Cournot Quantity Competition: Assumptions and Equilibrium

Introduction

In an oligopoly, each firm’s outcome depends partly on what its rivals do. In Cournot competition, firms choose quantities of a homogeneous product, and the market price adjusts to total quantity supplied. A firm choosing its own output therefore has to account for the output of its rivals.

The central idea is conditional optimization: when a firm chooses its quantity, it takes its rivals’ quantities as given. A Cournot equilibrium is reached when every firm’s quantity is a profit-maximizing choice given the quantities chosen by the others. This tutorial establishes the model’s assumptions and shows how to test that mutual consistency. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

The Cournot Model

Definition: In a Cournot quantity-competition model, each firm chooses how much of a homogeneous good to produce. Firms make their quantity choices simultaneously. The market price depends on total quantity, and each firm chooses a profit-maximizing quantity while treating its rivals’ quantities as fixed.

For \(n\) firms, let \(q_i\) denote firm \(i\)’s nonnegative quantity. Total market output is the sum of the firms’ quantities:

Q=\sum_{i=1}^{n}q_i.

With a common inverse market demand function \(P(Q)\), the price received by each firm is determined by total output. If firm \(i\) has cost function \(C_i(q_i)\), its profit is

\pi_i(q_i,q_{-i})=P\left(q_i+\sum_{j\ne i}q_j\right)q_i-C_i(q_i),

where \(q_{-i}\) denotes the quantities chosen by all firms other than firm \(i\). The first term is revenue: the market price multiplied by firm \(i\)’s own sales. It is not the price multiplied by total market output, because the firm receives revenue only on its own quantity.

The standard model assumes firms understand the relevant market demand and costs, seek to maximize their own profits, and can choose from specified feasible quantities. The product is homogeneous, so buyers regard the firms’ units as interchangeable and there is one market price. The basic model does not include firms choosing prices, coordinating output, or taking turns. If a problem adds a capacity limit or restricts quantities to whole units, that restriction is part of the feasible choices and must be respected.

Best Responses and Equilibrium

A firm’s best response is its profit-maximizing quantity for a particular set of rival quantities. While calculating that choice, the firm holds those rival quantities fixed and asks how its own profit would change if it chose a different feasible quantity. This is a way of evaluating the firm’s decision, not a claim that rivals’ quantities can never change.

Definition: A quantity profile \((q_1^*,\ldots,q_n^*)\) is a Cournot equilibrium if, for every firm \(i\), \(q_i^*\) maximizes firm \(i\)’s profit when the other firms’ quantities are held at their equilibrium values. Equivalently, no firm can raise its profit by changing its own quantity alone.

This definition gives a direct test. Fix the proposed quantities of all firms except one. Compare that firm’s profit at its proposed quantity with its profit at every other feasible quantity. Repeat the check for each firm. The profile is an equilibrium only if every firm passes its own check.

The equivalence between “best response” and “no profitable unilateral deviation” follows from the meaning of maximization. If a firm’s proposed quantity maximizes profit conditional on rivals’ quantities, then every alternative feasible quantity gives profit no greater than its proposed profit. Conversely, if no alternative quantity gives higher profit, the proposed quantity is a maximizer over the feasible choices. Thus a profile is an equilibrium precisely when this condition holds for all firms:

\pi_i(q_i^*,q_{-i}^*)\geq \pi_i(q_i,q_{-i}^*) \quad\text{for every feasible }q_i\text{ and every firm }i.

The condition concerns one firm changing its own output at a time, with other quantities held at the proposed values. It does not require that the firms’ combined profit be as large as possible. A profile can be an equilibrium even if all firms could earn more by jointly reducing or increasing output, provided no single firm would gain by deviating alone.

Worked Equilibrium Checks

Worked Example: Checking a Symmetric Quantity Profile

Two firms sell a homogeneous product. Each must choose an integer quantity from 0, 1, or 2 units. Inverse demand is \(P(Q)=12-Q\), and each firm’s cost is \(C_i(q_i)=3q_i\). Is \((q_1,q_2)=(2,2)\) a Cournot equilibrium?

Method. Calculate each firm’s profit at its proposed quantity and compare it with the profit from every other feasible quantity, holding the rival’s output fixed.

At the proposed profile, total output is \(Q=2+2=4\), so price is \(P(4)=12-4=8\). Each firm sells 2 units and has cost \(3(2)=6\). Therefore each earns \(8(2)-6=10\).

Now hold firm 2’s quantity at 2 and test firm 1’s choices:

If firm 1 chooses \(q_1=0\), then \(Q=2\), \(P=10\), and its profit is \(10(0)-3(0)=0\).
If firm 1 chooses \(q_1=1\), then \(Q=3\), \(P=9\), and its profit is \(9(1)-3(1)=6\).
If firm 1 chooses \(q_1=2\), then \(Q=4\), \(P=8\), and its profit is \(8(2)-3(2)=10\).

Firm 1’s proposed quantity gives the highest profit among its feasible choices. By symmetry, the same comparison applies to firm 2 when firm 1 produces 2 units. Thus neither firm can gain by changing its output alone, and \((2,2)\) is a Cournot equilibrium.

Check. Both firms earn 10 at the proposed profile, and the comparisons verify that neither has a more profitable feasible unilateral choice. A common error is to calculate a deviating firm’s price using only its own quantity; the correct price uses total output, including the rival’s fixed quantity.

Worked Example: Capacity Limits and Unequal Costs

Firm A can choose an integer quantity from 0 to 3, and firm B can choose an integer quantity from 0 to 2. They sell a homogeneous product with inverse demand \(P(Q)=20-Q\). Their costs are \(C_A(q_A)=4q_A\) and \(C_B(q_B)=7q_B\). Test whether \((q_A,q_B)=(3,2)\) is an equilibrium.

Method. Use each firm’s own cost and feasible quantity set, and check its profit against every alternative while holding its rival’s proposed quantity fixed.

At \((3,2)\), total output is \(Q=5\), so price is \(20-5=15\). Firm A earns revenue \(15(3)=45\) and incurs cost \(4(3)=12\), giving profit \(45-12=33\). Firm B earns revenue \(15(2)=30\) and incurs cost \(7(2)=14\), giving profit \(30-14=16\).

Hold firm B at \(q_B=2\). Firm A’s profits at all its feasible quantities are:

\(q_A=0\): \(Q=2\), \(P=18\), and profit \(18(0)-4(0)=0\).
\(q_A=1\): \(Q=3\), \(P=17\), and profit \(17(1)-4(1)=13\).
\(q_A=2\): \(Q=4\), \(P=16\), and profit \(16(2)-4(2)=24\).
\(q_A=3\): \(Q=5\), \(P=15\), and profit \(15(3)-4(3)=33\).

Firm A maximizes profit at 3 units. Now hold firm A at \(q_A=3\). Firm B’s profits are:

\(q_B=0\): \(Q=3\), \(P=17\), and profit \(17(0)-7(0)=0\).
\(q_B=1\): \(Q=4\), \(P=16\), and profit \(16(1)-7(1)=9\).
\(q_B=2\): \(Q=5\), \(P=15\), and profit \(15(2)-7(2)=16\).

Firm B maximizes profit at 2 units. Both quantities are best responses to the other firm’s proposed quantity, so \((3,2)\) is an equilibrium. The capacity limits matter: the checks cover the quantities each firm is allowed to choose, not quantities outside its feasible set.

Check. The two firms’ equilibrium profits are 33 and 16, respectively, and neither can improve by choosing another feasible quantity alone. A common mistake is to apply firm A’s cost to firm B or to ignore the firms’ different capacity limits.

Worked Example: The Role of a Fixed Operating Cost

Each of two firms can produce 0, 1, or 2 units of a homogeneous product. Inverse demand is \(P(Q)=12-Q\). A firm producing a positive quantity incurs cost \(C_i(q_i)=2q_i+3\), while a firm producing zero incurs zero cost. Test whether \((2,2)\) is an equilibrium.

Method. Include the fixed operating cost whenever a firm produces a positive quantity, then compare its proposed choice with both the other positive quantity and shutting down.

At \((2,2)\), total output is \(Q=4\) and price is \(12-4=8\). Each firm’s revenue is \(8(2)=16\), and its total cost is \(2(2)+3=7\). Each firm therefore earns \(16-7=9\).

Hold the rival at 2 units. If a firm produces 0, its profit is 0. If it produces 1, total output is 3, price is 9, revenue is \(9(1)=9\), cost is \(2(1)+3=5\), and profit is \(9-5=4\). If it produces 2, total output is 4, price is 8, revenue is 16, cost is 7, and profit is 9. Producing 2 gives the highest profit. Since the firms have identical demand conditions, costs, and feasible choices, the same comparison applies to each firm. Hence \((2,2)\) is an equilibrium.

Check. Each firm earns 9 at the proposed profile, compared with 4 from producing 1 and 0 from producing nothing. A common error is to omit the fixed cost when comparing positive quantities, or to subtract it when the firm shuts down, even though the problem specifies zero cost at zero output.

What the Assumptions Mean

Model feature	Meaning for the equilibrium test
Homogeneous output	All firms sell the same good at the common market price determined by total output.
Quantity is the strategic variable	Each firm chooses output, not a price or a product characteristic.
Simultaneous choice	Each firm chooses without first observing its rivals’ current choices.
Rivals’ quantities held fixed	When checking one firm’s choice, compare its profits across its own feasible quantities while leaving rivals’ quantities unchanged.
Mutual profit maximization	A profile is an equilibrium only if every firm’s quantity is a best response to the quantities of the others.

The simultaneous-choice assumption is compatible with using fixed rival quantities in a best-response calculation. The calculation asks what a firm would prefer given particular rival quantities; the equilibrium requirement then checks that those quantities are consistent with what the rivals themselves prefer. A firm does not assume that a rival will adjust immediately after it changes output during this unilateral comparison.

Cournot equilibrium is also different from maximizing total industry profit. Each firm evaluates its own revenue and costs, not the combined profit of all firms. Nor is a proposed profile an equilibrium merely because the firms happen to produce the stated quantities: each quantity must be optimal given the others’ quantities. These distinctions are useful whenever a problem asks whether an outcome is stable under unilateral changes.

Common Mistakes

Using the wrong quantity in the price calculation. The market price depends on \(Q\), the sum of all firms’ outputs, not on one firm’s output alone.
Applying the rival’s output as if it were the firm’s own. Revenue is \(P(Q)q_i\), while cost is \(C_i(q_i)\).
Changing two firms’ quantities in one deviation. The equilibrium test asks whether one firm can gain by changing its own quantity while other firms’ quantities stay fixed.
Checking only whether each firm earns a positive profit. Positive profit does not establish that the quantity is profit-maximizing; alternatives must be compared.
Ignoring feasibility or cost details. Respect any stated capacity or quantity restriction, and include fixed costs exactly as specified.
Confusing individual incentives with joint gains. An outcome need not maximize combined industry profit to satisfy the no-profitable-unilateral-deviation condition.

Key takeaway: To verify a Cournot equilibrium, calculate market price from total output, calculate each firm’s profit on its own output, and test whether each firm’s proposed quantity is a best response when its rivals’ quantities are held fixed.

Check Your Understanding

For each problem, use the stated feasible quantities and test the relevant firm-specific incentives.

Two firms face inverse demand \(P(Q)=18-Q\) and each has cost \(C_i(q_i)=5q_i\). At the proposed profile \((3,2)\), calculate the price and each firm’s profit. Then test whether firm 1 would gain by changing its quantity to 4 while firm 2 stays at 2. Is the profile an equilibrium? (Answer: \(P=13\); profits are 24 for firm 1 and 16 for firm 2; firm 1 earns 28 by choosing 4, so the profile is not an equilibrium.)
In a Cournot model, firm A chooses quantity before observing firm B’s current choice, and both sell a homogeneous product. What is each firm’s strategic variable, and what must be held fixed when checking one firm’s best response? (Answer: Each firm chooses quantity; the other firm’s quantity is held fixed.)
Two firms can each produce 0, 1, or 2 units. Inverse demand is \(P(Q)=10-Q\), and each firm has cost \(C_i(q_i)=2q_i\). At \((2,1)\), calculate the price and test firm B’s choices of 0, 1, and 2 while firm A remains at 2. Is the profile an equilibrium? (Answer: Price is 7; firm B’s profits at 0, 1, and 2 are 0, 5, and 8, respectively, so it would choose 2 and the profile is not an equilibrium.)
Two firms each choose 0, 1, or 2 units and have cost \(C_i(q_i)=4q_i\). Inverse demand is \(P(Q)=14-Q\). At \((1,1)\), calculate each firm’s profit, then test whether either firm would gain by changing its own quantity to 2. Is the profile an equilibrium? (Answer: Price is 12 and each firm earns 8; changing to 2 while the rival stays at 1 gives profit 14, so the profile is not an equilibrium.)
Suppose every firm in a proposed quantity profile is maximizing its own profit given the other firms’ quantities, but another feasible profile would give all firms higher combined profit. Does that fact alone invalidate the proposed Cournot equilibrium? (Answer: No. Cournot equilibrium requires no profitable unilateral deviation, not maximization of combined industry profit.)

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