Cournot Duopoly: Quantity Competition & Nash Equilibrium

Firm Quantities

Firm 1 output q₁ 30

Firm 2 output q₂ 30

Market Parameters

Choke price a 120

Marginal cost c 20

Market Outcome

Market price P

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Total output Q

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Firm 1 profit π₁

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Firm 2 profit π₂

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Cournot-Nash equilibrium q₁*=q₂*

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Firm 1 profit surface π₁(q₁,q₂)

Firm 1 reaction curve q₁=BR₁(q₂)

Firm 2 reaction curve q₂=BR₂(q₁)

Nash equilibrium point

How it works

Two firms simultaneously pick output quantities q₁, q₂. The market clears at a single linear inverse-demand price, and each firm has constant marginal cost c:

P(Q) = max(0, a − Q),   Q = q₁ + q₂
π₁(q₁,q₂) = (P(Q) − c)·q₁
π₂(q₁,q₂) = (P(Q) − c)·q₂

Each firm's best response maximizes its own profit given the rival's quantity (∂π₁/∂q₁ = 0):

BR₁(q₂) = max(0, (a − c − q₂) / 2)
BR₂(q₁) = max(0, (a − c − q₁) / 2)

The Cournot-Nash equilibrium is the quantity pair where both best responses hold simultaneously — the intersection of the two reaction lines:

q₁* = q₂* = (a − c) / 3
P* = a − 2(a−c)/3 = (a+2c)/3
π₁* = π₂* = (a−c)² / 9

Both firms end up producing more and earning less than a colluding monopoly would (which would set Q = (a−c)/2), because each ignores the effect of its own output on the rival's revenue — the classic externality behind oligopoly overproduction.

q₁ / q₂ sliders — set each firm's output directly and watch price, profits and the marker on the 3D profit surface update live.
a / c sliders — reshape the market: a higher choke price a means more willingness to pay; a higher marginal cost c squeezes both firms' margins and shifts the equilibrium.
Run best-response dynamics — starting from the current (q₁,q₂), firms alternate best-responding to each other (tâtonnement); the path is traced on the floor and spirals into the Nash equilibrium because each reaction line has slope −½, a stable process for linear demand.

Y-axis = Firm 1 profit π₁
floor plane = (q₁, q₂) quantity space

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