Deriving Cournot Best Responses with Linear Demand

Why a Cournot Best Response Depends on Rival Output

In Cournot quantity competition, each firm chooses its own output while treating the other firm's output as given. That choice affects the market price: a firm that produces more adds to total supply and lowers the price received on every unit sold. Its best response must therefore account for both the extra units it sells and the price reduction on its existing units.

This tutorial derives that choice for two firms selling a homogeneous product with linear inverse market demand and constant marginal costs. The focus is one firm's profit-maximizing quantity conditional on its rival's quantity—not the quantities at which both firms' best responses coincide.

Definition: In a two-firm Cournot model, firm $i$'s best-response quantity is the quantity $q_i\geq 0$ that maximizes its profit when the rival's quantity $q_j\geq 0$ is held fixed. A reaction function gives this best-response quantity as a function of the rival's quantity.

Model and Profit Function

Assume the firms sell the same product and face inverse market demand $P(Q)=a-bQ$, where $Q=q_i+q_j$, $a>0$, and $b>0$. Here $P$ is the market price and $q_i,q_j$ are the firms' nonnegative output quantities. Firm $i$ has constant marginal cost $c_i\geq 0$, so its variable cost is $c_iq_i$. Assume no fixed costs, capacity constraints, or other output restrictions. Each firm chooses quantity, not price.

The linear demand formula describes the relevant range where market price is nonnegative. If the rival's output is so high that firm $i$ cannot earn positive profit by producing, its best response is zero. The calculation below treats the rival's quantity as fixed throughout; it does not assume that the rival adjusts its output in response to firm $i$'s choice.

Firm $i$'s revenue is market price times its own sales, not market price times total market output. Substituting total quantity into price gives its profit:

\pi_i(q_i;q_j) =P(q_i+q_j)q_i-c_iq_i =\bigl[a-b(q_i+q_j)-c_i\bigr]q_i.

Expanding makes the effect of the firm's own quantity visible:

\pi_i(q_i;q_j) =(a-c_i-bq_j)q_i-bq_i^2.

The term $-bq_i^2$ matters. As firm $i$ raises output, its own output lowers the market price, including the price on units it would have sold anyway. This is why the best response is not found by treating price as fixed.

Deriving the Best Response

For a given rival quantity, profit is a quadratic function of $q_i$. Differentiate with respect to the firm's own quantity, then set the derivative equal to zero to locate an interior maximum:

\frac{\partial\pi_i}{\partial q_i} =a-c_i-bq_j-2bq_i=0.

Solving gives the unconstrained candidate $q_i=(a-c_i-bq_j)/(2b)$. But the firm cannot choose a negative quantity. The second derivative is $-2b<0$, so profit is strictly concave: any interior stationary point is the unique profit maximum. If that candidate is negative, profit is decreasing from $q_i=0$ onward, and the constrained maximum is zero.

\frac{\partial^2\pi_i}{\partial q_i^2}=-2b<0, \qquad BR_i(q_j)=\max\left\{0,\frac{a-c_i-bq_j}{2b}\right\}.

The response is positive only when $a-c_i-bq_j>0$, or equivalently $q_j<(a-c_i)/b$. At or above that threshold, firm $i$'s best response is zero. The threshold is relevant only when $a>c_i$; if $a\leq c_i$, the response is zero for every nonnegative rival output.

When the response is positive, it is a straight line in $q_j$. Its slope follows by differentiating the interior expression with respect to rival output:

\frac{dBR_i(q_j)}{dq_j}=-\frac{1}{2}.

Thus, within the positive-output range, one additional unit of rival output reduces firm $i$'s best-response quantity by one-half unit. This slope describes the response to rival output; it is not itself an equilibrium quantity.

Derivation: Expanding profit gives $(a-c_i-bq_j)q_i-bq_i^2$. Differentiating with respect to $q_i$ yields $a-c_i-bq_j-2bq_i$: the coefficient $2b$ arises because the firm's own output affects both the number of units sold and the price on those units. Setting this marginal change in profit to zero gives the interior best response. Strict concavity and the constraint $q_i\geq 0$ then give the maximum formula, including its zero-output case.

Write total output and price.
Use $Q=q_i+q_j$ and substitute it into $P=a-bQ$.

Construct the firm's profit.
Multiply market price by $q_i$, then subtract the firm's own cost $c_iq_i$.

Maximize with respect to own output.
Differentiate profit with respect to $q_i$, set the derivative to zero, and check the second derivative.

Enforce nonnegative output.
If the unconstrained quantity is not positive, use the zero-output corner instead.

Worked Applications

Worked Example: Finding an Interior Best Response

A hypothetical market has inverse demand $P=150-3Q$. Firm A has constant marginal cost of $30 per unit, and Firm B currently produces 20 units. Find Firm A's best-response quantity, the resulting market price, and Firm A's profit.

Method and profit function. Here $a=150$, $b=3$, $c_A=30$, and $q_B=20$. Firm A's profit at any chosen quantity $q_A$ is $\pi_A=[150-3(q_A+20)-30]q_A$. Simplifying the price-minus-cost term gives $\pi_A=(60-3q_A)q_A=60q_A-3q_A^2$.

Maximize. The first derivative is $60-6q_A$. Setting it equal to zero gives $6q_A=60$, so $q_A=10$. The second derivative is $-6<0$, confirming a maximum. The general formula gives the same result: $(150-30-3(20))/(2(3))=60/6=10$.

Price and profit check. Total output is $Q=10+20=30$, so $P=150-3(30)=60$. Firm A's revenue is $60(10)=600$, its variable cost is $30(10)=300$, and its profit is $600-300=\$300$. Substituting into its profit function also gives $60(10)-3(10^2)=600-300=\$300$.

The method avoids the common error of calculating demand using only the rival's quantity: the price calculation must include both firms' output.

Worked Example: Building a Reaction Function and Using Its Slope

A hypothetical market has inverse demand $P=200-4Q$. Firm C's constant marginal cost is $40 per unit. Derive its reaction function, state its positive-output range and slope, and calculate its best response and profit when the rival produces 25 units.

Derive the function. With rival output $q_R$, Firm C's profit is $\pi_C=[200-4(q_C+q_R)-40]q_C$. Expanding gives $\pi_C=(160-4q_R)q_C-4q_C^2$. Differentiating gives $\partial\pi_C/\partial q_C=160-4q_R-8q_C$. Setting this equal to zero yields $q_C=(160-4q_R)/8=20-\tfrac12q_R$. Applying the nonnegative-output constraint, $BR_C(q_R)=\max\{0,20-\tfrac12q_R\}$.

Range and slope. The response is positive when $20-\tfrac12q_R>0$, which means $q_R<40$. In that range its slope is $-\tfrac12$: an extra unit of rival output lowers C's best-response output by one-half unit.

Evaluate at 25 units. Firm C chooses $q_C=20-\tfrac12(25)=20-12.5=7.5$. Total output is $7.5+25=32.5$, so price is $200-4(32.5)=70$. Revenue is $70(7.5)=525$; cost is $40(7.5)=300$; profit is $525-300=\$225$. Direct substitution into the profit function gives $[160-4(25)](7.5)-4(7.5^2)=60(7.5)-4(56.25)=450-225=\$225$, confirming the result.

The method avoids a unit-slope mistake: the response is not one-for-one with rival output because the firm's own output changes price too.

Worked Example: Checking the Zero-Output Corner

A hypothetical market has inverse demand $P=96-2Q$. Firm D's constant marginal cost is $12 per unit, and the rival produces 44 units. Determine Firm D's best response and verify whether any positive output can improve its profit.

Apply the constrained formula. The unconstrained candidate is $(96-12-2(44))/(2(2))=(84-88)/4=-1$. Since output cannot be negative, the best response is $q_D=0$.

Verify the boundary. Firm D's profit at any nonnegative output is $\pi_D=[96-2(q_D+44)-12]q_D=(-4-2q_D)q_D$. At zero output, profit is zero. For any $q_D>0$, both $-4$ and $-2q_D$ are negative, so $(-4-2q_D)q_D<0$. Every positive output therefore produces a loss relative to choosing zero.

This check avoids reporting a negative best-response quantity. The negative stationary-point candidate signals that the constrained optimum is the zero-output corner; it is not a feasible production recommendation.

Common Exam Trap and Economic Interpretation

A frequent error is to hold the market price fixed while differentiating the firm's revenue. For example, with $P=a-b(q_i+q_j)$, an incorrect calculation might write $\pi_i=(a-bq_j-c_i)q_i$, then obtain $q_i=(a-c_i-bq_j)/b$. This omits the effect of $q_i$ on price. The correct profit includes the own-output term: $\pi_i=(a-c_i-bq_j)q_i-bq_i^2$. Its derivative includes $-2bq_i$, so the interior best response has denominator $2b$, not $b$.

For instance, in the first worked example, the incorrect calculation would give $(150-30-3(20))/3=60/3=20$ units instead of 10. At $q_A=20$, total output would be 40 and price would be $150-3(40)=30$, exactly equal to Firm A's $30 marginal cost. But an increase in output has also lowered the price on Firm A's existing units; the firm has no reason to ignore that lost revenue. The correct maximum, 10 units, reflects the full price effect.

This best-response calculation is useful because it converts a strategic choice into a function of the rival's output. Cost affects the intercept: a higher marginal cost lowers the firm's desired quantity, all else equal. Rival output affects the response through the market price, and the downward slope captures this strategic dependence under the stated model. These conclusions rely on homogeneous output, linear inverse demand, constant marginal cost, and firms choosing quantities while holding rivals' choices fixed. If demand, costs, or strategic choice differ, the derivation may change.

Key takeaway: With inverse demand $P=a-b(q_i+q_j)$ and constant marginal cost $c_i$, the firm's profit is $[a-c_i-bq_j]q_i-bq_i^2$. Its best response is $\max\{0,(a-c_i-bq_j)/(2b)\}$, with slope $-1/2$ wherever output is positive.

Check Your Understanding

For each problem, use the firm's own output in total market quantity and apply the nonnegative-output condition.

A hypothetical market has $P=120-2Q$. A firm's marginal cost is $20 and its rival produces 15 units. Find the firm's best response and the resulting market price. (Answer: 17.5 units; price $55.)
Inverse demand is $P=180-3Q$. A firm has marginal cost $36 and its rival produces 20 units. Find its best-response output and profit. (Answer: 14 units; profit $588.)
Inverse demand is $P=90-Q$. A firm's marginal cost is $30 and the rival produces 70 units. Find the best response and explain the profit comparison with any positive output. (Answer: 0 units; profit $0, while any positive output gives negative profit.)
Inverse demand is $P=240-5Q$. A firm's marginal cost is $40 and its rival produces 20 units. Find its best response, market price, and profit. (Answer: 10 units; price $90; profit $500.)
For $P=a-b(q_i+q_j)$, with $a>c_i$, $b>0$, and constant marginal cost $c_i$, state the reaction function, its positive-output condition, and its interior slope. (Answer: $BR_i(q_j)=\max\{0,(a-c_i-bq_j)/(2b)\}$; positive when $q_j<(a-c_i)/b$; slope $-1/2$.)

What You'll Learn