Cournot Versus Bertrand: The Role of Strategic Choice

Why the Strategic Choice Matters

Two firms can face the same customers, demand curve, and production costs yet reach different outcomes because they choose different strategic variables. In Cournot competition, each firm chooses a quantity, taking its rival’s quantity as given. In Bertrand competition, each firm chooses a price, taking its rival’s price as given. With a homogeneous good and identical constant marginal costs, the standard models predict different equilibrium prices and total output.

The comparison is meaningful only when the assumptions are matched. The analysis below uses two firms, a common linear market demand curve, and identical constant marginal costs. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Let inverse market demand be $P(Q)=a-bQ$ for total quantity $Q$ up to the choke quantity $a/b$, where $a>c$ and $b>0$. Equivalently, at a nonnegative price $p$, market demand is $D(p)=\max\{(a-p)/b,0\}$. Each of two firms has constant marginal cost $c$, no fixed cost, and no capacity constraint. The good is homogeneous, so consumers buy from the lower-priced firm; if prices tie, demand is split equally. Cournot firms choose quantities simultaneously; Bertrand firms choose prices simultaneously.

Cournot: Each Firm Chooses Quantity

Write $q_i$ for firm $i$'s quantity and $q_j$ for its rival's quantity. Total output is $Q=q_i+q_j$, so the market price is $a-b(q_i+q_j)$ when total output is below the choke quantity. Firm $i$'s profit is its revenue less its variable cost:

\pi_i(q_i,q_j)=\bigl(a-b(q_i+q_j)-c\bigr)q_i.

For a fixed rival quantity, differentiating with respect to $q_i$ gives the firm's marginal profit. Setting it equal to zero gives the best response when the resulting quantity is positive. The firm also cannot choose a negative quantity, so its best response is

q_i=\max\left\{0,\frac{a-c-bq_j}{2b}\right\}.

With identical firms, a symmetric equilibrium has $q_i=q_j=q$. When the equilibrium quantities are positive, the best-response equation becomes $q=(a-c-bq)/(2b)$. Solving gives the Cournot quantity per firm, total output, and market price:

q_C=\frac{a-c}{3b},\qquad Q_C=\frac{2(a-c)}{3b},\qquad P_C=a-bQ_C=\frac{a+2c}{3}.

The condition $a>c$ ensures positive output. At this equilibrium, each firm's profit is $(P_C-c)q_C$, or $(a-c)^2/(9b)$.

Worked Example: Calculating Both Outcomes from One Demand Curve

Two firms sell the same product. Inverse demand is $P(Q)=120-2Q$, and each firm has constant marginal cost $c=20$, no fixed cost, and enough capacity to serve demand. Calculate the Cournot equilibrium, then the Bertrand equilibrium under the assumptions above. Compare each firm's profit.

Method. Use the Cournot best response for simultaneous quantity choice. For price choice, apply the homogeneous-good Bertrand result and then find market demand at that price.

Cournot quantity per firm. Here $a=120$, $b=2$, and $a-c=100$. Thus

q_C=\frac{120-20}{3(2)}=\frac{100}{6}=\frac{50}{3}.

Total output is

Q_C=2\left(\frac{50}{3}\right)=\frac{100}{3}.

The market price is

P_C=120-2\left(\frac{100}{3}\right) =120-\frac{200}{3} =\frac{160}{3}\approx \$53.33.

The price-cost margin is $160/3-20=100/3$. Each firm's profit is therefore

\pi_C=\frac{100}{3}\cdot\frac{50}{3} =\frac{5000}{9}\approx \$555.56.

Bertrand price and output. With identical marginal cost, unlimited capacity, and the stated tie rule, the equilibrium price is $p_B=c=\$20$. At that price, demand is

Q_B=D(20)=\frac{120-20}{2}=50.

If the firms split demand equally at the tied price, each sells $25$ units and earns $(20-20)(25)=\$0$. The results are internally consistent: Cournot output is $100/3$, below Bertrand output of $50$; Cournot price is about $\$53.33$, above the Bertrand price of $\$20$; and Cournot profit is positive while Bertrand profit is zero.

Common mistake avoided: Do not calculate Bertrand profit using the Cournot price or quantity. In the Bertrand model, firms choose prices, and the equilibrium price under these assumptions is marginal cost.

Bertrand: Each Firm Chooses Price

In the homogeneous-good Bertrand model, a firm charging more than its rival loses its customers. A firm charging less than its rival serves the market, subject to its capacity. If both charge the same price, they split demand under the tie rule. These incentives explain why a price above marginal cost cannot be sustained as an equilibrium when both firms have the same cost and sufficient capacity.

At a common price $p$ with $c<p\le a$, demand is positive. A firm can charge a price just below $p$, attract the market's buyers, and earn a positive margin on the sales it wins. If the common price is $p>a$, demand is zero, so a small undercut may still leave the price above the choke price. In that case, the firm can instead choose a price strictly between $c$ and $a$; this attracts positive demand and yields a positive margin. Thus, the deviation argument must distinguish the two price ranges.

A common price below $c$ cannot be an equilibrium either: selling at that price would generate a loss on every unit. If the rival charges below cost, a firm can avoid those loss-making sales by not serving the market. At $p=c$, a firm cannot profitably undercut, because any units sold below $c$ would have negative per-unit margin. Charging above $c$ would lose demand to the rival charging $c$. With the tie rule, splitting demand at $c$ gives zero profit to each firm. Therefore, $p=c$ is the equilibrium price under the stated assumptions.

Theorem: In a two-firm Bertrand model with a homogeneous good, identical constant marginal cost $c$, no capacity constraints, and simultaneous price choice, the equilibrium price is $p_B=c$. For the demand curve above, equilibrium market output is $Q_B=(a-c)/b$, and both firms earn zero profit. The result need not hold unchanged with capacity constraints, differentiated goods, or unequal marginal costs.

Why the Cournot Formula Holds

The Cournot result follows from how one firm's quantity affects both its own sales and the market price. Holding the rival's quantity $q_j$ fixed, firm $i$'s revenue is $\bigl(a-b(q_i+q_j)\bigr)q_i$. Differentiating profit with respect to $q_i$ gives

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

For an interior best response, set this derivative equal to zero:

a-bq_j-2bq_i-c=0 \quad\Longrightarrow\quad q_i=\frac{a-c-bq_j}{2b}.

The second derivative is $\partial^2\pi_i/\partial q_i^2=-2b<0$, so this stationary point maximizes profit. At a symmetric equilibrium, substitute $q_j=q_i=q$, obtaining $2bq=a-c-bq$, or $3bq=a-c$. Hence $q_C=(a-c)/(3b)$. This derivation requires an interior solution; if the parameters or rival's quantity make the best response nonpositive, the firm instead chooses zero.

Fix the rival's quantity.
In Cournot, the firm treats $q_j$ as given while choosing $q_i$.

Account for the price effect.
Increasing $q_i$ raises the firm's sales but lowers the price received on all units sold.

Find mutual best responses.
In a symmetric equilibrium, each firm's quantity must satisfy its best-response condition when the rival chooses the same quantity.

Comparing the Equilibrium Predictions

Under the shared linear-demand and cost assumptions, the predictions can be compared directly. The Cournot price is above marginal cost by $(a-c)/3$; Bertrand price equals marginal cost. Cournot output is $2(a-c)/(3b)$, whereas Bertrand output is $(a-c)/b$. The Bertrand quantity is therefore $3/2$ times the Cournot quantity:

\frac{Q_B}{Q_C} = \frac{(a-c)/b}{2(a-c)/(3b)} =\frac{3}{2}.

The models do not disagree about the demand curve or the production cost. They differ in what each firm chooses and how its rival's choice affects the payoff. In Cournot, an additional unit contributes to total market supply and reduces the price. In Bertrand, a firm with the lower price captures demand, so a firm facing a rival price above cost has an incentive to undercut. That incentive pushes the equilibrium price to cost in the standard model.

These are model-specific predictions, not a rule that every real market must produce one of these exact outcomes. For example, a capacity limit can prevent a low-price firm from serving all demand, while product differentiation can leave some customers with a reason to buy from a higher-priced firm. Those changes alter the incentives used in the comparison.

Worked Example: Correcting an Invalid Bertrand Undercut

Suppose inverse demand is $P(Q)=100-Q$, each firm has constant marginal cost $c=20$, and both firms charge $p=110$. A claim says this cannot be an equilibrium because one firm can undercut by $\$1$, charge $\$109$, and attract the market. Assess the claim and give a profitable deviation.

Method. Check whether the proposed undercut produces positive demand. The choke price is $a=\$100$. At $p=110$, demand is $\max\{100-110,0\}=0$; at $\$109$, demand is also $\max\{100-109,0\}=0$. So the stated $\$1$ undercut does not attract buyers and is not a valid profitable-deviation calculation.

However, the claim's conclusion is correct for a different reason. A firm can choose a price strictly between cost and the choke price, such as $\$60$. At that price, it is cheaper than its rival's $\$110$, and market demand is

$$ D(60)=100-60=40. $$

The deviating firm earns a per-unit margin of $60-20=\$40$, so its profit is $40\times40=\$1{,}600$, compared with zero at the original price of $\$110$. The common price of $\$110$ cannot be an equilibrium, but a small undercut alone does not establish that fact. The corrected reasoning uses a price in $(c,a)$ to ensure positive demand.

Common mistake avoided: A price reduction is not automatically a profitable undercut. Check the demand at the new price, especially when the starting price is above the choke price.

Worked Example: Comparing Outcomes with a Different Market Scale

A homogeneous product has inverse demand $P(Q)=90-Q$. Two firms have identical constant marginal cost $c=30$, no fixed costs, and sufficient capacity. Calculate total output, price, and each firm's profit in the symmetric Cournot equilibrium. Then calculate total output and each firm's profit in the Bertrand equilibrium.

Cournot. Here $a-c=90-30=60$ and $b=1$. Each firm's equilibrium quantity is

q_C=\frac{60}{3(1)}=20.

Thus $Q_C=20+20=40$, and the market price is $P_C=90-40=\$50$. Each firm's margin is $50-30=\$20$, so each earns $20\times20=\$400$.

Bertrand. The equilibrium price is $p_B=c=\$30$. Demand at that price is $Q_B=90-30=60$. Each firm earns zero because the price equals marginal cost; if demand is split equally, each sells $30$ units and earns $(30-30)\times30=\$0$.

The check is consistent with the general comparison: Bertrand output is $60$, which is $3/2$ of Cournot output $40$. The Bertrand price is lower, and Cournot firms earn positive profit while Bertrand firms earn zero in this model.

Common Exam Traps

Using the wrong strategic variable. The Cournot best response is a quantity choice; the Bertrand equilibrium argument is about price undercutting. Do not apply one model's choice rule in the other.
Forgetting the choke price. If a common Bertrand price exceeds $a$, a small undercut may still leave demand at zero. Use a deviation into $(c,a)$ to show that such a price cannot persist.
Comparing models with different assumptions. A price-equals-cost conclusion depends on homogeneity, identical costs, and sufficient capacity. A capacity constraint or product differentiation changes the relevant incentives.
Calling zero Bertrand profit a zero-output result. At $p=c<a$, demand is positive. Firms sell the demanded quantity but earn no economic profit because price equals marginal cost.

Key takeaway: With two identical-cost firms, linear demand, a homogeneous good, and no capacity constraints, Cournot quantity choice gives $Q_C=2(a-c)/(3b)$ and $P_C=(a+2c)/3$. Bertrand price choice gives $p_B=c$ and $Q_B=(a-c)/b$. The difference comes from strategic choice and the assumptions that support each equilibrium.

Check Your Understanding

For each problem, use the stated assumptions and keep the demand and cost conditions matched when comparing the models.

Two firms face $P(Q)=84-2Q$, with identical marginal cost $c=12$, no fixed costs, and sufficient capacity. Find symmetric Cournot output per firm, total Cournot output, Cournot price, and each firm's profit. Then give the Bertrand price, total output, and each firm's profit. (Answer: Cournot $q_i=12$, $Q_C=24$, $P_C=\$36$, and profit $\$288$ per firm; Bertrand $p_B=\$12$, $Q_B=36$, and zero profit per firm.)
Inverse demand is $P(Q)=150-3Q$, and each firm's marginal cost is $c=30$. Under the symmetric Cournot model, calculate each firm's output, total output, price, and profit. (Answer: $q_i=40/3$, $Q_C=80/3$, $P_C=\$70$, and profit $1600/3$, or about $\$533.33$, per firm.)
In a Bertrand market, inverse demand is $P(Q)=90-2Q$, marginal cost is $c=15$, and both firms charge $\$105$. Explain why undercutting by $\$1$ does not attract demand, then give a price in $(c,a)$ that does attract demand and calculate the deviating firm's profit. (Answer: At $\$104$, demand is zero because the choke price is $\$90$; choosing $\$50$ gives demand $20$, margin $\$35$, and profit $\$700$.)
Two firms face $P(Q)=110-Q$, with identical marginal cost $c=20$. Calculate symmetric Cournot total output and price, then compare with Bertrand total output and price. (Answer: Cournot $q_i=30$, $Q_C=60$, $P_C=\$50$; Bertrand $p_B=\$20$, $Q_B=90$.)
For $P(Q)=100-2Q$ and $c=40$, find each firm's symmetric Cournot quantity and profit. Then find Bertrand price and market output. (Answer: Cournot $q_i=10$, $Q_C=20$, $P_C=\$60$, and profit $\$200$ per firm; Bertrand $p_B=\$40$ and $Q_B=30$.)

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