x 6 x 24 = 72.
Q: What profit does each firm earn?
A: Profit = (52 - 28) x 6 = 144 per firm.
Q: If Firm 2 produces 10 units, how much should Firm 1 produce?
A: Use Firm 1's reaction function: q1 = 9 - 0.5(10) = 4 units.
Cournot sits between monopoly and perfect competition on the spectrum of market structures. Comparing all three for the same demand and cost curves is a common exam exercise. Cournot also connects to Bertrand competition (firms choose prices instead of quantities, yielding a very different outcome) and Stackelberg leadership (one firm moves first, gaining a strategic advantage).
The concept of a reaction function reappears in game theory more broadly, and the Nash equilibrium concept here is the same one used in simultaneous-move games throughout the course.
Cournot duopoly, Cournot-Nash equilibrium, reaction function, best response function, oligopoly, duopoly, marginal revenue, simultaneous quantity competition, deadweight loss, market power, Nash equilibrium, Bertrand competition, Stackelberg model, linear demand, symmetric firms, profit maximisation, ECON 302, intermediate microeconomics, Penn State