μMicroeconomics LabGRAPHICAL EXPLORATIONSOffer Curve Lab ↗

EXPLORE · ADJUST · EXPLAIN

Best responses meet at equilibrium

Change each firm’s marginal cost and watch their quantity choices respond to one another.

Two quantity-setting firms

Firms choose quantities simultaneously in the game. The alternating steps are a way to find their mutual best response; they do not change the game’s timing.

Quantity best-response diagram

Firm 1 chooses q₁ on the horizontal axis; firm 2 chooses q₂ on the vertical axis.

Cournot best response curves, adjustment path and Nash equilibrium0021.421.442.842.864.264.285.685.6107107Nash equilibriumFirm 1 quantity · q₁Firm 2 quantity · q₂
BR₁(q₂)BR₂(q₁)Equilibrium
Firm 1 · q₁*33.33
Firm 2 · q₂*13.33
Market price53.33
Total quantity46.67
Firm 1 profit1,111.11
Firm 2 profit177.78
Both firms produce. Each firm’s equilibrium quantity is the best response to the other’s. Raise only one marginal cost to see the other firm expand.

Derive and calculate the equilibrium

1 · Optimize against a given rival quantity

Best response on the positive-price branch
P=max⁡{0,A−B(q1+q2)}πi=(P−ci)qi,qi≥0∂πi∂qi=A−ci−2Bqi−Bqj=0∂2πi∂qi2=−2B<0\begin{aligned}P&=\max\{0,A-B(q_1+q_2)\}\\\pi_i&=(P-c_i)q_i,\qquad q_i\ge0\\\frac{\partial\pi_i}{\partial q_i}&=A-c_i-2Bq_i-Bq_j=0\\\frac{\partial^2\pi_i}{\partial q_i^2}&=-2B<0\end{aligned}

These derivatives apply on the positive-price branch. The zero-output boundary prevents negative quantities.

Each firm takes the other quantity as given
BR1(q2)=max⁡ ⁣{0,100−20−1q22}BR2(q1)=max⁡ ⁣{0,100−40−1q12}\begin{aligned}\mathrm{BR}_1(q_2)&=\max\!\left\{0,\frac{100-20-1q_2}{2}\right\}\\\mathrm{BR}_2(q_1)&=\max\!\left\{0,\frac{100-40-1q_1}{2}\right\}\end{aligned}

2 · Solve the two conditions together

Interior equilibrium: both outputs must be positive
q1∗=A−2c1+c23Bq2∗=A−2c2+c13B\boxed{q_1^*=\frac{A-2c_1+c_2}{3B}}\qquad\boxed{q_2^*=\frac{A-2c_2+c_1}{3B}}

Use these expressions only if both are positive. Otherwise apply the appropriate boundary:

Boundary cases
c2≥A+c12, c1<A:(q1∗,q2∗)=(A−c12B,0)c1≥A+c22, c2<A:(q1∗,q2∗)=(0,A−c22B)c1,c2≥A:(q1∗,q2∗)=(0,0)\begin{aligned}c_2\ge\frac{A+c_1}{2},\ c_1<A:\quad&\\(q_1^*,q_2^*)&=\left(\frac{A-c_1}{2B},0\right)\\[5pt]c_1\ge\frac{A+c_2}{2},\ c_2<A:\quad&\\(q_1^*,q_2^*)&=\left(0,\frac{A-c_2}{2B}\right)\\[5pt]c_1,c_2\ge A:\quad&\\(q_1^*,q_2^*)&=(0,0)\end{aligned}

3 · Check the mutual best responses

Verify the equilibrium
BR1(13.3333)=33.3333=q1∗BR2(33.3333)=13.3333=q2∗P∗=A−B(q1∗+q2∗)=53.3333π1∗=(P∗−c1)q1∗=1,111.1111π2∗=(P∗−c2)q2∗=177.7778\begin{aligned}\mathrm{BR}_1(13.3333)&=33.3333=q_1^*\\\mathrm{BR}_2(33.3333)&=13.3333=q_2^*\\P^*&=A-B(q_1^*+q_2^*)=53.3333\\\pi_1^*&=(P^*-c_1)q_1^*=1,111.1111\\\pi_2^*&=(P^*-c_2)q_2^*=177.7778\end{aligned}

No fixed costs are included. Where a zero-cost firm has multiple best responses at zero price, the curves select its smallest best response.