Cournot Oligopoly

Two firms compete while watching each other's output. Each picks its best quantity against the other.

My best quantity depends on how much my rival makes. That relationship is the reaction function. Where the two reaction functions cross sits the Cournot-Nash equilibrium. Add more firms and price and quantity move toward competition. With just one firm, it is monopoly again.

For example, say the neighborhood has two snack 🏪 shops. If one makes lots of tteokbokki, the other does better making little. Watching each other, they settle on quantities that stop moving. Grow to ten shops, and the price falls close to cost.

qi=A−c−Q−i2B(reaction)q_i = \frac{A - c - Q_{-i}}{2B} \quad \text{(reaction)}
QN=NN+1A−cB,PN=A+NcN+1Q_N = \frac{N}{N+1}\frac{A-c}{B}, \qquad P_N = \frac{A+Nc}{N+1}

Part of these learning paths

Examples & Exercises

In a duopoly with demand P=10−QP = 10 - Q and c=2c = 2, derive each firm's reaction function and compute the Cournot-Nash equilibrium by hand.

q1=A−c−q22,q2=A−c−q12q_1 = \frac{A - c - q_2}{2}, \qquad q_2 = \frac{A - c - q_1}{2}
N=2N = 2 — the reaction functions cross at the Nash equilibrium.
Show proof

  1. Firm 1's profit is (10−q1−q2−2)q1(10 - q_1 - q_2 - 2)q_1.

    Differentiating in q1q_1: 8−2q1−q2=08 - 2q_1 - q_2 = 0, so q1=(8−q2)/2q_1 = (8 - q_2)/2.

    Symmetrically, q2=(8−q1)/2q_2 = (8 - q_1)/2.

  2. By symmetry q1=q2=q∗q_1 = q_2 = q^*: q∗=(8−q∗)/2q^* = (8 - q^*)/2, so q∗=8/3q^* = 8/3. Total output Q=16/3Q = 16/3 and price P=10−16/3=14/3≈4.67P = 10 - 16/3 = 14/3 \approx 4.67 — between the monopoly price 66 and the competitive price 22.

Show that the Cournot model gives the monopoly solution at N=1N = 1 and converges to the competitive outcome as N→∞N \to \infty.

Compute the numbers for N=8N = 8.

QN=NN+1A−cB,PN=A+NcN+1Q_N = \frac{N}{N+1}\frac{A-c}{B}, \qquad P_N = \frac{A + Nc}{N+1}
N=8N = 8 — price hugs marginal cost.
Show proof

  1. At N=1N = 1, Q1=12A−cBQ_1 = \frac{1}{2}\frac{A-c}{B} and P1=A+c2P_1 = \frac{A+c}{2} — exactly the monopoly solution. As N→∞N \to \infty, QN→A−cB=QcQ_N \to \frac{A-c}{B} = Q_c and PN→cP_N \to c — the competitive outcome.

  2. At N=8N = 8: Q=89⋅8=64/9≈7.11Q = \frac{8}{9}\cdot 8 = 64/9 \approx 7.11, P=10+169=26/9≈2.89P = \frac{10 + 16}{9} = 26/9 \approx 2.89. With just 8 firms the price is already close to marginal cost 22 — how the Cournot model shows competition as the limit in the number of firms.

11. Market Power